Existence and Uniqueness of Mild Solutions for Fractional Impulsive Evolution Equations of Mixed Type with Nonlocal and Delay Conditions in Banach Spaces

Abstract

In this paper, based on the Schauder fixed point theorem, the (generalized) Darbo fixed point theorem, and the (generalized) Banach contraction mapping principle, we study the mixed-type fractional impulse evolution equation with non-local and delay terms, and obtain the existence and uniqueness theorems under whether the operator is compact or not. The order of the derivative in this paper is $0<\alpha <1$, this fractional order introduces a series of problems concerning compactness, continuity, and convergence. We overcome these problems using methods such as Hölder inequality and Minkowski inequality. Moreover, under the condition of the non-compact measure, the non-negative constant is extended to an unbounded Lebesgue-integrable function. In addition, when obtaining the uniqueness of the solution through the (generalized) Banach contraction mapping principle, the non-negative constant L in the Lipschitz condition is extended to an unbounded Lebesgue integrable function. Finally, a case study is conducted to demonstrate the validity of the theoretical results.

1. Introduction

In recent years, the study of various solution problems for fractional evolution equations has attracted considerable attention. These include problems with delay terms [1], convergence analysis [2,3], nonlocal conditions [4], multiple solutions [5,6], and others. The main tools employed in these theoretical studies include the monotone iteration method [7], the contraction mapping principle [8], the Schauder fixed-point theorem [9,10], the k-set contractive mapping principle [4], the Krasnoselskii fixed-point theorem [11], and iterative methods [12], among others. Compared with traditional integer-order differential models, fractional differential models exhibit distinct properties, including non-locality (i.e., memory effects) [13], as well as complexity and diversity [14]. Consequently, systems described by fractional differential equations offer advantages such as high accuracy, high efficiency, and robustness. Owing to these superior properties, fractional differential equations have found extensive applications across various engineering disciplines, including viscoelastic materials [15], signal processing [16], control systems [17], circuit analysis [18], market equilibrium [19] and others.

Theoretical research on evolution equations is essential, and many mathematicians are currently engaged in foundational studies of fractional evolution equations. Gou et al. [8] explored an integro-differential equation within the framework of Caputo fractional derivatives. By employing the contraction mapping fixed-point theorem, they established the attainability of solutions under Lipschitz continuity conditions. In reference [4], Kumar and colleagues examined the attainability of fractional differential equations involving the Atangana–Baleanu operator, specifically addressing systems with non-instantaneous impulsive effects and nonlocal boundary constraints. Their analysis rigorously proved well-posedness under Lipschitzian continuity assumptions via fixed-point methods. Similarly, Ravichandran et al. [11] demonstrated the well-posedness of solutions for their system through fixed-point analysis under Lipschitz continuity constraints. Thus, rigorous investigation of fractional differential models carries substantial academic and applied weight, as it simultaneously advances the mathematical foundations of calculus and provides innovative analytical tools for addressing complex phenomena across scientific and engineering disciplines.

The fractional derivative with the Mittag–Leffler kernel has attracted the interest of the mathematical community in various research directions. For example, Liang [20] investigated the new Hermite–Hadamard type inequalities under Mittag–Leffler kernel fractional integral operators. Abdeljawad [21] established the well-posedness theorems for the nonlocal Cauchy problems governed by Mittag–Leffler integral operators and give out the illustrative examples. Hammad [22] estimated the controllability of nonlocal Mittag–Leffler kernel fractional order stochastic systems. Priya [23] investigated soil pollution due to microplastics using the Mittag–Leffler kernel fractional-order model. Dhayal and Zhu [24] explore a class of Mittag–Leffler kernel fractional order stochastic system and analyze a constructive result for the solution of the system. In [25], Chen pioneered a fractional stochastic discrete-time chaos framework utilizing Mittag–Leffler memory operators to capture financial market anomalies. Akram [26] develops a rigorous framework for deriving closed-form solutions to Mittag–Leffler kernel fractional differential equations under Pythagorean fuzzy uncertainty. Kalamani [27] rigorously validates the well-posedness (existence, uniqueness) and controllability of fractional-order systems within Mittag–Leffler kernel differentiation frameworks. In [28], the deterministic mathematical model of monkeypox virus is constructed using classical and Mittag–Leffler kernel non-integer order differential equations. El-hady et al. [29] discussed stability results for Atangana–Baleanu–Caputo fractional equations with impulse influences. Hussain et al. [30] investigated the controllability of fractional equations with delay terms of the Atangana–Baleanu–Caputo type. Although Mittag–Leffler kernel fractional-order derivatives have been used in various industries, the theoretical landscape surrounding fractional differential systems with such kernels remains underdeveloped. Current investigations into the solution of initial-boundary value problems are still in their early stages, with fundamental aspects of solution existence, regularity, and structural stability requiring systematic consolidation through rigorous mathematical frameworks.

Currently, most of these results have been obtained under the Lipschitz condition and the non-compact measure condition, whereas only a few have been established under the extended Lipschitz condition and the generalized non-compact measure condition. In [8], Gou studied the following integro-differential equation under Caputo fractional derivative frame

$$\left\{\begin{array}{l}{}^{c}D_t^{\alpha }u\left(t\right)+Au\left(t\right)=f(t,u\left(t\right))+{\int }_0^{t}q(t-s)g(s,u\left(s\right))ds,t\ge 0,t\ne t_k,\\ {\Delta u|}_{t=t_k}=I_k\left(u\left(t_k^{-}\right)\right),k=1,2,\cdots ,m,\\ u\left(0\right)=u_0\in E,\end{array}\right.$$

where ${}^{c}D_t^{\alpha }$ is the Caputo fractional derivative of order $\alpha \in (0,1]$, $f,g:[0,\infty )\times E\to E$, and $q:I\to E$ are continuous, $t\in I=[0,T], 0<T\le +\infty$, $u_0\in E$, $I_k:E\to E$, $0=t_0<t_1<\dots <t_m<t_{m+1}=T,$$\Delta u_k{|}_{t=t_k}=u\left(t_k^{+}\right)-u\left(t_k^{-}\right)$, $u\left(t_k^{+}\right)={\lim }_{m\to 0}u(t_k+m)$ and $u\left(t_k^{-}\right)={\lim }_{m\to 0}u(t_k-m)$ represent the right and left limits of $u\left(t\right)$ at $t=t_k$, respectively. $A:D\left(A\right)\subseteq E\to E$ is a closed liner operator and $-A$ generates a uniformly bounded $C_0-$ semigroup $T\left(t\right)(t\ge 0)$ in $E,$ the authors obtained the existence results by the fixed point theorem of condensation mapping under the Lipschitz condition. In [31], El-hady et al. studied the following fractional differential equation

$$\left\{\begin{array}{l}{}^{ABC}D_t^{\alpha }u\left(t\right)=f(t,u\left(t\right),{\int }_0^{t}Z(t,s,u\left(s\right))ds,t\in (s_i,t_{i+1}),0<\alpha \le 1,\\ u\left(t\right)=H_i(t,u\left(t\right)),t\in (t_i,s_i],i=1,2,\cdots ,m,\\ u\left(b\right)-u^{\prime }\left(b\right)={\int }_b^{T}h(s,u\left(s\right))ds,\end{array}\right.$$

where ${}^{ABc}D_t^{\alpha }$ is the Atangana–Baleanu fractional derivative of order $\alpha \in (0,1]$, and $0=s_0<t_1\le t_2<\dots <t_m\le s_m\le s_{m+1}=T$, $f:[b,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$; $h:[b,T]\times \mathbb{R}\to \mathbb{R}$, $H_i:[t_i,s_i]\times \mathbb{R}\to \mathbb{R}$ is continuous. The authors obtained the existence results by the fixed point theorem of condensation mapping under the Lipschitz condition. In [32], Hamza et al. studied following fractional differential equation

$$\left\{\begin{array}{l}{}^{ABC}D_t^{\alpha }u\left(t\right)=f(t,u\left(t\right),u\left(\tau \left(t\right)\right)),1<\alpha \le 2,t\in (0,T],\\ u\left(t\right)=\zeta \left(t\right),u^{\prime }\left(t\right)={\zeta }^{\prime }\left(t\right),t\in [-\delta ,0],\end{array}\right.$$

where ${}^{ABc}D_t^{\alpha }$ is the Atangana–Baleanu fractional derivative of order $\alpha \in (1,2]$, $\zeta \in C^1\left([-\delta ,0]\right)$ is a prescribed initial function, $f:[0,T]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous, and the delay function $\tau \left(t\right)\le t.$ The authors obtained the existence results by the fixed point theorem of condensation mapping under the Lipschitz condition. In [4], the authors researched the following Atangana–Baleanu fractional differential equations with non-transient impulse and nonlocal conditions

$$\left\{\begin{array}{l}{}^{ABc}D_t^{\alpha }u\left(t\right)=Au\left(t\right)+f(t,u\left(t\right)),t\in {\cup }_{k=0}^m(t_k,t_{k+1}],\\ u\left(t\right)={\gamma }_k(t,u\left(t\right)),t\in {\cup }_{k=0}^m(t_k,t_{k+1}],\\ u\left(0\right)=x_0-g\left(x\right),\end{array}\right.$$

where ${}^{ABc}D_t^{\alpha }$ is the Atangana–Baleanu fractional derivative of order $\alpha \in (0,1]$, $A:D\left(A\right)\subseteq E\to E$ is a closed liner operator and $-A$ generates a uniformly bounded $C_0-$ semigroup $T\left(t\right)(t\ge 0)$ in $E,$$T\left(t\right)(t\ge 0)$ is the solution operator on a Banach space $(E,\parallel ·\parallel ), J=[0,T]$, $T>0$ is a constant, $0<t_1<t_2<\dots <t_m<t_{m+1}=T$, $s_0=0$ and $s_k\in (t_k,t_{k+1})$ for each $k=1,2,\dots ,m$, $f:{\cup }_{k=0}^m(t_k,t_{k+1}]\times E\to E$ satisfies Carathedory condition, i.e., $f(·,x):J\to X$ is Lebesgue measurable and $f(t,·):E\to E$ is continuous, $g:PC(J,E)\to E$ satisfies Lipschitz condition. In [11], the authors investigated the following initial-value problem

$$\left\{\begin{array}{l}{}^{ABc}D_t^{\alpha }u\left(t\right)=f\left(t,u\left(t\right),{\int }_0^{t}h(t,s,u\left(s\right))ds,{\int }_0^T\mathsf{\ell }(t,s,u\left(s\right))ds\right),0<\alpha <1,\\ u\left(0\right)=x_0,\end{array}\right.$$

where $t\in [0,1]=J$, ${}^{ABc}D_t^{\alpha }$ is the Atangana–Baleanu fractional derivative of order $\alpha \in (0,1]$, $f\in C[0,1]$, $f(0,x\left(0\right),0,{\int }_0^T\mathsf{\ell }(0,s,u\left(s\right)))=0$, $T>0$, $h,\mathsf{\ell }:J\times E\to E$ satisfy Lipschitz condition, $f:J\times E\times E\times E\to E$ is a continuous function and satisfy the Lipschitz condition, the authors got the existence and uniqueness of this system under the Lipschitz condition.

To date, most existing results have been obtained under standard Lipschitz conditions, with only a few under generalized Lipschitz conditions. Moreover, a defining feature of impulsive dynamical systems is their ability to rigorously incorporate discontinuous state transitions, thereby offering a refined framework for modeling evolutionary processes subject to abrupt perturbations. In this study, we investigate the well-posedness of hybrid Atangana–Baleanu–Caputo fractional differential equations in Banach spaces, incorporating delays, nonlocal boundary constraints, and impulsive discontinuities. The specific equation is as follows:

$$\left\{\begin{array}{l}{}^{ABc}D_{\kappa }^{\alpha }u\left(\kappa \right)=Au\left(\kappa \right)+\Xi \left(\kappa ,u\left(\kappa \right),u\left({\tau }_1\left(\kappa \right)\right),\dots ,u\left({\tau }_{n-2}\left(\kappa \right)\right),\left(\overline{\mathcal{H}}u\right)\left(\kappa \right),\left(\overline{\Lambda }u\right)\left(\kappa \right)\right),\kappa \in J^{\prime },\\ {\Delta u\left(\kappa \right)|}_{\kappa ={\kappa }_i}=I_i\left(u\left({\kappa }_i\right)\right),i=1,2,\cdots ,m,\\ u\left(0\right)=u_0+g\left(u\right).\end{array}\right.$$

(1)

where $J=[0,T]$, $J^{\prime }=J\setminus \left\{{\kappa }_i\right\}$, $\alpha \in (0,1),$$u_0\in E$, $g:PC(J,E)\to E$ is continuous, $I_i:E\to E(i=1,2,\cdots ,m)$ is a impulse term and continuous, and $0={\kappa }_0<{\kappa }_1<\cdots <{\kappa }_m<{\kappa }_{m+1}=T$, $I_k\left(u\left({\kappa }_i\right)\right)=u\left({\kappa }_i^{+}\right)-u\left({\kappa }_i^{-}\right),i=1,2,\cdots ,m$, $u\left({\kappa }_i^{-}\right)={\lim }_{\lambda \to 0}u({\kappa }_i-\lambda )$ and $u\left({\kappa }_i^{+}\right)={\lim }_{\lambda \to 0}u({\kappa }_i+\lambda )$ denote the left and right limits of $u\left(\kappa \right)$ at $\kappa ={\kappa }_i$, respectively, nonlinear integral operator $\overline{\mathcal{H}}$ and $\overline{\Lambda }$ is defined as follows:

$$\left(\overline{\mathcal{H}}u\right)\left(\kappa \right)={\int }_0^{\kappa }\hslash (\kappa ,s,u\left(s\right))ds,\left(\overline{\Lambda }u\right)\left(\kappa \right)={\int }_0^T\rho (\kappa ,s,u\left(s\right))ds,$$

where $\hslash :{\Theta }_1\times E\to E$ and $\rho :{\Theta }_2\times E\to E$ are continuous, here ${\Theta }_1=\left\{(\kappa ,s)\right|0\le s\le \kappa \le T\}$, ${\Theta }_2=\left\{(\kappa ,s)\right|0\le s,\kappa \le T\}$, $F:J\times E^{n+1}\to E$ is continuous, delay term ${\tau }_i:J\to J$ is continuous and $0\le {\tau }_i\left(\kappa \right)\le \kappa (i=1,2,\dots ,n-2),\kappa \in J,$ and ${}^{ABc}D_{\kappa }^{\alpha }$ is the Atangana–Baleanu Caputo type fractional derivative of order $\alpha .$

In this study, we investigates the existence and uniqueness of mild solutions for fractional impulsive evolution equations with nonlocal and delay conditions in Banach spaces. Theorem 1 establishes the existence of a mild solution for Equation (1) under the assumption that the semigroup is compact. Theorems 2 and 3 provide the existence of a unique solution for Equation (1) under generalized Lipschitz conditions. Theorems 4 and 5 demonstrate the existence of at least one solution for Equation (1) under the generalized measure of noncompactness under the assumption that the semigroup is not compact. Compared with [4,8,11,31], the non-negative constants L of Lipschitz conditions in Section 3.2 and non-compactness measure conditions in Section 3.3 are generalized to the unbounded Lebesgue integrable functions. Compared with [11,31], the equations in this article involve the closed linear operator A. Therefore, this manuscript can be regarded as a generalization of these several reference papers. Compared with [11,31], the equation we are studying contains impulse terms or involves Fredholm operators.

2. Preliminaries and Lemmas

In this article, E represents a Banach space, $C(J,E)=\{u:J\to E\mathrm{be}\mathrm{continuous}\}$ is also a Banach space, whose norm is defined as ${\parallel u\parallel }_C=\max \{\parallel u\left(\kappa \right)\parallel :\kappa \in J\}$, $PC(J,E)=\{u:u:J\to E\mathrm{is}\mathrm{continuous}\mathrm{at}\kappa \ne {\kappa }_k, \mathrm{is}\mathrm{left}\mathrm{continuous}\mathrm{at}\kappa ={\kappa }_i, \mathrm{and} u\left({\kappa }_i^{+}\right)\mathrm{exist},i=1,2,\dots ,m\}$, and the norm is ${\parallel u\parallel }_{PC}={\sup }_{0\le \kappa \le T}\parallel u\left(\kappa \right)\parallel$. $J_0=[{\kappa }_0,{\kappa }_1], J_1=({\kappa }_1,{\kappa }_2],\dots ,J_m=({\kappa }_m,{\kappa }_{m+1}]$, for any $p\ge 1$, $L^p(J,E)$ is Bochner integrable function space of E value. Let $T_R=\{u\in E:\parallel u\parallel \le R\}$ and $B_R={\{u\in PC(J,E):\parallel u\parallel }_{PC}\le R\}$ for $R>0$.

Definition 1 

([33]). Given E is a real complete space under the given norm, $Y\subset E$ is any bounded set, the Kuratowski measure of noncompactness of Y is characterized by

$$\omega \left(Y\right)=\inf \{\epsilon >0:Y={\cup }_{i=1}^m{E}_i,diam\left(E_i\right)\le ϵ,i=1,2,\dots ,m\}.$$

If $U,V\subset X$ are two bounded set, then the assertions are valid:

$\left(1\right)$ If $V\subset U$, then $\omega \left(U\right)\le \omega \left(V\right)$;

$\left(2\right)$$\omega \left(U\right)=\omega \left(\overline{U}\right)=\omega \left(ConvU\right)$, where $\overline{U}$ and $ConvU$ represent the closure and convex hull;

$\left(3\right)$U is relatively compact is equivalent to $\omega \left(U\right)=0$;

$\left(4\right)$$\omega \left(\lambda U\right)=\left|\lambda \right|\omega \left(U\right)$, λ is real number;

$\left(5\right)$$\omega (U+V)\le \omega \left(U\right)+\omega \left(V\right)$, where $U+V=\{x+y:x\in U,y\in V\}$;

$\left(6\right)$$\omega (U\cup V)=\max \left\{\omega \right(U),\omega (V\left)\right\}$.

Lemma 1 

([34]). Let $B\subset PC(J,E)$ form a totally bounded and uniformly equicontinuous family of functions over $J_k(k=0,1,\dots ,m)$. Then $\omega \left(B\right(\kappa \left)\right)$ is continuous on $J_k$ and

$${\omega }_{PC}\left(B\right)=\underset{\kappa \in J}{\sup }\omega \left(B\left(t\right)\right),$$

where $B\left(\kappa \right)=\left\{u\right(\kappa ):u\in B\}$, $\kappa \in J$, ${\omega }_{PC}\left(B\right)$ denotes the Kuratowski measure of noncompactness of the set B in $PC(J,E)$ and $\omega \left(B\right(\kappa \left)\right)$ represents the Kuratowski measure of noncompactness of $B\left(t\right)$ in E.

Lemma 2 

([35]). Assume $B\subset PC(J,X)$ is bounded and is equicontinuous on $J_k(k=0,1,\dots ,m)$, then $m\left(\kappa \right)=\omega \left(B\right(\kappa \left)\right)$ is continuous on $J_k(k=0,1,\dots ,m)$, and

$$\omega \left({\int }_{J}B\left(s\right)ds\right)\le {\int }_{J}m\left(s\right)ds,$$

where ω represents the Kuratowski noncompactness measure defined over a closed bounded set.

Lemma 3 

([35]). Let $B=\left\{u_n\right\}\subset PC(J,E)$ be countable. Assume that there exists $\psi \in L^1(J,{\mathbb{R}}^{+})$ making that $\parallel u_n\left(\kappa \right)\parallel \le \psi \left(\kappa \right)$$a.e.t\in J,n=1,2,\cdots ,$ then $\omega \left(B\right(t\left)\right)$ represents Lebesgue integrable on J and

$$\omega \left(\left\{{\int }_J{u}_n\left(\kappa \right)d\kappa :n\in \mathbb{N}\right\}\right)\le 2{\int }_J\omega \left(B\left(\kappa \right)\right)d\kappa .$$

Lemma 4 

([36]). The set $B\subset C(J,E)$ is assumed to be bounded. Hence, one can find a countable subset $B_0$ of B such that $\omega \left(B_0\right)=\omega \left(B\right)$.

Lemma 5 

([37]). Let G be a closed and convex subset of a real Banach space X, and let the operator $\aleph :G\to G$ be bounded and continuous. For any bounded subset $\mathrm{\Omega }\subset G$, suppose

$${\tilde{\aleph }}^1\left(\mathrm{\Omega }\right)=\aleph \left(\mathrm{\Omega }\right),{\tilde{\aleph }}^n\left(\mathrm{\Omega }\right)=\aleph \left(\overline{Conv}{\tilde{\aleph }}^{n-1}\left(\mathrm{\Omega }\right)\right),n=2,3,\dots$$

Assume there exist a constant $0\le \eta <1$ and a positive integer $n_0$, making that for any bounded subset $\mathrm{\Omega }\subset E,$

$$\omega \left({\tilde{\aleph }}^{n_0}\left(\mathrm{\Omega }\right)\right)\le \eta \omega \left(\mathrm{\Omega }\right),$$

then ℵ has a fixed point in G.

Lemma 6 

([37]). The set $\mathrm{\Omega }\subset PC(J,X)$ is assumed to be bounded and equicontinuous on $J_k(k=0,1,\dots ,m)$, then $\overline{Co}\mathrm{\Omega }\subset PC(J,X)$ is bounded and equicontinuous on $J_k(k=0,1,\dots ,m)$.

Lemma 7 

([38]). For any $R>0$, the function $f:J\times T_R^{n+1}\to X$ is bounded and uniformly continuous, $\mathrm{\Omega }\subset PC(J,X)$ is bounded and equicontinuous in $J_i(i=0,1,\dots ,m)$, then

$$\{f(\kappa ,u\left(\kappa \right),u\left({\tau }_1\left(\kappa \right)\right),\dots ,u\left({\tau }_{n-1}\left(\kappa \right)\right),\left(\overline{\mathcal{H}}u\right)\left(t\right),\left(\overline{\Lambda }u\right)\left(\kappa \right)):u\in \mathrm{\Omega }\}$$

is bounded and equicontinuous in $J_i(i=0,1,\dots ,m)$.

Lemma 8 

([37]). Suppose that $0<\mathsf{\ell }<1$ and $h>0$ are constants. Let

$$S_m={\mathsf{\ell }}^m+C_m^1{\mathsf{\ell }}^{m-1}h+{\displaystyle \frac{C_m^2{\mathsf{\ell }}^{m-2}{h}^2}{2!}}+\dots +{\displaystyle \frac{h^m}{m!}},m\in \mathbb{N},$$

where $C_m^k={\displaystyle \frac{m!}{k!(m-k)!}}$, then $S_m=o\left({\displaystyle \frac{1}{m^s}}\right)(m\to +\infty )$, where $s>1$ is any real number.

Definition 2 

([39]). Assume $x:(0,\infty )\to {\mathbb{R}}^1$, and $\alpha >0$, then the Riemann–Liouville fractional integral of order α is given by

$$I_{\kappa }^{\alpha }x\left(\kappa \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}x\left(s\right)ds,$$

(2)

provided the right-hand side is pointwise defined on $(0,\infty )$.

Definition 3 

([40]). Let $\alpha \in [0, 1]$ and $x^{\prime }\in H^{\prime }(a,b)$, where $a\le b,$ then the Atangana–Baleanu Caputo derivative is

$${(}^{ABc}{D}_{\kappa }^{\alpha }x)\left(\kappa \right)={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{\int }_0^{\kappa }{x}^{\prime }\left(s\right)E_{\alpha }\left(-\alpha {\displaystyle \frac{{(\kappa -s)}^{\alpha }}{1-\alpha }}\right)ds,$$

where B is defined as a normalization function satisfying $B\left(0\right)=B\left(1\right)=1$, and $E_{\alpha }$ denotes the well-known Mittag–Leffler function about a variable α.

Lemma 9 

([41]). For $0\le \alpha \le 1$, then

$$\begin{array}{ll}\left(I_{\kappa }^{\alpha }{(}^{ABC}{D}_{\kappa }^{\alpha }x)\right)\left(\kappa \right)& =x\left(\kappa \right)-x\left(0\right)E_{\alpha }\left(\lambda {\kappa }^{\alpha }\right)-{\displaystyle \frac{\alpha }{1-\alpha }}x\left(0\right)E_{\alpha ,\alpha +1}\left(\lambda {\kappa }^{\alpha }\right)\\ & =x\left(t\right)-x\left(0\right).\end{array}$$

Remark 1. 

Unlike classical fractional derivatives, the AB formulation’s Mittag–Leffler kernel enables precise modeling of power-law memory decay in viscoelastic materials and anomalous diffusion processes, and the commonly Riemann–Liouville and Caputo derivatives have a singular kernel that they does not have the function of fully describing memory.

For the convenience of the following statement, we give some introduction to the operator A.

The symbol A represents a closed linear operator, $-A$ is a finite generator of Banach space $(E,\parallel .\parallel )$ for the semigroup ${\Im }_{\alpha }\left(\kappa \right)(\kappa >0)$, $\parallel {\Im }_{\alpha }\left(\kappa \right)\parallel \le M{e}^{\varpi \kappa }$ and let $M^{★}={\sup }_{\kappa \in [0,+\infty )}\parallel {\Im }_{\alpha }\left(\kappa \right)\parallel$.

Lemma 10 

([42]). Let $\alpha \in (0,1)$, $\varrho \in L^p(J,E)$ with $p>{\displaystyle \frac{1}{\alpha }}$, for linear Cauchy problems with specified initial data in the framework of integral-form solutions

$$\left\{\begin{array}{l}{}^{ABc}D_{\kappa }^{\alpha }u\left(\kappa \right)=Au\left(\kappa \right)+\varrho \left(\kappa \right),\kappa \ge a\ge 0,\\ u\left(a\right)=u_0+g\left(u\right),\end{array}\right.$$

(3)

then the solution to Equation (3) can be formulated as

$$\begin{array}{cc}\hfill u\left(\kappa \right)=& \mathcal{G}{\Im }_{\alpha }(\kappa -a)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_a^{\kappa }{(\kappa -s)}^{\alpha -1}\varrho \left(s\right)ds\hfill \\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_a^{\kappa }{\Re }_{\alpha }(\kappa -s)\varrho \left(s\right)ds,\kappa \in [a,\infty ),\hfill \end{array}$$

where $\mathcal{G}$ and $\mathcal{K}$ are linear operators:

$$\mathcal{G}=\mu {(\mu I-A)}^{-1},\mathcal{K}=-\gamma A{(\mu I-A)}^{-1}\mathrm{and}\mu ={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }},\gamma ={\displaystyle \frac{\alpha }{1-\alpha }},$$

$${\Im }_{\alpha }\left(t\right)=E_{\alpha }(-\mathcal{K}{t}^{\alpha })={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{st}{s}^{\alpha -1}{(s^{\alpha }I-\mathcal{K})}^{-1}ds,$$

$${\overline{\Im }}_{\alpha }\left(t\right)=E_{\alpha }(-\mathcal{K}{t}^{\alpha })={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{st}{s}^{\alpha -2}{(s^{\alpha }I-\mathcal{K})}^{-1}ds,$$

$${\Re }_{\alpha }\left(t\right)=t^{\alpha -1}{E}_{\alpha ,\alpha }(-\mathcal{K}{t}^{\alpha })={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{st}{(s^{\alpha }I-\mathcal{K})}^{-1}ds,$$

and Γ is a certain path lying on ${\Sigma }_{(\alpha ,\omega )}$, for more detail see [4].

Remark 2. 

Combing the formula $\left(2.5\right)$ of [42], the derivative definition of Mittag–Leffler kernel, the derivation process of Theorem 3.1 of [8], the definition of this type should be appropriate.

Lemma 11 

([8]). In the context of semigroup theory, operators ${\Im }_{\alpha }\left(\kappa \right)(\kappa \ge 0)$ and ${\Re }_{\alpha }\left(\kappa \right)(\kappa \ge 0)$ satisfy the following conditions:

$\left(1\right)$ Given an arbitrary fixed $\kappa \ge 0,$${\Im }_{\alpha }\left(\kappa \right)$ and ${\Re }_{\alpha }\left(\kappa \right)$ exhibit both linearity and boundedness, i.e., one can find a constant $M^{*}>0$, for every admissible $\kappa \ge 0,u\in E$, such that

$$\parallel {\Im }_{\alpha }\left(\kappa \right)u\parallel \le M^{★}\parallel u\parallel ,\parallel {\Re }_{\alpha }\left(\kappa \right)u\parallel \le {\displaystyle \frac{\alpha M^{★}}{\Gamma (\alpha +1)}}\parallel u\parallel ;$$

$\left(2\right)$ For every $u\in E,$$\kappa \to {\Im }_{\alpha }\left(\kappa \right)u$ and $t\to {\Re }_{\alpha }\left(\kappa \right)u$ are continuous mapping from $[0,\infty )$ into $E;$

$\left(3\right)$ The operators ${\Im }_{\alpha }\left(\kappa \right)(\kappa \ge 0)$ and ${\Re }_{\alpha }\left(\kappa \right)(\kappa \ge 0)$ exhibit strong continuity over the interval $[0,\infty ).$

Lemma 12. 

Assume a function $u\in PC(J,E)$ is a mild solution for the problem (1), then u satisfies the following formula,

$$u\left(\kappa \right)=\left\{\begin{array}{l}\mathcal{G}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds,\kappa \in [0,{\kappa }_1],\\ {\mathcal{G}}^2{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}{\mathcal{G}}^2(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\Im }_{\alpha }(\kappa -{\kappa }_1){\int }_0^{{\kappa }_1}{({\kappa }_1-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^3}{B\left(\alpha \right)}}{\int }_0^{{\kappa }_1}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds+\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_1)I_1(u\left({\kappa }_1\right)\\ +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds,\kappa \in ({\kappa }_1,{\kappa }_2],\\ \dots \\ {\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \sum _{i=1}^{\nu }}{g}^{\nu +1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_k}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds,t\in ({\kappa }_i,{\kappa }_{i+1}],i=2,\dots ,m,\end{array}\right.$$

(4)

for

$$\overline{\varrho }(s,u\left(s\right))=\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\mathcal{H}u\right)\left(s\right),(\Lambda u)\left(s\right)\right).$$

Proof. 

By Lemma 10, for $\kappa \in [0,{\kappa }_1]$, one has

$$\begin{array}{cc}\hfill u\left(\kappa \right)=& \mathcal{G}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\hfill \\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds.\hfill \end{array}$$

(5)

By (5), then

$$\begin{array}{cc}\hfill u\left({\kappa }_1\right)=& \mathcal{G}{\Im }_{\alpha }\left({\kappa }_1\right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{{\kappa }_1}{({\kappa }_1-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\hfill \\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{{\kappa }_1}{\Re }_{\alpha }({\kappa }_1-s)\overline{\varrho }(s,u\left(s\right))ds.\hfill \end{array}$$

(6)

By impulse term ${\Delta u\left(\kappa \right)|}_{\kappa ={\kappa }_i}=I_k\left(u\left({\kappa }_i\right)\right)(i=1,2,\cdots ,m)$ in Equations (1) and (6), one has

$$u\left({\kappa }_1^{+}\right)=u\left({\kappa }_1^{-}\right)+I_i\left(u\left({\kappa }_i^{-}\right)\right)=u\left({\kappa }_1\right)+I_1\left(u\left({\kappa }_1\right)\right).$$

Take $a={\kappa }_1$ in Lemma 10, for $\kappa \in ({\kappa }_1,{\kappa }_2]$, one has

$$\begin{array}{ll}u\left(\kappa \right)& =\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_1)\left(u\left({\kappa }_1^{+}\right)\right)+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\\ & =\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_1)(u\left({\kappa }_1\right)+I_1\left(u\left({\kappa }_1\right)\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\\ & =\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_1)\left(\mathcal{G}{\Im }_{\alpha }\left({\kappa }_1\right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{{\kappa }_1}{({\kappa }_1-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\right.\\ & \left.+{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{{\kappa }_1}{\Re }_{\alpha }({\kappa }_1-s)\overline{\varrho }(s,u\left(s\right))ds+I_1\left(u\left({\kappa }_1\right)\right)\right)+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\\ & ={\mathcal{G}}^2{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}{\mathcal{G}}^2(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\Im }_{\alpha }(\kappa -{\kappa }_1){\int }_0^{{\kappa }_1}{({\kappa }_1-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^3}{B\left(\alpha \right)}}{\int }_0^{{\kappa }_1}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds+\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_1)I_1(u\left({\kappa }_1\right)\\ & +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_1}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds.\end{array}$$

(7)

Follow the steps above in turn, for $\kappa \in ({\kappa }_{i-1},{\kappa }_i]$, one arrives

$$\begin{array}{ll}u\left(\kappa \right)& ={\mathcal{G}}^k{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\left({\mathcal{G}}^k{\Im }_{\alpha }(\kappa -{\kappa }_1){\int }_0^{{\kappa }_1}{({\kappa }_1-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\right.\\ & +{\mathcal{G}}^{k-1}{\Im }_{\alpha }(\kappa -{\kappa }_2){\int }_{{\kappa }_1}^{{\kappa }_2}{({\kappa }_2-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds+\dots \\ & \left.+{\mathcal{G}}^2{\Im }_{\alpha }(\kappa -{\kappa }_{k-1}){\int }_{{\kappa }_{k-2}}^{{\kappa }_{k-1}}{({\kappa }_{k-1}-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\right)\\ & +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}\left({\mathcal{G}}^{k+1}{\int }_0^{{\kappa }_1}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds+{\mathcal{G}}^k{\int }_{{\kappa }_1}^{{\kappa }_2}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds+\dots \right.\\ & \left.+{\mathcal{G}}^3{\int }_{{\kappa }_{k-2}}^{{\kappa }_{k-1}}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\right)\\ & +{\mathcal{G}}^{k-1}{\Im }_{\alpha }(\kappa -{\kappa }_1)I_1\left(u\left({\kappa }_1\right)\right)+{\mathcal{G}}^{k-2}{\Im }_{\alpha }(\kappa -{\kappa }_2)I_2\left(u\left({\kappa }_2\right)\right)+\dots +\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_{k-1})I_2\left(u\left({\kappa }_{k-1}\right)\right)\\ & +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_k}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{t_k}^{\kappa }{\Re }_{\alpha }(t-s)\overline{\varrho }(s,u\left(s\right))ds\\ & ={\mathcal{G}}^k{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \sum _{i=1}^k}{g}^{k-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)\\ & +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_{k-1}}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_{k-1}}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds.\end{array}$$

(8)

As a result, by (8), one has

$$\begin{array}{ll}u\left({\kappa }_k\right)& ={\mathcal{G}}^k{\Im }_{\alpha }\left({\kappa }_k\right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+1-i}{\Im }_{\alpha }({\kappa }_k-{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(t_k-s)\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \sum _{i=1}^k}{g}^{k-i}{\Im }_{\alpha }({\kappa }_k-{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)\\ & +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_{i-1}}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds+{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_{i-1}}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds,\end{array}$$

(9)

hence, for $\kappa \in ({\kappa }_i,{\kappa }_{i+1}]$, by $\left(9\right)$, one arrives

$$\begin{array}{ll}u\left(\kappa \right)& ={\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(u\left(t_i\right)\right)+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds,t\in ({\kappa }_i,{\kappa }_{i+1}],i=2,3,\dots .\end{array}$$

(10)

Hence, Equation (1) is mathematically interchangeable with the integral formulation (4), establishing their equivalence under the prescribed analytical framework. □

Now we define an operator $\aleph :PC(J,E)\to PC(J,E)$, for any $u\in PC(J,E)$, define $\aleph u$ by the right of the Equation (4). Obviously, u is the solutions of Equation $\left(4\right)$ in $PC(J,E)$ iff u is a fixed point of the operator ℵ in $PC(J,E)$.

Definition 4. 

A function $u\in PC(J,E)$ is said to be a mild solution of the problem (1) if u satisfies (4), that is, u is a fixed point of the operator ℵ in $PC(J,E)$.

3. Main Results

To streamline the presentation, we formally define assumptions that will govern our subsequent analysis.

$\left(H_1\right)$ There exist integrable functions ${\hslash }_1(\kappa ,·),{\rho }_1(\kappa ,·),{\zeta }_{\hslash }(\kappa ,s),{\zeta }_{\rho }(\kappa ,s)\in L(J,{\mathbb{R}}^{+}),{\mathbb{R}}^{+}=[0,+\infty )$, subject to

$$\begin{gathered} {\hslash }_0={\sup }_{\kappa \in [0,T]}{\int }_0^{\kappa }{\hslash }_1(\kappa ,s)ds<+\infty ,{\rho }_0={\sup }_{\kappa \in [0,T]}{\int }_0^T{\rho }_1(\kappa ,s)ds<+\infty , \\ {\overline{\hslash }}_0={\max }_{\kappa \in [0,T]}{\int }_0^{\kappa }{\zeta }_{\hslash }(\kappa ,s)ds<+\infty ,{\overline{\rho }}_0={\max }_{\kappa \in [0,T]}{\int }_0^T{\zeta }_{\rho }(\kappa ,s)ds<+\infty , \end{gathered}$$

(11)

$$\parallel \hslash (\kappa ,s,u)\parallel \le {\hslash }_1(\kappa ,s)\parallel u\parallel +{\zeta }_{\hslash }(\kappa ,s),(\kappa ,s)\in {\Theta }_1,u\in E,$$

(12)

$$\parallel \rho (\kappa ,s,u)\parallel \le {\rho }_1(\kappa ,s)\parallel u\parallel +{\zeta }_{\rho }(\kappa ,s),(\kappa ,s)\in {\Theta }_2,u\in E.$$

(13)

$\left(H_2\right)$ Functions $I_k:E\to E$ are continuous and there exist constants $d_{I_k}>0,{\zeta }_{I_k}>0$, satisfies

$$\parallel I_k\left(u\right)\parallel \le d_{I_k}\parallel u\parallel +{\zeta }_{I_k},u\in E,k=1,2,\dots ,m.$$

(14)

$\left(H_3\right)$ Continuity of $g:PC(J,E)\to E$ with there existing constants $d_1>0,{\zeta }_1>0$, for all $u\in PC(J,E)$, satisfies

$$\parallel g\left(u\right)\parallel \le d_1{\parallel u\parallel }_{PC}+{\zeta }_1.$$

(15)

$\left(H_4\right)$$\mathcal{G}$ and $\mathcal{K}$ are bounded operators, that is, the operator admits bounds $l,\overline{l}$ fulfilling

$$\parallel \mathcal{G}\parallel \le l,\parallel \mathcal{K}\parallel \le \overline{l}.$$

$\left(H_5\right)$ Nonlinear term $\Xi :J\times T_R^{n+1}\to E$ is bounded and uniformly continuous,

$$\gamma =\underset{\parallel {\chi }_1\parallel +\parallel {\chi }_2\parallel +\parallel {\chi }_3\parallel +\dots +\parallel {\chi }_{n+1}\parallel \to +\infty }{\mathrm{lim\; sup}}\left(\underset{s\in J}{\sup }{\displaystyle \frac{\parallel \Xi (s,{\chi }_1,{\chi }_2,{\chi }_3,\dots ,{\chi }_{n+1})\parallel }{\parallel {\chi }_1\parallel +\parallel {\chi }_2\parallel +\parallel {\chi }_3\parallel +\dots +\parallel {\chi }_{n+1}\parallel }}\right)<+\infty ,$$

(16)

and

$$K=\tilde{l}{M}^{★}{d}_1+\widehat{d}\gamma \left(n-1+{\hslash }_0+{\rho }_0\right)+l^m{M}^{★}{\displaystyle \sum _{k=1}^m}{d}_{I_k}<1,$$

(17)

where

$$\overline{d}={\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\},$$

$$\tilde{l}=\max \{l,l^{m+1}\},\widehat{d}=\overline{d}\left({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+T\right).$$

Remark 3. 

Condition $(H_1$)–($H_5)$ is mainly used to prove that the operator mapping from a bounded region to another bounded region is bounded.

3.1. The Existence Theorem of Mild Solution Under the Compact Resolvent Operator Framework

Theorem 1. 

Suppose $\left\{{\Im }_{\alpha }\left(\kappa \right)\right\}$ is compact, and the conditions of $\left(H_1\right)$–$\left(H_5\right)$ hold, then the model $\left(1\right)$ has at least one global mild solution in $B_{R^{★}}\subset PC(J,E)$.

Proof. 

Our first objective is to construct a parameter $R^{★}>0$ verifying $\aleph :B_{R^{★}}\to B_{R^{★}}$. By $\left(17\right)$ of condition $\left(H_5\right)$, there exists ${\gamma }^{\prime }>\gamma$, satisfies

$$K^{\prime }=l{M}^{★}{d}_1+d{\gamma }^{\prime }(n-1+h_0+{\rho }_0)+l{M}^{★}{\displaystyle \sum _{k=1}^m}{d}_{I_k}<1.$$

(18)

By $\left(16\right)$, there exists $r_0>0$, for $\parallel {\chi }_1\parallel +\parallel {\chi }_2\parallel +\dots +\parallel {\chi }_{n+1}\parallel >r_0$ filling

$$\parallel \Xi (s,{\chi }_1,{\chi }_2,\dots ,{\chi }_{n+1})\parallel <{\gamma }^{\prime }(\parallel {\chi }_1\parallel +\parallel {\chi }_2\parallel +\dots +\parallel {\chi }_{n+1}\parallel ),s\in J,$$

(19)

then for $s\in J,{\chi }_1,{\chi }_2,\dots ,{\chi }_{n+1}\in E$, we have

$$\parallel \Xi (s,{\chi }_1,{\chi }_2,\dots ,{\chi }_{n+1})\parallel \le {\gamma }^{\prime }(\parallel {\chi }_1\parallel +\parallel {\chi }_2\parallel +\dots +\parallel {\chi }_{n+1}\parallel )+M,$$

(20)

where

$$M=\underset{s\in J,\parallel {\chi }_1\parallel +\parallel {\chi }_2\parallel +\dots +\parallel {\chi }_{n+1}\parallel \le r_0}{\sup }\parallel \Xi (s,{\chi }_1,{\chi }_2,\dots ,{\chi }_{n+1})\parallel <+\infty .$$

(21)

For fixed $R>0$, $\forall u\in B_R={\{u\in PC(J,E):\parallel u\parallel }_{PC}\le R\},\kappa \in J$, we have

$$\parallel u\left({\tau }_i\left(\kappa \right)\right){\parallel \le \parallel u\parallel }_{PC}\le R,i=1,2,\dots ,n-2.$$

(22)

By $\left(H_1\right)$, one has

$$\begin{array}{ll}\parallel \left(\overline{\mathcal{H}}u\right)\left(\kappa \right)\parallel =∥{\int }_0^{\kappa }\hslash (\kappa ,\tau ,u\left(\tau \right))d\tau ∥& \le {\int }_0^{\kappa }∥\hslash (\kappa ,\tau ,u(\tau \left)\right)∥d\tau \\ & \le {\int }_0^{\kappa }({\hslash }_1(\kappa ,\tau ){\parallel u\parallel }_{PC}+{\zeta }_{\hslash }(\kappa ,\tau ))d\tau \\ & \le {\hslash }_{0}R+{\overline{\hslash }}_0,t\in J,\end{array}$$

(23)

where ${\hslash }_0,{\overline{\hslash }}_0$ are as $\left(H_1\right)$. Similarly,

$$\parallel \left(\overline{\Lambda }u\right)\left(\kappa \right)\parallel =∥{\int }_0^T\rho (\kappa ,s,u\left(s\right))ds∥\le {\int }_0^T∥\rho (\kappa ,s,u(s\left)\right)∥ds\le {\rho }_{0}R+{\overline{\rho }}_0,\kappa \in J,$$

(24)

where ${\rho }_0,{\overline{\rho }}_0$ are as $\left(H_1\right)$. Hence, for $\forall \kappa \in J$, $u\in B_R$, one has

$$\begin{array}{l}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel \\ \le {\gamma }^{\prime }(\parallel u\left(s\right)\parallel +\parallel u\left({\tau }_1\right)\left(s\right)\parallel +\dots +\parallel u\left({\tau }_{n-2}\right)\left(s\right)\parallel +\parallel \left(\overline{\mathcal{H}}u\right)\left(s\right)\parallel \\ +\parallel \left(\overline{\Lambda }u\right)\left(s\right)\parallel )+M\\ \le {\gamma }^{\prime }\left((n-1)R+{\hslash }_{0}R+{\overline{\hslash }}_0+{\rho }_{0}R+{\overline{\rho }}_0\right)+M\\ ={\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M,\end{array}$$

(25)

where M is a constant, ${\gamma }^{\prime }$ is a constant and ${\gamma }^{\prime }>\gamma$. For $u\in B_R$, by the definition of operator ℵ and Formula $\left(25\right)$, for every $\kappa \in [{\kappa }_0,{\kappa }_1]$, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)\parallel \\ \le l{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )\\ +{\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le l{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )\\ +{\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le l{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(1-\alpha ),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ \times \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds.\end{array}$$

Continuing to simplify above formula, we have

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)\parallel \\ \le l{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(1-\alpha ),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ \times \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}(d_{I_i}\parallel u\parallel +{\zeta }_{I_i})\\ \le l{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ \times \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}(d_{I_i}\parallel u\parallel +{\zeta }_{I_i})\\ \le l{M}^{★}\left(\parallel u_0\parallel +d_1{\parallel u\parallel }_{PC}+{\zeta }_1\right)+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}(d_{I_i}R+{\zeta }_{I_i})\\ +\overline{d}({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+T)\left({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M\right).\end{array}$$

After further simplifying, we have

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)\parallel \\ \le \tilde{l}{M}^{★}\parallel u_0\parallel +\overline{l}{M}^{★}{\zeta }_1+\widehat{d}\left({\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M\right)+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}{\zeta }_{I_i}\\ +\left\{\tilde{l}{M}^{★}{d}_1+\widehat{d}{\gamma }^{\prime }\left(n-1+{\hslash }_0+{\rho }_0\right)+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}{d}_{I_i}\right\}R\\ \le C+K^{\prime }R\le R,\end{array}$$

(26)

where $C>0$ is a constant, $K^{\prime }$ is in (18), $\overline{d}$ is from $\left(H_5\right)$, $\tilde{l}=\max \{l,l^{m+1}\},\widehat{d}=\overline{d}({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+T)$. By the definition of operator ℵ, formula $\left(25\right)$ and the conditions $\left(H_1\right)$–$\left(H_5\right)$, for $u\in B_R$, $\kappa \in ({\kappa }_i,{\kappa }_{i+1}]$, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)\parallel \\ \le l^{m+1}{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )\\ +{\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{m\alpha l^{m+2}+\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +l{M}^{★}{\displaystyle \sum _{0<{\kappa }_i<\kappa }}\parallel I_i\left(u\left({\kappa }_i\right)\right)\parallel \\ \le l{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )\\ +{\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +l^k{M}^{★}{\displaystyle \sum _{i=1}^m}(d_{I_i}\parallel u\parallel +{\zeta }_{I_i})\\ \le l^{m+1}{M}^{★}(\parallel u_0\parallel +\parallel g(u\parallel )+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ \times \parallel \Xi \left(\kappa ,u\left(\kappa \right),u\left({\tau }_1\left(\kappa \right)\right),\dots ,u\left({\tau }_{n-2}\left(\kappa \right)\right),\left(\overline{\mathcal{H}}u\right)\left(\kappa \right),\left(\overline{\Lambda }u\right)\left(\kappa \right)\right)\parallel ds+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}(d_{I_i}\parallel u\parallel +{\zeta }_{I_i})\\ \le l^{m+1}{M}^{★}\left(\parallel u_0\parallel +d_1{\parallel u\parallel }_{PC}+{\zeta }_1\right)+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}(d_{I_i}R+{\zeta }_{I_i})\\ +\overline{d}({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+T)\left({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M\right)\\ \le \overline{l}{M}^{★}\parallel u_0\parallel +\overline{l}{M}^{★}{\zeta }_1+\widehat{d}\left({\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M\right)+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}{\zeta }_{I_i}\\ +\left\{\overline{l}{M}^{★}{d}_1+\widehat{d}{\gamma }^{\prime }\left(n-1+{\hslash }_0+{\rho }_0\right)+l^m{M}^{★}{\displaystyle \sum _{i=1}^m}{d}_{I_i}\right\}R\\ \le C+K^{\prime }R\le R,\end{array}$$

(27)

where $C>0$ is as (26), $K^{\prime }$ is in (18), $\overline{d}$ is from $\left(H_5\right)$, $\overline{l}$ is from $\left(26\right)$, $\widehat{d}=\overline{d}({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+T)$. By $\left(18\right)$, $\left(26\right)$ and $\left(27\right)$, we only need take

$$R^{★}>{(1-K^{\prime })}^{-1}C,$$

(28)

then for any $u\in B_{R^{★}}$, we have $\parallel \aleph u\parallel \le R^{★}$, that is, $\aleph :B_{R^{★}}\to B_{R^{★}}$.

As evidenced by the above arguments, $\aleph :B_{R^{★}}\to B_{R^{★}}$ is norm-bounded in the operator topology. We proceed to demonstrate the continuity of $\aleph :B_{R^{★}}\to B_{R^{★}}$ within the operator topology. Let $u_n\to u$ in $B_{R^{★}}$, then $\parallel u_n{\parallel }_{PC}\le R^{★}(n=1,2,\dots )$, ${\parallel u\parallel }_{PC}\le R^{★},$ combine the definition of $\left(17\right)$, for any $\kappa \in ({\kappa }_0,{\kappa }_1]$, one arrives

$$\begin{array}{l}\parallel \aleph u_n\left(\kappa \right)-\aleph u\left(\kappa \right)\parallel \le \parallel \mathcal{G}{\Im }_{\alpha }\left(\kappa \right)\parallel \parallel g\left(u_n\right)-g\left(u\right)\parallel \\ +∥{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}∥{\int }_0^{\kappa }\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le l{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel \\ +{\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le l{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(1-\alpha ),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\right.\\ \left.\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)-\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le l{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel \parallel \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\right.\\ \left.\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)-\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le \tilde{l}{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel \\ +\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le I+II,\end{array}$$

(29)

where $I=\tilde{l}{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel +l^m{M}^{★}{\sum }_{i=1}^m\parallel I_i\left(u_n\left({\kappa }_i\right)\right)-I_i\left(u\left({\kappa }_i\right)\right)\parallel$,

$$\begin{array}{ll}II=& \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\times \parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ & -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds,\end{array}$$

$\overline{d}$ is as Formula $\left(16\right)$.

For any $\kappa \in ({\kappa }_i,{\kappa }_{i+1}]$, one has

$$\begin{array}{l}\parallel \aleph u_n\left(\kappa \right)-\aleph u\left(\kappa \right)\parallel \le \parallel {\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)\parallel \parallel g\left(u_n\right)-g\left(u\right)\parallel \\ +∥{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥\parallel {\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right)\right.\\ \left.\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)-\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_{{\kappa }_k}^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\\ -\Xi s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\parallel ds\\ +\parallel {\displaystyle \frac{\alpha }{B\left(\alpha \right)}}\parallel {\displaystyle \sum _{i=1}^k}{\parallel \mathcal{G}\parallel }^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}∥{\int }_{{\kappa }_k}^{\kappa }\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \sum _{i=1}^k}\parallel {\mathcal{G}}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)\parallel I_i\left(u_n\left({\kappa }_i\right)\right)-I_i\left(u\left({\kappa }_i\right)\right)\parallel \\ \le l^{m+1}{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel +l^m{M}^{★}{\displaystyle \sum _{i=1}^m}∥I_i\left(u_n\left({\kappa }_i\right)\right)-I_k\left(u\left({\kappa }_i\right)\right)∥\\ +{\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le l^{m+1}{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel +l^m{M}^{★}{\displaystyle \sum _{i=1}^m}\parallel I_i\left(u_n\left({\kappa }_i\right)\right)-I_i\left(u\left({\kappa }_i\right)\right)\parallel \\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\right.\\ \left.\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)-\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ =l^{m+1}{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel +l^m{M}^{★}{\displaystyle \sum _{i=1}^m}\parallel I_i\left(u_n\left({\kappa }_i\right)\right)-I_k\left(u\left({\kappa }_i\right)\right)\parallel \\ +\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le I+II,\end{array}$$

(30)

where $I,II$ are as $\left(29\right)$, $\overline{d}$ is as Formula $\left(16\right)$.

From $\left(H_2\right)$ and $\left(H_3\right)$, we know that g and $I_k$ are continuous, then we easily get

$$I=\tilde{l}{M}^{★}\parallel g\left(u_n\right)-g\left(u\right)\parallel +l^m{M}^{★}{\displaystyle \sum _{i=1}^m}\parallel I_i\left(u_n\left({\kappa }_i\right)\right)-I_i\left(u\left({\kappa }_i\right)\right)\parallel \to 0,n\to \infty ,$$

(31)

By $\left(22\right)$–$\left(25\right)$, we have

$$\begin{array}{ll}II& =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ & -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ & \le 2\overline{d}\left({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+T\right)T({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)\mho +{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M)\\ & =2\widehat{d}T({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)\mho +{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M)\\ & <+\infty ,\end{array}$$

(32)

where $\widehat{d}$ is from formula $\left(16\right)$. From $\Xi :J\times T_R^{n+1}\to E$ is bounded and uniformly continuous, it is easy to see that for any fixed $s\in J,$

$$\begin{array}{l}\parallel ({(\kappa -s)}^{\alpha -1}+1)[\Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)]\parallel \to 0,n\to \infty ,\end{array}$$

(33)

then by $\left(32\right)$, $\left(33\right)$ and the Lebesgue control convergence theorem, we have

$$\begin{array}{ll}II& =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \Xi \left(s,u_n\left(s\right),u_n\left({\tau }_1\left(s\right)\right),\dots ,u_n\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}{u}_n\right)\left(s\right),\left(\overline{\Lambda }{u}_n\right)\left(s\right)\right)\\ & -\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ & \to 0,n\to \infty .\end{array}$$

(34)

Formulas $\left(28\right)$–$\left(34\right)$ imply that $\aleph :B_{R^{★}}\to B_{R^{★}}$ is continuous. By the above proof, we have $\aleph \left(B_{R^{★}}\right)$ is bounded. Now we prove the set $\aleph \left(B_{R^{★}}\right)\subset PC(J,E)$ is the precompactness set. Given an arbitrary but fixed $\kappa (0<\kappa \le T)$ and any $ϵ$: $0<ϵ<\kappa$, let $u\in B_{R^{★}}$, one defines

$${\aleph }_{ϵ}u\left(\kappa \right)=\left\{\begin{array}{l}\mathcal{G}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa -ϵ}{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{\kappa -ϵ}{\Re }_{\alpha }(t-s)\overline{\varrho }(s,u\left(s\right))ds,\kappa \in (0,{\kappa }_1],\\ {\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(u\left({\kappa }_i^{-}\right)\right)+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa -ϵ}{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa -ϵ}{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u\left(s\right))ds,\kappa \in ({\kappa }_i,{\kappa }_{i+1}],i=1,2,\dots ,m,\end{array}\right.$$

(35)

for

$$\overline{\varrho }(s,u\left(s\right))=\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right).$$

According to the compactness of ${\Im }_{\alpha }\left(t\right)$, for any $ϵ(0<ϵ<\kappa )$, the set $\{\left({\aleph }_{ϵ}u\right)\left(t\right):u\in B_{R^{★}}\}$ is precompact on E, for any $u\in B_{R^{★}},\kappa \in J$, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)-\left({\aleph }_{ϵ}u\right)\left(\kappa \right)\parallel \\ \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{\kappa -ϵ}^{\kappa }{(\kappa -s)}^{\alpha -1}∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ +{\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_{\kappa -ϵ}^{\kappa }∥{\Re }_{\alpha }(\kappa -s)\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{\kappa -ϵ}^{\kappa }{(\kappa -s)}^{\alpha -1}∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{\kappa -ϵ}^{\kappa }∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ \le {\displaystyle \frac{l}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}\max \{\overline{l}(1-\alpha ),\alpha l{M}^{★}\}\\ \times {\int }_{\kappa -ϵ}^{\kappa }({(\kappa -s)}^{\alpha -1}+1)∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ =\tilde{d}{\int }_{\kappa -ϵ}^{\kappa }({(\kappa -s)}^{\alpha -1}+1)∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ \le \tilde{d}[{\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R^{★}+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M]{\int }_{\kappa -ϵ}^{\kappa }({(\kappa -s)}^{\alpha -1}+1)ds\\ ={\displaystyle \frac{\tilde{d}}{\alpha }}[{\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R^{★}+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M]({ϵ}^{\alpha }+ϵ),\end{array}$$

(36)

in which $\tilde{d}={\displaystyle \frac{l}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}\max \{\overline{l}(1-\alpha ),\alpha l{M}^{★}\}$, hence, for any $\kappa \in J$, $\{(\aleph u)\left(\kappa \right):u\in B_{R^{★}}\}$ is totally bounded, that is, for any $\kappa \in J$, the set $\{(\aleph u)\left(\kappa \right):u\in B_{R^{★}}\}$ is precompact in E. By Arzela–Ascoli theorem, we have $\{\aleph u:u\in B_{R^{★}}\}$ is to be relatively compact on $PC(J,E)$. Hence, $\aleph :B_{R^{★}}\to B_{R^{★}}$ is completely continuous operator. As a result, Equation $\left(1\right)$ gives rise to at least one global mild solution on $B_{R^{★}}\subset PC(J,E)$ by the Schauder fixed point theorem. The proof is complete. □

Remark 4. 

Theorem 1 is completed under the condition of ${\Im }_{\alpha }\left(t\right)$ being compact; hence, we do not need the non-compactness measure conditions and the Lipschitz conditions.

3.2. The Well-Posedness of Mild Solutions Under the Weakened Lipschitz Conditions

The conditions for this section are as follows.

$\left(C_1\right)$ Assume that $\Xi :J\times E^{n+1}\to E$ has a continuous property, and we can find non-negative and is integrable functions with respect to Lebesgue measure $L_i\in L^p(J,{\mathbb{R}}^{+})(i=1,2,\dots ,n+1)$ with $p>{\displaystyle \frac{1}{\alpha }}$, for any $u_i,v_i\in E,\kappa \in J$, such that

$$\parallel \Xi (\kappa ,u_1,u_2,\dots ,u_{n+1})-\Xi (\kappa ,v_1,v_2,\dots ,v_{n+1})\parallel \le {\displaystyle \sum _{i=1}^{n+1}}{L}_i\left(\kappa \right)\parallel u_i-v_i\parallel .$$

(37)

$\left(C_2\right)$ There exist integrable functions $L_{\hslash }(\kappa ,·),L_{\rho }(\kappa ,·)\in L(J,{\mathbb{R}}^{+}),{\mathbb{R}}^{+}=[0,+\infty )$, for any $u,v\in E,$ such that

$$L_{{\hslash }_0}={\max }_{\kappa \in [0,T]}{\int }_0^{\kappa }{L}_{\hslash }(\kappa ,s)ds<+\infty ,L_{{\rho }_0}={\max }_{\kappa \in [0,T]}{\int }_0^T{L}_{\rho }(\kappa ,s)ds<+\infty ,$$

$$\parallel \hslash (\kappa ,s,u)-\hslash (\kappa ,s,v)\parallel \le L_{\hslash }(\kappa ,s)\parallel u-v\parallel ,(\kappa ,s)\in {\Theta }_1,$$

(38)

$$\parallel \rho (\kappa ,s,u)-\rho (\kappa ,s,v)\parallel \le L_{\rho }(\kappa ,s)\parallel u-v\parallel ,(\kappa ,s)\in {\Theta }_2.$$

(39)

$\left(C_3\right)$ It is possible to select parameters $L_{I_i}>0$, for any $u,v\in E$, subject to

$$\parallel I_i\left(u\right)-I_i\left(v\right)\parallel \le L_{I_i}\parallel u-v\parallel ,i=1,2,\dots ,m.$$

(40)

$\left(C_4\right)$ It is possible to select a parameter $L_g>0$, for any $u,v\in PC(J,E)$, such that

$$\parallel g\left(u\right)-g\left(v\right)\parallel \le L_g{\parallel u-v\parallel }_{PC}.$$

(41)

$\left(C_5\right)$ For the Lebesgue integrable function $L_i\in L^p(J,R^{+})(i=1,2,\dots ,n+1)$ in $\left(C_1\right)$, the constants in $\left(C_1\right)$–$\left(C_4\right)$ and $\left(H_4\right)$ satisfy

$$\varrho =M^{★}\tilde{l}\left(L_g+{\displaystyle \sum _{i=1}^m}{L}_{I_k}\right)+\overline{d}\iota <1,$$

(42)

where $\overline{d},\tilde{l}$ are as $\left(16\right)$ and

$$\iota =l_0{\left({\int }_0^T{L}^p\left(s\right)ds\right)}^{{\displaystyle \frac{1}{p}}},$$

where $L_{{\hslash }_0},L_{{\rho }_0}$ are from $\left(C_2\right)$,

$$L\left(\kappa \right)={\displaystyle \sum _{i=1}^{n-1}}{L}_i\left(\kappa \right)+L_{{\hslash }_0}{L}_n\left(\kappa \right)+L_{{\rho }_0}{L}_{n+1}\left(\kappa \right),l_0=2^{{\displaystyle \frac{q-1}{q}}}{\left({\displaystyle \frac{T^{(\alpha -1)q+1}}{(\alpha -1)q+1}}+T\right)}^{{\displaystyle \frac{1}{q}}},{\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1.$$

$\left(C_6\right)$ The constants $L_{I_k}$ in $\left(C_3\right)$ and l in $\left(H_4\right)$, satisfy

$$l^m{M}^{★}{\displaystyle \sum _{i=1}^m}{L}_{I_i}=j<1.$$

Lemma 13. 

Let $0<\alpha <1$, suppose that $L\in L^p(J,{\mathbb{R}}^{+})$ with $p>{\displaystyle \frac{1}{\alpha }}$, then for every $\kappa \in J$

$${\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)L\left(s\right)ds\le l_0{\left({\int }_0^T{L}^p\left(s\right)ds\right)}^{{\displaystyle \frac{1}{p}}}.$$

Proof. 

By the Hölder inequality and Minkowski inequality, one has

$$\begin{array}{ll}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)L\left(s\right)ds& \le {\left({\int }_0^{\kappa }{({(\kappa -s)}^{\alpha -1}+1)}^{q}ds\right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_0^{\kappa }{L}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}\\ & \le 2^{{\displaystyle \frac{q-1}{q}}}{\left({\int }_0^{\kappa }\left({(\kappa -s)}^{q(\alpha -1)}+1^q\right)ds\right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_0^T{L}^p\left(s\right)ds\right)}^{{\displaystyle \frac{1}{p}}}\\ & =2^{{\displaystyle \frac{q-1}{q}}}{\left({\displaystyle \frac{T^{(\alpha -1)q+1}}{(\alpha -1)q+1}}+T\right)}^{{\displaystyle \frac{1}{q}}}{\left({\int }_0^T{L}^p\left(s\right)ds\right)}^{{\displaystyle \frac{1}{p}}}\\ & =l_0{\left({\int }_0^T{L}^p\left(s\right)ds\right)}^{{\displaystyle \frac{1}{p}}},\end{array}$$

where $l_0=2^{{\displaystyle \frac{q-1}{q}}}{\left({\displaystyle \frac{T^{(\alpha -1)q+1}}{(\alpha -1)q+1}}+T\right)}^{{\displaystyle \frac{1}{q}}}$, ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$. □

Theorem 2. 

Suppose conditions $\left(C_1\right)$–$\left(C_5\right)$ and $\left(H_4\right)$ hold, then the model $\left(1\right)$ admits exactly one globally mild solution in the phase space $PC(J,E)$, for every $u_0\in PC(J,E)$, the sequence of iterations $\left\{u_n\right\}$ be defined by

$$u_n\left(\kappa \right)=\left\{\begin{array}{l}\mathcal{G}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u\right))+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u_{n-1}\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^t{\Re }_{\alpha }(t-s)\overline{\varrho }(s,u_{n-1}\left(s\right))ds,\kappa \in (0,{\kappa }_1],\\ {\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(u_{n-1}\right))+{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\overline{\varrho }(s,u_{n-1}\left(s\right))ds\\ +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(t-s)\overline{\varrho }(s,u_{n-1}\left(s\right))ds\\ +{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(u\left({\kappa }_i^{-}\right)\right)+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{(\kappa -s)}^{\alpha -1}\overline{\varrho }(s,u_{n-1}\left(s\right))ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{\Re }_{\alpha }(\kappa -s)\overline{\varrho }(s,u_{n-1}\left(s\right))ds,\kappa \in ({\kappa }_i,{\kappa }_{i+1}),i=1,2,\dots ,m,\end{array}\right.$$

(43)

for

$$\overline{\varrho }(s,u\left(s\right))=\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right),$$

uniformly converges to a unique global mild solution $u^{★}\left(t\right)$ of the model $\left(1\right)$ about the norm of E on $t\in J$ and for ϱ that defined by $\left(42\right)$, one has

$$\parallel u_n-u^{★}\parallel =o\left({\varrho }^n\right),n\to +\infty .$$

(44)

Proof. 

For every $\kappa \in [{\kappa }_0,{\kappa }_1]({\kappa }_0=0),u,v\in PC(J,E)$, by Lemma 13, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)-(\aleph v)\left(\kappa \right)\parallel \\ \le l{M}^{★}\parallel g\left(u\right)-g\left(v\right)\parallel \\ +{\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\right.\\ \left.\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }∥{\Re }_{\alpha }(\kappa -s)∥\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\right.\\ \left.\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ \le l{M}^{★}{L}_g{\parallel u-v\parallel }_{PC}+{\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}(L_1{\left(s\right)\parallel u-v\parallel }_{PC}\\ +L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}\\ +L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}+L_{n+1}\left(s\right){\parallel \Lambda u-\Lambda v\parallel }_{PC})ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }(L_1{\left(s\right)\parallel u-v\parallel }_{PC}+L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots \\ +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}+L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}+L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds\\ \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(1-\alpha ),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1{\left(s\right)\parallel u-v\parallel }_{PC}\\ +L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}+L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}\\ +L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds+M^{★}l{\displaystyle \sum _{k=1}^m}{L}_{I_k}{\parallel u-v)\parallel }_{PC}+l{M}^{★}{L}_g{\parallel u-v\parallel }_{PC}\\ =\tilde{l}{M}^{★}{L}_g{\parallel u-v\parallel }_{PC}+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1{\left(s\right)\parallel u-v\parallel }_{PC}+L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots \\ +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}+L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}+L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds\\ \le M^{★}\tilde{l}{L}_g{\parallel u-v\parallel }_{PC}\\ +\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\left(L_1\left(s\right)+L_2\left(s\right)+\dots +L_{{\hslash }_0}{L}_n\left(s\right)+L_{{\rho }_0}{L}_{n+1}\left(s\right)\right){\parallel u-v\parallel }_{PC}ds\\ \le M^{★}\tilde{l}{L}_g{\parallel u-v\parallel }_{PC}+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1){L\left(s\right)ds·\parallel u-v\parallel }_{PC}\\ \le \left(M^{★}\tilde{l}{L}_g+\overline{d}\iota \right){\parallel u-v\parallel }_{PC}\le \varrho {\parallel u-v\parallel }_{PC},\end{array}$$

(45)

where $\overline{d},\tilde{l},\iota ,\varrho ,L\left(s\right)$ are from the condition $\left(C_5\right)$, $L_{{\hslash }_0},L_{{\rho }_0}$ are from $\left(C_2\right)$.

For any $\kappa \in ({\kappa }_i,{\kappa }_{i+1}],u,v\in PC(J,E)$, as the same way with $\left(45\right)$, by Lemma 13, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)-(\aleph v)\left(\kappa \right)\parallel \\ \le l^{m+1}{M}^{★}\parallel g\left(u\right)-g\left(v\right)\parallel \\ +{\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\right.\\ \left.\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{m\alpha l^{m+2}+\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }∥{\Re }_{\alpha }(\kappa -s)∥\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\right.\\ \left.\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ +l^m\parallel {\displaystyle \sum _{0<{\kappa }_i<t}}{\Im }_{\alpha }(\kappa -{\kappa }_i)\left(I_k\left(u\left({\kappa }_k\right)\right)-I_k\left(v\left({\kappa }_k\right)\right)\right)\parallel \\ \le l^{m+1}{M}^{★}{L}_g{\parallel u-v\parallel }_{PC}+{\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}(L_1{\left(s\right)\parallel u-v\parallel }_{PC}\\ +L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}\\ +L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}+L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds\\ +{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }(L_1{\left(s\right)\parallel u-v\parallel }_{PC}+L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots \\ +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}+L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}+L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds\\ +M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}\parallel u\left({\kappa }_i\right)-v\left({\kappa }_i\right)\parallel \\ \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1{\left(s\right)\parallel u-v\parallel }_{PC}\\ +L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}+L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}\\ +L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds+M^{★}l{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\parallel u-v)\parallel }_{PC}+l{M}^{★}{L}_g{\parallel u-v\parallel }_{PC}\\ =l^{m+1}{M}^{★}{L}_g{\parallel u-v\parallel }_{PC}+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1{\left(s\right)\parallel u-v\parallel }_{PC}+L_2{\left(s\right)\parallel u-v\parallel }_{PC}+\dots \\ +L_{n-1}{\left(s\right)\parallel u-v\parallel }_{PC}+L_n\left(s\right)\parallel \overline{\mathcal{H}}u-\overline{\mathcal{H}}{v\parallel }_{PC}+L_{n+1}\left(s\right){\parallel \overline{\Lambda }u-\overline{\Lambda }v\parallel }_{PC})ds\\ +M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\parallel u-v)\parallel }_{PC}\\ \le M^{★}{l}^{m+1}{L}_g{\parallel u-v\parallel }_{PC}+M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\parallel u-v\parallel }_{PC}+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)L\left(s\right){ds·\parallel u-v\parallel }_{PC}\\ \le M^{★}\tilde{l}{L}_g{\parallel u-v\parallel }_{PC}+M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\parallel u-v\parallel }_{PC}+\overline{d}{l}_0{\left({\int }_0^T{L}^p\left(s\right)ds\right)}^{{\displaystyle \frac{1}{p}}}{·\parallel u-v\parallel }_{PC}\\ \le \left(M^{★}\tilde{l}\left(L_g+{\displaystyle \sum _{i=1}^m}{L}_{I_k}\right)+\overline{d}\iota \right){\parallel u-v\parallel }_{PC}\\ =\varrho {\parallel u-v\parallel }_{PC},\end{array}$$

(46)

where $\overline{d},\tilde{l},\iota ,\varrho ,L\left(s\right)$ are from the condition $\left(C_5\right)$. Hence, by $\left(45\right)$ and $\left(46\right)$, we have ${\parallel \aleph u-\aleph v\parallel }_{PC}\le \varrho {\parallel u-v\parallel }_{PC}$. According to $\left(C_5\right)$, ℵ admits a unique fixed point $u^{★}\in PC(J,E)$ by using of the Banach compression fixed point theorem, that is, a mild solution is admitted for model $\left(1\right)$, and subsequently $\left(44\right)$ holds. □

Remark 5. 

Theorem 2 is proved by the Banach compression principle under the generalized Lipschitz conditions.

Theorem 3. 

Suppose the condition $\left(C_1\right)$–$\left(C_3\right)$ and $\left(C_6\right),\left(H_4\right)$ hold, $g\left(u\right)=0$, $L_{n+1}\left(\kappa \right)\equiv 0$ in $\left(C_1\right)$, then the model $\left(4\right)$ admits a unique global mild solution in $PC(J,E)$, that means model $\left(1\right)$ admits a unique global mild solution; moreover, for any $u_0\in PC(J,E)$, the iterative sequence $\left\{u_n\right\}$ defined by (43) uniformly converges to a unique global mild solution $u^{★}\left(t\right)$ of the model $\left(4\right)$ about the norm of E on $t\in J$; that means model $\left(1\right)$ admits a unique mild solution, and for any real number $s>1$, one has

$$\parallel u_n-u^{★}\parallel =o\left({\displaystyle \frac{1}{n^s}}\right),n\to +\infty .$$

Proof. 

For every $\kappa \in [{\kappa }_0,{\kappa }_1]({\kappa }_0=0),u,v\in PC(J,E)$, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)-(\aleph v)\left(\kappa \right)\parallel \\ \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\right.\\ \left.\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}(L_1\left(s\right){∥u-v∥}_{PC}+L_2\left(s\right){∥u-v∥}_{PC}+\dots +L_{n-1}\left(s\right){∥u-v∥}_{PC}\\ +L_n\left(s\right){∥\overline{\mathcal{H}}u-\overline{\mathcal{H}}v∥}_{PC})ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }(L_1\left(s\right){∥u-v∥}_{PC}+L_2\left(s\right){∥u-v∥}_{PC}+\dots +L_{n-1}\left(s\right){∥u-v∥}_{PC}\\ +L_n\left(s\right)∥\left(\overline{\mathcal{H}}u\right)\left(s\right)-\left(\overline{\mathcal{H}}v\right)\left(s\right)∥)ds\\ \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(1-\alpha ),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right)+L_2\left(s\right)+\dots \\ +L_{{\hslash }_0}{L}_n\left(s\right)){\parallel u-v\parallel }_{PC}ds\\ =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\overline{B}{\left(s\right)\parallel u\left(s\right)-v\left(s\right)\parallel ds+j\parallel u-v\parallel }_{PC},\end{array}$$

(47)

where $\overline{d}$ is shown in $\left(C_5\right)$, $j=l^m{M}^{★}{\sum }_{i=1}^m{L}_{I_i}<1$ in $\left(C_6\right),$ $B\left(\kappa \right)=L_1\left(\kappa \right)+L_2\left(\kappa \right)+L_3\left(\kappa \right)+\dots +L_{n-1}\left(\kappa \right)+L_n\left(\kappa \right)L_{{\hslash }_0}$.

For any $\kappa \in ({\kappa }_i,{\kappa }_{i+1}](i=1,2,\dots ,m),u,v\in PC(J,E)$, one has

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)-(\aleph v)\left(\kappa \right)\parallel \\ \le {\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ +{\displaystyle \frac{m\alpha l^{m+2}+\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }\parallel {\Re }_{\alpha }(\kappa -s)\parallel \parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\right.\\ \left.\left(\overline{\Lambda }u\right)\left(s\right)\right)-\Xi \left(s,v\left(s\right),v\left({\tau }_1\left(s\right)\right),\dots ,v\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}v\right)\left(s\right),\left(\overline{\Lambda }v\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_k)(I_k\left(u\left({\kappa }_k\right)\right)-I_k\left(v\left({\kappa }_k\right)\right))∥\\ \le {\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}(L_1\left(s\right){∥u-v∥}_{PC}+L_2\left(s\right){∥u-v∥}_{PC}+\dots +L_{n-1}\left(s\right){∥u-v∥}_{PC}\\ +L_n\left(s\right){∥u-v∥}_{PC})ds\\ +{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }(L_1\left(s\right){∥u-v∥}_{PC}+L_2\left(s\right){∥u-v∥}_{PC}+\dots +L_{n-1}\left(s\right){∥u-v∥}_{PC}\\ +L_n\left(s\right)∥\left(\overline{\mathcal{H}}u\right)\left(s\right)-\left(\overline{\mathcal{H}}v\right)\left(s\right)∥)ds+M^{★}{l}^m{\displaystyle \sum _{i=1}^k}{L}_{I_i}{∥u-v∥}_{PC}\\ \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right)+L_2\left(s\right)+\dots \\ +L_{{\hslash }_0}{L}_n\left(s\right)){\parallel u-v\parallel }_{PC}ds+M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\parallel u-v)\parallel }_{PC}\\ =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\overline{B}\left(s\right)\parallel u\left(s\right)-v\left(s\right){\parallel ds+j\parallel u-v\parallel }_{PC},\\ \end{array}$$

(48)

where $\overline{d},\overline{B}\left(t\right),j$ are from $\left(47\right)$.

Then, by $\left(47\right)$ and $\left(48\right)$, for any $\kappa \in J,u,v\in PC(J,E)$, one has

$$\parallel (\aleph u)\left(\kappa \right)-(\aleph v)\left(\kappa \right)\parallel \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\overline{B}{\left(s\right)\parallel u\left(s\right)-v\left(s\right)\parallel ds+j\parallel u-v\parallel }_{PC}.$$

(49)

By $0<j<1$, there exists $0<{ϵ}_1<1$, such that $0<\mathsf{\ell }={ϵ}_1+j<1$. From $\overline{B}\left(\kappa \right)=L_1\left(\kappa \right)+L_2\left(\kappa \right)+L_3\left(\kappa \right)+\dots +L_{n-1}\left(\kappa \right)+L_n\left(\kappa \right)L_{{\hslash }_0}\in L^p(J,R)$, there has a continuous functions $\varphi :J\to R^{+}$, such that

$$\overline{d}{l}_0{\left({\int }_0^T{|\overline{B}\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}<{ϵ}_1.$$

(50)

By the definition of the operator ℵ and $\left(49\right),\left(50\right)$ and Lemma 13, for any $\kappa \in J,u,v\in PC(J,E)$, one arrives

$$\begin{array}{l}\parallel (\aleph u)\left(\kappa \right)-(\aleph v)\left(\kappa \right)\parallel \\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)|\overline{B}\left(s\right)-\varphi \left(s\right)|\parallel u\left(s\right)-v\left(s\right)\parallel ds+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\left|\varphi \left(s\right)\right|\parallel u\left(s\right)-v\left(s\right)\parallel ds\\ { +j\parallel u-v\parallel }_{PC}\\ \le \overline{d}{l}_0{\left({\int }_0^T{|\overline{B}\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}{\parallel u-v\parallel }_{PC}+\overline{d}♭l_0{\left({\int }_0^{\kappa }ds\right)}^{{\displaystyle \frac{1}{q}}}{\parallel u-v\parallel }_{PC}{+j\parallel u-v\parallel }_{PC}\\ \le \overline{d}{l}_0{\left({\int }_0^T{|\overline{B}\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}{\parallel u-v\parallel }_{PC}+\overline{d}♭l_0\left({\int }_0^{\kappa }ds\right){\parallel u-v\parallel }_{PC}{+j\parallel u-v\parallel }_{PC}\\ \le {ϵ}_1{\parallel u-v\parallel }_{PC}+\overline{♭}{t\parallel u-v\parallel }_{PC}{+j\parallel u-v\parallel }_{PC}\\ \le (\mathsf{\ell }+\overline{♭}\kappa ){\parallel u-v\parallel }_{PC}=\left(C_1^0\mathsf{\ell }+C_1^1{\displaystyle \frac{\overline{♭}\kappa }{1!}}\right){\parallel u-v\parallel }_{PC},\end{array}$$

where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, $♭=\max \left\{\right|\varphi \left(\kappa \right)|:\kappa \in [0,T]\},\overline{♭}=\overline{d}♭l_0$. Let us prove it by mathematical induction. For any natural number $n\in N,\kappa \in J$, one has

$$\parallel \left({\aleph }^{n}u\right)\left(\kappa \right)-\left({\aleph }^{n}v\right)\left(\kappa \right)\parallel \le \left({\mathsf{\ell }}^n+C_n^1{\mathsf{\ell }}^{n-1}\left(\overline{♭}\kappa \right)+{\displaystyle \frac{C_n^2{\mathsf{\ell }}^{n-2}{\left(\overline{♭}\kappa \right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}\kappa \right)}^n}{n!}}\right){\parallel u-v\parallel }_{PC}.$$

(51)

Assume Formula (51) holds when $n=i$, that is, the following Formula $\left(52\right)$ holds as follows:

$$\parallel \left({\aleph }^{i}u\right)\left(\kappa \right)-\left({\aleph }^{i}v\right)\left(\kappa \right)\parallel \le \left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}\kappa \right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}\kappa \right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}\kappa \right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}.$$

(52)

Now we prove the Formula (51) hold when $n=i+1$. By the definition of the operator ℵ, and the conditions $\left(C_1\right)$–$\left(C_4\right)$ and $\left(C_6\right),\left(H_5\right)$, for any $\kappa \in J$, one arrives

$$\begin{array}{l}\parallel \left({\aleph }^{i+1}u\right)\left(\kappa \right)-\left({\aleph }^{i+1}v\right)\left(\kappa \right)\parallel =\parallel \aleph \left({\aleph }^{i}u\right)\left(\kappa \right)-\aleph \left({\aleph }^{i}v\right)\left(\kappa \right)\parallel \\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1))L_1\left(s\right)\parallel \left({\aleph }^{i}u\right)\left(s\right)-{\aleph }^{i}v)\left(s\right)\parallel \\ +L_2\left(s\right)\parallel ({\aleph }^{i}u\left({\tau }_1\right)\left(s\right)-{\aleph }^{i}v\left({\tau }_1\left(s\right)\right)\parallel +\dots +L_{n-1}\left(s\right)\parallel ({\aleph }^{i}u\left({\tau }_{n-2}\right)\left(s\right)-{\aleph }^{i}v\left({\tau }_{n-2}\left(s\right)\right)\parallel \\ +L_n\left(s\right)∥(\overline{\mathcal{H}}{\aleph }^{i}u\left(s\right)-\left(\overline{\mathcal{H}}{\aleph }^{i}v\right)\left(s\right)∥)ds\\ +M^{★}l{\displaystyle \sum _{i=1}^m}{L}_{I_i}\parallel {\aleph }^{i}u\left({\kappa }_i\right)-{\aleph }^{i}v\left({\kappa }_i\right)\parallel \\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)|\overline{B}\left(s\right)-\varphi \left(s\right)|ds\parallel \left({\aleph }^{i}u\right)-{\aleph }^{i}v)\parallel +♭\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\parallel \left({\aleph }^{i}u\right)\left(s\right)-\left({\aleph }^{i}v\right)\left(s\right)\parallel ds\\ +j\parallel \left({\aleph }^{i}u\right)-{\aleph }^{i}v)\parallel \\ \le \overline{d}{l}_0{\left({\int }_0^T{|\overline{B}\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}ds\parallel \left({\aleph }^{i}u\right)-{\aleph }^{i}v)\parallel +♭\overline{d}{l}_0{\left({\int }_0^{\kappa }ds\right)}^{{\displaystyle \frac{1}{q}}}\parallel \left({\aleph }^{i}u\right)-{\aleph }^{i}v)\parallel \\ +j\parallel \left({\aleph }^{i}u\right)-{\aleph }^{i}v)\parallel \\ \le \overline{d}{l}_0{\left({\int }_0^T{|\overline{B}\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}ds\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}\\ +\overline{♭}{\int }_0^t\left({\mathsf{\ell }}^i+C_k^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}ds\\ +j\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}\\ \le {ϵ}_1\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}\\ +\overline{♭}{\int }_0^t\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}ds\\ +j\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}\\ \le \mathsf{\ell }\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){\parallel u-v\parallel }_{PC}\\ +\overline{♭}{\int }_0^t\left({\mathsf{\ell }}^i+C_i^1{\mathsf{\ell }}^{i-1}\left(\overline{♭}s\right)+{\displaystyle \frac{C_i^2{\mathsf{\ell }}^{i-2}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♭}s\right)}^i}{i!}}\right){ds\parallel u-v\parallel }_{PC}\\ \le \left(C_{i+1}^0{\mathsf{\ell }}^{i+1}+C_{i+1}^1{\mathsf{\ell }}^i\left(\overline{♭}s\right)+{\displaystyle \frac{C_{i+1}^2{\mathsf{\ell }}^{i-1}{\left(\overline{♭}s\right)}^2}{2!}}+\dots +C_{i+1}^{i+1}{\displaystyle \frac{{\left(\overline{♭}s\right)}^{i+1}}{(i+1)!}}\right){\parallel u-v\parallel }_{PC},\end{array}$$

(53)

hence, for any natural number $n\in \mathbb{N}$, the Formula $\left(51\right)$ holds. Then, let natural number $n\in \mathbb{N}$ be arbitrary, one has

$$\parallel \left({\aleph }^{n}u\right)-\left({\aleph }^{n}v\right){\parallel }_{PC}\le \left({\mathsf{\ell }}^n+C_n^1{\mathsf{\ell }}^{n-1}(♭T)+{\displaystyle \frac{C_n^2{\mathsf{\ell }}^{n-2}{(♭T)}^2}{2!}}+\dots +{\displaystyle \frac{{(♭T)}^n}{n!}}\right){\parallel u-v\parallel }_{PC}.$$

By Lemma 8, for $s>1$, one has

$$\parallel {\aleph }^{n}u-{\aleph }^n{v\parallel }_{PC}=o\left({\displaystyle \frac{1}{n^s}}\right){\parallel u-v\parallel }_{PC},n\to \infty .$$

(54)

Hence, by the generalized Banach compression fixed point theorem, ℵ possesses a solitary fixed point $u^{★}$ on $PC(J,E)$, then the model (1) admits a unique global mild solution $u^{★}\left(t\right)$, and $\left(54\right)$ holds. □

Remark 6. 

Theorem 3 is proved by the generalized Banach compression principle under the condition of $g=0$ and $L_{n+1}\left(\kappa \right)\equiv 0$ in $\left(C_5\right)$. Moreover, the condition $\left(C_6\right)$ of Theorem 3 is weaker than the condition $\left(C_5\right)$ of Theorem 2. Hence, compared with related literature, this part generalizes the constants L of the Lipschitz conditions to the Lebesgue integrable functions $L_i\in L^p(J,R^{+})(i=1,2,\dots ,n+1)$ with $p>{\displaystyle \frac{1}{\alpha }}$.

3.3. The Attainability of Mild Solutions Is Established via Non-Compactness Criteria

For this part, we need to add the following conditions.

$\left(H_6\right)$ There exist integrable functions $L_{\hslash }(\kappa ,·),L_{\rho }(\kappa ,·)\in L(J,{\mathbb{R}}^{+}),{\mathbb{R}}^{+}=[0,+\infty )$, for every bounded set $S\subset E$, satisfies

$$\begin{array}{c}{L}_{{\hslash }_0}={\max }_{t\in [0,T]}{\int }_0^{\kappa }{L}_{\hslash }(\kappa ,s)ds<+\infty ,L_{{\rho }_0}={\max }_{\kappa \in [0,T]}{\int }_0^T{L}_{\rho }(\kappa ,s)ds<+\infty ,\\ \omega (\hslash (\kappa ,s,S))\le L_{\hslash }(\kappa ,s)\omega \left(S\right),(\kappa ,s)\in {\Theta }_1\omega \left(\rho (\kappa ,s,S)\right)\le L_{\rho }(\kappa ,s)\omega \left(S\right),(\kappa ,s)\in {\Theta }_2.\end{array}$$

(55)

$\left(H_7\right)$ For appropriately chosen $L_{I_k}>0(k=0,1,\dots ,m)$, given any bounded set $S\subset E$, satisfies

$$\omega \left(I_k\left(S\right)\right)\le L_{I_k}\omega \left(S\right),k=0,1,\dots ,m.$$

(56)

$\left(H_8\right)$ There exists a constant $L_g>0$, for any bounded set $S^{★}\subset PC(J,E)$, satisfies

$$\omega \left(g\left(S^{★}\right)\right)\le L_g{\omega }_{PC}\left(S^{★}\right).$$

(57)

$\left(H_9\right)$ There exist integrable function $L_i\in L^p(J,{\mathbb{R}}^{+})(i=1,2,\dots ,n+1)$ with $p>{\displaystyle \frac{1}{\alpha }}$, for any bounded set $S_i\subset E$, $\kappa \in J$, such that

$$\omega (\Xi (\kappa ,S_1,S_2,S_3,\dots ,S_{n+1}))\le {\displaystyle \sum _{i=1}^{n+1}}{L}_i\left(t\right)\omega \left(S_i\right).$$

(58)

Theorem 4. 

Suppose the condition $\left(H_1\right)$–$\left(H_9\right)$ and $\left(C_5\right)$ hold, then the model $\left(1\right)$ admits at least one global mild solution in $B_{R^{★}}\subset PC(J,E)$.

Proof. 

As demonstrated in the study of Theorem 1, we have $\aleph :\overline{Co}\aleph \left(B_{R^{★}}\right)\to \overline{Co}\aleph \left(B_{R^{★}}\right)\subset B_{R^{★}}$ is bounded and continuous operator,

Next we show the set $\aleph B_{R^{★}}$ is equicontinuous. In fact, for any $\forall u\in B_{R^{★}},$${\kappa }^{\prime },{\kappa }^{″}\in J_0,{\kappa }^{\prime }\le {\kappa }^{″}$, one arrives

$$\begin{array}{l}\parallel \aleph u\left({\kappa }^{″}\right)-\aleph u\left({\kappa }^{\prime }\right)\parallel =\parallel \mathcal{G}{\Im }_{\alpha }\left({\kappa }^{″}\right)(u_0+g\left(u\right))-\mathcal{G}{\Im }_{\alpha }\left({\kappa }^{\prime }\right)(u_0+g\left(u\right))\\ +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{{\kappa }^{″}}{({\kappa }^{″}-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ -{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{{\kappa }^{\prime }}{({\kappa }^{\prime }-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{{\kappa }^{″}}{\Re }_{\alpha }({\kappa }^{″}-s)\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ -{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{{\kappa }^{\prime }}{\Re }_{\alpha }({\kappa }^{\prime }-s)\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ \le l\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel \parallel u_0+g\left(u\right)\parallel \\ +∥{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_0^{{\kappa }^{\prime }}({({\kappa }^{″}-s)}^{\alpha -1}-{({\kappa }^{\prime }-s)}^{\alpha -1})∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ +∥{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_{{\kappa }^{\prime }}^{{\kappa }^{″}}{({\kappa }^{″}-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ +∥{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}∥{\int }_0^{{\kappa }^{\prime }}∥{\Re }_{\alpha }({\kappa }^{″}-s)-{\Re }_{\alpha }({\kappa }^{\prime }-s)∥\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}∥{\int }_{{\kappa }^{\prime }}^{{\kappa }^{″}}∥{\Re }_{\alpha }({\kappa }^{″}-s)∥\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ \le \tilde{l}\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel (\parallel u_0\parallel +d_1\parallel u\parallel +{\zeta }_1)+{{\rm Y}}_1+{{\rm Y}}_2+{{\rm Y}}_3+{{\rm Y}}_4.\end{array}$$

(59)

For any $\forall u\in B_{R^{★}},$${\kappa }^{\prime },{\kappa }^{″}\in J_i,{\kappa }^{\prime }\le {\kappa }^{″}$, one reaches

$$\begin{array}{l}\parallel \aleph u\left({\kappa }^{″}\right)-\aleph u\left({\kappa }^{\prime }\right)\parallel =\parallel {\mathcal{G}}^{k+1}{\Im }_{\alpha }\left({\kappa }^{″}\right)(u_0+g\left(u\right))-{\mathcal{G}}^{k+1}{\Im }_{\alpha }\left({\kappa }^{\prime }\right)(u_0+g\left(u\right))\\ +{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}({\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds+{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_k}^{{\kappa }^{″}}{({\kappa }^{″}-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ -{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_k}^{{\kappa }^{\prime }}{({\kappa }^{\prime }-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_k}^{{\kappa }^{″}}{\Re }_{\alpha }({\kappa }^{″}-s)\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ -{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_k}^{{\kappa }^{\prime }}{\Re }_{\alpha }({\kappa }^{\prime }-s)\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ +{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)-{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }({\kappa }^{\prime }-t_k)I_k\left(u\left({\kappa }_k\right)\right)\parallel \\ \le l^{m+1}\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel \parallel u_0+g\left(u\right)\parallel \\ +∥{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥\parallel {\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}({\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\parallel +∥{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_{{\kappa }_k}^{{\kappa }^{\prime }}({({\kappa }^{″}-s)}^{\alpha -1}-{({\kappa }^{\prime }-s)}^{\alpha -1})\\ \times ∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ +∥{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_{{\kappa }^{\prime }}^{{\kappa }^{″}}{({\kappa }^{″}-s)}^{\alpha -1}\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ +∥{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}∥{\int }_{{\kappa }_k}^{{\kappa }^{\prime }}∥{\Re }_{\alpha }({\kappa }^{″}-s)-{\Re }_{\alpha }({\kappa }^{\prime }-s)∥\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}∥{\int }_{{\kappa }^{\prime }}^{{\kappa }^{″}}∥{\Re }_{\alpha }({\kappa }^{″}-s)∥\parallel \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ +∥{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)-{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)∥\\ \le \tilde{l}\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel (\parallel u_0\parallel +d_1\parallel u\parallel +{\zeta }_1)+{{\rm Y}}_0+{{\rm Y}}_1+{{\rm Y}}_2+{{\rm Y}}_3+{{\rm Y}}_4+{{\rm Y}}_5\\ \le \tilde{l}\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel (\parallel u_0\parallel +d_1{R}^{★}+{\zeta }_1)+{{\rm Y}}_0+{{\rm Y}}_1+{{\rm Y}}_2+{{\rm Y}}_3+{{\rm Y}}_4+{{\rm Y}}_5.\end{array}$$

(60)

First, since ${\Im }_{\alpha }(.)$ is bounded and strongly continuous (Lemma 11), one arrives $\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel \to 0$ as ${\kappa }^{\prime }\to {\kappa }^{″},$ this means $\tilde{l}\parallel {\Im }_{\alpha }\left({\kappa }^{″}\right)-{\Im }_{\alpha }\left({\kappa }^{\prime }\right)\parallel (\parallel u_0\parallel +d_1{R}^{★}+{\zeta }_1)\to 0$. From (25), for $u\in B_{R^{★}}$, ${\kappa }^{\prime },{\kappa }^{″}\in J_i$, we get

$$\begin{array}{ll}{{\rm Y}}_0& =∥{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}({\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\\ & \times \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\\ & \le {\displaystyle \frac{\overline{l}(1-\alpha )l^{m+1}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R^{★}+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M){\displaystyle \sum _{i=1}^m}\parallel {\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)\parallel {\int }_0^T{(\kappa -s)}^{\alpha -1}ds\\ & ={\displaystyle \frac{l^m\wp }{\alpha }}{\displaystyle \sum _{i=1}^m}\parallel {\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)\parallel ,\end{array}$$

(61)

where $\wp ={\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R^{★}+{\gamma }^{\prime }({\zeta }_{\hslash }T+{\zeta }_{\rho }T)+M)$. Similarly, from (25), for $u\in B_{R^{★}}$, ${\kappa }^{\prime },{\kappa }^{″}\in J_i$, we have

$$\begin{array}{ll}{{\rm Y}}_1& \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}∥{\int }_0^{{\kappa }^{\prime }}\left({({\kappa }^{\prime }-s)}^{\alpha -1}-{({\kappa }^{″}-s)}^{\alpha -1}\right)\\ & \times \Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)ds\parallel \\ & \le \wp {\int }_0^{{\kappa }^{\prime }}({({\kappa }^{\prime }-s)}^{\alpha -1}-{({\kappa }^{″}-s)}^{\alpha -1})ds\\ & \le {\displaystyle \frac{\wp }{\alpha }}\left(({\left({\kappa }^{″}\right)}^{\alpha }-{\left({\kappa }^{\prime }\right)}^{\alpha })+{({\kappa }^{″}-{\kappa }^{\prime })}^{\alpha }\right),\end{array}$$

(62)

where ℘ is from $\left(61\right)$. Since ${\kappa }^{\alpha }$ has uniformly continuous property on J, we have ${\left({\kappa }^{″}\right)}^{\alpha }-{\left({\kappa }^{\prime }\right)}^{\alpha }\to 0$ as ${\kappa }^{\prime }\to {\kappa }^{″}$, simply, we have ${({\kappa }^{″}-{\kappa }^{\prime })}^{\alpha }\to 0$ as ${\kappa }^{\prime }\to {\kappa }^{″}$, that means ${\lim }_{{\kappa }^{\prime }\to {\kappa }^{″}}{{\rm Y}}_1=0$. Moreover,

$$\begin{array}{ll}{{\rm Y}}_2& ={\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }^{\prime }}^{{\kappa }^{″}}{({\kappa }^{″}-s)}^{\alpha -1}∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\right.\left.\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)\parallel ds\\ & \le {\displaystyle \frac{\wp }{\alpha }}{({\kappa }^{″}-{\kappa }^{\prime })}^{\alpha }\to 0as{\kappa }^{\prime }\to {\kappa }^{″},\end{array}$$

(63)

that means ${\lim }_{{\kappa }^{\prime }\to {\kappa }^{″}}{{\rm Y}}_2=0$. In addition,

$$\begin{array}{ll}{{\rm Y}}_3& \le {\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{{\kappa }^{\prime }}\parallel {\Re }_{\alpha }({\kappa }^{″}-s)-{\Re }_{\alpha }({\kappa }^{\prime }-s)\parallel ∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ & \le \overline{\wp }{\int }_0^T\parallel {\Re }_{\alpha }({\kappa }^{″}-s)-{\Re }_{\alpha }({\kappa }^{\prime }-s)\parallel ds\\ & \le \overline{\wp }T\parallel {\Re }_{\alpha }({\kappa }^{″}-s)-{\Re }_{\alpha }({\kappa }^{\prime }-s)\parallel ,\end{array}$$

(64)

where $\overline{\wp }={\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}({\gamma }^{\prime }(n-1+{\hslash }_0+{\rho }_0)R^{★}+{\gamma }^{\prime }({\overline{\hslash }}_0+{\overline{\rho }}_0)+M)$, since ${\Re }_{\alpha }(.)$ is bounded and strongly continuous (Lemma 11), one arrives $\parallel {\Re }_{\alpha }({\kappa }^{″}-s)-{\Re }_{\alpha }({\kappa }^{\prime }-s)\parallel \to 0$ uniformly for $s\in [0,T]$ as $t^{\prime }\to t^{″},$ this means ${\lim }_{{\kappa }^{\prime }\to {\kappa }^{″}}{{\rm Y}}_3=0$. Subsequently,

$$\begin{array}{ll}{{\rm Y}}_4& ={\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_{{\kappa }^{\prime }}^{{\kappa }^{″}}\parallel {\Re }_{\alpha }({\kappa }^{″}-s)\parallel ∥\Xi \left(s,u\left(s\right),u\left({\tau }_1\left(s\right)\right),\dots ,u\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}u\right)\left(s\right),\left(\overline{\Lambda }u\right)\left(s\right)\right)∥ds\\ & \le {\displaystyle \frac{M^{★}\overline{\wp }}{\Gamma \left(\alpha \right)}}({\kappa }^{″}-{\kappa }^{\prime })\to 0as{\kappa }^{\prime }\to {\kappa }^{″},\end{array}$$

(65)

where $\overline{\wp }$ is from $\left(64\right)$. Finally, for any $\forall u\in B_{R^{★}},$${\kappa }^{\prime },{\kappa }^{″}\in J_k,{\kappa }^{\prime }\le {\kappa }^{″},$

$$\begin{array}{ll}{{\rm Y}}_5& =\parallel {\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)-{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)I_i\left(u\left({\kappa }_i\right)\right)\parallel \\ & \le l^m∥{\displaystyle \sum _{i=1}^k}({\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i))I_i\left(u\left({\kappa }_i\right)\right)∥\\ & \le l^m{\displaystyle \sum _{i=1}^m}∥{\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)∥\left(d_{I_i}{R}^{★}+{\zeta }_{I_i}\right).\end{array}$$

(66)

Combining boundedness with strong continuity of ${\Im }_{\alpha }(.)$, one has $l^m{\sum }_{i=1}^m\parallel {\Im }_{\alpha }({\kappa }^{″}-{\kappa }_i)-{\Im }_{\alpha }({\kappa }^{\prime }-{\kappa }_i)\parallel (d_{I_i}{R}^{★}+{\zeta }_{I_i})\to 0$.

Hence, by $\left(59\right)$–$\left(66\right)$, $\aleph \left(B_{R^{★}}\right)$ is equicontinuous in $J_i$ for $i=1,2,\dots ,m$, then by Lemma 6, we know that $\overline{Co}\aleph B_{R^{★}}$ is also equicontinuous in $J_i$ for $i=1,2,\dots ,m$.

Then, by Lemma 6, $\overline{Co}\aleph \left(B_{R^{★}}\right)\subset B_{R^{★}}$ forms a uniformly bounded and uniformly equicontinuous family on $J_k$ for $k=1,2,\dots ,m$. According to the demonstrate the definition of the operator ℵ, the conditions $\left(H_1\right)$–$\left(H_9\right)$ and the properties of non-compact measures, for any bounded set $\mathrm{\Omega }\subset \overline{Co}\aleph \left(B_{R^{★}}\right)\subset B_{R^{★}},\kappa \in [{\kappa }_0,{\kappa }_1]$, by $\left(H_4\right),\left(H_5\right),\left(H_8\right)$, Lemmas 2 and 11, one has

$$\begin{array}{l}\omega (\aleph \mathrm{\Omega }\left(\kappa \right))=\omega \{\mathcal{G}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(\mathrm{\Omega }\right))\\ +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\}\\ \le M^{★}l\omega \left(g\left(\mathrm{\Omega }\right)\right)\\ +{\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)\right)ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)\right)ds\\ \le M^{★}l{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}\left((1-\alpha )\right),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right)\omega \left(\mathrm{\Omega }\left(s\right)\right)\\ +L_2\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_1\left(s\right)\right)\right)+\dots +L_{n-1}\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right)\right)+L_n\left(s\right)\omega \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right)+L_{n+1}\left(s\right)\omega \left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right))ds\\ =M^{★}l{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ +\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right)\omega \left(\mathrm{\Omega }\left(s\right)\right)+L_2\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_1\left(s\right)\right)\right)+\dots +L_{n-1}\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right)\right)\\ +L_n\left(s\right)\omega \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right)+L_{n+1}\left(s\right)\omega \left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right))ds\\ \le M^{★}\tilde{l}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\left(L_1\left(s\right)+L_2\left(s\right)+\dots +L_{{\hslash }_0}{L}_n\left(s\right)+L_{{\rho }_0}{L}_{n+1}\left(s\right)\right)ds\times \omega \left(\mathrm{\Omega }\left(s\right)\right)\\ =M^{★}\tilde{l}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)L\left(s\right)ds\times \omega \left(\mathrm{\Omega }\left(s\right)\right)\\ \le \left\{M^{★}\tilde{l}{L}_g+\overline{d}\iota \right\}{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ =\varrho {\omega }_{PC}\left(\mathrm{\Omega }\right),\end{array}$$

(67)

where $\iota ,L\left(s\right)$ are from the condition $\left(C_5\right)$, $L_{{\hslash }_0},L_{{\rho }_0}$ are from $\left(C_2\right)$.

For $\kappa \in ({\kappa }_i,{\kappa }_{i+1}](i=1,2,\dots ,m)$, by $\left(H_4\right),\left(H_5\right),\left(H_7\right),\left(H_8\right)$, Lemmas 2 and 11, one has

$$\begin{array}{l}\omega (\aleph \mathrm{\Omega }\left(\kappa \right))=\omega \{{\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(\mathrm{\Omega }\right))\\ +{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_k}^{\kappa }{(\kappa -s)}^{\alpha -1}\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_k}^{\kappa }{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_k)I_k\left(\mathrm{\Omega }\left({\kappa }_k\right)\right)\}\\ \le M^{★}{l}^{m+1}\omega \left(g\left(\mathrm{\Omega }\right)\right)\\ +{\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)\right)ds\\ +{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)\right)ds\\ +\omega \left({\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(\mathrm{\Omega }\left(t_i\right)\right)\right)\\ \le M^{★}{l}^{m+1}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right)\omega \left(\mathrm{\Omega }\left(s\right)\right)\\ +L_2\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_1\left(s\right)\right)\right)+\dots +L_{n-1}\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right)\right)+L_n\left(s\right)\omega \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right)+L_{n+1}\left(s\right)\omega \left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right))ds\\ +M^{★}{l}^m\left({\displaystyle \sum _{0<{\kappa }_i<\kappa }}{L}_{I_i}\omega \left(\mathrm{\Omega }\left({\kappa }_i\right)\right)\right)\\ =M^{★}{l}^{m+1}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ +\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right)\omega \left(\mathrm{\Omega }\left(s\right)\right)+L_2\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_1\left(s\right)\right)\right)+\dots +L_{n-1}\left(s\right)\omega \left(\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right)\right)\\ +L_n\left(s\right)\omega \left(H\mathrm{\Omega }\right)\left(s\right)+L_{n+1}\left(s\right)\omega (\Lambda \mathrm{\Omega })\left(s\right))ds+M^{★}{l}^m\left({\displaystyle \sum _{0<{\kappa }_i<\kappa }}{L}_{I_i}\omega \left(\mathrm{\Omega }\left(t_k\right)\right)\right)\\ \le M^{★}\tilde{l}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\left(L_1\left(s\right)+L_2\left(s\right)+\dots +L_{{\hslash }_0}{L}_n\left(s\right)+L_{{\rho }_0}{L}_{n+1}\left(s\right)\right)ds\times \omega \left(\mathrm{\Omega }\left(s\right)\right)\\ =M^{★}\tilde{l}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)L\left(s\right)ds\times \omega \left(\mathrm{\Omega }\left(s\right)\right)+M^{★}{l}^m\left({\displaystyle \sum _{i=1}^m}{L}_{I_i}\omega \left(\mathrm{\Omega }\left({\kappa }_i\right)\right)\right)\\ =M^{★}\tilde{l}{L}_g{\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)L\left(s\right)ds\times \omega \left(\mathrm{\Omega }\left(s\right)\right)+M^{★}{l}^m\left({\displaystyle \sum _{i=1}^m}{L}_{I_i}\omega \left(\mathrm{\Omega }\left({\kappa }_i\right)\right)\right)\\ \le \left\{M^{★}\tilde{l}\left(L_g+{\displaystyle \sum _{i=1}^m}{L}_{I_i}\right)+\overline{d}\iota \right\}{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ =\varrho {\omega }_{PC}\left(\mathrm{\Omega }\right),\end{array}$$

(68)

where $\iota ,L\left(s\right)$ are from the condition $\left(C_5\right)$, $L_{{\hslash }_0},L_{{\rho }_0}$ are from $\left(C_2\right)$. By Lemma 5, for any bounded set $\mathrm{\Omega }\subset \overline{Co}\aleph \left(B_{R^{★}}\right)\subset B_{R^{★}}$, by (67), (68) and Lemma 1, one arrives

$${\omega }_{PC}(\aleph \left(\mathrm{\Omega }\right))\le \varrho {\omega }_{PC}\left(\mathrm{\Omega }\right).$$

According to $\left(42\right)$, one has $0<\varrho <1$, then by the Darbo fixed point theorem, one achieves the operator ℵ admits at least one fixed point $u^{★}\in B_{R^{★}}$, that is, $u^{★}$ is admitted a global mild solution of the model $\left(1\right)$ on $B_{R^{★}}\subset PC(J,E)$, and the validation is finished. □

Remark 7. 

Theorem 4 is finished based on the Darbo fixed point theorem.

Theorem 5. 

Suppose the conditions $\left(H_1\right)$–$\left(H_9\right),\left(C_6\right)$ hold, $g:PC(J,E)\to E$ is compact; moreover, $L_{n+1}\left(t\right)\equiv 0$ in $\left(H_9\right)$, then the model $\left(1\right)$ has at least one global mild solution in $B_{R^{★}}\subset PC(J,E)$.

Proof. 

According to Theorem 1, we have $\aleph :B_{R^{★}}\to B_{R^{★}}$ is a bounded continuous operator. From Theorem 4, we have $\overline{Co}\aleph \left(B_{R^{★}}\right)$ is equicontinuous on $J_k$, for ∀$\mathrm{\Omega }\subset \overline{Co}\aleph \left(B_{R^{★}}\right)$, then by Lemmas 5 and 6, we have ${\tilde{\aleph }}^n\left(\mathrm{\Omega }\right)(n=1,2,\dots ,)$ is bounded and equicontinuous on $J_k$, and it follows from Lemma 1 that

$${\omega }_{PC}\left({\tilde{\aleph }}^n\left(\mathrm{\Omega }\right)\right)=\underset{t\in J}{\max }\omega (\left({\tilde{\aleph }}^n\left(\mathrm{\Omega }\right)\right)\left(\kappa \right),n=1,2,\dots .$$

(69)

Now we show there is a constant $0\le \eta <1$ and an integer number $n_0>0$, for any $\mathrm{\Omega }\subset \overline{Co}\aleph \left({\mathrm{\Omega }}_{R^{★}}\right)$, such that

$${\omega }_{PC}\left({\tilde{\aleph }}^{n_0}\left(\mathrm{\Omega }\right)\right)\le \eta {\omega }_{PC}\left(\mathrm{\Omega }\right).$$

By non-compactness measure and the conditions of $\left(H_1\right)$–$\left(H_9\right)$, for any $\kappa \in [{\kappa }_0,{\kappa }_1]$, one has

$$\begin{array}{ll}\omega (\aleph \mathrm{\Omega }\left(\kappa \right))& =\omega \{\mathcal{G}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(\mathrm{\Omega }\right))\\ & +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\}\\ & \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega (\Xi (s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\\ & \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds+{\displaystyle \frac{\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }({\kappa }_2-s)\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\right.\right.\\ & .\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds\\ & \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega (\Xi (s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\\ & \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds+{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\right.\right.\\ & .\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds\\ & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(1-\alpha ),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ & \times \left(L_1\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_2\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+\dots +L_{n-1}\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_n\left(s\right){\omega }_{PC}\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\right)ds\\ & =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_2\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+\dots \\ & +L_{n-1}\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_n\left(s\right){\omega }_{PC}\left(\overline{\mathcal{H}}\mathrm{\Omega }\right))ds\\ & \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)B_1\left(s\right)\omega \left(\mathrm{\Omega }\left(s\right)\right)ds+j{\omega }_{PC}\left(\mathrm{\Omega }\right),\end{array}$$

(70)

where $\overline{d}$ is from the condition $\left(C_5\right)$, $j=l^m{M}^{★}{\sum }_{i=1}^m{L}_{I_i}<1$ in $\left(C_6\right),$$B_1\left(s\right)=L_1\left(s\right)+L_2\left(s\right)+\dots +L_{n-1}\left(s\right)+L_{{\hslash }_0}{L}_n\left(s\right)$.

For any $\kappa \in ({\kappa }_i,{\kappa }_{i+1}]$, we have

$$\begin{array}{ll}\omega (\aleph \mathrm{\Omega }\left(\kappa \right))& =\omega \{{\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(\kappa \right)(u_0+g\left(\mathrm{\Omega }\right))\\ & +{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\right.\\ & \left.\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ & +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{(\kappa -s)}^{\alpha -1}\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ & +{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ & +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)\right)ds\\ & +{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{I}_i\left(\mathrm{\Omega }\left(t_i\right)\right)\}\\ & \le {\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega (\Xi (s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\\ & \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds+{\displaystyle \frac{m\alpha l^{m+2}+\alpha l^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }({\kappa }_2-s)\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\right.\right.\\ & .\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds+\omega \left({\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(\mathrm{\Omega }\left({\kappa }_i\right)\right)\right)\\ & \le {\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega (\Xi (s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\\ & \left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds+{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\omega \left(\Xi \left(s,\mathrm{\Omega }\left(s\right),\mathrm{\Omega }\left({\tau }_1\left(s\right)\right),\dots ,\mathrm{\Omega }\left({\tau }_{n-2}\left(s\right)\right),\right.\right.\\ & .\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\left(s\right),\left(\overline{\Lambda }\mathrm{\Omega }\right)\left(s\right)))ds+\omega \left({\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(\mathrm{\Omega }\left({\kappa }_i\right)\right)\right)\\ & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ & \times \left(L_1\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_2\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+\dots +L_{n-1}\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_n\left(s\right){\omega }_{PC}\left(\overline{\mathcal{H}}\mathrm{\Omega }\right)\right)ds\\ & +M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\omega }_{PC}\left(\mathrm{\Omega }\right)\\ & =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)(L_1\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_2\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+\dots \\ & +L_{n-1}\left(s\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+L_n\left(s\right){\omega }_{PC}\left(\overline{\mathcal{H}}\mathrm{\Omega }\right))ds+M^{★}{l}^m{\displaystyle \sum _{0<{\kappa }_i<\kappa }}{L}_{I_i}{\omega }_{PC}\left(\mathrm{\Omega }\right),\\ & \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)B_1\left(s\right)\omega \left(\mathrm{\Omega }\left(s\right)\right)ds+M^{★}{l}^m{\displaystyle \sum _{i=1}^m}{L}_{I_i}{\omega }_{PC}\left(\mathrm{\Omega }\right),\end{array}$$

(71)

where $\overline{d},L\left(s\right),j$ is as $\left(70\right)$. Then, by $\left(70\right)$ and $\left(71\right)$, for any $\kappa \in J,u,v\in PC(J,E)$, one arrives

$$\omega (\aleph \mathrm{\Omega }\left(\kappa \right))\le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)B_1\left(s\right)\omega (\aleph \mathrm{\Omega }\left(\kappa \right))ds+j\omega (\aleph \mathrm{\Omega }\left(\kappa \right)).$$

(72)

By $0<j<1$, there exists $0<{ϵ}_0<1$, such that $0<\mathsf{\ell }={ϵ}_0+j<1$. From $B_1\left(\kappa \right)=L_1\left(\kappa \right)+L_2\left(\kappa \right)+L_3\left(\kappa \right)+\dots +L_{n-1}\left(\kappa \right)+L_n\left(\kappa \right)L_{{\hslash }_0}\in L^p(J,\mathbb{R})$, A continuous functions can be constructed $\varphi :J\to {\mathbb{R}}^{+}$, subject to

$$\overline{d}{l}_0{\left({\int }_0^T{|B_1\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}<{ϵ}_0.$$

By $\left(71\right)$ and $\left(72\right)$, one arrive

$$\begin{array}{l}\omega \left(\left({\tilde{\aleph }}^1\left(\mathrm{\Omega }\right)\right)\left(\kappa \right)\right)\\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)|B_1\left(s\right)-\varphi \left(s\right)|\omega \left(\mathrm{\Omega }\left(s\right)\right)ds+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\left|\varphi \left(s\right)\right|\omega \left(\mathrm{\Omega }\left(s\right)\right)ds\\ +j\omega \left(\mathrm{\Omega }\left(s\right)\right)\\ \le \overline{d}{l}_0{\left({\int }_0^T{|B_1\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}\omega \left(\mathrm{\Omega }\left(s\right)\right)+\overline{d}♮l_0{\left({\int }_0^{\kappa }\omega \left(\mathrm{\Omega }\left(s\right)\right)ds\right)}^{{\displaystyle \frac{1}{p}}}+j\omega \left(\mathrm{\Omega }\left(s\right)\right)\\ \le \overline{d}{l}_0{\left({\int }_0^T{|B_1\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}\omega \left(\mathrm{\Omega }\left(s\right)\right)+\overline{d}♮l_0{\left({\int }_0^{\kappa }ds\right)}^{{\displaystyle \frac{1}{p}}}+j\omega \left(\mathrm{\Omega }\left(s\right)\right)\\ \le {ϵ}_0{\parallel u-v\parallel }_{PC}+\overline{♮}\kappa \omega \left(\mathrm{\Omega }\left(s\right)\right)+j\omega \left(\mathrm{\Omega }\left(s\right)\right)\\ \le (\mathsf{\ell }+\overline{♭}\kappa )\omega \left(\mathrm{\Omega }\left(s\right)\right)=\left(C_1^0\mathsf{\ell }+C_1^1{\displaystyle \frac{\overline{♮}\kappa }{1!}}\right)\omega \left(\mathrm{\Omega }\left(s\right)\right),\kappa \in J,\end{array}$$

(73)

where $♮=\max \left\{\right|\varphi \left(\kappa \right)|:\kappa \in J\},\overline{♮}=\overline{d}♮l_0.$

Assume for natural number $j\in N$, one has

$$\omega \left(\left({\tilde{\aleph }}^j\right)\right)\left(\kappa \right)\le \left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+{\displaystyle \frac{C_j^2{\mathsf{\ell }}^{j-2}{\left(\overline{♮}\kappa \right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right),\kappa \in J,$$

(74)

according to the definition of operation, the condition $\left(H_1\right)$–$\left(H_9\right)$, by the same steps as (70) and (71), for $\kappa \in [{\kappa }_0,{\kappa }_1]$, one arrives

$$\begin{array}{l}\omega \left({\tilde{\aleph }}^{j+1}\left(\mathrm{\Omega }\right)\right)\left(\kappa \right)=\omega \{\mathcal{G}{\Im }_{\alpha }\left(\kappa \right)\left(u_0+g\left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\right)\\ +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_0^{\kappa }{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds\}\\ \le {\displaystyle \frac{l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega \left(\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\right.\right.\\ \left.\left.\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)\right)ds\\ +{\displaystyle \frac{\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\omega \left(\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\right.\right.\\ \left.\left.\left.\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)\right)ds\right)\\ \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}\left((1-\alpha )\right),M^{★}\alpha l^2\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ \times \left(L_1\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)\right.+L_2\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left({\tau }_1\left(s\right)\right)+\dots +L_{n-2}\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left({\tau }_{n-2}\left(s\right)\right)\\ \left.+L_{n-1}\left(s\right)\omega \left(H{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds\\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)B_1\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds+j{\omega }_{PC}\left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)|B_1\left(s\right)-\varphi \left(s\right)|\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\left|\varphi \left(s\right)\right|\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +j{\omega }_{PC}\left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\\ \le \overline{d}{l}_0{\left({\int }_0^T{|B_1\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}\omega \left({\tilde{\aleph }}^j\right)\left(\mathrm{\Omega }\right))+\overline{d}♮l_0{\int }_0^{\kappa }\omega \left({\tilde{\aleph }}^j\right)ds\\ +j\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ \le {ϵ}_0\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{♮}{\int }_0^{\kappa }\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)ds\\ +j\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ \le \mathsf{\ell }\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ +\left(C_j^0{\mathsf{\ell }}^j{\displaystyle \frac{\overline{♮}\kappa }{1!}}+C_j^1{\mathsf{\ell }}^{j-1}{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^2}{2!}}+\dots +C_j^j{\mathsf{\ell }}^0{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^{j+1}}{(j+1)!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ \le \left(C_{j+1}^0{\mathsf{\ell }}^{j+1}+C_{j+1}^1{\mathsf{\ell }}^j\left(\overline{♮}\kappa \right)+{\displaystyle \frac{C_{j+1}^2{\mathsf{\ell }}^{j-1}{\left(\overline{♮}\kappa \right)}^2}{2!}}+\dots +C_{j+1}^{j+1}{\mathsf{\ell }}^0{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^{j+1}}{(j+1)!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right).\end{array}$$

(75)

For $\kappa \in ({\kappa }_i,{\kappa }_{i+1}](i=1,2,\dots ,m)$, one arrives

$$\begin{array}{l}\omega \left({\tilde{\aleph }}^{j+1}\left(\mathrm{\Omega }\right)\right)\left(\kappa \right)=\omega \{{\mathcal{G}}^{k+1}{\Im }_{\alpha }\left(t\right)\left(u_0+g\left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\right)\\ +{\displaystyle \frac{\mathcal{K}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+2-i}{\Im }_{\alpha }(\kappa -{\kappa }_i){\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{({\kappa }_i-s)}^{\alpha -1}\\ \times \Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\mathcal{K}\mathcal{G}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_{{\kappa }_k}^{\kappa }{(\kappa -s)}^{\alpha -1}\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds+{\displaystyle \frac{\alpha }{B\left(\alpha \right)}}{\displaystyle \sum _{i=1}^k}{\mathcal{G}}^{k+3-i}{\int }_{{\kappa }_{i-1}}^{{\kappa }_i}{\Re }_{\alpha }(\kappa -s)\\ \times \Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds\\ +{\displaystyle \frac{\alpha {\mathcal{G}}^2}{B\left(\alpha \right)}}{\int }_{{\kappa }_i}^{\kappa }{\Re }_{\alpha }(\kappa -s)\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\right.\\ \left.\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds+{\displaystyle \sum _{i=1}^k}{g}^{k+1-i}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\kappa }_i\right)\right)\}\\ \le {\displaystyle \frac{m{M}^{★}{l}^{m+1}\overline{l}+l\overline{l}(1-\alpha )}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -s)}^{\alpha -1}\omega \left(\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\right.\right.\\ \left.\left.\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)\right)ds\\ +{\displaystyle \frac{m\alpha l^{m+2}{M}^{★}+\alpha l^2{M}^{★}}{B\left(\alpha \right)\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }\omega \left(\Xi \left(s,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right),\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_1\left(s\right)\right),\dots ,\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\tau }_{n-2}\left(s\right)\right),\right.\right.\\ \left.\left.\left(\overline{\mathcal{H}}\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right),\left(\overline{\Lambda }\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)\right)ds+\omega \left({\displaystyle \sum _{0<{\kappa }_i<t}}\mathcal{G}{\Im }_{\alpha }(\kappa -{\kappa }_i)I_i\left(\overline{co}{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left({\kappa }_i\right)\right)\right)\\ \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)\\ \times \left(L_1\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)\right.+L_2\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left({\tau }_1\left(s\right)\right)+\dots +L_{n-2}\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left({\tau }_{n-2}\left(s\right)\right)\\ \left.+L_{n-1}\left(s\right)\omega \left(H{\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left(s\right)\right)ds+M^{★}{l}^m{\displaystyle \sum _{0<{\kappa }_i<\kappa }}{L}_{I_i}\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\right)\left({\kappa }_i\right)\\ =\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)B_1\left(s\right)\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds+M^{★}{l}^m\left({\displaystyle \sum _{k=1}^m}{L}_{I_k}\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s_k\right)\right)\right)\\ \le \overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)|B_1\left(s\right)-\varphi \left(s\right)|\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds+\overline{d}{\int }_0^{\kappa }({(\kappa -s)}^{\alpha -1}+1)|\varphi \left(s\right)|\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds\\ +j{\parallel u-v\parallel }_{PC}\\ \le \overline{d}{l}_0{\left({\int }_0^T{|B\left(s\right)-\varphi \left(s\right)|}^{p}ds\right)}^{{\displaystyle \frac{1}{p}}}\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)+\overline{d}♮l_0\left({\int }_0^{\kappa }\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)ds\right)+j\omega \left({\tilde{\aleph }}^j\left(\mathrm{\Omega }\right)\left(s\right)\right)\\ +j\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{(♮\kappa )}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ \le {ϵ}_0\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)+\overline{♮}{\int }_0^{\kappa }\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)ds\\ +j\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ \le \mathsf{\ell }\left({\mathsf{\ell }}^j+C_j^1{\mathsf{\ell }}^{j-1}\left(\overline{♮}\kappa \right)+\dots +{\displaystyle \frac{{\left(\overline{♮}t\right)}^j}{j!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ +\left(C_j^0{\mathsf{\ell }}^j{\displaystyle \frac{\overline{♮}\kappa }{1!}}+C_j^1{\mathsf{\ell }}^{j-1}{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^2}{2!}}+\dots +C_j^j{\mathsf{\ell }}^0{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^{j+1}}{(j+1)!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right)\\ \le \left(C_{j+1}^0{\mathsf{\ell }}^{j+1}+C_{j+1}^1{\mathsf{\ell }}^j\left(\overline{♮}\kappa \right)+{\displaystyle \frac{C_{j+1}^2{\mathsf{\ell }}^{j-1}{\left(\overline{♮}\kappa \right)}^2}{2!}}+\dots +C_{j+1}^{j+1}{\mathsf{\ell }}^0{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^{j+1}}{(j+1)!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right),\kappa \in J,\end{array}$$

(76)

where $L\left(s\right)$ is from Formula $\left(70\right)$.

Hence, by principle of mathematical induction, (75) and (76), for any natural number $n\in \mathbb{N}$ and $\kappa \in J$, one has

$$\omega \left(\left({\tilde{\aleph }}^n\right)\left(\mathrm{\Omega }\right)\right)\left(\kappa \right)\le \left({\mathsf{\ell }}^n+C_n^1{\mathsf{\ell }}^{n-1}\left(\overline{♮}\kappa \right)+{\displaystyle \frac{C_n^2{\mathsf{\ell }}^{n-2}{\left(\overline{♮}\kappa \right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♮}\kappa \right)}^n}{n!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right),$$

(77)

for any natural number $n\in \mathbb{N}$. By $\left(69\right)$, one has

$${\omega }_{PC}\left(\left({\tilde{\aleph }}^n\right)\left(\mathrm{\Omega }\right)\right)\le \left({\mathsf{\ell }}^n+C_n^1{\mathsf{\ell }}^{n-1}\left(\overline{♮}T\right)+{\displaystyle \frac{C_n^2{\mathsf{\ell }}^{n-2}{\left(\overline{♮}T\right)}^2}{2!}}+\dots +{\displaystyle \frac{{\left(\overline{♮}T\right)}^n}{n!}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right),$$

(78)

by Lemma 8, for any $s>1$, as $n\to \infty$,

$${\omega }_{PC}\left(\left({\tilde{\aleph }}^n\right)\left(\mathrm{\Omega }\right)\right)=o\left({\displaystyle \frac{1}{n^s}}\right){\omega }_{PC}\left(\mathrm{\Omega }\right).$$

Hence, there exist $0<\eta <1$ and natural number $n_0\in \mathbb{N}$ satisfies ${\omega }_{PC}\left({\tilde{\aleph }}^{n_0}\left(\mathrm{\Omega }\right)\right)\le \eta {\omega }_{PC}\left(\mathrm{\Omega }\right)$ from Lemma 5, that is, the model $\left(1\right)$ has at least one global mild solution on $B_{R^{★}}\subset PC(J,E)$. □

Remark 8. 

Theorem 5 is completed based on the generalized Darbo fixed point theorem. Theorem 4 is without the condition $L_{n+1}=0$, but need the condition $\left(C_5\right)$, and Theorem 5 need the condition $\left(C_6\right)$. The condition $\left(C_6\right)$ of Theorem 5 is weaker than $\left(C_5\right)$ of Theorem 4. Compared with previous relevant literature, this part generalizes the L of the Lipschitz constants to unbounded Lebesgue integrable functions $L_i\left(s\right)(i=1,2,\dots ,i+1)$.

4. An Illustration

Example 1. 

Consider the following Mittag–Leffler kernel fractional-derivative model

$$\left\{\begin{array}{l}{}^{ABc}D_{\kappa }^{\alpha }u(x,\kappa )-a\left(\kappa \right)\Delta u(x,\kappa )={\displaystyle \frac{u(x,\kappa )}{1+8\sqrt{\kappa }}}+{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\sin \left(u(x,\kappa -{\tau }_1)\right)\\ +{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\cos \left(u(x,\kappa -{\tau }_2)\right)+{\displaystyle \frac{1}{2+{\kappa }^2}}{\int }_0^{\kappa }\kappa u(x,s)\cos sds+{\displaystyle \frac{{\kappa }^{-\frac{1}{2}}}{1+e^{2\kappa }}}{\int }_0^1{\displaystyle \frac{s}{3}}u(x,s)ds,\\ \kappa \in (0,1),\kappa \ne {\displaystyle \frac{1}{3}},\\ ▵u(x,\kappa ){|}_{\kappa ={\kappa }_i}=I\left(u(x,s)\right)=u\left(x,{(1/3)}^{+}\right)-u\left(x,(1/3^{-}\right),i=1,\\ u(x,0)=\phi \left(x\right)+g\left(u(x,\kappa )\right),0<x<1,\\ u(0,\kappa )=u(\pi ,\kappa )=0,(x,\kappa )\in (0,1)\times J^{\prime },\end{array}\right.$$

(79)

where $\alpha ={\displaystyle \frac{3}{4}}$. Let $E=L^2\left([0,1]\right)$, $Au\left(\kappa \right)=-a\left(\kappa \right)\overline{\Delta }u$, $A:D\left(A\right)\subset X\to X$$u(·,\kappa )=u\left(\kappa \right)$, $D\left(A\right)=\{u\in E:u^{″}\in E,u\left(0\right)=u\left(1\right)=0\},$ then A is infinitesimal generator of the bounded linear operator ${\Im }_{\alpha }\left(\kappa \right)$, and ${\tau }_1\left(\kappa \right)=\kappa -{\tau }_1\le \kappa ,{\tau }_2\left(\kappa \right)=\kappa -{\tau }_2\le \kappa ,$$\left(\mathcal{H}u\right)\left(\kappa \right)={\int }_0^{\kappa }\kappa u(x,s)\cos sds$, $(\Lambda u(x,s))\left(\kappa \right)={\int }_0^1{\displaystyle \frac{s}{3}}u(x,s)ds$, $g\left(u(x,s)\right)={\displaystyle \frac{1}{8}}(u(x,s)+\cos u(x,s)),$ $\Xi (\kappa ,x_1,x_2,x_3,x_4,x_5)={\displaystyle \frac{x_1}{1+8\sqrt{\kappa }}}+{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\sin x_2+{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\cos x_3+{\displaystyle \frac{1}{2+{\kappa }^2}}{x}_4+{\displaystyle \frac{{\kappa }^{-\frac{1}{2}}}{1+e^{2\kappa }}}{x}_5$, $I\left(u(x,s)\right)={\displaystyle \frac{1}{8}}u(x,s)$, then the model $\left(79\right)$ can be written by the following abstract form

$$\left\{\begin{array}{l}{}^{ABc}D_{\kappa }^{\alpha }u\left(\kappa \right)+Au\left(t\right)={\displaystyle \frac{u\left(t\right)}{1+8\sqrt{\kappa }}}+{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\sin \left(u(\kappa -{\tau }_1)\right)\\ +{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\cos \left(u(\kappa -{\tau }_2)\right)+{\displaystyle \frac{1}{2+{\kappa }^2}}{\int }_0^{\kappa }\kappa u\left(s\right)\cos sds+{\displaystyle \frac{{\kappa }^{-\frac{1}{2}}}{1+e^{2\kappa }}}{\int }_0^1{\displaystyle \frac{s}{3}}u\left(s\right)ds,\\ ▵u\left(\kappa \right){|}_{\kappa =1/3}=I\left(u(1/3)\right),k=1,\\ u\left(0\right)=u_0+g\left(u\right),\end{array}\right.$$

(80)

hence, for any $u_i,v_i\in E(i=1,2,3,4,5),\kappa \in (0,1]$, we easily get

$$\parallel \hslash (\kappa ,s,u)\parallel =\parallel \kappa u(x,s)\parallel \le \kappa \parallel u\parallel ={\hslash }_1(\kappa ,s)\parallel u\parallel +{\zeta }_{\hslash }(\kappa ,s),(\kappa ,s)\in {\Theta }_1,u\in E,$$

$$\parallel \rho (\kappa ,s,u)\parallel =\parallel {\displaystyle \frac{s}{3}}u(x,s)\parallel \le {\displaystyle \frac{s}{3}}\parallel u\parallel ={\rho }_1(\kappa ,s)\parallel u\parallel +{\zeta }_{\rho }(\kappa ,s),(\kappa ,s)\in {\Theta }_2,u\in E.$$

Obviously, ${\hslash }_1(\kappa ,s)=\kappa$, ${\zeta }_{\hslash }(\kappa ,s)=0$, ${\rho }_1(\kappa ,s)={\displaystyle \frac{s}{3}},{\zeta }_{\rho }(\kappa ,s)=0,$ then ${\hslash }_0=\sup {}_{\kappa \in [0,1]}\int _0^{\kappa }{\hslash }_1(\kappa ,s)ds=1$, ${\rho }_0=\sup {}_{\kappa \in [0,1]}\int _0^1{\rho }_1(\kappa ,s)ds={\displaystyle \frac{1}{6}}$, ${\overline{\hslash }}_0={\overline{\rho }}_0=0$, hence, $\left(H_1\right)$ holds. Moreover, $\parallel I\left(u\right)\parallel =\parallel {\displaystyle \frac{1}{8}}u\parallel \le {\displaystyle \frac{1}{8}}\parallel u\parallel$, then, $d_I={\displaystyle \frac{1}{8}},{\zeta }_1=0.$ Hence, the condition $\left(H_2\right)$ holds. $\parallel g\left(u\right)\parallel ={\displaystyle \frac{1}{8}}\parallel (u+\cos u)\parallel \le {\displaystyle \frac{1}{8}}\parallel u\parallel +{\displaystyle \frac{1}{8}},$ then $d_1={\displaystyle \frac{1}{8}}$, hence, the condition $\left(H_3\right)$ holds. Moreover

$$\begin{array}{ll}\gamma & =\underset{\parallel x_1\parallel +\parallel x_2\parallel +\parallel x_3\parallel +\parallel x_4\parallel <R;\parallel x_5\parallel \to +\infty }{\lim \sup }\left(\underset{s\in J}{\sup }{\displaystyle \frac{\frac{x_1}{1+8\sqrt{\kappa }}+\frac{1}{6}{\kappa }^{-\frac{1}{2}}{e}^{-\kappa }\sin x_2+\frac{1}{6}{\kappa }^{-\frac{1}{2}}{e}^{-\kappa }\cos x_3+\frac{1}{2+{\kappa }^2}{x}_4+\frac{{\kappa }^{-\frac{1}{2}}}{1+e^{2\kappa }}{x}_5}{\parallel x_1\parallel +\parallel x_2\parallel +\parallel x_3\parallel +\parallel x_4\parallel +\parallel x_5\parallel }}\right)\\ & ={\displaystyle \frac{1}{2}}<+\infty ,\end{array}$$

and take $p=q=2,M^{★}={\displaystyle \frac{1}{5}}$, $l={\displaystyle \frac{1}{4}},\overline{l}={\displaystyle \frac{1}{5}},m=1$, then

$$\tilde{l}=\max \{l,l^{m+1}\}={\displaystyle \frac{1}{4}},$$

$$\begin{array}{ll}\overline{d}& ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}\\ & ={\displaystyle \frac{1}{\Gamma \left(\frac{3}{4}\right)B\left(\frac{3}{4}\right)}}\max \{{\displaystyle \frac{1}{4}}\times {\displaystyle \frac{1}{5}}({\displaystyle \frac{1}{5}}\times {\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{4}}),{\displaystyle \frac{1}{5}}\times {\displaystyle \frac{3}{4}}{\left({\displaystyle \frac{1}{4}}\right)}^2({\displaystyle \frac{1}{4}}+1)\}\approx 0.1483,\end{array}$$

$$\widehat{d}=\overline{d}\left({\displaystyle \frac{1}{\alpha }}{T}^{\alpha }+1\right)\approx 0.1483({\displaystyle \frac{4}{3}}+1)\approx 0.3460,$$

hence

$$\begin{array}{ll}K& =\tilde{l}{M}^{★}{d}_1+\widehat{d}\gamma \left(n-1+{\hslash }_0+{\rho }_0\right)+l^m{M}^{★}{\displaystyle \sum _{k=1}^m}{d}_{I_k}\\ & <{\displaystyle \frac{1}{4}}\times {\displaystyle \frac{1}{5}}\times {\displaystyle \frac{1}{8}}+0.3460\times {\displaystyle \frac{1}{2}}\left(3+1+{\displaystyle \frac{1}{6}}\right)+{\displaystyle \frac{1}{4}}\times {\displaystyle \frac{1}{5}}\times {\displaystyle \frac{1}{8}}\approx 0.7271<1,\end{array}$$

then, all the condition of Theorem 1 hold, the Mittag–Leffler kernel fractional-derivative model (79) at least has one fixed point by the Schauder fixed point theorem.

Example 2. 

Continue to consider model (80), for any $u_i,v_i\in E(i=1,2,3,4,5),\kappa \in (0,1]$, we easily get

$$\begin{array}{l}\parallel \Xi (\kappa ,u_1,u_2,u_3,u_4,u_5)-\Xi (\kappa ,v_1,v_2,v_3,v_4,v_5)\parallel \\ \le {\displaystyle \frac{1}{1+8\sqrt{\kappa }}}\parallel u_1-v_1\parallel +{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\parallel u_2-v_2\parallel +{\displaystyle \frac{1}{6}}{\kappa }^{-{\displaystyle \frac{1}{2}}}{e}^{-\kappa }\parallel u_3-v_3\parallel +{\displaystyle \frac{1}{2+{\kappa }^2}}\parallel \mathcal{H}{u}_4-\mathcal{H}{v}_4\parallel +{\displaystyle \frac{{\kappa }^{-\frac{1}{2}}}{1+e^{2\kappa }}}\parallel \Lambda u_5-\Lambda v_5\parallel ,\end{array}$$

$$\parallel \hslash (\kappa ,s,u)-\hslash (\kappa ,s,v)\parallel \le \parallel u-v\parallel ,\kappa ,s\in J=[0,1],\kappa \ne {\displaystyle \frac{1}{3}},u,v\in E,$$

$$\parallel \rho (\kappa ,s,u)-\rho (\kappa ,s,v)\parallel \le {\displaystyle \frac{1}{3}}\parallel u-v\parallel ,\kappa ,s\in J=[0,1],\kappa \ne {\displaystyle \frac{1}{3}},u,v\in E,$$

$$\parallel g\left(u\right)-g\left(v\right)\parallel \le {\displaystyle \frac{1}{4}}\parallel u-v\parallel ,u,v\in E,$$

$$\parallel I\left(u\right)-I\left(v\right)\parallel \le {\displaystyle \frac{1}{8}}\parallel u-v\parallel ,u,v\in E.$$

By simple calculation, we have

$$\begin{array}{ll}\overline{d}& ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)B\left(\alpha \right)}}\max \{l\overline{l}(m{M}^{★}{l}^m+(1-\alpha )),M^{★}\alpha l^2(m{l}^m+1)\}\\ & ={\displaystyle \frac{1}{\Gamma \left(\frac{3}{4}\right)B\left(\frac{3}{4}\right)}}\max \{M^{★}l+{\displaystyle \frac{1}{4}},M^{★}{\displaystyle \frac{1}{4}}{l}^2(l+1)\},\end{array}$$

$$\tilde{l}=\max \{l,l^2\},$$

Taking $p=q=2,M^{★}={\displaystyle \frac{1}{5}}$, $l={\displaystyle \frac{1}{4}}$, we have

$$l_0=2^{{\displaystyle \frac{q-1}{q}}}{\left({\displaystyle \frac{T^{(\alpha -1)q+1}}{(\alpha -1)q+1}}+T\right)}^{{\displaystyle \frac{1}{q}}}=6^{\sqrt{2}},\tilde{l}={\displaystyle \frac{1}{4}},\overline{d}={\displaystyle \frac{3}{10\Gamma \left(\frac{3}{4}\right)B\left(\frac{3}{4}\right)}}\approx 0.27,$$

$$L\left(\kappa \right)={\displaystyle \sum _{i=1}^3}{L}_i\left(\kappa \right)+L_{{\hslash }_0}{L}_4\left(\kappa \right)+L_{{\rho }_0}{L}_5\left(\kappa \right)={\displaystyle \frac{1}{8+8\sqrt{\kappa }}}+{\displaystyle \frac{4{\kappa }^{-\frac{1}{2}}{e}^{-\kappa }}{6}}+{\displaystyle \frac{1}{2+t^2}}+{\displaystyle \frac{1}{{\kappa }^{\frac{1}{2}}(1+e^{2\kappa })}},$$

$$\iota =l_0{\left({\int }_0^1{L}^2\left(s\right)ds\right)}^{{\displaystyle \frac{1}{2}}}=6^{\sqrt{2}}{\left({\int }_0^1{\left({\displaystyle \frac{1}{8+8\sqrt{\kappa }}}+{\displaystyle \frac{{\kappa }^{-\frac{1}{2}}{e}^{-\kappa }}{6}}+{\displaystyle \frac{1}{2+{\kappa }^2}}+{\displaystyle \frac{1}{4{\kappa }^{\frac{1}{2}}(1+e^{2\kappa })}}\right)}^{2}d\kappa \right)}^{{\displaystyle \frac{1}{2}}}\approx 2.84,$$

then we have

$$\varrho =M^{★}\tilde{l}\left(L_g+L_I\right)+\overline{d}\iota =M^{★}\tilde{l}\left({\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{8}}\right)+2.84\overline{d}={\displaystyle \frac{3}{8}}{M}^{★}\tilde{l}+2.84\overline{d}\approx 0.7855<1.$$

(81)

Then by Theorem 2, we have model $\left(79\right)$ admits a unique global mild solution $u^{★}\in PC(J,E)$, then the sequence defined by Theorem (12) converges uniformly to the only global mild solution $u^{★}\left(t\right)$ on the norm of E on the interval $[0, 1]$, and for any real number $s>1$, one has

$$\parallel u_n-u^{★}\parallel =o\left({\displaystyle \frac{1}{n^s}}\right),n\to +\infty .$$

5. Conclusions

In the case of compact and non-compact resolvent operators, the existence and uniqueness are obtained for Mittag–Leffler kernel fractional differential equations of mixed type with delay subject to nonlocal and impulsive boundary value conditions. The equations we studied are completed in the framework of Mittag–Leffler kernel fractional differentiation; this type of fractional differentiation has the advantage of having non-singular kernel and can fully describe memory, and at the same time, our equation is a generalization of many fractional differential equations compared with the references; the order of the derivative being $0<\alpha <1$, this fractional order will bring a series of difficulties in compactness, continuity, convergence property and so on and we overcome these difficulties by the Hölder inequality, Minkowski inequality and the like. The results of Theorem 1 is obtained under the condition of semigroup being compact; moreover, mathematical induction is used in the process of proving the existence and uniqueness of solutions in Theorems 3 and 5. The uniqueness result is obtained by the Banach compression mapping principle, generalized Banach compression mapping principle in Theorems 2 and 3, respectively. The constant L in the Lipschitz conditions are generalized to Lebesgue integrable functions. Subsequently, the non-compactness measure is used to prove the existence of Theorems 4 and 5 under the semigroup being non-compact based on the Darbo fixed point theorem and generalized Darbo fixed point theorem, respectively. Theorems 4 and 5 are completed under conditions $\left(C_5\right)$ and $\left(C_6\right)$, respectively, and condition $\left(C_6\right)$ is weaker than condition $\left(C_5\right)$, that means the equation is generalized; at the same time, the constants L in the noncompactness measure is also generalized to Lebesgue integrable functions in this process. Hence, the main innovation of this article is to generalize the types of equations and weakens the conditions; furthermore, we extend the constants L in the Lipschitz conditions and the noncompactness measure conditions to Lebesgue integrable functions $L_i\left(s\right)(i=1,2,\dots ,i+1)$.

6. Future Work

In the future, we plan to conduct further research on the methods for solving the numerical solutions of the equation, and moreover, we will integrate with the actual system and apply the theory to practice.