Fractional Neutral Integro-Differential Equations with Nonlocal Initial Conditions

Abstract

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions.

1. Introduction

Fractional derivatives, with their capacity to describe nonlocal and long-range dependencies, offer a more precise representation of the dynamic behaviors observed in complex systems compared to traditional integer derivatives [1]. Consequently, fractional differential equations find extensive practical applications across various fields [2,3,4,5,6,7].

The types of fractional differential equations studied so far include fractional ordinary differential equations, fractional partial differential equations, neutral differential equations, functional differential equations, and impulsive differential equations. Among these, fractional partial differential equations involve various types such as Laplace, diffusion, wave, Schrödinger, Navier–Stokes, Heisenberg, Langevin, and Fokker–Planck equations.

It is noteworthy that fractional diffusion equations effectively capture the characteristics of anomalous diffusion phenomena, including those with long-tail distributions. Mu et al. [8] have conducted a comprehensive study on the existence and regularity of solutions to fractional diffusion equations:

$$\left\{\begin{array}{c}{\partial }_t^{\alpha }u(x,t)=Au(x,t)+f(x,t),(x,t)\in \Omega \times (0,b),\hfill \\ {\displaystyle \sum _{i,j=1}^n}{a}_{ij}\left(x\right)\frac{\partial }{\partial x_j}u(x,t)=0,(x,t)\in \partial \Omega \times (0,b),\hfill \\ u(x,0)=u_0\left(x\right),x\in \Omega .\hfill \end{array}\right.$$

(1)

Here, ${\partial }_t^{\alpha }$ is the $\alpha$-order partial Caputo derivative with respect to t, $\alpha \in (0,1)$, $\Omega \subset {\mathbb{R}}^n$ with smooth boundary $\partial \Omega$, $T>0$, f is weighted H$\ddot{\mathrm{o}}$lder continuous;

$$Au(x,t)=\sum _{i,j=1}^n{\partial }_{x_i}\left[a_{ij}\left(x\right){\partial }_{x_j}u(x,t)\right]-p\left(x\right)u(x,t),$$

(2)

$a_{ij}$ are real-valued functions that satisfy

$$a_{ij}\in L^{\infty }(\Omega ),1\le i,j\le n,$$

$$\sum _{i,j=1}^n{a}_{ij}\left(x\right){\vartheta }_i{\vartheta }_j\ge \varsigma {\left|\vartheta \right|}^2,\vartheta \in {\mathbb{R}}^n,\mathrm{a}.\mathrm{e}.x\in \Omega ,$$

(3)

with some $\varsigma >0$, $p\left(x\right)$ is also a real-valued function satisfying

$$p\in L^{\infty }(\Omega ),p\left(x\right)\ge p_0>0,\mathrm{a}.\mathrm{e}.x\in \Omega ,$$

By incorporating the integral term, the fractional diffusion equation extends its capacity to model a broader spectrum of complex diffusion phenomena in practical settings [9].

Moreover, with the rise of studies on the dynamic behavior of delay systems, fractional neutral equations have also attracted wide attention [10,11,12,13,14]. Among these studies, Bedi et al. [13] considered a neutral fractional differential equations system with the Atangana–Baleanu–Caputo derivatives, establishing some controllability results. Zhou et al. [14] consider the neutral evolution equation

$$\left\{\begin{array}{c}{}^c{D}^q[x\left(t\right)-h(t,x_t)]+Bx\left(t\right)=f(t,x_t),t\in (0,T],\hfill \\ x_0\left(\varsigma \right)+\left(g(x_{t_1},\dots ,x_{t_n})\right)\left(\varsigma \right)=\phi \left(\varsigma \right),\varsigma \in [-r,0],\hfill \end{array}\right.$$

where ${}^c{D}_t^{\alpha }$ is the $\alpha$-order Caputo derivative, $\alpha \in (0,1)$, $t_i$ is a monotonically increasing sequence on $(0,T]$ for $i=1,\dots ,n$, $-B$ generates an analytic semigroup on a Banach space, $g,h,\phi$ are given, and the delay term $x_t\left(\varsigma \right)=x(t+\varsigma )$ for $\varsigma \in [-r,0]$.

We find that there are no integral terms in References [10,11,12,13,14]. Based on the above study, this paper will investigate the existence of mild solutions to the fractional neutral integro-differential diffusion equations with nonlocal initial conditions:

$$\left\{\begin{array}{c}{}^c{\partial }_t^{\alpha }[u(x,t)+H(x,t,u(x,t))]=Au(x,t)+I_t^{\beta }f(x,t,u(x,t)),(x,t)\in \Omega \times J^{\prime },\hfill \\ {\displaystyle \sum _{i,j=1}^n}{a}_{ij}\left(x\right)\frac{\partial }{\partial x_j}u(x,t)=0,(x,t)\in \partial \Omega \times J^{\prime },\hfill \\ u(x,0)+h\left(u\right)=u_1\left(x\right),x\in \Omega ,\hfill \end{array}\right.$$

(4)

where $J^{\prime }=(0,T]$, $T>0$, $I_t^{\beta }$ is the $\beta$-order partial Riemann–Liouville integral with respect to t, $\beta >0$, h, $u_1$, and H and f are given functions satisfying some assumptions. Moreover, $Au(x,t)$ satisfies Equation (2), and $a_{ij}$ are real-valued functions satisfying Equation (3) and

$$a_{ij}\in C^1\left(\overline{\Omega }\right),1\le i,j\le n,$$

$p\left(x\right)$ is also a real-valued function satisfying

$$p\in C\left(\overline{\Omega }\right),p\left(x\right)\ge p_0>0,x\in \overline{\Omega }.$$

Due to the presence of neutral terms H, integral terms $I_t^{\beta }f$, and nonlocal terms h, the representation of mild solutions to Equation (4) becomes highly challenging. We utilize Laplace transforms, the resolvent family, and probability density functions to find the solution to the equivalent integral equation of Equation (4) and investigate the boundedness, strong continuity, and compactness related to the resolvent family. Finally, we discuss the existence of admissible solutions under weaker conditions.

The resolvent family serves as a potent instrument for examining certain fractional differential equations, as documented in References [15,16,17,18]. Ref. [15] offers adequate integral estimations necessary for constructing a class of $\alpha$-times resolvent families. However, due to the simultaneous inclusion of derivative and integral terms in this paper, the single-parameter resolvent family is deemed unsuitable. Mu et al. [19] demonstrated the existence of mild solutions for fractional diffusion equations with Dirichlet boundary conditions using the $(\alpha ,\beta )$-resolvent family, which is also relevant to Equation (4).

This article is structured as follows: Section 2 furnishes the essential background required for the ensuing discussions, encompassing topics such as boundedness, strong continuity, compactness, and the definition of mild solutions in the context of the resolvent family. Section 3 presents a sequence of research findings concerning the existence of mild solutions under less stringent conditions.

2. Preliminaries

In this paper, let $X=L^2(\Omega )$ with the norm $\parallel ·\parallel$, $J=[0,T]$, let $Z\subset \mathbb{R}$, $1\le p\le \infty$, for measurable functions $F:Z\to \mathbb{R}$. We define the norm

$${\parallel F\parallel }_{L^{p}Z}=\left\{\begin{array}{c}{\left({\displaystyle {\int }_Z{\left|F\left(t\right)\right|}^{p}dt}\right)}^{\frac{1}{p}},1\le p<\infty ,\hfill \\ \underset{\mu \left(\overline{Z}\right)=0}{inf}\{\underset{t\in Z-\overline{Z}}{sup}\left|F\left(t\right)\right|\},p=\infty ,\hfill \end{array}\right.$$

where $\mu \left(\overline{Z}\right)$ is the Lebesgue measure on $\overline{Z}$, $L^p(Z,\mathbb{R})$ denotes the Banach space that consists of all measurable functions F with ${\parallel F\parallel }_{L^{p}Z}<\infty$.

We introduce several definitions and notation that is consistently utilized throughout this paper.

Definition 1

([1]). For $\beta >0$ and a function $f\in L^1[0,\infty )$, the β-order Riemann–Liouville fractional integral of f with respect to $t>0$ is defined as follows:

$$I_t^{\beta }f\left(t\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}f\left(s\right)ds.$$

(5)

Definition 2

([1]). Given α within the interval $(0,1)$ and an absolutely continuous function f on $[0,\infty )$, we define the α-order Riemann–Liouville fractional derivative of f with respect to $t>0$ as

$$D_t^{\alpha }f\left(t\right)=\frac{1}{\Gamma (1-\alpha )}\frac{d}{dt}{\int }_0^{t}f\left(s\right){(t-s)}^{-\alpha }ds.$$

(6)

Definition 3

([1]). For $\alpha \in (0,1)$ and an absolutely continuous function f defined on $[0,\infty )$, its α-order Caputo derivative with respect to $t>0$ can be written as

$${}^c{D}_t^{\alpha }f\left(t\right)=D_t^{\alpha }(f\left(t\right)-f\left(0\right)).$$

(7)

Remark 1.

We can find the following two properties in [1]

(i) 

For $\alpha ,\beta \ge 0$, $I_t^{\alpha }{I}_t^{\beta }=I_t^{\alpha +\beta }$;

(ii) 

For $m>0$ and $f\in L^1(0,m)$, $D_t^{\alpha }{I}_t^{\alpha }f=f$.

In addition, if f is an abstract function taking values in Banach spaces, the integrals and derivatives presented in Equations (5)–(7) should be interpreted in the sense of Bochner.

Definition 4

([20]). Define the function $h\left(s\right)$ on the measure space $(S,\mathcal{J},m)$ with values in X. It is termed Bochner m-integrable if there exists a sequence $\left\{h_n\left(s\right)\right\}$ approximating $h\left(s\right)$ such that

$$\underset{n\to \infty }{lim}{\int }_S\parallel h\left(s\right)-h_n\left(s\right)\parallel m\left(ds\right)=0.$$

For any set $B\in \mathcal{J}$, the Bochner m-integral of $h\left(s\right)$ on B is

$${\int }_{B}h\left(s\right)m\left(ds\right)=s-\underset{n\to \infty }{lim}{\int }_S{C}_B\left(s\right)h_n\left(s\right)m\left(ds\right),$$

$C_B$ is the characteristic function of set B.

Definition 5

([21]). Consider W as a metric space, and let U be a bounded subset of W. The Kuratowski measure of noncompactness is then defined as follows:

$$\nu \left(U\right)=inf\{\delta >0|U=\bigcup _{i=1}^m{U}_i,\mathit{diam}{U}_i\le \delta \}.$$

Lemma 1

([22]). If R is a Banach space, $Z_i\subset R$ for $i=1,2$. Then, $\nu \left(Z_i\right)$ satisfies

(i) 

$\nu \left(Z_i\right)=0$ if and only if $Z_i$ is relatively compact;

(ii) 

$\nu \left(Z_i\right)=\nu \left(\overline{Z_i}\right)$;

(iii) 

if $Z_1\subset Z_2$, then $\nu \left(Z_1\right)\le \nu \left(Z_2\right)$;

(iv) 

$\nu (Z_1+Z_2)\le \nu \left(Z_1\right)+\nu \left(Z_2\right)$;

(v) 

$\nu \left(c{Z}_i\right)=\left|c\right|\nu \left(Z_i\right)$ for any $c\in \mathbb{R}$;

(vi) 

$\nu \left(Z_i\right)\ge 0$.

Let $C(J,X)$ represent the Banach space of all continuous functions mapping from J to X, which is endowed with a norm

$${\parallel ·\parallel }_{\infty }=\underset{t\in J}{sup}\left\{\right|·\left(t\right)\left|\right\}.$$

Let $B_k=\{u\in C(J,X)|\parallel u\parallel \le k\}$. It is evident that $B_k$ is a bounded, closed, and convex subset in $C(J,X)$.

Consider the operator $A:D\left(A\right)\subset X\to X$, where $D\left(A\right)=H_0^1(\Omega )\cap H^2(\Omega )$. For details on these spaces, see [23]. We define $\left(Au\right)\left(t\right)\left(x\right)=Au(x,t)$, establishing that A generates an analytic semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ on X. For simplicity, and without loss of generality, we assume $0\in \rho \left(A\right)$ and the semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is uniformly bounded. Additionally, we define the fractional power $A^{\mu }$ as a closed linear operator on its domain $D\left(A^{\mu }\right)$ for $\mu \in (0,1]$, and it satisfies $N\ge 1$ for some $\parallel T\left(t\right)\parallel \le N$.

Set $u\left(t\right)\left(x\right)=u(x,t)$, $H\left(t\right)\left(x\right)=H(x,t)$, and $f(t,u(t\left)\right)\left(x\right)=f(x,t,u(x,t\left)\right)$. Consequently, Equation (4) can be recast as an abstract problem incorporating nonlocal initial conditions:

$$\left\{\begin{array}{c}{}^c{D}_t^{\alpha }(u\left(t\right)+H(t,u\left(t\right)))=Au\left(t\right)+I_t^{\beta }f(t,u\left(t\right)),t\in J^{\prime },\hfill \\ u\left(0\right)+h\left(u\right)=u_1,\hfill \end{array}\right.$$

(8)

where ${}^c{D}_t^{\alpha }$ represents the $\alpha$-order Caputo derivative and $I_t^{\beta }$ denotes the $\beta$-order Riemann–Liouville integral. $u_1\in X$, $h:C(J,X)\to X$, and $H,f:J\times X\to X$.

Remark 2.

If $\alpha \in (0,1)$, Bajlekova ([24], chapter 2) considered the special case where $\beta =0$ and $H(t,u(t\left)\right)=f(t,u(t\left)\right)=0$ in Equation (8):

$$\left\{\begin{array}{c}{}^c{D}_t^{\alpha }u\left(t\right)=Au\left(t\right),t>0,\hfill \\ u\left(0\right)=x_0\in X_1,\hfill \end{array}\right.$$

(9)

where $X_1$ is a Banach space. An α-times resolvent family ${\left\{S_{\alpha }\left(t\right)\right\}}_{t\ge 0}$ was applied to obtain the uniquely solvable result of Equation (9), with the solution $u\left(t\right)=S_{\alpha }\left(t\right)x_0$. Furthermore, a necessary and sufficient condition is given for Equation (9) to be well-posed.

Lemma 2.

If $u\in C(J,X)$, ${}^c{D}_t^{\alpha }u\in C(J^{\prime },X)$, $u\left(t\right)\in D\left(A\right)$ for $t\in J^{\prime }$ and u satisfies Equation (8), then we have

$$\begin{array}{cc}\hfill u\left(t\right)& =S_{\alpha ,1-\alpha }\left(t\right)\left[u_1-h\left(u\right)+H(0,u\left(0\right))\right]-H(t,u\left(t\right))\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))ds+{\int }_0^t{S}_{\alpha ,\beta }(t-s)f(s,u\left(s\right))ds,\hfill \end{array}$$

(10)

where

$$\Psi \left(t\right)={\int }_0^{\infty }\alpha \theta {\zeta }_{\alpha }\left(\theta \right)T\left(t^{\alpha }\theta \right)d\theta .$$

(11)

The family ${\left\{S_{\alpha ,\beta }\left(t\right)\right\}}_{t\ge 0}$ is the resolvent family generated by the operator A and

$${\lambda }^{-\beta }{({\lambda }^{\alpha }I-A)}^{-1}x={\int }_0^{\infty }{e}^{-\lambda t}{S}_{\alpha ,\beta }\left(t\right)xdt$$

(12)

for all $x\in X$.

Proof. 

By Definition 1, Definition 3, and Remark 1, Equation (8) can be rewritten as

$$\begin{array}{cc}\hfill u\left(t\right)& =u_1-h\left(u\right)-H(t,u\left(t\right))+H(0,u\left(0\right))+\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}Au\left(s\right)ds\hfill \\ \hfill & +\frac{1}{\Gamma (\alpha +\beta )}{\int }_0^t{(t-s)}^{\alpha +\beta -1}f(s,u\left(s\right))ds.\hfill \end{array}$$

(13)

Applying the Laplace transform to Equation (13), let

$$\phi \left(\lambda \right)={\int }_0^{\infty }{e}^{-\lambda t}u\left(t\right)dt,{\phi }_1\left(\lambda \right)={\int }_0^{\infty }{e}^{-\lambda t}f(t,u\left(t\right))dt,{\phi }_2\left(\lambda \right)={\int }_0^{\infty }{e}^{-\lambda t}H(t,u\left(t\right))dt.$$

Then,

$$\begin{array}{cc}\hfill \phi \left(\lambda \right)& =\frac{1}{\lambda }(u_1-h\left(u\right)+H(0,u\left(0\right)))-{\phi }_2\left(\lambda \right)+\frac{1}{{\lambda }^{\alpha }}A\phi \left(\lambda \right)+\frac{1}{{\lambda }^{\alpha +\beta }}{\phi }_1\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^{\alpha -1}{({\lambda }^{\alpha }I-A)}^{-1}(u_1-h\left(u\right)+H(0,u\left(0\right)))-{\lambda }^{\alpha }{({\lambda }^{\alpha }I-A)}^{-1}{\phi }_2\left(\lambda \right)\hfill \\ \hfill & +{\lambda }^{-\beta }{({\lambda }^{\alpha }I-A)}^{-1}{\phi }_1\left(\lambda \right)\hfill \\ \hfill & =:I_1-I_2+I_3.\hfill \end{array}$$

(14)

For $I_2$, refer to [2], we have

$$\begin{array}{c}\hfill I_2={\int }_0^{\infty }{e}^{-\lambda t}\left[H(t,u\left(t\right))-{\int }_0^t{(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))ds\right]dt,\end{array}$$

where ${\zeta }_{\alpha }\left(\theta \right)$ is the probability density function defined on $(0,\infty )$, and

$${\zeta }_{\alpha }\left(\theta \right)\ge 0,$$

$${\int }_0^{\infty }{\theta }^{\nu }{\zeta }_{\alpha }\left(\theta \right)d\theta =\frac{\Gamma (1+\nu )}{\Gamma (1+\alpha \nu )},\nu >-1.$$

(15)

Then, taking the inverse Laplace transform on both sides of Equation (14), we obtain that

$$\begin{array}{cc}\hfill u\left(t\right)& =S_{\alpha ,1-\alpha }\left(t\right)\left[u_1-h\left(u\right)+H(0,u\left(0\right))\right]-H(t,u\left(t\right))\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))ds+{\int }_0^t{S}_{\alpha ,\beta }(t-s)f(s,u\left(s\right))ds,\hfill \end{array}$$

the proof is completed. □

Based on previous work, we define the mild solution of Equation (8) as follows.

Definition 6.

The function u, belonging to $C(J,X)$, is termed a mild solution to Equation (8) if it fulfills the conditions outlined in u and satisfies Equation (10).

At the end of this section, we list some lemmas that need to be used in this article.

Lemma 3

([14,19]). $\Psi \left(t\right)$ and $S_{\alpha ,\beta }\left(t\right)$ are bounded, that is,

$$\parallel \Psi \left(t\right)\parallel \le \frac{M_{\omega }}{\Gamma \left(\alpha \right)},$$

and

$$\parallel S_{\alpha ,\beta }\left(t\right)\parallel \le M{t}^{\alpha +\beta -1},t>0,$$

(16)

where $M,C>0$ are constants.

Lemma 4

([14,19]). The families ${\left\{R_{\alpha ,\beta }\left(t\right)\right\}}_{t\ge 0}$ and ${\{\Psi \left(t\right)\}}_{t\ge 0}$ exhibit strong continuity.

Lemma 5

([19]). Should $T\left(t\right)$ be compact for any $t>0$, then $R_{\alpha ,\beta }\left(t\right)$ and $\Psi \left(t\right)$ also exhibit compactness for $t>0$, where $R_{\alpha ,\beta }\left(t\right)=t^{1-\alpha -\beta }{S}_{\alpha ,\beta }\left(t\right)$.

Lemma 6

([14]). For $x\in X$ and $t\in J$,

$$A\Psi \left(t\right)x=A^{1-\rho }\Psi \left(t\right)A^{\rho }x,$$

(17)

and for $0<t<\infty$,

$$\parallel A^{\eta }\Psi \left(t\right)\parallel \le C{t}^{-\alpha \eta },$$

(18)

where $\eta \in (0,2)$ and $\rho \in (0,1)$; $C>0$ is a constant.

Lemma 7

([25]). Assuming $G:J\to X$ is measurable and $\parallel G\parallel$ satisfies Lebesgue integrability conditions, then G qualifies as Bochner integrable.

Lemma 8

([26]). Let D be a bounded, convex, and closed subset of a Banach space, with $0\in D$. Consider a continuous mapping $N:D\to D$. If, for any subset $V\subset D$, the conditions $V=\overline{\mathit{conv}}N\left(V\right)$ or $V=N\left(V\right)\cup \left\{0\right\}$ imply $\nu \left(V\right)=0$, then it follows that N possesses a fixed point.

Lemma 9

([27]). Given B, a closed bounded and convex subset of a Banach space, and assuming F, a completely continuous mapping B into itself, it follows that F possesses a fixed point within B.

3. Main Results

To demonstrate the existence of a mild solution to Equation (8), it is essential to outline the requisite assumptions.

$\left({\mathrm{H}}_1\right)$ For any value of $t>0$, $T\left(t\right)$ possesses compactness;

$\left({\mathrm{H}}_2\right)$ For almost every $t\in J$, the functions $f(t,z)$ and $H(t,z)$ are continuous with respect to z in X, and for every $z\in X$, they are strongly measurable with respect to t over J;

$\left({\mathrm{H}}_2^{\prime }\right)$ For each element $z\in X$ and for the majority of elements $t\in J$, the functions $f(t,z)$ and $H(t,z)$ are strongly measurable;

$\left({\mathrm{H}}_3\right)$ The function $\varpi \in C(J,{\mathbb{R}}^{+})$ exists such that for every $z\in X$ and the majority of $t\in J$, the inequality $\parallel f(t,z)\parallel \le \varpi \left(t\right)\parallel z\parallel$ holds;

$\left({\mathrm{H}}_4\right)$ The mapping $h:C(J,X)\to X$ demonstrates complete continuity, and a constant $L>0$ can be identified such that for all $u\in C(J,X)$, $\parallel h\left(u\right)\parallel \le {L\parallel u\parallel }_{\infty }$;

$\left({\mathrm{H}}_4^{\prime }\right)$ In the case of the function $h:C(J,X)\to X$, a constant $L>0$ is established, ensuring that $\parallel h\left(u\right)\parallel \le {L\parallel u\parallel }_{\infty }$ for any u;

$\left({\mathrm{H}}_5\right)$ The function $H:J\times X\to X$ exhibits continuity. Furthermore, specific constants $\rho \in (0,1)$ and ${\xi }_1,{\xi }_2>0$ are identified, ensuring that $H\in D\left(A^{\rho }\right)$. Additionally, for any pair of elements $z_1,z_2\in X$ and any $t\in J$, the function $A^{\rho }H(t,z)$ demonstrates strong measurability with respect to t within J. It is also established that

$$\parallel A^{\rho }H(t,z_1)-A^{\rho }H(t,z_2)\parallel \le {\xi }_1\parallel z_1-z_2\parallel ,$$

(19)

$$\parallel A^{\rho }H(t,z)\parallel \le {\xi }_2\parallel z\parallel ;$$

(20)

$\left({\mathrm{H}}_6\right)$ For every bounded subset $X_1$ of X and each $t\in J$, there are constant $a>0$ and a function $\varpi \in C(J,{\mathbb{R}}^{+})$ such that the measures satisfy $\nu \left(h\left(X_1\right)\right)\le a\nu \left(X_1\right)$, $\nu \left(A^{\rho }H(t,X_1)\right)\le \varpi \left(t\right)\nu \left(X_1\right)$, and $\nu \left(f(t,X_1)\right)\le \varpi \left(t\right)\nu \left(X_1\right)$ for $0<\rho <1$;

$\left({\mathrm{H}}_7\right)$ There exists a constant $L^{\prime }>0$ such that for any $y_1,y_2\in B_{k_1}$, the inequalities

$$\parallel h\left(y_1\right)-h\left(y_2\right)\parallel \le L^{\prime }{\parallel y_1-y_2\parallel }_{\infty }$$

and

$$\parallel f(t,y_1\left(t\right))-f(t,y_2\left(t\right))\parallel \le \varpi \left(t\right)\parallel y_1-y_2{\parallel }_{\infty },$$

hold, where

$$k_1=\frac{\alpha \rho M(\alpha +\beta )(\parallel u_1\parallel +\parallel h\left(0\right)\parallel )}{\alpha \rho (\alpha +\beta )(1-M{L}^{\prime }-(M+1){\xi }_2\parallel A^{-\rho }\parallel )-C(\alpha +\beta ){\xi }_2{T}^{\alpha \rho }-\alpha \rho M{T}^{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }}.$$

Theorem 1.

Under the conditions specified by assumptions $\left(H_1\right)$ to $\left(H_5\right)$, and assuming that $H(t,u(t\left)\right)$ is compact for $t>0$, the following inequality holds:

$$ML+M{\xi }_2\parallel A^{-\rho }\parallel +{\xi }_2\parallel A^{-\rho }\parallel +\frac{C{\xi }_2}{\alpha \rho }{T}^{\alpha \rho }+\frac{M{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }<1,$$

(21)

where $C>0$ denotes a constant. Then, Equation (8) has a mild solution.

Proof. 

We choose $k_0$ such that

$$k_0=\frac{M\parallel u_1\parallel }{1-ML-M{\xi }_2\parallel A^{-\rho }\parallel -{\xi }_2\parallel A^{-\rho }\parallel -\frac{C{\xi }_2}{\alpha \rho }{T}^{\alpha \rho }-\frac{M{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }}.$$

(22)

For any $u\in B_{k_0}$, by Lemma 3, $\left({\mathrm{H}}_4\right)$, and $\left({\mathrm{H}}_5\right)$,

$$\parallel S_{\alpha ,1-\alpha }\left(t\right)\left[u_1-h\left(u\right)+H(0,u\left(0\right))\right]\parallel \le M\left(\parallel u_1\parallel +L{k}_0+{\xi }_2{k}_0\parallel A^{-\rho }\parallel \right),$$

(23)

and

$$\parallel H(t,u\left(t\right))\parallel \le {\xi }_2{k}_0\parallel A^{-\rho }\parallel .$$

(24)

Furthermore, given $\left({\mathrm{H}}_2\right)$, the function $f(t,u(t\left)\right)$ is measurable over the interval J. Additionally, considering the stipulations of Lemma 3 and assumption $\left({\mathrm{H}}_3\right)$,

$${\int }_0^t\parallel S_{\alpha ,\beta }(t-s)f(s,u\left(s\right))\parallel ds\le \frac{M{k}_0{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty },$$

(25)

so $\parallel S_{\alpha ,\beta }(t-s)f(s,u\left(s\right))\parallel$ is Lebesgue integrable with respect to $s\in J$ and $t\in J$. Then, according to Lemma 7, $S_{\alpha ,\beta }(t-s)f(s,u\left(s\right))$ achieves Bochner integrability with respect to $s\in J$ and for each $t\in J$. According to $\left({\mathrm{H}}_5\right)$, $A^{\rho }H(t,u\left(t\right))$ is strongly measurable. Given that ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is analytic, it follows that for any $t\in J^{\prime }$ and $\theta >0$, $s\to {(t-s)}^{\alpha -1}AT\left({(t-s)}^{\alpha }\theta \right)$ is continuous in the uniform operator topology over the interval $[0,t)$. Consequently, ${(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))$ also maintains continuity in $[0,t)$.

By Equations (17), (18), and (20), for $t\in J$ and $u\in B_{k_0}$,

$${\int }_0^t\parallel {(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))\parallel ds\le \frac{C{\xi }_2{k}_0}{\alpha \rho }{T}^{\alpha \rho },$$

(26)

similarly, ${(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))$ is Bochner integrable with respect to both $s\in J$ and $t\in J$. We are now in a position to define an operator G on $B_{k_0}$ as follows:

$$\begin{array}{cc}\hfill \left(Gu\right)\left(t\right)& =S_{\alpha ,1-\alpha }\left(t\right)\left[u_1-h\left(u\right)+H(0,u\left(0\right))\right]-H(t,u\left(t\right))\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))ds+{\int }_0^t{S}_{\alpha ,\beta }(t-s)f(s,u\left(s\right))ds,\hfill \end{array}$$

where $u\in B_{k_0}$. If (21) holds, G maps $B_{k_0}$ into itself. We will demonstrate that the operator G possesses a fixed point within $B_{k_0}$.

Firstly, we establish that the operator G is completely continuous. Assume that $u_n,u\in B_{k_0}$ and $u_n\to u$ as $n\to \infty$, then

$$\begin{array}{cc}\hfill & \parallel \left(G{u}_n\right)\left(t\right)-\left(Gu\right)\left(t\right)\parallel \hfill \\ \hfill & \le \parallel S_{\alpha ,1-\alpha }\left(t\right)\left[h\left(u_n\right)-h\left(u\right)\right]\parallel +\parallel H(t,u_n\left(t\right))-H(t,u\left(t\right))\parallel \hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha -1}\parallel A\Psi (t-s)\parallel \parallel H(s,u_n\left(s\right))-H(s,u\left(s\right))\parallel ds\hfill \\ \hfill & +{\int }_0^t\parallel S_{\alpha ,\beta }(t-s)\parallel \parallel f(s,u_n\left(s\right))-f(s,u\left(s\right))\parallel ds\hfill \\ \hfill & =:I_4+I_5+I_6+I_7.\hfill \end{array}$$

Clearly, in accordance with Equation (16), $\left({\mathrm{H}}_2\right)$, and $\left({\mathrm{H}}_4\right)$, the terms $I_4,I_5\to 0$. Referencing Lemma 6 and Equation (19), we have

$$\begin{array}{cc}\hfill I_6& \le {\int }_0^t{(t-s)}^{\alpha -1}\parallel A^{1-\rho }\Psi (t-s)\parallel \parallel A^{\rho }H(s,u_n\left(s\right))-A^{\rho }H(s,u\left(s\right))\parallel ds\hfill \\ \hfill & \le C{\xi }_1{\parallel u_n-u\parallel }_{\infty }{\int }_0^t{(t-s)}^{\alpha \rho -1}ds\hfill \\ \hfill & \le \frac{C{\xi }_1{k}_0}{\alpha \rho }{T}^{\alpha \rho },\hfill \end{array}$$

Then, applying the Lebesgue dominated convergence theorem [28] and assumption $\left({\mathrm{H}}_2\right)$, $I_6,I_7\to 0$ as $n\to \infty$[19]. Consequently, this establishes the continuity of the operator G on $B_{k_0}$.

Next, we will demonstrate that the set $\left\{Gu\right|u\in B_{k_0}\}$ is relatively compact. To establish this, it is necessary to show that the set is uniformly bounded and equicontinuous, and that for any $t\in J$, $\left\{\left(Gu\right)\left(t\right)\right|u\in B_{k_0}\}$ possesses the relative compactness in X. We have established that $\parallel \left(Gu\right)\left(t\right)\parallel \le k_0$, as derived from Equations (23)–(26), this confirms that the set $\left\{Gu\right|u\in B_{k_0}\}$ is uniformly bounded. Considering $u\in B_{k_0}$ and any interval where $0\le t_1<t_2\le T$, then

$$\parallel \left(Gu\right)\left(t_2\right)-\left(Gu\right)\left(t_1\right)\parallel \le T_1+T_2+T_3+T_4+T_5+T_6.$$

where

$$\begin{array}{cc}\hfill & T_1=\parallel \left[S_{\alpha ,1-\alpha }\left(t_2\right)-S_{\alpha ,1-\alpha }\left(t_1\right)\right]\left[u_1-h\left(u\right)+H(0,u\left(0\right))\right]\parallel ,\hfill \\ \hfill & T_2=\parallel H(t_2,u\left(t_2\right))-H(t_1,u\left(t_1\right))\parallel ,\hfill \\ \hfill & T_3={\int }_0^{t_1}\parallel S_{\alpha ,\beta }(t_2-s)-S_{\alpha ,\beta }(t_1-s)\parallel \parallel f(s,u\left(s\right))\parallel ds,\hfill \\ \hfill & T_4={\int }_{t_1}^{t_2}\parallel S_{\alpha ,\beta }(t_2-s)f(s,u\left(s\right))\parallel ds,\hfill \\ \hfill & T_5={\int }_0^{t_1}\parallel {(t_2-s)}^{\alpha -1}A\Psi (t_2-s)-{(t_1-s)}^{\alpha -1}A\Psi (t_1-s)\parallel \parallel H(s,u\left(s\right))\parallel ds,\hfill \\ \hfill & T_6={\int }_{t_1}^{t_2}{(t_2-s)}^{\alpha -1}\parallel A\Psi (t_2-s)\parallel \parallel H(s,u\left(s\right))\parallel ds.\hfill \end{array}$$

We can prove that $T_1,T_3\to 0$ as $t_1\to t_2$. Given that $u\in B_{k_0}$ and in light of assumption $\left({\mathrm{H}}_2\right)$, $T_2\to 0$ as $t_1\to t_2$. Applying Equation (16) together with $\left({\mathrm{H}}_3\right)$ to $T_4$, we find that

$$T_4\le {\int }_{t_1}^{t_2}M{(t_2-s)}^{\alpha +\beta -1}{k}_0{\parallel \varpi \parallel }_{\infty }ds=\frac{M{k}_0{\parallel \varpi \parallel }_{\infty }}{\alpha +\beta }{(t_2-t_1)}^{\alpha +\beta },$$

therefore, $T_4\to 0$ as $t_1\to t_2$. Since

$$\begin{array}{cc}\hfill T_5& \le {\int }_0^{t_1}\left[{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}\right]\parallel A\Psi (t_2-s)H(s,u\left(s\right))\parallel ds\hfill \\ \hfill & +{\int }_0^{t_1}{(t_1-s)}^{\alpha -1}\parallel A\Psi (t_2-s)-A\Psi (t_1-s)\parallel \parallel H(s,u\left(s\right))\parallel ds\hfill \\ \hfill & =:T_{51}+T_{52},\hfill \end{array}$$

by Equation (18) and $\left({\mathrm{H}}_5\right)$,

$$\begin{array}{cc}\hfill T_{51}& \le {\int }_0^{t_1}\left[{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}\right]\parallel A^{1-\rho }\Psi (t_2-s)\parallel \parallel A^{\rho }H(s,u\left(s\right))\parallel ds\hfill \\ \hfill & \le C{\xi }_2{k}_0{\int }_0^{t_1}\left[{(t_2-s)}^{\alpha -1}-{(t_1-s)}^{\alpha -1}\right]ds{\int }_0^{t_1}{(t_1-s)}^{\alpha (\rho -1)}ds\hfill \\ \hfill & =\frac{t_2^{\alpha }-t_1^{\alpha }-{(t_2-t_1)}^{\alpha }}{\alpha }·\frac{C{\xi }_2{k}_0}{\alpha (\rho -1)+1}{t}_1^{\alpha (\rho -1)+1}\hfill \\ \hfill & \le C{t}_1^{\alpha (\rho -1)+1}{\xi }_2{k}_0\left[t_2^{\alpha }-t_1^{\alpha }-{(t_2-t_1)}^{\alpha }\right],\hfill \end{array}$$

where $C>0$ is a constant, then $T_{51}\to 0$ as $t_1\to t_2$. If $t_1=0$ and $t_2$ is within $J^{\prime }$, it is clear that $T_{52}\to 0$ as $t_1\to t_2$. Additionally, for $t_1>0$ and a sufficiently small $ϵ>0$,

$$\begin{array}{cc}\hfill T_{52}& \le {\int }_0^{t_1-ϵ}{(t_1-s)}^{\alpha -1}\parallel \Psi (t_2-s)-\Psi (t_1-s)\parallel \parallel AH(s,u\left(s\right))\parallel ds\hfill \\ \hfill & +{\int }_{t_1-ϵ}^{t_1}{(t_1-s)}^{\alpha -1}\parallel \Psi (t_2-s)-\Psi (t_1-s)\parallel \parallel AH(s,u\left(s\right))\parallel ds\hfill \\ \hfill & \le {\xi }_2{k}_0\parallel A^{1-\rho }\parallel {\int }_0^{t_1-ϵ}{(t_1-s)}^{\alpha -1}ds\underset{s\in [0,t_1-ϵ]}{sup}\parallel \Psi (t_2-s)-\Psi (t_1-s)\parallel \hfill \\ \hfill & +C{\xi }_2{k}_0\parallel A^{1-\rho }\parallel {\int }_{t_1-ϵ}^{t_1}{(t_1-s)}^{\alpha -1}ds\hfill \\ \hfill & =\frac{{\xi }_2{k}_0}{\alpha }(t_1^{\alpha }-{ϵ}^{\alpha })\parallel A^{1-\rho }\parallel \underset{s\in [0,t_1-ϵ]}{sup}\parallel \Psi (t_2-s)-\Psi (t_1-s)\parallel +C{\xi }_2{k}_0{ϵ}^{\alpha }\parallel A^{1-\rho }\parallel ,\hfill \end{array}$$

since $\Psi \left(t\right)$ is continuous in the uniform operator topology for $t>0$, then $T_{52}\to 0$ independently with $u\in B_{k_0}$ as $t_1\to t_2$ and $ϵ\to 0$; hence, $T_5\to 0$ as $t_1\to t_2$. Furthermore, according to Equation (18) and $\left({\mathrm{H}}_5\right)$, $T_6\le \frac{C{\xi }_2{k}_0}{\alpha \rho }{(t_2-t_1)}^{\alpha \rho }$, $C>0$ is a constant, so $T_6\to 0$ as $t_1\to t_2$, ensuring the equicontinuity of $\left\{Gu\right|u\in B_{k_0}\}$.

Finally, to establish the relative compactness of the set $\left\{\left(Gu\right)\left(t\right)\right|u\in B_{k_0}\}$ in X, according to ([19], theorem 2.11) and due to the compactness of $H(t,u(t\left)\right)$, it is necessary to demonstrate that the set $\left\{\left(\mathcal{J}u\right)\left(t\right)\right|u\in B_{k_0}\}$ is also relatively compact in X, where

$$\left(\mathcal{J}u\right)\left(t\right)={\int }_0^t{(t-s)}^{\alpha -1}A\Psi (t-s)H(s,u\left(s\right))ds.$$

Obviously, $\left(\mathcal{J}u\right)\left(0\right)$ possesses the relative compactness in X. Let $0<t\le T$ be fixed, for arbitrary values $0<p_1<t$ and $p_2>0$, we define an operator $\mathcal{V}$ on $B_{k_0}$ as

$$\left(\mathcal{V}u\right)\left(t\right)={\int }_0^{t-p_1}{\int }_{p_2}^{\infty }\alpha \theta {(t-s)}^{\alpha -1}A{\zeta }_{\alpha }\left(\theta \right)T\left({(t-s)}^{\alpha }\theta \right)H(s,u\left(s\right))d\theta ds,$$

then,

$$\left(\mathcal{V}u\right)\left(t\right)=T\left(p_1^{\alpha }{p}_2\right){\int }_0^{t-p_1}{\int }_{p_2}^{\infty }\alpha \theta {(t-s)}^{\alpha -1}A{\zeta }_{\alpha }\left(\theta \right)T({(t-s)}^{\alpha }\theta -p_1^{\alpha }{p}_2)H(s,u\left(s\right))d\theta ds.$$

By $\left({\mathrm{H}}_1\right)$, $T\left(p_1^{\alpha }{p}_2\right)(p_1^{\alpha }{p}_2>0)$ is compact, then for arbitrary $0<p_1<t$ and $p_2>0$, $\left\{\left(\mathcal{V}u\right)\left(t\right)\right|u\in B_{k_0}\}$ is relatively compact in X. For any $u\in B_{k_0}$, as indicated by $\parallel T\left(t\right)\parallel \le N$, Equation (15), and $\left({\mathrm{H}}_5\right)$,

$$\begin{array}{cc}\hfill & \parallel \left(\mathcal{J}u\right)\left(t\right)-\left(\mathcal{V}u\right)\left(t\right)\parallel \hfill \\ \hfill & \le \alpha {\int }_0^t{\int }_0^{p_2}\parallel \theta {(t-s)}^{\alpha -1}A{\zeta }_{\alpha }\left(\theta \right)T\left({(t-s)}^{\alpha }\theta \right)H(s,u\left(s\right))\parallel d\theta ds\hfill \\ \hfill & +\alpha {\int }_{t-p_1}^t{\int }_{p_2}^{\infty }\parallel \theta {(t-s)}^{\alpha -1}A{\zeta }_{\alpha }\left(\theta \right)T\left({(t-s)}^{\alpha }\theta \right)H(s,u\left(s\right))\parallel d\theta ds\hfill \\ \hfill & \le \alpha N{\xi }_2{k}_0\parallel A^{1-\rho }\parallel {\int }_0^t{(t-s)}^{\alpha -1}ds{\int }_0^{p_2}\theta {\zeta }_{\alpha }\left(\theta \right)d\theta \hfill \\ \hfill & +\alpha N{\xi }_2{k}_0\parallel A^{1-\rho }\parallel {\int }_{t-p_1}^t{(t-s)}^{\alpha -1}ds{\int }_0^{\infty }\theta {\zeta }_{\alpha }\left(\theta \right)d\theta \hfill \\ \hfill & \le N{T}^{\alpha }{\xi }_2{k}_0\parallel A^{1-\rho }\parallel {\int }_0^{p_2}\theta {\zeta }_{\alpha }\left(\theta \right)d\theta +\frac{N{p}_1^{\alpha }}{\Gamma (1+\alpha )}{\xi }_2{k}_0\parallel A^{1-\rho }\parallel .\hfill \end{array}$$

thus, it is possible to identify relatively compact sets arbitrarily close to $\left\{\left(\mathcal{J}u\right)\left(t\right)\right|u\in B_{k_0}\}$, where $t>0$. Consequently, $\left\{\left(\mathcal{J}u\right)\left(t\right)\right|u\in B_{k_0}\}$ itself is relatively compact in X. Following this, $\left\{\left(Gu\right)\left(t\right)\right|u\in B_{k_0}\}$ also achieves relative compactness in X. According to the Arzela–Ascoli theorem [20], the set $\left\{Gu\right|u\in B_{k_0}\}$ is relatively compact. The compactness, combined with the continuity of G establishes that G is completely continuous. Applying Lemma 9, we find that G has a fixed point on $B_{k_0}$, implying that Equation (8) admits a mild solution. The proof is thereby complete. □

Theorem 2.

Under the conditions set forth by assumptions $\left(H_2\right)$–$\left(H_3\right)$, $\left(H_4^{\prime }\right)$, and $\left(H_5\right)$–$\left(H_6\right)$, if the inequality

$$Ma+\parallel A^{-\rho }{\parallel \parallel \varpi \parallel }_{\infty }+C{T}^{\alpha \rho }{\parallel \varpi \parallel }_{\infty }+\frac{M{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }<1,$$

holds, and if Equation (21) is satisfied, then Equation (8) admits a mild solution.

Proof. 

Based on Theorem 1, we establish a proof that demonstrates the continuity of $G:B_{k_0}\to B_{k_0}$. Additionally, the set $\left\{Gu\right|u\in B_{k_0}\}$ is uniformly bounded and equicontinuous, where $k_0$ satisfies Equation (22). Consider a subset V of $B_{k_0}$ such that $V\subset \overline{G\left(V\right)}\cup \left\{0\right\}$. Due to the boundedness and equicontinuity of V, it follows that $v\left(t\right)=\nu \left(V\right(t\left)\right)$ is continuous for any $t\in J$. By employing Lemma 1, Equation (16), Lemma 6, $\left({\mathrm{H}}_5\right)$, and $\left({\mathrm{H}}_6\right)$, we have

$$\begin{array}{cc}\hfill {\parallel v\parallel }_{\infty }& \le \underset{t\in J}{sup}\parallel \nu (\left(GV\right)\left(t\right)\cup \left\{0\right\})\parallel \hfill \\ \hfill & \le \underset{t\in J}{sup}\parallel \nu \left(\left(GV\right)\left(t\right)\right)\parallel \hfill \\ \hfill & \le \nu \left(h\left(V\right)\right)\underset{t\in J}{sup}\parallel S_{\alpha ,1-\alpha }\left(t\right)\parallel +\parallel A^{-\rho }\parallel \varpi \left(t\right)\nu \left(V\left(t\right)\right)\hfill \\ \hfill & +\underset{t\in J}{sup}{\int }_0^t{(t-s)}^{\alpha -1}\parallel A^{1-\rho }\Psi (t-s)\parallel \varpi \left(s\right)\nu \left(V\left(s\right)\right)ds\hfill \\ \hfill & +\underset{t\in J}{sup}{\int }_0^t\parallel S_{\alpha ,\beta }(t-s)\parallel \varpi \left(s\right)\nu \left(V\left(s\right)\right)ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {Ma\parallel v\parallel }_{\infty }+\parallel A^{-\rho }{\parallel \parallel \varpi \parallel }_{\infty }{\parallel v\parallel }_{\infty }+\underset{t\in J}{sup}{\int }_0^{t}C{(t-s)}^{\alpha \rho -1}\varpi \left(s\right)v\left(s\right)ds\hfill \\ \hfill & +M\underset{t\in J}{sup}{\int }_0^t{(t-s)}^{\alpha +\beta -1}\varpi \left(s\right)v\left(s\right)ds\hfill \\ \hfill & \le \left(Ma+\parallel A^{-\rho }{\parallel \parallel \varpi \parallel }_{\infty }+C{T}^{\alpha \rho }{\parallel \varpi \parallel }_{\infty }+\frac{M{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }\right){\parallel v\parallel }_{\infty },\hfill \end{array}$$

since

$$Ma+\parallel A^{-\rho }{\parallel \parallel \varpi \parallel }_{\infty }+C{T}^{\alpha \rho }{\parallel \varpi \parallel }_{\infty }+\frac{M{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }<1,$$

then, we have ${\parallel v\parallel }_{\infty }=0$, which indicates that $v\left(t\right)=\nu \left(V\right(t\left)\right)=0$, ensuring that $V\left(t\right)$ is relatively compact in X. By applying the Arzela–Ascoli theorem [20], $\left\{Gu\right|u\in B_{k_0}\}$ possesses the relative compactness. This confirms that $\nu \left(V\right)=0$. Lemma 8 ensures that G has a fixed point on $B_{k_0}$. In other words, Equation (8) has a mild solution. Hence, the proof is complete. □

Theorem 3.

Suppose that $\left(H_2^{\prime }\right)$, $\left(H_3\right)$,$\left(H_4^{\prime }\right)$, $\left(H_5\right)$, and $\left(H_7\right)$ hold, then Equation (8) is guaranteed to has a unique mild solution provided that

$$M{L}^{\prime }+(1+M){\xi }_2\parallel A^{-\rho }\parallel +\frac{C{\xi }_2}{\alpha \rho }{T}^{\alpha \rho }+\frac{M{T}^{\alpha +\beta }}{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }<1$$

(27)

and

$$M{L}^{\prime }+{\xi }_1\parallel A^{-\rho }\parallel +\frac{C{\xi }_1}{\alpha \rho }{T}^{\alpha \rho }+\frac{M{T}^{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }}{\alpha +\beta }<1.$$

(28)

Proof. 

In the proof of Theorem 1, it is not necessary to utilize the continuity conditions in $\left({\mathrm{H}}_2\right)$ and complete continuity conditions in $\left({\mathrm{H}}_4\right)$ while we proved G maps $B_{k_0}$ into itself. Therefore, by $\left({\mathrm{H}}_2^{\prime }\right)$, $\left({\mathrm{H}}_3\right)$,$\left({\mathrm{H}}_4^{\prime }\right)$, and $\left({\mathrm{H}}_5\right)$, given that $u\in B_{k_1}$, we can demonstrate that $S_{\alpha ,1-\alpha }\left(t\right)\left(u_1-h\left(u\right)+H(0,u\left(0\right))\right)$ exist, $S_{\alpha ,\beta }(t-s)f(s,u\left(s\right))$ and ${(t-s)}^{\alpha -1}A\Psi$$(t-s)H(s,u(s\left)\right)$ are Bochner integrable with respect to $s\in J$ and $t\in J$, and G maps $B_{k_1}$ into itself. Now, we only need to prove that G has a unique fixed point on $B_{k_1}$.

For arbitrary $u,v\in B_{k_1}$ and $t\in J$, by Equation (16), Lemma 6, $\left({\mathrm{H}}_5\right)$, and $\left({\mathrm{H}}_7\right)$, we obtain

$$\begin{array}{cc}\hfill & \parallel \left(Gu\right)\left(t\right)-\left(Gv\right)\left(t\right)\parallel \hfill \\ \hfill & \le \parallel S_{\alpha ,1-\alpha }\left(t\right)\left[h\left(v\right)-h\left(u\right)\right]\parallel +\parallel H(t,v\left(t\right))-H(t,u\left(t\right))\parallel \hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha -1}\parallel A^{1-\rho }\Psi (t-s)\parallel \parallel A^{\rho }H(s,u\left(s\right))-A^{\rho }H(s,v\left(s\right))\parallel ds\hfill \\ \hfill & +{\int }_0^t\parallel S_{\alpha ,\beta }(t-s)\parallel \parallel f(s,u\left(s\right))-f(s,v\left(s\right))\parallel ds\hfill \\ \hfill & \le M{L}^{\prime }{\parallel u-v\parallel }_{\infty }+{\xi }_1\parallel A^{-\rho }{\parallel \parallel u-v\parallel }_{\infty }+C{\xi }_1{\parallel u-v\parallel }_{\infty }{\int }_0^t{(t-s)}^{\alpha \rho -1}ds\hfill \\ \hfill & {+M\parallel u-v\parallel }_{\infty }{\int }_0^t{(t-s)}^{\alpha +\beta -1}\varpi \left(s\right)ds\hfill \\ \hfill & \le \left(M{L}^{\prime }+{\xi }_1\parallel A^{-\rho }\parallel +\frac{C{\xi }_1{T}^{\alpha \rho }}{\alpha \rho }+\frac{M{T}^{\alpha +\beta }{\parallel \varpi \parallel }_{\infty }}{\alpha +\beta }\right){\parallel u-v\parallel }_{\infty }.\hfill \end{array}$$

It follows from Equation (28) that G is a contraction mapping. Therefore, by applying the Banach fixed-point theorem, we conclude that Equation (8) has a unique mild solution, the proof is complete. □

4. An Example

We consider fractional neutral integro-differential equations with nonlocal initial conditions:

$$\left\{\begin{array}{c}{\partial }_t^{\frac{1}{3}}\left(u(x,t)+H(x,t,u(x,t\left)\right)\right)={\partial }_x^{2}u(x,t)+I_t^{\frac{1}{2}}f(x,t,u(x,t)),x\in [0,1],t\in (0,T],\hfill \\ u_x^{\prime }(0,t)=u_x^{\prime }(\pi ,t)=0,t\in [0,T],\hfill \\ u(x,0)+{\displaystyle \sum _{i=0}^n}{\int }_0^{\pi }k(x,s)u(s,t_i)ds=u_0\left(x\right),x\in [0,1],\hfill \end{array}\right.$$

(29)

where ${\partial }_t^{\frac{1}{3}}$ and $I_t^{\frac{1}{2}}$ are the Caputo fractional partial derivative and the Riemann–Liouville integral, respectively. $k(x,s)\in L^2(\left[0,\pi \right]\times \left[0,\pi \right])$.

Let $X=L^2\left([0,1]\right)$ and $Au=u^{″}$ with the domain

$$D\left(A\right)=\{u(·)\in X|u\mathrm{and}{u}^{\prime }\mathrm{as}\mathrm{absolutely}\mathrm{continuous},\mathrm{and}{u}^{″}\in X,u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0.\}.$$

Then, A generates a strongly continuous continuous semigroup, which is denoted by ${\left\{T\left(t\right)\right\}}_{t\ge 0}$. Then, ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ is a compact semigroup ($\left({\mathrm{H}}_1\right)$ holds), continuous in uniform operator topology, and $\parallel T\left(t\right)\parallel \le e^{-t},t\ge 0$. Moreover, the resolvent family

$$\begin{array}{c}\hfill S_{\frac{1}{3},\frac{1}{2}}\left(t\right)=\sum _{n=1}^{\infty }{t}^{-\frac{1}{6}}{E}_{\frac{1}{3},\frac{5}{6}}(-n^2{t}^{\frac{1}{3}})(u,y_n)y_n,t\ge 0,\end{array}$$

and

$$\begin{array}{c}\hfill \parallel S_{\frac{1}{3},\frac{1}{2}}\left(t\right)\parallel \le \frac{t^{-\frac{1}{6}}}{\Gamma \left(\frac{5}{6}\right)},t>0,\end{array}$$

where $E_{\gamma ,\eta }(·)$ is the Mittag–Leffler function for $\gamma$, $\eta >0$ [19].

Set $u\left(t\right)x=u(x,t)$, $H(t,u(t\left)\right)x=H(x,t,u(x,t\left)\right)$, and $f(t,u(t\left)\right)x=f(x,t,u(x,t\left)\right)$, where $x\in [0,1]$, $t\in [0,T]$, and $g:C\left(\right[0,T],X)\to X$ is given by

$$\left(gu\right)\left(x\right)=\sum _{i=0}^n{\int }_0^{1}k(x,s)u(s,t_i)ds,x\in [0,1].$$

Then, (31) can be written as the following problem in X:

$$\left\{\begin{array}{c}{D}_t^{\frac{1}{3}}\left(u\left(t\right)+H(t,u(t\left)\right)\right)=Au\left(t\right)+I_t^{\frac{1}{2}}f(t,u\left(t\right)),t\in (0,T],\hfill \\ u\left(0\right)+g\left(u\right)=u_0.\hfill \end{array}\right.$$

(30)

If $f(t,u(t\left)\right)=sinu\left(t\right)$, $H(t,u(t\left)\right)=cosu\left(t\right)$, $L=L^{\prime }=(n+1){\left[{\int }_0^{\pi }{\int }_0^{\pi }{k}^2(x,s)dsdx\right]}^{\frac{1}{2}}$, and note that g is completely continuous, then $\left({\mathrm{H}}_2\right)$–$\left({\mathrm{H}}_7\right)$ and $\left({\mathrm{H}}_2^{\prime }\right)$–$\left({\mathrm{H}}_4^{\prime }\right)$ are satisfied, where $\varpi \left(t\right)=1$. By Theorem 1, or 2, or 3, (8) has a mild solution if the inequalities in the theorems hold.

5. Discussion

This paper defines the mild solutions of fractional neutral equations with Neumann boundary conditions through the Laplace transform, a resolvent family ${\left\{S_{\alpha ,\beta }\left(t\right)\right\}}_{t\ge 0}$, and the function $\Psi \left(t\right)$. It also establishes several sufficient conditions for the existence of mild solutions to the equations. Importantly, it demonstrates deriving (16) from

$$S_{\alpha ,\beta }\left(t\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}{s}^{\alpha -1}\Psi \left(s\right)ds$$

in [19]. Furthermore, the study achieves the same results as those reported in [19] without the use of path integration.

On the other hand, the probability density function ${\zeta }_{\alpha }\left(\theta \right)$ plays a significant role in studying solutions of fractional differential equations. Building on the findings presented in this article, further investigation into the solution of differential equations of the form

$${}^c{D}_t^{\alpha }u(x,t)+\sum _{i=1}^m{b}_i^c{D}_t^{{\alpha }_i}u(x,t)=Au(x,t)+f(x,t),$$

(31)

which involve multiple fractional derivatives, is warranted, where $t>0$, $x\in {\mathbb{R}}^n$, $0<{\alpha }_m<\cdots <{\alpha }_1<\alpha \le 1$, $b_i>0$, $i=1,\dots ,m$, A generates an analytic semigroup with boundedness. Bazhlekova established the fundamental properties, primarily complete monotonicity, of the Prabhakar-type generalizations for multinomial Mittag–Leffler functions. These properties were investigated through the use of Laplace transform and Bernstein functions in studying Equation (31), resulting in several derived estimates.

As the results in [29,30], $\mathcal{L}\left[{\zeta }_{\alpha }\left(\theta \right)\right]=E_{\alpha }(-z)$, and the multinomial Mittag–Leffler function

$$E_{(a_1,a_2,\dots ,a_n),b}(z_1,z_2,\dots ,z_n)=\sum _{k=0}^{\infty }\sum _{\genfrac{}{}{0pt}{}{l_1+\cdots +l_n=k}{l_1\ge 0,\dots l_n\ge 0}}\frac{k!}{{\displaystyle \prod _{i=1}^n}{l}_i!}\frac{{\displaystyle \prod _{i=1}^n}{z}_i^{l_i}}{\Gamma (b+{\displaystyle \sum _{i=1}^n}{a}_i{l}_i)}.$$

(32)

Since the Laplace transform in ${\mathbb{R}}^n$ is defined as

$$\mathcal{L}\left[\phi \left(\mathbf{t}\right)\right]=\widehat{\phi }\left(\mathbf{s}\right)={\int }_{{\mathbb{R}}_{+\cdots +}^n}{e}^{-\mathbf{st}}\phi \left(\mathbf{t}\right)d\mathbf{t},$$

(33)

where $\mathbf{s}\in {\mathbb{C}}^n$, ${\mathbb{R}}_{+\cdots +}^n=\{\mathbf{t}=(t_1,\dots ,t_n)\in {\mathbb{R}}^n,t_j>0,j=1,\dots ,n\}$. Then, we will consider obtaining the multinomial form of ${\zeta }_{\alpha }\left(\theta \right)$ by Equations (32) and (33) to investigate solutions of Equation (31).