Existence of Mild Solutions for Fractional Integrodifferential Equations with Hilfer Derivatives

Abstract

In this paper, we study the existence of solutions for fractional integrodifferential equations with Hilfer derivatives. We establish some new existence theorems for mild solutions by using Schaefer’s fixed-point theorem, a measure of noncompactness, and the resolvent operators associated with almost sectorial operators. Our results improve and extend many known results in the relevant references by removing some strong assumptions. Furthermore, we propose new nonlocal initial conditions for Hilfer evolution equations and study the existence of mild solutions to nonlocal problems.

1. Introduction

Fractional calculus tools are quite effective in modeling anomalous diffusion processes, as fractional-order operators can characterize long memory processes. Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering. In recent years, there have been significant developments in ordinary and partial differential equations involving fractional derivatives (see the monographs [1,2,3,4] and the relevant literature [5,6,7,8,9,10,11,12,13,14,15,16,17]). Many researchers have pointed out that fractional calculus is more suitable for characterizing materials and processes with genetic and memory properties than integer calculus. However, due to the nonlocality and singularity of fractional differential operators, the theoretical research of fractional differential equations is very difficult. Therefore, the study of fractional differential equations has important theoretical significance and practical value.

There are two main motivations for studying fractional evolution equations. One is that many mathematical models in physics and fluid mechanics are characterized by fractional partial differential equations. The other is that many types of fractional partial differential equations that describe physical phenomena and fluid characteristics, such as fractional diffusion equations, wave equations, Navier–Stokes equations, Rayleigh–Stokes equations, Fokker–Planck equations, Schrödinger equations, and so on, can be abstracted as fractional evolution equations [4]. Therefore, the study of fractional evolution equations is of great significance in terms of both theory and practical application.

Hilfer [1] proposed a new fractional derivative called the general Riemann–Liouville fractional derivative or the Hilfer fractional derivative, which is described as an interpolation of the Riemann–Liouville and Caputo fractional derivatives in the theoretical modeling of the broadband dielectric relaxation spectroscopy of glasses [1,18]. In addition to defining the order of the Hilfer fractional derivative, Hilfer also introduced a new parameter $q\in [0,1]$ such that $q=0$ gives the Riemann–Liouville derivative, $q=1$ gives the Caputo fractional derivative, and $0<q<1$ gives the interpolation of the two derivatives. For more detailed explanations of the mathematics and physics of fractional derivatives and integrals, we refer the reader to [1,18,19].

In recent years, fractional differential equations with Hilfer fractional derivatives have aroused the interest of many scholars. Many authors have studied the existence and controllability of initial value problems for evolution equations with Hilfer fractional derivatives (see [6,9,10,11,12,13,14,15,16,17,20]). However, much less is known about Hilfer fractional integrodifferential equations with almost sectorial operators.

Consider the initial value problem of fractional integrodifferential equations with almost sectorial operators:

$$\left\{\begin{array}{c}{\displaystyle {}^{H}D_{0+}^{p,q}x\left(t\right)=Ax\left(t\right)+F(t,x\left(t\right))+{\int }_0^{t}h(t,s,x\left(s\right))ds,t\in (0,T],}\hfill \\ I_{0+}^{(1-p)(1-q)}x\left(0\right)=x_0,\hfill \end{array}\right.$$

(1)

where ${}^{H}D_{0+}^{p,q}$ is the Hilfer fractional derivative of order $0<p<1$ and type $0\le q\le 1$, A is an almost sectorial operator in the Banach space X, $I_{0+}^{(1-p)(1-q)}$ is the Riemann–Liouville fractional integral of order $(1-p)(1-q)$, and $F:[0,T]\times X\to X$ and $h:\Lambda \times X\to X$ $(\Lambda =\{(t,s)\in [0,T]\times [0,T]:t\ge s\left\}\right)$ are two functions that are introduced later, with $x_0\in X,T\in (0,\infty )$.

In this paper, we study the existence of solutions for fractional integrodifferential Equation (1) with Hilfer derivatives. Furthermore, we extend our methods to researching the existence of mild solutions for nonlocal problems.

The novelty and important aspects of this manuscript are as follows:

(i)

For Hilfer fractional integrodifferential equations with almost sectorial operators, existence theorems are established in the case where the nonlinear terms on the right-hand side of the equation are not continuous.

(ii)

New nonlocal initial conditions are proposed for Hilfer evolution equations and applied to the investigation of existence, which arises from physical problems.

(iii)

Our results improve and extend many known results in the relevant references.

The remainder of this paper is organized as follows. In Section 2, we introduce some preliminaries about fractional calculus, almost sectorial operators, the measure of noncompactness, and the resolvent operators. In Section 3, we state the basic hypotheses of this article and prove some lemmas. For two cases in which the almost sectorial operators are compact and noncompact, we establish two new existence theorems of mild solutions for the Cauchy problem via extended Carathéodory conditions in Section 4. Our main theorems essentially improve and generalize some known results in the literature. In Section 5, we propose a new nonlocal condition and provide some existence criteria for the mild solutions of nonlocal problems in two cases where the almost sectorial operators are compact and noncompact.

2. Preliminaries

We begin by stating some basic facts about almost sectorial operators, fractional calculus, the measure of noncompactness, and the resolvent operators. For more details, we refer the reader to [1,2,3,4,21,22].

Let A be a linear operator from X to itself. Denote by $D\left(A\right)$ the domain of A and by $\sigma \left(A\right)$ its spectrum, while $\rho \left(A\right):=\mathbb{C}-\sigma \left(A\right)$ is the resolvent set of A. Let $S_{\nu }^0$ with $0<\nu <\pi$ be the open sector $\{z\in \mathbb{C}\setminus \{0\}:|argz|<\nu \}$, and $S_{\nu }$ be its closure, i.e., $S_{\nu }=\{z\in \mathbb{C}\setminus \left\{0\right\}:|argz|\le \nu \}\cup \left\{0\right\}$.

Definition 1.

Assume that $0<k<1$ and $0<\omega <{\displaystyle \frac{\pi }{2}}$. By ${\mathrm{\Theta }}_{\omega }^{-k}\left(X\right)$, we denote a family of all linear closed operators $A:D\left(A\right)\subset X\to X$ which satisfy

(i)

$\sigma \left(A\right)\subset S_{\omega }=\{z\in \mathbb{C}\setminus \left\{0\right\}:|argz|\le \omega \}\cup \left\{0\right\}$;

(ii)

$\forall \nu \in (\omega ,\pi )$, there exists a constant $C_{\nu }$ such that

$${\parallel R(z;A)\parallel }_{\mathcal{L}\left(X\right)}\le C_{\nu }{\left|z\right|}^{-k},\mathrm{for}\mathrm{all}z\in \mathbb{C}\setminus S_{\nu },$$

where $R(z;A)={(zI-A)}^{-1},z\in \rho \left(A\right)$ the resolvent of A for $z\in \rho \left(A\right)$.

The linear operator A is called an almost sectorial operator on X if $A\in {\mathrm{\Theta }}_{\omega }^{-k}\left(X\right)$.

Define the power of A as

$$A^{\beta }={\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_{\vartheta }}{z}^{\beta }R(z;A)dz,\beta >1-k,$$

where ${\Gamma }_{\vartheta }=\left\{{\mathbb{R}}^{+}{e}^{i\vartheta }\right\}\bigcup \left\{{\mathbb{R}}^{+}{e}^{-i\vartheta }\right\}$ is an appropriate path oriented counterclockwise and $\omega <\vartheta <\nu$. Then, the linear power space $X_{\vartheta }:=D\left(A^{\vartheta }\right)$ can be defined and $X_{\vartheta }$ is a Banach space with the graph norm ${\parallel x\parallel }_{\vartheta }=\left|A^{\vartheta }x\right|,x\in D\left(A^{\vartheta }\right)$.

Next, let us introduce the semigroup associated with A. Denote the semigroup ${\left\{Q\left(t\right)\right\}}_{t\ge 0}$ by

$$Q\left(t\right)=e^{-tz}\left(A\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_{\vartheta }}{e}^{-tz}R(z;A)dz,t\in S_{{\displaystyle \frac{\pi }{2}}-\omega }^0,$$

where ${\Gamma }_{\vartheta }=\left\{{\mathbb{R}}^{+}{e}^{i\rho }\right\}\bigcup \left\{{\mathbb{R}}^{+}{e}^{-i\vartheta }\right\}$ and $\omega <\vartheta <\nu <{\displaystyle \frac{\pi }{2}}-|argt|$ is oriented counterclockwise.

Lemma 1

(see [22]). Assume that $0<k<1$ and $0<\omega <{\displaystyle \frac{\pi }{2}}$. Set $A\in {\mathrm{\Theta }}_{\omega }^{-k}\left(X\right)$. Then,

(i)

$Q(s+t)=Q\left(s\right)Q\left(t\right)$, for any $s,t\in S_{{\displaystyle \frac{\pi }{2}}-\omega }^0$;

(ii)

there exists a constant $C_0>0$ such that ${\parallel Q\left(t\right)\parallel }_{\mathcal{L}\left(X\right)}\le C_0{t}^{k-1}$, for any $t>0$.

Definition 2

([2]). The fractional integral of order p for a function $x\in L^1(J,X)$ is defined as

$$I_{0+}^{p}x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(p\right)}}{\int }_0^t{(t-s)}^{p-1}x\left(s\right)ds,p>0,t>0,$$

where $\Gamma (·)$ is the gamma function.

Definition 3

([1]). Let $0<p<1$ and $0\le q\le 1$. The Hilfer fractional derivative of order p and type q for a function $x\in L^1(J,X)$ is defined as

$${}^{H}D_{0+}^{p,q}x\left(t\right)=I_{0+}^{q(1-p)}{\displaystyle \frac{d}{dt}}{I}_{0+}^{(1-p)(1-q)}x\left(t\right).$$

Remark 1.

(i) 

If $q=0$, $0<p<1$, then

$${}^{H}D_{0+}^{p,0}x\left(t\right)={\displaystyle \frac{d}{dt}}{I}_{0+}^{1-p}x\left(t\right)=:{}^{L}D_{0+}^{p}x\left(t\right),$$

where ${}^{L}D_{0+}^p$ is a Riemann–Liouville derivative.

(ii) 

If $q=1$, $0<p<1$, then

$${}^{H}D_{0+}^{p,1}x\left(t\right)=I_{0+}^{1-p}{\displaystyle \frac{d}{dt}}x\left(t\right)=:{}^{C}D_{0+}^{p}x\left(t\right),$$

where ${}^{C}D_{0+}^p$ is a Caputo derivative.

Definition 4

([23]). The Wright function $W_p\left(\theta \right)$ is defined as follows:

$$W_p\left(\theta \right)=\sum _{n=1}^{\infty }{\displaystyle \frac{{(-\theta )}^{n-1}}{(n-1)!\Gamma (1-pn)}},0<p<1,\theta \in \mathbb{C},$$

which satisfies

$${\int }_0^{\infty }{\theta }^{\delta }{W}_p\left(\theta \right)d\theta ={\displaystyle \frac{\Gamma (1+\delta )}{\Gamma (1+p\delta )}},for\delta >-1.$$

Assume that D B is a nonempty subset of X. Kuratowski’s measure of noncompactness $\chi$ is said to be

$$\chi \left(D\right)=inf\left\{d>0:\Omega \subset \bigcup _{i=1}^n{M}_i\mathrm{and}\mathrm{diam}\left(M_i\right)\le d\right\},$$

where the diameter of $M_i$ is given by diam$\left(M_i\right)=sup\left\{\right|x-y|:x,y\in M_i\}$, $i=1,\dots ,n.$

Proposition 1.

Let $D_1$ and $D_2$ be two bounded sets of a Banach space X. Then, the following properties hold:

(i) 

$\chi \left(D_1\right)=0$ if and only if $D_1$ is relatively compact in X;

(ii) 

$\chi \left(D_1\right)\le \chi \left(D_2\right)$ if $D_1\subseteq D_2$;

(iii) 

$\chi \left\{\right\{x\}\cup D\}=\chi \left(D\right)$ for every $x\in X$ and every nonempty subset $B\subset X$;

(iv) 

$\chi \{D_1+D_2\}\le \chi \left(D_1\right)+\chi \left(D_2\right)$, where $D_1+D_2=\{x+y:x\in D_1,y\in D_2\}$;

(v) 

$\chi \{D_1\cup D_2\}\le max\{\chi \left(D_1\right),\chi \left(D_2\right)\}$;

(vi) 

$\chi \left(\kappa D\right)\le \left|\kappa \right|\chi \left(D\right)$ for any $\kappa \in \mathbb{R}$.

Let $W\subset C(J,X)$. Assume that

$${\int }_0^{t}W\left(s\right)ds=\left\{{\int }_0^{t}u\left(s\right)ds:u\in W\right\},\mathrm{for}t\in J,$$

where $W\left(s\right)=\left\{u\right(s)\in X:u\in W\}$.

Proposition 2

([24]).  Let $W\subset C(J,X)$ be bounded and equicontinuous. Then, $\overline{co}W\subset C(J,X)$ is also bounded and equicontinuous.

Proposition 3

([25]).  Let $W\subset C(J,X)$ be bounded and equicontinuous. Then, $t\to \chi \left(W\right(t\left)\right)$ is continuous on J, and

$$\chi \left(W\right)=\underset{t\in J}{max}\chi \left(W\left(t\right)\right),\chi \left({\int }_0^{t}W\left(s\right)ds\right)\le {\int }_0^t\chi \left(W\left(s\right)\right)ds,\mathrm{for}t\in J.$$

Proposition 4

([26]).  Let W be bounded. Then, for each $\epsilon >0$, there is a sequence ${\left\{y_n\right\}}_{n=1}^{\infty }\subset W$ such that

$$\chi \left(W\right)\le 2\chi \left({\left\{y_n\right\}}_{n=1}^{\infty }\right)+\epsilon .$$

Proposition 5

([24]).  Let X be a Banach space, and let ${\left\{y_n\left(t\right)\right\}}_{n=1}^{\infty }:[0,\infty )\to X$ be a continuous function family. Assume that there exists $\xi \in L^1[0,\infty )$ such that

$$|y_n\left(t\right)|\le \xi \left(t\right),t\in [0,\infty ),n=1,2,\dots .$$

Then, $\chi \left({\left\{y_n\left(t\right)\right\}}_{n=1}^{\infty }\right)$ is integrable on $[0,\infty )$, and

$$\chi \left(\left\{{\int }_0^t{y}_n\left(s\right)ds:n=1,2,\dots \right\}\right)\le 2{\int }_0^t\chi \left(\{y_n\left(s\right):n=1,2,\dots \}\right)ds.$$

Lemma 2

([10]).  Problem (1) is equivalent to the following integral equation:

$$\begin{array}{cc}\hfill x\left(t\right)=& {\displaystyle \frac{x_0}{\Gamma \left(q\right(1-p)+p)}}{t}^{(q-1)(1-p)}\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(p\right)}}{\int }_0^t{(t-s)}^{p-1}[Ax\left(s\right)+\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)]ds,t\in (0,T].\hfill \end{array}$$

(2)

Lemma 3.

Assume that $x\left(t\right)$ satisfies the integral Equation (2). Then,

$$x\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)x_0+{\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,t\in (0,T],$$

(3)

where

$${\mathcal{F}}_{p,q}\left(t\right)=I_{0+}^{q(1-p)}{\mathcal{G}}_p\left(t\right),{\mathcal{G}}_p\left(t\right)=t^{p-1}{\mathcal{T}}_p\left(t\right),and{\mathcal{T}}_p\left(t\right)={\int }_0^{\infty }p\theta W_p\left(\theta \right)Q\left(t^p\theta \right)d\theta .$$

Proof. 

This proof is similar to that in [10], so we omit it. □

Definition 5.

If $x\in C\left(\right(0,T],X)$ satisfies

$$x\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)x_0+{\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,t\in (0,T],$$

then $x\left(t\right)$ is called a mild solution for the initial value problem (1).

Lemma 4

([11]).  Let the operator ${\left\{Q\left(t\right)\right\}}_{t>0}$ be compact. Then, the operators ${\left\{{\mathcal{F}}_{p,q}\left(t\right)\right\}}_{t>0}$ and ${\left\{{\mathcal{T}}_p\left(t\right)\right\}}_{t>0}$ are also compact.

Lemma 5

([27]).  Let the operator ${\left\{Q\left(t\right)\right\}}_{t>0}$ be compact. Then, ${\left\{Q\left(t\right)\right\}}_{t>0}$ is equicontinuous.

Lemma 6

([20]).  ${\mathcal{T}}_p\left(t\right)$, ${\mathcal{G}}_p\left(t\right)$, and ${\mathcal{F}}_{p,q}\left(t\right)$ are linear operators for any fixed $t>0$. For any $t>0,x\in X$,

$$|{\mathcal{T}}_p\left(t\right)x|\le l_1{t}^{p(k-1)}\left|x\right|,|{\mathcal{G}}_p\left(t\right)x|\le l_1{t}^{pk-1}\left|x\right|,\mathrm{and}|{\mathcal{F}}_{p,q}\left(t\right)x|\le l_2{t}^{-1+q-pq+pk}\left|x\right|,$$

where

$$l_1={\displaystyle \frac{C_0\Gamma \left(k\right)}{\Gamma \left(pk\right)}},l_2={\displaystyle \frac{C_0\Gamma \left(k\right)}{\Gamma \left(q\right(1-p)+pk)}}.$$

Lemma 7

([11]). Assume that ${\left\{Q\left(t\right)\right\}}_{t>0}$ is equicontinuous. Then, ${\left\{{\mathcal{T}}_p\left(t\right)\right\}}_{t>0}$, ${\left\{{\mathcal{G}}_p\left(t\right)\right\}}_{t>0}$, and ${\left\{{\mathcal{F}}_{p,q}\left(t\right)\right\}}_{t>0}$ are strongly continuous; that is, for any $x\in X$ and $\sigma >\tau >0$,

$$\begin{array}{cc}\hfill & |{\mathcal{T}}_p\left(\sigma \right)x-{\mathcal{T}}_p\left(\tau \right)x|\to 0,|{\mathcal{G}}_p\left(\sigma \right)x-{\mathcal{G}}_p\left(\tau \right)x|\to 0,\hfill \\ \hfill & |{\mathcal{F}}_{p,q}\left(\sigma \right)x-{\mathcal{F}}_{p,q}\left(\tau \right)x|\to 0,as\sigma \to \tau .\hfill \end{array}$$

3. Lemmas

In the following, we always suppose that $A\in {\mathrm{\Theta }}_{\omega }^{-k}\left(X\right)$, $0<k<1$, and $0<\omega <{\displaystyle \frac{\pi }{2}}$. We first state the basic hypotheses of this article:

(H1)

${\left\{Q\left(t\right)\right\}}_{t>0}$ is equicontinuous.

(H2)

For each $t\in [0,T]$, the function $F(t,·):X\to X$ is continuous; for each $x\in X$, the function $F(·,x):[0,T]\to X$ is strongly measurable.

(H3)

There exists a function $m_1\in L((0,T],{\mathbb{R}}^{+})$ such that

$$\left|F(t,x)\right|\le m_1\left(t\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in (0,T]\mathrm{and}\mathrm{any}x\in X,$$

where $m_1\left(t\right)$ satisfies

$$I_{0+}^{pk}{m}_1\in C((0,T],{\mathbb{R}}^{+}),\underset{t\to 0+}{lim}{t}^{1-q+pq-pk}{I}_{0+}^{pk}{m}_1\left(t\right)=0.$$

(H4)

For each $(t,s)\in \Lambda$, the function $h(t,s,·):X\to X$ is continuous; for each $x\in X$, the function $h(·,·,x):\Lambda \to X$ is strongly measurable. Moreover, there exists a function $G:\Lambda \to {\mathbb{R}}^{+}$ such that

$$\left|h\right(t,s,x\left)\right|\le G(t,s),\mathrm{for}\mathrm{almost}\mathrm{all}(t,s)\in \Lambda ,\mathrm{and}\mathrm{any}x\in X.$$

(H5)

There exists a function $m_2\in L((0,T],{\mathbb{R}}^{+})$ such that

$$\left|{\int }_0^{t}G(t,s)ds\right|\le m_2\left(t\right),\mathrm{for}\mathrm{almost}\mathrm{all}(t,s)\in \Lambda ,$$

where $m_2\left(t\right)$ satisfies

$$I_{0+}^{pk}{m}_2\in C((0,T],{\mathbb{R}}^{+}),\underset{t\to 0+}{lim}{t}^{1-q+pq-pk}{I}_{0+}^{pk}{m}_2\left(t\right)=0.$$

From (H3) and (H5), we know that $t^{1-q+pq-pk}{I}_{0+}^{pk}\left(m_1\left(t\right)+m_2\left(t\right)\right)$ is bounded on $[0,T]$. Therefore, there exists a constant $r>0$ such that

$$l_2\left|x_0\right|+l_1\underset{t\in [0,T]}{sup}\left\{t^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\right\}\le r.$$

where

$$l_1={\displaystyle \frac{C_0\Gamma \left(k\right)}{\Gamma \left(pk\right)}},l_2={\displaystyle \frac{C_0\Gamma \left(k\right)}{\Gamma \left(q\right(1-p)+pk)}}.$$

Let

$$\begin{array}{cc}\hfill & C_p((0,T],X)=\left\{x\in C([0,T],X):\underset{t\to 0+}{lim}{t}^{1-q+pq-pk}\left|x\left(t\right)\right|\mathrm{exists}\mathrm{and}\mathrm{is}\mathrm{finite}\right\},\hfill \end{array}$$

with the norm

$${\parallel x\parallel }_p=\underset{t\in (0,T]}{sup}\{t^{1-q+pq-pk}\left|x\left(t\right)\right|\}.$$

Then, $(C_p((0,T],X),\parallel ·{\parallel }_p)$ is a Banach space.

For any $x\in C_p((0,T],X)$, define an operator $\Psi$ as follows:

$$(\Psi x)\left(t\right)=\left({\Psi }_{1}x\right)\left(t\right)+\left({\Psi }_{2}x\right)\left(t\right),$$

where

$$\begin{array}{cc}\hfill & \left({\Psi }_{1}x\right)\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)x_0,\hfill \\ \hfill & \left({\Psi }_{2}x\right)\left(t\right)={\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,\mathrm{for}t\in (0,T].\hfill \end{array}$$

Clearly, problem (1) has a mild solution $x^{*}\in C_p((0,T],X)$ if and only if $\Psi$ has a fixed point $x^{*}\in C_p((0,T],X)$.

For any $u\in C\left(\right[0,T],X)$, let

$$x\left(t\right)=t^{-1+q-pq+pk}u\left(t\right),t\in (0,T].$$

Then, $x\in C_p((0,T],X)$. Define an operator $\Phi$ as

$$(\Phi u)\left(t\right)=\left({\Phi }_{1}u\right)\left(t\right)+\left({\Phi }_{2}u\right)\left(t\right),$$

where

$$\begin{array}{cc}\hfill \left({\Phi }_{1}u\right)\left(t\right)& =\left\{\begin{array}{cccc}\hfill & t^{1-q+pq-pk}\left({\Psi }_{1}x\right)\left(t\right),\hfill & \hfill & \mathrm{for}t\in (0,T],\hfill \\ \hfill & 0,\hfill & \hfill & \mathrm{for}t=0,\hfill \end{array}\right.\hfill \\ \hfill \left({\Phi }_{2}u\right)\left(t\right)& =\left\{\begin{array}{cccc}\hfill & t^{1-q+pq-pk}\left({\Psi }_{2}x\right)\left(t\right),\hfill & \hfill & \mathrm{for}t\in (0,T],\hfill \\ \hfill & 0,\hfill & \hfill & \mathrm{for}t=0.\hfill \end{array}\right.\hfill \end{array}$$

Set

$$S_r=\{u\in C([0,T],X):\parallel u\parallel \le r\}.$$

and

$${\tilde{S}}_r=\{x\in C_p{((0,T],X):\parallel x\parallel }_p\le r\}.$$

Obviously, $S_r$ and ${\tilde{S}}_r$ are closed, convex, and nonempty subsets of $C\left(\right[0,T],X)$ and $C_p((0,T],X)$, respectively.

Lemma 8.

Suppose that (H1)–(H5) hold. Then, the set $\left\{(\Phi u)\left(t\right),u\in S_r\right\}$ is equicontinuous.

Proof. Claim I. 

$\left\{{\Phi }_{1}u,u\in S_r\right\}$ is equicontinuous.

• Step 1. For $\tau =0$, $\sigma \in (0,T]$, we have

  $$\begin{array}{cc}\hfill |\left({\Phi }_{1}u\right)\left(\sigma \right)-\left({\Phi }_{1}u\right)\left(0\right)|\le & |{\sigma }^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(\sigma \right)x_0-0|\to 0,\mathrm{as}\sigma \to 0.\hfill \end{array}$$

  (4)

  In fact,

  $$\begin{array}{cc}\hfill {\sigma }^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(\sigma \right)x_0& ={\displaystyle \frac{{\sigma }^{1-q+pq-pk}}{\Gamma \left(q\right(1-p\left)\right)}}{\int }_0^{\sigma }{(\sigma -s)}^{q(1-p)-1}{s}^{p-1}{\mathcal{T}}_p\left(s\right)x_{0}ds\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma \left(q\right(1-p\left)\right)}}{\int }_0^1{(1-z)}^{q(1-p)-1}{z}^{p-1}{\sigma }^{p(1-k)}{\mathcal{T}}_p\left(\sigma z\right)x_{0}dz.\hfill \end{array}$$

  Since ${lim}_{\sigma \to 0+}{\sigma }^{p(1-k)}{\mathcal{T}}_p\left(\sigma z\right)x_0=0$ and ${\int }_0^1{(1-z)}^{q(1-p)-1}{z}^{p-1}dz$ exists, (4) holds.
• Step 2. For any $\tau ,\sigma \in (0,T]$ and $\tau <\sigma$, we have

  $$\begin{array}{cc}\hfill & |\left({\Phi }_{1}u\right)\left(\sigma \right)-\left({\Phi }_{1}u\right)\left(\tau \right)|\hfill \\ \hfill \le & |{\sigma }^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(\sigma \right)x_0-{\tau }^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(\tau \right)x_0|\hfill \\ \hfill \le & |{\sigma }^{1-q+pq-pk}\left|\right|{\mathcal{F}}_{p,q}\left(\sigma \right)x_0-{\mathcal{F}}_{p,q}\left(\tau \right)x_0|\hfill \\ \hfill & +|{\sigma }^{1-q+pq-pk}-{\tau }^{1-q+pq-pk}\left|\right|{\mathcal{F}}_{p,q}\left(\tau \right)x_0|\to 0,\mathrm{as}\sigma \to \tau .\hfill \end{array}$$

  (5)

  Hence, $\left\{{\Phi }_{1}u,u\in S_r\right\}$ is equicontinuous.

Claim II. $\left\{{\Phi }_{2}u,u\in S_r\right\}$ is equicontinuous.

Let $x\left(t\right)=t^{-1+q-pq+pk}u\left(t\right)$ for any $u\in S_r$, $t\in (0,T]$. Then, $x\in {\tilde{S}}_r,$

• Step 1. For $\tau =0$, $0<\sigma <T$, we obtain

  $$\begin{array}{cc}\hfill & |\left({\Phi }_{2}u\right)\left(\sigma \right)-\left({\Phi }_{2}u\right)\left(0\right)|\hfill \\ \hfill =& \left|{\sigma }^{1-q+pq-pk}{\int }_0^{\sigma }{\mathcal{G}}_p(\sigma -s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|\to 0,\mathrm{as}\sigma \to 0.\hfill \end{array}$$

  In fact, by Lemma 6 and (H3)–(H5), we have

  $$\begin{array}{cc}\hfill & \left|{\sigma }^{1-q+pq-pk}{\int }_0^{\sigma }{\mathcal{G}}_p(\sigma -s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|\hfill \\ \hfill \le & l_1{\sigma }^{1-q+pq-pk}{\int }_0^{\sigma }{(\sigma -s)}^{pk-1}\left(m_1\left(s\right)+{\int }_0^{s}G(s,\mu )d\mu \right)ds\hfill \\ \hfill \le & l_1{\sigma }^{1-q+pq-pk}{\int }_0^{\sigma }{(\sigma -s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\to 0,\mathrm{as}\sigma \to 0.\hfill \end{array}$$

  (6)
• Step 2. For $0<\tau <\sigma \le T$, we obtain

  $$\begin{array}{cc}\hfill & |\left({\Phi }_{2}u\right)\left(\sigma \right)-\left({\Phi }_{2}u\right)\left(\tau \right)|\hfill \\ \hfill \le & \left|{\tau }^{1-q+pq-pk}{\int }_{\tau }^{\sigma }{(\sigma -s)}^{p-1}{\mathcal{T}}_p(\sigma -s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|\hfill \\ \hfill +& |{\tau }^{1-q+pq-pk}{\int }_0^{\tau }\left({(\sigma -s)}^{p-1}-{(\tau -s)}^{p-1}\right)\hfill \\ \hfill \times & {\mathcal{T}}_p(\sigma -s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds|\hfill \\ \hfill +& \left|{\tau }^{1-q+pq-pk}{\int }_0^{\tau }{(\tau -s)}^{p-1}\left({\mathcal{T}}_p(\sigma -s)-{\mathcal{T}}_p(\tau -s)\right)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|\hfill \\ \hfill +& |{\sigma }^{1-q+pq-pk}-{\tau }^{1-q+pq-pk}|\left|{\int }_0^{\sigma }{(\sigma -s)}^{p-1}{\mathcal{T}}_p(\sigma -s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|\hfill \\ \hfill \le & l_1{\tau }^{1-q+pq-pk}|{\int }_0^{\sigma }{(\sigma -s)}^{pk-1}(m_1\left(s\right)+m_2\left(s\right))ds-{\int }_0^{\tau }{(\tau -s)}^{pk-1}(m_1\left(s\right)+m_2\left(s\right))ds|,\hfill \\ \hfill +& 2l_1{\tau }^{1-q+pq-pk}{\int }_0^{\tau }({(\tau -s)}^{p-1}-{(\sigma -s)}^{p-1}){(\sigma -s)}^{p(k-1)}(m_1\left(s\right)+m_2\left(s\right))ds,\hfill \\ \hfill +& {\tau }^{1-q+pq-pk}\left|{\int }_0^{\tau }{(\tau -s)}^{p-1}\left({\mathcal{T}}_p(\sigma -s)-{\mathcal{T}}_p(\tau -s)\right)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|,\hfill \\ \hfill +& |{\sigma }^{1-q+pq-pk}-{\tau }^{1-q+pq-pk}|\left|l_1{\int }_0^{\sigma }{(\sigma -s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\right|\hfill \\ \hfill =:& I_1+I_2+I_3+I_4,\hfill \end{array}$$

  One can deduce that ${lim}_{\sigma \to \tau }{I}_1=0$, since $I_{0+}^{pk}(m_1+m_2)\in C((0,T],(0,T])$. Noting that

  $$\begin{array}{cc}\hfill & ({(\tau -s)}^{p-1}-{(\sigma -s)}^{p-1}){(\sigma -s)}^{p(k-1)}\left(m_1\left(s\right)+m_2\left(s\right)\right)\hfill \\ \hfill \le & {(\tau -s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right),\mathrm{for}s\in [0,\tau ),\hfill \end{array}$$

  the Lebesgue dominated convergence theorem implies that

  $${\int }_0^{\tau }\left({(\tau -s)}^{p-1}-{(\sigma -s)}^{p-1}\right){(\sigma -s)}^{p(k-1)}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\to 0,\mathrm{as}\sigma \to \tau .$$

  So, $I_2\to 0$ as $\sigma \to \tau$. By (H3), for $\epsilon >0$, we have

  $$\begin{array}{cc}\hfill I_3\le & {\tau }^{1-q+pq-pk}{\int }_0^{\tau -\epsilon }{(\tau -s)}^{p-1}\hfill \\ \hfill & \times \parallel {\mathcal{T}}_p(\sigma -s)-{\mathcal{T}}_p(\tau -s){\parallel }_{\mathcal{L}\left(X\right)}\left|\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)\right|ds\hfill \\ \hfill & +{\tau }^{1-q+pq-pk}{\int }_{\tau -\epsilon }^{\tau }{(\tau -s)}^{p-1}\hfill \\ \hfill & \times \parallel {\mathcal{T}}_p(\sigma -s)-{\mathcal{T}}_p(\tau -s){\parallel }_{\mathcal{L}\left(X\right)}\left|\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)\right|ds\hfill \\ \hfill \le & {\tau }^{1-q+pq-pk}{\int }_0^{\tau }{(\tau -s)}^{p-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\underset{s\in [0,\tau -\epsilon ]}{sup}{\parallel {\mathcal{T}}_p(\sigma -s)-{\mathcal{T}}_p(\tau -s)\parallel }_{\mathcal{L}\left(X\right)}\hfill \\ \hfill & +2l_1{\tau }^{1-q+pq-pk}{\int }_{\tau -\epsilon }^{\tau }{(\tau -s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\hfill \\ \hfill \le & {\tau }^{1-q+pq-pk}{\int }_0^{\tau }{(\tau -s)}^{p-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\underset{s\in [0,\tau -\epsilon ]}{sup}{\parallel {\mathcal{T}}_p(\sigma -s)-{\mathcal{T}}_p(\tau -s)\parallel }_{\mathcal{L}\left(X\right)},\hfill \\ \hfill & +2l_{1}L{\tau }^{1-q+pq-pk}|{\int }_0^{\tau }{(\tau -s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\hfill \\ \hfill & -{\int }_0^{\tau -\epsilon }{(\tau -\epsilon -s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds|,\hfill \\ \hfill & +2l_{1}L{\tau }^{1-q+pq-pk}{\int }_0^{\tau -\epsilon }({(\tau -\epsilon -s)}^{pk-1}-{(\tau -s)}^{pk-1})\left(m_1\left(s\right)+m_2\left(s\right)\right)ds\hfill \\ \hfill =:& I_{31}+I_{32}+I_{33}.\hfill \end{array}$$

  From (H1) and Lemma 7, we easily obtain that $I_{31}\to 0$ as $\sigma \to \tau$. Similar to the process of proving that $I_1\to 0$ and $I_2\to 0$, we have $I_{32}\to 0$ and $I_{33}\to 0$ as $\epsilon \to 0$. Thus, $I_3\to 0$ as $\sigma \to \tau$. Obviously, $I_4\to 0$ as $\sigma \to \tau$. Hence, $\left\{{\Phi }_{2}u,u\in S_r\right\}$ is equicontinuous. Therefore, $\left\{\Phi u,u\in S_r\right\}$ is equicontinuous. □

Lemma 9.

Suppose that (H1)–(H5) hold. Then, $\Phi S_r\subset S_r$.

Proof. 

Let $x\left(t\right)=t^{-1+q-pq+pk}u\left(t\right)$, and let $u\in S_r$, $t\in (0,T]$. Then, $x\in {\tilde{S}}_r$.

By Lemma 8, we can find that $\Phi S_r\subset C([0,T],X)$. For $t>0$ and any $u\in S_r$,

$$\begin{array}{cc}\hfill \left|\right(\Phi u\left)\right(t\left)\right|\le & \left|t^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(t\right)x_0\right|\hfill \\ & +\left|t^{1-q+pq-pk}{\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds\right|\hfill \\ \hfill \le & l_2\left|x_0\right|+l_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(F(s,x\left(s\right))+{\int }_0^{s}G(s,\mu )d\mu \right)ds\hfill \\ \hfill \le & l_2\left|x_0\right|+l_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds<r.\hfill \end{array}$$

(7)

In the case where $t=0$, we know that $\left|\right(\Phi u\left)\right(0\left)\right|=0<r$. Thus, $\Phi S_r\subset S_r$. □

Lemma 10.

Suppose that (H1)–(H5) hold. Then, $\Phi$ is continuous.

Proof. 

Let ${\left\{u_n\right\}}_{n=1}^{\infty }\in S_r$ and $u_n\to u\in S_r$ as $n\to \infty$. Thus,

$$\underset{n\to \infty }{lim}{u}_n\left(t\right)=u\left(t\right),\mathrm{and}\underset{n\to \infty }{lim}{t}^{-1+q-pq+pk}{u}_n\left(t\right)=t^{-1+q-pq+pk}u\left(t\right),\mathrm{for}t\in (0,T].$$

Let $x\left(t\right)=t^{-1+q-pq+pk}u\left(t\right)$ and $x_n\left(t\right)=t^{-1+q-pq+pk}{u}_n\left(t\right)$, $t\in (0,T]$. Then $x,x_n\in {\tilde{S}}_r$.

By (H3) and (H5), we obtain

$$\underset{n\to \infty }{lim}F(t,x_n\left(t\right))=\underset{n\to \infty }{lim}F(t,t^{-1-q+pq-pk}{u}_n\left(t\right))=F(t,t^{-1-q+pq-pk}u\left(t\right))=F(t,x\left(t\right)),$$

and

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}{\int }_0^{t}h(t,\mu ,x_n\left(\mu \right))d\mu =\underset{n\to \infty }{lim}{\int }_0^{t}h(t,\mu ,{\mu }^{-1-q+pq-pk}{u}_n\left(\mu \right))d\mu \hfill \\ \hfill & ={\int }_0^{t}h(t,\mu ,{\mu }^{-1-q+pq-pk}u\left(\mu \right))d\mu ={\int }_0^{t}h(t,\mu ,x\left(\mu \right))d\mu .\hfill \end{array}$$

On the other hand, for each $t\in (0,T]$,

$$\begin{array}{cc}\hfill & {(t-s)}^{pk-1}|\left(F(s,x_n\left(s\right))+{\int }_0^{s}h(s,\mu ,x_n\left(\mu \right))d\mu \right)-\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)|\hfill \\ \hfill \le & 2{(t-s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right).\hfill \end{array}$$

By the Lebesgue dominated convergence theorem, we obtain

$$\begin{array}{cc}\hfill & {\int }_0^t{(t-s)}^{pk-1}|\left(F(s,x_n\left(s\right))+{\int }_0^{s}h(s,\mu ,x_n\left(\mu \right))d\mu \right)\hfill \\ \hfill & -\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)|ds\to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

Hence, for $t\in [0,T]$,

$$\begin{array}{cc}\hfill |(\Phi u_n)\left(t\right)-(\Phi u)\left(t\right)|\le & t^{1-q+pq-pk}{\int }_0^t\left|{\mathcal{G}}_p(t-s)\right(\left(F(s,x_n\left(s\right))+{\int }_0^{s}h(s,\mu ,x_n\left(\mu \right))d\mu \right)\hfill \\ \hfill & -\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)\left)\right|ds\hfill \\ \hfill \le & l_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}|\left(F(s,x_n\left(s\right))+{\int }_0^{s}h(s,\mu ,x_n\left(\mu \right))d\mu \right)\hfill \\ \hfill & -\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)|ds\to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

Hence, $\parallel \Phi u_n-\Phi u\parallel \to 0$ as $n\to \infty$. Therefore, $\Phi$ is continuous. The proof is complete. □

4. Existence

4.1. Compact Operator Case

Assume that the operator $Q\left(t\right)$ is compact for any $t>0$.

Theorem 1.

Suppose that (H2)–(H5) hold. Then, the initial value problem (1) has at least one mild solution in ${\tilde{S}}_r$.

Proof. 

By Lemmas 9 and 10, $\Phi S_r\subset S_r$ and $\Phi$ is continuous. By Lemma 8, the set $\left\{\Phi u,u\in S_r\right\}$ is equicontinuous. It remains to prove that $\left\{(\Phi u)\left(t\right),u\in S_r\right\}$ is relatively compact in X for $t\in [0,T]$. Clearly, $\left\{(\Phi u)\left(0\right),u\in S_r\right\}$ is relatively compact in X. We only need to consider the case where $t>0$. For any $\epsilon >0,\delta >0$, we define ${\Phi }_{\epsilon ,\delta }$ on $S_r$ as

$$\begin{array}{cc}\hfill \left({\Phi }_{\epsilon ,\delta }u\right)\left(t\right):=& t^{1-q+pq-pk}\left({\Psi }_{\epsilon ,\delta }x\right)\left(t\right)\hfill \\ \hfill =& t^{1-q+pq-pk}({\mathcal{F}}_{p,q}\left(t\right)x_0+{\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }p\theta {(t-s)}^{p-1}{W}_p\left(\theta \right)\hfill \\ \hfill & \times Q\left({(t-s)}^p\theta \right)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)d\theta ds).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \left({\Phi }_{\epsilon ,\delta }u\right)\left(t\right)=& t^{1-q+pq-pk}({\mathcal{F}}_{p,q}\left(t\right)x_0+Q\left({\epsilon }^p\delta \right){\int }_0^{t-\epsilon }{\int }_{\delta }^{\infty }p\theta {(t-s)}^{p-1}{W}_p\left(\theta \right)\hfill \\ \hfill & \times Q({(t-s)}^p\theta -{\epsilon }^p\delta )\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)d\theta ds).\hfill \end{array}$$

Since $Q\left(t\right)$ is compact for $t>0$, by Lemma 4, ${\mathcal{F}}_{p,q}\left(t\right)$ is compact. Moreover, $Q\left({\epsilon }^p\delta \right)$ is compact. Therefore, the set $\{\left({\Phi }_{\epsilon ,\delta }u\right)\left(t\right),u\in S_r\}$ is relatively compact in X. In addition, for every $u\in S_r$, we have

$$\begin{array}{cc}\hfill & |(\Phi u)\left(t\right)-\left({\Phi }_{\epsilon ,\delta }u\right)\left(t\right)|\hfill \\ \hfill \le & t^{1-q+pq-pk}\hfill \\ \hfill & \times \left|{\int }_0^t{\int }_0^{\delta }p\theta {(t-s)}^{p-1}{W}_p\left(\theta \right)Q\left({(t-s)}^p\theta \right)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)d\theta ds\right|\hfill \\ \hfill & +t^{1-q+pq-pk}\hfill \\ \hfill & \times \left|{\int }_{t-\epsilon }^t{\int }_{\delta }^{\infty }p\theta {(t-s)}^{p-1}{W}_p\left(\theta \right)Q\left({(t-s)}^p\theta \right)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)d\theta ds\right|\hfill \\ \hfill \le & p{C}_0{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left|\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)\right|ds{\int }_0^{\delta }{\theta }^k{W}_p\left(\theta \right)d\theta \hfill \\ \hfill & +p{C}_0{t}^{1-q+pq-pk}{\int }_{t-\epsilon }^t{(t-s)}^{pk-1}\left|\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)\right|ds{\int }_0^{\infty }{\theta }^k{W}_p\left(\theta \right)d\theta \hfill \\ \hfill \le & p{C}_0{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(F(s,x\left(s\right))+{\int }_0^{s}G(s,\mu )d\mu \right)ds{\int }_0^{\delta }{\theta }^k{W}_p\left(\theta \right)d\theta \hfill \\ \hfill & +p{C}_0{t}^{1-q+pq-pk}{\int }_{t-\epsilon }^t{(t-s)}^{pk-1}\left(F(s,x\left(s\right))+{\int }_0^{s}G(s,\mu )d\mu \right)ds{\int }_0^{\infty }{\theta }^k{W}_p\left(\theta \right)d\theta \hfill \\ \hfill \le & p{C}_0{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds{\int }_0^{\delta }{\theta }^k{W}_p\left(\theta \right)d\theta \hfill \\ \hfill & +p{C}_0{t}^{1-q+pq-pk}{\int }_{t-\epsilon }^t{(t-s)}^{pk-1}\left(m_1\left(s\right)+m_2\left(s\right)\right)ds{\int }_0^{\infty }{\theta }^k{W}_p\left(\theta \right)d\theta \hfill \\ \hfill \to & 0,\mathrm{as}\epsilon \to 0,\delta \to 0.\hfill \end{array}$$

Hence, $\left\{(\Phi u)\left(t\right),u\in S_r\right\}$ is also a relatively compact set in X for $t\in [0,T]$. By the Ascoli–Arzela theorem, $\left\{\Phi u,u\in S_r\right\}$ is relatively compact. Therefore, $\Phi$ is a completely continuous operator. By the Schauder fixed-point theorem, $\Phi$ has at least one fixed point $u^{*}\in S_r$. Let $x^{*}\left(t\right)=t^{-1+q-pq+pk}{u}^{*}\left(t\right)$. Thus,

$$x^{*}\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)x_0+{\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x^{*}\left(s\right))+{\int }_0^{s}h(s,\mu ,x^{*}\left(\mu \right))d\mu \right)ds,t\in (0,T],$$

which implies that $x^{*}$ is a mild solution of (1) in ${\tilde{S}}_r$. The proof is complete. □

Corollary 1.

Suppose that (H2) and (H3) hold. Then, the initial value problem of the fractional evolution equations

$$\left\{\begin{array}{c}{}^{H}D_{0+}^{p,q}x\left(t\right)=Ax\left(t\right)+F(t,x\left(t\right)),t\in (0,T],\hfill \\ I_{0+}^{(1-p)(1-q)}x\left(0\right)=x_0,\hfill \end{array}\right.$$

(8)

has at least one mild solution in ${\tilde{S}}_r$.

Corollary 2.

Suppose that (H4) and (H5) hold. Then, the initial value problem of the fractional integrodifferential equations

$$\left\{\begin{array}{c}{\displaystyle {}^{H}D_{0+}^{p,q}x\left(t\right)=Ax\left(t\right)+{\int }_0^{t}h(t,s,x\left(s\right))ds,t\in (0,T],}\hfill \\ I_{0+}^{(1-p)(1-q)}x\left(0\right)=x_0,\hfill \end{array}\right.$$

(9)

has at least one mild solution in ${\tilde{S}}_r$.

Remark 2.

Corollary 1 improves upon the main results in [11] by removing a very strong constraint condition

$$\underset{t\in [0,T]}{sup}\left(t^{(1-pk)(1-q)}\left|{\mathcal{F}}_{p,q}\left(t\right)x_0\right|+t^{(1-pk)(1-q)}{\int }_0^t{(t-s)}^{pk-1}{l}_1\left(s\right)ds\right)\le r.$$

Corollary 1 also improves and extends some related results, such as those in [3,20] and other relevant literature.

4.2. Noncompact Operator Case

Next, we consider the case where the almost sectorial operator A generates a noncompact semigroup $Q\left(t\right)(t>0)$ on X. The following assumptions are required:

(H6)

For any bounded set $E\subset X$, there exists a constant $\vartheta >0$ such that

$$\chi \left(F\right(t,E\left)\right)\le \vartheta \chi \left(E\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in [0,T].$$

(H7)

For any bounded set $E\subset X$, there exists a function $\ell :\Lambda \to {\mathbb{R}}^{+}$ such that

$$\chi \left(h(t,s,E)\right)\le {(s/t)}^{1-q+pq-pk}\ell (t,s)\chi \left(E\right),\mathrm{for}\mathrm{a}.\mathrm{e}.(t,s)\in \Lambda ,t>0,$$

where ℓ satisfies ${sup}_{t\in [0,T]}{\int }_0^t\ell (t,s)ds=:\eta$.

Theorem 2.

Suppose that (H1)-(H7) hold. Then, the initial value problem (1) has at least one mild solution in ${\tilde{S}}_r$.

Proof. 

In view of the proof of Theorem 1, we only need to show that there exists a set $\widehat{\mathrm{\Delta }}\subset S_r$ such that $\left\{(\Phi u)\left(t\right),u\in \widehat{\mathrm{\Delta }}\right\}$ is relatively compact in X for $t\in [0,T]$.

For any bounded subset $E_0\subset S_r$, let

$${\Phi }^1\left(E_0\right)=\Phi \left(E_0\right),{\Phi }^n\left(E_0\right)=\Phi \left(\overline{co}\left({\Phi }^{n-1}\left(E_0\right)\right)\right),n=2,3,\dots .$$

By Propositions 3–5, for any $\epsilon >0$, there is a sequence ${\left\{u_n^{\left(1\right)}\right\}}_{n=1}^{\infty }\subset E_0$ such that

$$\begin{array}{cc}\hfill & \chi \left({\Phi }^1\left(E_0\left(t\right)\right)\right)=\chi (\Phi \left(E_0\left(t\right)\right))\hfill \\ \hfill \le & 2\chi \left(t^{1-q+pq-pk}{\int }_0^t{\mathcal{G}}_p(t-s)\right(f\left(s,{\left\{s^{-\left(1-q+pq-pk\right)}{u}_n^{\left(1\right)}\left(s\right)\right\}}_{n=1}^{\infty }\right)\hfill \\ \hfill & +{\int }_0^{s}h(s,v,{\left\{v^{-\left(1-q+pq-pk\right)}{u}_n^{\left(1\right)}\left(v\right)\right\}}_{n=1}^{\infty })dv\left)ds\right)+\epsilon \hfill \\ \hfill \le & 4L_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(\chi \right(f\left(s,{\left\{s^{-\left(1-q+pq-pk\right)}{u}_n^{\left(1\right)}\left(s\right)\right\}}_{n=1}^{\infty }\right)\hfill \\ \hfill & +{\int }_0^{s}h(s,v,{\left\{v^{-\left(1-q+pq-pk\right)}{u}_n^{\left(1\right)}\left(v\right)\right\}}_{n=1}^{\infty })dv\left)\right)ds+\epsilon \hfill \\ \hfill \le & 4L_1{t}^{1-q+pq-pk}(\vartheta \chi \left(E_0\right){\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)}ds\hfill \\ \hfill & +2\chi \left(E_0\right){\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)}\left({\int }_0^s\ell (s,v)dv\right)ds)+\epsilon \hfill \\ \hfill \le & 4L_1(\vartheta +2\eta )t^{1-q+pq-pk}\chi \left(E_0\right){\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)}ds+\epsilon \hfill \\ \hfill \le & 4L_1(\vartheta +2\eta )t^{pk}\chi \left(E_0\right){\displaystyle \frac{\Gamma \left(pk\right)\Gamma (q-pq+pk))}{\Gamma (2pk+q-pq)}}+\epsilon .\hfill \end{array}$$

Due to the arbitrariness of $\epsilon >0$, we obtain

$$\begin{array}{c}\hfill \chi \left({\Phi }^1\left(E_0\left(t\right)\right)\right)\le 4L_1(\vartheta +2\eta )t^{pk}{\displaystyle \frac{\Gamma \left(pk\right)\Gamma (q-pq+pk))}{\Gamma (2pk+q-pq)}}\chi \left(E_0\right).\end{array}$$

By Propositions 3–5, for any $\epsilon >0$, there is a sequence ${\left\{u_n^{\left(2\right)}\right\}}_{n=1}^{\infty }\subset \overline{co}\left({\Phi }^1\left(E_0\right)\right)$ such that

$$\begin{array}{cc}\hfill & \chi \left({\Phi }^2\left(E_0\left(t\right)\right)\right)=\chi (\Phi \left(\overline{co}\left({\Phi }^1\left(E_0\left(t\right)\right)\right)\right))\hfill \\ \hfill \le & 2\chi \left(t^{1-q+pq-pk}{\int }_0^t{\mathcal{G}}_p(t-s)\right(f\left(s,{\left\{s^{-\left(1-q+pq-pk\right)}{u}_n^{\left(2\right)}\left(s\right)\right\}}_{n=1}^{\infty }\right)\hfill \\ \hfill & +{\int }_0^{s}h(s,v,{\left\{v^{-\left(1-q+pq-pk\right)}{u}_n^{\left(2\right)}\left(v\right)\right\}}_{n=1}^{\infty })dv\left)ds\right)+\epsilon \hfill \\ \hfill \le & 4L_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}\left(\chi \right(f\left(s,{\left\{s^{-\left(1-q+pq-pk\right)}{u}_n^{\left(2\right)}\left(s\right)\right\}}_{n=1}^{\infty }\right)\hfill \\ \hfill & +{\int }_0^{s}h(s,v,{\left\{v^{-\left(1-q+pq-pk\right)}{u}_n^{\left(2\right)}\left(v\right)\right\}}_{n=1}^{\infty })dv\left)\right)ds+\epsilon \hfill \\ \hfill \le & 4L_1\vartheta t^{1-q+pq-pk}({\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)}\chi \left(u_n^{\left(2\right)}\left(s\right)\right)ds\hfill \\ \hfill & +2{\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)}\left({\int }_0^s\ell (s,v)dv\right)\chi \left(u_n^{\left(2\right)}\left(s\right)\right)ds)+\epsilon \hfill \\ \hfill \le & 4L_1(\vartheta +2\eta )t^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)}\chi \left(u_n^{\left(2\right)}\left(s\right)\right)ds+\epsilon \hfill \\ \hfill \le & {\left(4L_1(\vartheta +2\eta )\right)}^2{t}^{1-q+pq-pk}\chi \left(E_0\right){\displaystyle \frac{\Gamma \left(pk\right)\Gamma (q-pq+pk))}{\Gamma (2pk+q-pq)}}\hfill \\ \hfill & \times {\int }_0^t{(t-s)}^{pk-1}{s}^{-\left(1-q+pq-pk\right)+pk}ds+\epsilon \hfill \\ \hfill =& {\left(4L_1(\vartheta +2\eta )\right)}^2{t}^{2pk}{\displaystyle \frac{{\Gamma }^2\left(pk\right)\Gamma (q-pq+pk))}{\Gamma (3pk+q-pq)}}\chi \left(E_0\right)+\epsilon .\hfill \end{array}$$

Through mathematical induction, it can be shown that

$$\begin{array}{c}\hfill \chi ({\Phi }^{\overline{n}}(E_0\left(t\right)))\le {\left(4L_1(\vartheta +2\eta )\right)}^{\overline{n}}{t}^{\overline{n}pk}{\displaystyle \frac{{\Gamma }^{\overline{n}}\left(pk\right)\Gamma (q-pq+pk))}{\Gamma ((\overline{n}+1)pk+q-pq)}}\chi \left(E_0\right),\mathrm{for}\mathrm{every}\overline{n}\in {\mathbb{N}}^{+}.\end{array}$$

Because

$$\underset{\overline{n}\to \infty }{lim}{(4L_1(\vartheta +2\eta )T^{pk})}^{\overline{n}}{\displaystyle \frac{{\Gamma }^{\overline{n}}\left(pk\right)\Gamma (q-pq+pk))}{\Gamma ((\overline{n}+1)pk+q-pq)}}=0,$$

there exists a $\widehat{n}\in {\mathbb{N}}^{+}$ such that

$$\begin{array}{cc}\hfill & {\left(4L_1(\vartheta +2\eta )\right)}^{\widehat{n}}{t}^{\widehat{n}pk}{\displaystyle \frac{{\Gamma }^{\widehat{n}}\left(pk\right)\Gamma (q-pq+pk))}{\Gamma ((\widehat{n}+1)pk+q-pq)}}\hfill \\ \hfill \le & {\left(4L_1(\vartheta +2\eta )\right)}^{\widehat{n}}{T}^{\widehat{n}pk}{\displaystyle \frac{{\Gamma }^{\widehat{n}}\left(pk\right)\Gamma (q-pq+pk))}{\Gamma ((\widehat{n}+1)pk+q-pq)}}=\ell <1\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \chi ({\Phi }^{\widehat{n}}(E_0(t)))\le \ell \chi (E_0).\end{array}$$

In addition, by Proposition 2, we know that ${\Phi }^{\widehat{n}}\left(E_0\left(t\right)\right)$ is bounded and equicontinuous. From Proposition 3, we obtain

$$\chi \left({\Phi }^{\widehat{n}}\left(E_0\right)\right)=\underset{t\in [0,a]}{max}\chi \left({\Phi }^{\widehat{n}}\left(E_0\left(t\right)\right)\right).$$

So,

$$\chi ({\Phi }^{\widehat{n}}\left(E_0\right))\le \ell \chi \left(E_0\right).$$

Let

$${\mathrm{\Delta }}_0=S_r,{\mathrm{\Delta }}_1=\overline{co}({\Phi }^{\widehat{n}}({\mathrm{\Delta }}_0)),\dots ,{\mathrm{\Delta }}_n=\overline{co}({\Phi }^{\widehat{n}}({\mathrm{\Delta }}_{n-1})),n=2,3,\dots .$$

In view of [28], we find the following:

(i)

  ${\mathrm{\Delta }}_0\supset {\mathrm{\Delta }}_1\supset {\mathrm{\Delta }}_2\supset ...\supset {\mathrm{\Delta }}_{n-1}\supset {\mathrm{\Delta }}_n\supset \dots ;$

(ii)

${lim}_{n\to \infty }\chi \left({\mathrm{\Delta }}_n\right)=0.$

Then, $\widehat{\mathrm{\Delta }}={\bigcap }_{n=0}^{\infty }{\mathrm{\Delta }}_n$ is a nonempty, convex, and compact subset in $S_r$.

Thanks to the method in [28], we can show that $\Phi (\widehat{\mathrm{\Delta }})\subset \widehat{\mathrm{\Delta }}.$ Due to ${\Phi }^1({\mathrm{\Delta }}_0)=\Phi ({\mathrm{\Delta }}_0)\subset {\mathrm{\Delta }}_0,$ we have $\overline{co}({\Phi }^1({\mathrm{\Delta }}_0))\subset {\mathrm{\Delta }}_0.$ Thus,

$$\begin{array}{cc}\hfill {\Phi }^2({\mathrm{\Delta }}_0)=\Phi (\overline{co}({\Phi }^1({\mathrm{\Delta }}_0)))\subset & \Phi ({\mathrm{\Delta }}_0)={\Phi }^1({\mathrm{\Delta }}_0),\hfill \\ \hfill {\Phi }^3({\mathrm{\Delta }}_0)=\Phi (\overline{co}({\Phi }^2({\mathrm{\Delta }}_0)))\subset & \Phi (\overline{co}({\Phi }^1({\mathrm{\Delta }}_0)))={\Phi }^2({\mathrm{\Delta }}_0),\hfill \\ \hfill \cdots & \\ \hfill {\Phi }^{\widehat{n}}({\mathrm{\Delta }}_0)=\Phi (\overline{co}({\Phi }^{\widehat{n}-1}({\mathrm{\Delta }}_0)))\subset & \Phi (\overline{co}({\Phi }^{\widehat{n}-2}({\mathrm{\Delta }}_0)))={\Phi }^{\widehat{n}-1}({\mathrm{\Delta }}_0).\hfill \end{array}$$

Therefore, ${\mathrm{\Delta }}_1=\overline{co}({\Phi }^{\widehat{n}}({\mathrm{\Delta }}_0))\subset \overline{co}({\Phi }^{\widehat{n}-1}({\mathrm{\Delta }}_0)),$ so $\Phi ({\mathrm{\Delta }}_1)\subset \Phi (\overline{co}({\Phi }^{\widehat{n}-1}({\mathrm{\Delta }}_0)))={\Phi }^{\widehat{n}}({\mathrm{\Delta }}_0)\subset \overline{co}({\Phi }^{\widehat{n}}({\mathrm{\Delta }}_0))={\mathrm{\Delta }}_1.$ By using the same method, we can show $\Phi ({\mathrm{\Delta }}_n)\subset {\mathrm{\Delta }}_n(n=0,1,2,\dots ).$ By [28], we obtain $\Phi (\widehat{\mathrm{\Delta }})\subset {\bigcap }_{n=0}^{\infty }\Phi ({\mathrm{\Delta }}_n)\subset {\bigcap }_{n=0}^{\infty }{\mathrm{\Delta }}_n=\widehat{\mathrm{\Delta }}.$ Thus, $\Phi (\widehat{\mathrm{\Delta }})$ is compact, which shows that $\Phi$ is a completely continuous operator. By Schauder’s fixed-point theorem, $\Phi$ has at least one fixed point $u^{*}\in \widehat{\mathrm{\Delta }}\subset S_r$. Set $x^{*}(t)=t^{-1+q-pq+pk}{u}^{*}(t)$. Thus, $x^{*}$ is a mild solution of (1) in ${\tilde{S}}_r$. The proof is complete. □

Corollary 3.

Suppose that (H1)–(H3), (H6), and (H7) hold. Then, the initial value problem of the fractional evolution Equation (8) has at least one mild solution in ${\tilde{S}}_r$.

Corollary 4.

Suppose that (H1) and (H4)–(H7) hold. Then, the initial value problem of the fractional evolution Equation (9) has at least one mild solution in ${\tilde{S}}_r$.

Remark 3.

By using the methods of this paper, we can study the initial value problem of fractional neutral integrodifferential equations

$$\left\{\begin{array}{c}{\displaystyle {}^{H}D_{0+}^{p,q}[x\left(t\right)+k(t,x_t)]=Ax\left(t\right)+F(t,x_t)+{\int }_0^{t}h(t,s,x_s)ds,t\in (0,T],}\hfill \\ I_{0+}^{(1-p)(1-q)}x\left(t\right)=\varphi \left(t\right),t\in (-\gamma ,0],\hfill \end{array}\right.$$

(10)

where $\varphi \in C\left(\right[-\gamma ,0],X)$, $x_t\left(\theta \right)=x(t+\theta ),\theta \in [-\gamma ,0]$.

5. Extension to Nonlocal Problems

In the following, we consider the nonlocal Cauchy problem for the fractional integrodifferential equations

$$\left\{\begin{array}{c}{\displaystyle {}^{H}D_{0+}^{p,q}x\left(t\right)=Ax\left(t\right)+F(t,x\left(t\right))+{\int }_0^{t}h(t,s,x\left(s\right))ds,t\in (0,1],}\hfill \\ {\displaystyle I_{0+}^{(1-p)(1-q)}x\left(0\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}x\left(t_k\right),k=1,2,\dots ,m,}\hfill \end{array}\right.$$

(11)

where $g:\Lambda \times X\to X$$(\Lambda =\{(t,s)\in [0,1]\times [0,1]:t\ge s\left\}\right)$ is a function, $a_i\in \mathbb{R}(i=1,2,\dots ,m)$ satisfying ${\sum }_{k=1}^m{\sigma }_k\ne 1$, and $t_k(k=1,2,\dots ,m)$ are known points with $0\le t_1\le t_2\le \dots \le t_m<1$.

The research background of nonlocal problems of differential equations comes from physics. Byszewski [7] proposed nonlocal conditions to prove the existence and uniqueness of the classical and mild solutions of a semi-linear evolution Cauchy problem. As Lakshmikantham and Byszewski [8] pointed out, nonlocal conditions may be more useful for describing certain physical phenomena than standard initial conditions.

In this section, we propose a new nonlocal condition and provide a suitable definition of the mild solutions of problem (1) by introducing a bounded operator $\mathcal{B}$. Meanwhile, we obtain some existence criteria for the mild solutions of (11) for two cases where the semigroups are compact and noncompact.

Next, for convenience, we set

$$g\left(x\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}x\left(t_k\right).$$

Lemma 11.

Assume that $x\left(t\right)$ is a solution of problem (11). Then,

$$x\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)g\left(x\right)+{\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,t\in (0,1],$$

(12)

where ${\mathcal{G}}_p\left(t\right)$ and ${\mathcal{F}}_{p,q}\left(t\right)$ are defined as in Lemma 3.

Define an operator $\mathcal{B}:X\to X$ as follows:

$$\mathcal{B}={\left(I-\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(t_k\right)\right)}^{-1}.$$

(13)

Under appropriate conditions, $\mathcal{B}$ exists and is bounded.

Lemma 12.

Assume that one of the following conditions holds:

(i) 

There exists ${\sigma }_k\in \mathbb{R}$ such that

$$l_2\sum _{k=1}^m\left|{\sigma }_k\right|<1;$$

(14)

(ii) 

The operator $Q\left(t\right)$$(t>0)$ is compact. The Cauchy problem of the homogeneous evolution equations

$$\left\{\begin{array}{c}{}^{H}D_{0+}^{p,q}x\left(t\right)=Ax\left(t\right),t\in (0,1],\hfill \\ {\displaystyle I_{0+}^{(1-p)(1-q)}x\left(0\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}x\left(t_k\right),k=1,2,\dots ,m,}\hfill \end{array}\right.$$

(15)

has no non-trivial solutions.

Then, the operator $\mathcal{B}$ defined in (13) exists and is bounded.

Proof. 

(i) From Lemma 6 and (14), we have

$$∥\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(t_k\right)∥\le l_2\sum _{k=1}^m\left|{\sigma }_k\right|<1.$$

Therefore, using Neumann’s theorem, $\mathcal{B}$ exists and is bounded.

(ii) From Lemma 11, the solution of (15) satisfies

$$x\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)g\left(x\right),$$

and then

$$g\left(x\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}x\left(t_k\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(t_k\right)g\left(x\right).$$

In view of this, $g\left(x\right)=0$ if $x=0$. By Lemma 4, ${\mathcal{F}}_{p,q}\left(t_k\right)$ is compact for each $t_k>0$, $k=1,2,\dots ,m$. Then, ${\sum }_{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(t_k\right)$ is also compact. Since problem (13) has no non-trivial solutions, by Fredholm’s alternative theorem, we know that $\mathcal{B}$ exists. □

Let

$$F\left(t\right)={\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,t\in (0,1].$$

(16)

From (12), we obtain

$$x\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)g\left(x\right)+F\left(t\right),t\in (0,1],$$

which implies that

$$g\left(x\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}{\mathcal{F}}_{p,q}\left(t_k\right)g\left(x\right)+\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}F\left(t_k\right),$$

i.e.,

$$g\left(x\right)=\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}\mathcal{B}\left(F\left(t_k\right)\right).$$

(17)

Thus, according to Lemmas 11 and (17), the mild solutions of Equation (1) can be defined as follows.

Definition 6.

If a function $x\in C\left(\right(0,1],X)$ satisfies the equation

$$x\left(t\right)={\mathcal{F}}_{p,q}\left(t\right)\sum _{k=1}^m{\sigma }_k{t}_k^{1-q+pq-pk}\mathcal{B}\left(F\left(t_k\right)\right)+F\left(t\right),t\in (0,1],$$

(18)

where

$$F\left(t_k\right)={\int }_0^{t_k}{\mathcal{G}}_p(t_k-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,$$

(19)

$$F\left(t\right)={\int }_0^t{\mathcal{G}}_p(t-s)\left(F(s,x\left(s\right))+{\int }_0^{s}h(s,\mu ,x\left(\mu \right))d\mu \right)ds,t\in (0,1],$$

(20)

we call $x\left(t\right)$ a mild solution of problem (1).

By using a method similar to the proof of Theorems 1 and 2, we obtain the following results under condition (i) or condition (ii) from Lemma 12.

Theorem 3.

Assume that the operator $Q\left(t\right)$ is compact for any $t>0$. Furthermore, suppose that (H2)–(H5) hold. Then, the initial value problem (11) has at least one mild solution in ${\tilde{S}}_r$.

Theorem 4.

Suppose that (H1)–(H7) hold. Then, the initial value problem (11) has at least one mild solution in ${\tilde{S}}_r$.