Existence of Hilfer Fractional Evolution Inclusions with Almost Sectorial Operators

Abstract

In this paper, we mainly focus on the the existence of Hilfer fractional evolution inclusions with almost sectorial operators. For two cases in which the almost sector operators are compact and noncompact, we obtain existence criteria for mild solutions, which extend and improve some related results in the literature.

1. Introduction

Many phenomena in nature have uncertainty, and there are many methods to describe uncertainty, among which differential inclusion is one of the most commonly used methods. Differential inclusion theory is an important branch of nonlinear analysis theory, and the existence of solutions for differential inclusion and the controllability of systems are fundamental concepts and focused topics within this theory. Compared with systems described by general differential equations, differential inclusion systems have greater versatility. They not only play a very important role in many disciplines, such as physics, engineering, medicine and biology, but also have close connections with other branches of mathematics, such as optimization and optimal control. Therefore, in the past thirty years, differential inclusion systems have become a major research topic—for example, see [1,2,3,4,5,6] and the references therein.

On the other hand, fractional calculus has become an important research subject in the field of mathematics because of its wide applicability. However, with the deepening of scientific research, people have found that traditional integer-order calculus is inadequate in dealing with some complex phenomena. These phenomena include fractal structures, chaotic systems, viscoelastic materials, etc. Their evolution processes often exhibit nonlinearity and nonlocality, making it difficult to describe them accurately using traditional integer-order calculus. Against this backdrop, fractional calculus emerged accordingly. By extending the order of calculus from integers to real numbers, fractional calculus provides new mathematical tools for these complex phenomena. Fractional calculus not only inherits the advantages of traditional calculus but also possesses greater flexibility and adaptability, enabling it to better describe and explain many phenomena in nature.

Fractional differential inclusions refer to differential inclusions containing derivatives of any order. In disciplines such as biology, economics, and optimal control, differential inclusion models are established to conduct theoretical analysis and research on some practical problems. 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, theoretical research on fractional differential inclusions is very difficult. Therefore, the study of fractional differential inclusions has important theoretical significance and practical value. For more details, readers can refer to [7,8,9,10] and the recent references.

In the past decade, the study of fractional differential inclusions has become a major topic, and a number of studies have focused on the existence and controllability of fractional evolution equations. For example, Agarwal et al. [11] obtained the existence results for functional differential inclusions involving Riemann–Liouville fractional derivatives. In [12], Wang and Zhou investigated the existence and controllability of fractional evolution inclusions with Caputo derivatives.

Hilfer [7] 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 [7]. In addition to defining the order of the Hilfer fractional derivative, he also introduced a new parameter, the type $q\in [0,1]$, 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. Many authors have studied the well-posedness and controllability of initial value problems for evolution equations with Hilfer fractional derivatives (see [13,14,15,16,17,18,19]). However, much less is known about the fractional evolution inclusions with Hilfer fractional derivatives. The purpose of this paper is to establish some criteria for the existence of Hilfer fractional evolution inclusions with almost sectorial operators.

Consider the following Hilfer fractional evolution inclusions

$$\left\{\begin{array}{c}{}^{H}D_{0+}^{p,q}y\left(t\right)\in Ay\left(t\right)+G(t,y\left(t\right)),t\in (0,T],\hfill \\ I_{0+}^{(1-p)(1-q)}y\left(0\right)=y_0,\hfill \end{array}\right.$$

(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 Banach space X, $I_{0+}^{(1-p)(1-q)}$ is a Riemann–Liouville fractional integral of order $(1-p)(1-q)$, and $G:J\times X\to \mathcal{P}\left(X\right):=2^X\setminus \{⌀\}$ is a nonempty, convex, bounded, and closed multivalued map, $J=[0,T],T\in (0,\infty )$.

In this paper, we obtain some existence results regarding mild solutions for the inclusion (1). The novelties and most important aspects of this manuscript are listed below.

(i)

We generalize prior results on evolution inclusions for Riemann–Liouville and Caputo derivatives to the Hilfer case ($0<p<1,0\le q\le 1$), which unifies both derivatives.

(ii)

For compact semigroups case, the existence of mild solutions is proved by using Bohnenblust–Karlin’s fixed-point theorem and the properties of multivalued maps. For the noncompact semigroup case, we use Kuratowski’s measure of noncompactness to handle this problem under weaker conditions.

(iii)

We identify and resolve a flaw in prior research (Theorem 3.1 of [20]), where incorrect operator definitions led to invalid conclusions. We rigorously redefine the solution operators to ensure continuity at $t=0$. Our results improve and extend some known results in the relevant references.

The remaining part of the paper is organized as follows. In Section 2, we introduce some preliminaries on multivalued maps, fractional calculus, almost sectorial operators, the measure of noncompactness, and the resolvent operators. In Section 3, we state some basic hypotheses for this article. For two cases in which the almost sector operators are compact and noncompact, we establish two new solvable results of mild solutions. Our main theorems essentially improve and generalize some known results in the literature. Finally, an example is given to illustrate the theory.

2. Preliminaries

We shall begin by stating some basic facts about multivalued maps. For more details, we refer to [21,22].

Let X be a Banach space with the norm $|·|$. Suppose that $\mathbb{R}=(-\infty ,\infty )$ and $J=[0,T],T\in {\mathbb{R}}^{+}$. Denote $C(J,X)$ as the Banach space of all continuous functions from J to X with the norm $\parallel u\parallel ={sup}_{t\in J}\left|u\left(t\right)\right|<\infty$. By $\mathcal{L}\left(X\right)$, we denote the space of all bounded linear operators from X to X with the usual operator norm ${\parallel ·\parallel }_{\mathcal{L}\left(X\right)}$.

Let $F:X\to \mathcal{P}\left(X\right)$ be a multivalued map. If $F\left(x\right)$ is convex (closed) for all $x\in X$, then we call F convex (closed) valued. If ${sup}_{x\in \Omega }\left\{sup\{\parallel y\parallel :y\in F(x\left)\right\}\right\}<\infty$ for any bounded set $\Omega$ of X, then F is bounded on bounded sets.

Definition 1.

Assume that the set $F\left(x_0\right)$ is a nonempty closed subset of X for each $x_0\in X$, and there exists an open neighborhood V of $x_0$ such that $F\left(V\right)\subseteq \Omega$ for each open set Ω of X containing $F\left(x_0\right)$. Then, F is called upper semicontinuous (u.s.c. for short) on X.

Definition 2.

If $F(\Omega )$ is relatively compact for every bounded subset Ω of X, then F is called completely continuous.

If the multivalued map F is completely continuous with nonempty values, then F is u.s.c. if and only if F has a closed graph, i.e., $x_n\to x_{*}$, $y_n\to y_{*}$, $y_n\in F{x}_n$, implying $y_{*}\in F{x}_{*}$. G has a fixed point if there is a $x\in X$ such that $x\in F\left(x\right)$.

Lemma 1

([23]). Let J be a compact real interval; $BCC\left(X\right)$ be the set of all nonempty, bounded, closed, and convex subsets of X; and G be a multivalued map satisfying that $G:J\times X\to BCC\left(X\right)$ is measurable to t for each fixed $x\in X$ and u.s.c. to x for each $t\in J$, and, for each $x\in C(J,X)$, the set $S_{G,x}=\left\{g\in L^1(J,X):g\left(t\right)\in G(t,x\left(t\right)),t\in J\right\}$ is nonempty. Let $\mathcal{F}$ be linear continuous from $L^1(J,X)$ to $C(J,X)$, and the operator

$$\mathcal{F}\circ S_G:C(J,X)\to BCC\left(C(J,X)\right),x\mapsto (\mathcal{F}\circ S_G)\left(x\right)=\mathcal{F}\left(S_{G,x}\right),$$

is a closed graph operator in $C(J,X)\times C(J,X)$.

Lemma 2

([24]). Let $\mathcal{D}$ be a nonempty, convex, bounded, and closed subset of X. Assume that $F:\mathcal{D}\to \mathcal{P}\left(X\right)$ is u.s.c. with closed, convex values, such that $F\left(\mathcal{D}\right)\subseteq \mathcal{D}$ and $F\left(\mathcal{D}\right)$ is compact. Then, F has a fixed point.

We also state some definitions of almost sectorial operators. For more details, we refer to [25,26,27].

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 3.

Assume that $0<k<1$ and $0<\omega <{\displaystyle \frac{\pi }{2}}$. By ${\Theta }_{\omega }^{-k}\left(X\right)$, 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 $A\in {\Theta }_{\omega }^{-k}\left(X\right)$.

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 3

([25,26]). Assume that $0<k<1$ and $0<\omega <{\displaystyle \frac{\pi }{2}}$. Set $A\in {\Theta }_{\omega }^{-k}\left(X\right)$. 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 $t>0$.

Next, we introduce some basic concepts of fractional calculus. For more details, we refer to [7,8]

Definition 4

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

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

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

Definition 5.

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

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

Remark 1.

(i) If $q=0$, $0<p<1$, then

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

where ${{}^{L}D}_{0+}^p$ is the Riemann–Liouville derivative.

(ii) If $q=1$, $0<p<1$, then

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

where ${{}^{C}D}_{0+}^p$ is the Caputo derivative.

Definition 6

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

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

which satisfies

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

Next, we recall the measure of noncompactness and its properties.

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

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

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

([10]). Let $D_1$ and $D_2$ be two bounded sets of a Banach space X. Then,

(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)

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

Let $W\subset C(J,X)$. We define

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

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

Proposition 2

([29]). If $W\subset C(J,X)$ is bounded and equicontinuous, then $\overline{co}W\subset C(J,X)$ is also bounded and equicontinuous.

Proposition 3

([10]). Set $W\subset C(J,X)$ is 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

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

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

Proposition 5

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

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

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

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

Based on our previous work [10], we introduce some definitions and lemmas.

Definition 7.

A function $y\in C\left(\right(0,T],X)$ is said to be a mild solution of the inclusion (1) if $I_{0+}^{(1-p)(1-q)}y\left(0\right)=y_0$ and there exists $g\in L^1(J,X)$ such that $g\left(t\right)\in G(t,y(t\left)\right)$ on $t\in J$ and

$$y\left(t\right)={\mathcal{H}}_{p,q}\left(t\right)y_0+{\int }_0^t{\mathcal{J}}_p(t-s)g\left(s\right)ds,t\in (0,T],$$

where

$${\mathcal{H}}_{p,q}\left(t\right)=I_{0+}^{q(1-p)}{\mathcal{J}}_p\left(t\right),{\mathcal{J}}_p\left(t\right)=t^{p-1}{\mathcal{K}}_p\left(t\right),and{\mathcal{K}}_p\left(t\right)={\int }_0^{\infty }p\theta M_p\left(\theta \right)Q\left(t^p\theta \right)d\theta .$$

Lemma 4

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

Lemma 5

([20]). 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

([19]). ${\mathcal{K}}_p\left(t\right)$, ${\mathcal{J}}_p\left(t\right)$, and ${\mathcal{H}}_{p,q}\left(t\right)$ are linear operators for any fixed $t>0$. For any $t>0$ and $y\in X$,

$$|{\mathcal{K}}_p\left(t\right)y|\le L_1{t}^{p(k-1)}\left|y\right|,|{\mathcal{J}}_p\left(t\right)y|\le L_1{t}^{pk-1}\left|y\right|,\mathrm{and}|{\mathcal{H}}_{p,q}\left(t\right)y|\le L_2{t}^{-1+q-pq+pk}\left|y\right|,$$

where

$$\begin{array}{c}\hfill L_1={\displaystyle \frac{C_0\Gamma \left(k\right)}{\Gamma \left(pk\right)}},L_2={\displaystyle \frac{C_0\Gamma \left(k\right)}{\Gamma \left(q\right(1-p)+pk)}}.\end{array}$$

(2)

Lemma 7

([15]). Let ${\left\{Q\left(t\right)\right\}}_{t>0}$ be equicontinuous. ${\left\{{\mathcal{K}}_p\left(t\right)\right\}}_{t>0}$, ${\left\{{\mathcal{J}}_p\left(t\right)\right\}}_{t>0}$, and ${\left\{{\mathcal{H}}_{p,q}\left(t\right)\right\}}_{t>0}$ are strongly continuous; that is, for any $y\in X$ and $\sigma >\tau >0$,

$$\begin{array}{cc}\hfill & |{\mathcal{K}}_p\left(\sigma \right)y-{\mathcal{K}}_p\left(\tau \right)y|\to 0,|{\mathcal{J}}_p\left(\sigma \right)y-{\mathcal{J}}_p\left(\tau \right)y|\to 0,\hfill \\ \hfill & |{\mathcal{H}}_{p,q}\left(\sigma \right)y-{\mathcal{H}}_{p,q}\left(\tau \right)y|\to 0,as\sigma \to \tau .\hfill \end{array}$$

3. Existence

In the following, we always assume that $A\in {\Theta }_{\omega }^{-k}\left(X\right)$, $0<k<1$, and $0<\omega <{\displaystyle \frac{\pi }{2}}$, $y_0\in D\left(A^{\vartheta }\right)$ with $\vartheta >1-k$.

Let

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

Clearly, $(C_p((0,T],X),\parallel ·{\parallel }_p)$ is a Banach space with the norm

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

We first state some basic hypotheses for this article as follows.

(H1)

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

(H2)

$G:J\times X\to BCC\left(X\right)$ is a multivalued map such that

(i)

the map $t\mapsto G(t,u)$ is measurable;

(ii)

the map $u\mapsto G(t,u)$ is u.s.c.;

(iii)

for each $y\in C_p(J,X)$, the set

$$S_{G,y}=\left\{g\in L^1(J,X):g\left(t\right)\in G(t,y\left(t\right)),t\in J\right\}$$

is nonempty.

(H3)

There exists a function $m\in L^1((0,T],{\mathbb{R}}^{+})$ such that

$$sup\left\{\right|g|:g(t)\in G(t,y\left(t\right)\left)\right\}\le m\left(t\right),\mathrm{for}\mathrm{a}.\mathrm{e}.t\in (0,T],\mathrm{and}\mathrm{any}y\in X,$$

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

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

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

$$L_2\left|y_0\right|+L_1\underset{t\in [0,T]}{sup}\left\{t^{1-q+pq-pk}{I}_{0+}^{pk}m\left(t\right)\right\}\le r,$$

i.e.,

$$L_2\left|y_0\right|+L_1\underset{t\in [0,T]}{sup}\left\{t^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}m\left(s\right)ds\right\}\le r,$$

where $L_1$, $L_2$ are defined as in (2).

Set

$${\Omega }_r=\{u\in C(J,X):\parallel u\parallel \le r\}.$$

and

$${\tilde{\Omega }}_r=\{y\in C_p{(J,X):\parallel y\parallel }_p\le r\}.$$

Clearly, ${\Omega }_r$ and ${\tilde{\Omega }}_r$ are nonempty, closed, and convex subsets of $C(J,X)$ and $C_p(J,X)$, respectively.

3.1. Compact Operator Case

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

Theorem 1.

Suppose that (H1), (H2), and (H3) hold. Then, (1) is solvable on J.

Proof. 

By (H2), the set $S_{G,y}$ is nonempty. So, for any $y\in C_p(J,X)$, there exists a $g\in S_{G,y}$. Define an operator $\Psi$ as follows:

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

where

$$\left({\Psi }_{1}y\right)\left(t\right)={\mathcal{H}}_{p,q}\left(t\right)y_0,\left({\Psi }_{2}y\right)\left(t\right)={\int }_0^t{\mathcal{J}}_p(t-s)g\left(s\right)ds,\mathrm{for}t\in (0,T].$$

We can easily prove that

$$\begin{array}{c}\hfill \underset{t\to 0+}{lim}{t}^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0=0.\end{array}$$

(3)

In fact,

$$\begin{array}{cc}\hfill t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0& ={\displaystyle \frac{t^{1-q+pq-pk}}{\Gamma \left(q\right(1-p\left)\right)}}{\int }_0^t{(t-s)}^{q(1-p)-1}{s}^{p-1}{\mathcal{K}}_p\left(s\right)y_{0}ds\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma \left(q\right(1-p\left)\right)}}{\int }_0^1{(1-v)}^{q(1-p)-1}{v}^{p-1}{t}^{p(1-k)}{\mathcal{K}}_p\left(tv\right)y_{0}dv.\hfill \end{array}$$

By Proposition 2.3 in [9], we have ${lim}_{t\to 0+}{t}^{p(1-k)}{\mathcal{K}}_p\left(tv\right)y_0=0$ for all $v\in [0,1]$. In view of ${\int }_0^1{(1-v)}^{q(1-p)-1}{v}^{p-1}dz$, (3) holds.

Moreover, from Lemma 6 and (H2), we have

$$\begin{array}{cc}\hfill \left|t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g\left(s\right)ds\right|& \le L_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}m\left(s\right)ds\hfill \\ & \to 0\mathrm{as}t\to 0.\hfill \end{array}$$

(4)

For each $y\in C_p(J,X)$, let us set $u\left(t\right)=t^{1-q+pq-pk}y\left(t\right)$ for $t\in (0,T]$ and $u\left(0\right)={lim}_{t\to 0+}{t}^{1-q+pq-pk}y\left(t\right)$. From the continuity of factor $t^{1-q+pq-pk}$, we have $u\in C(J,X)$. For each $y\in C_p(J,X)$, since there is a $g\in S_{G,y}$, it follows that there is also a $\tilde{g}\in S_{G,u}$ for $u\in C(J,X)$ associated with each y, and

$$S_{G,u}=\left\{\tilde{g}\in L^1(J,X):\tilde{g}\left(t\right)\in G(t,u\left(t\right)),t\in J\right\}.$$

By $\Phi$, we define an operator

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

where

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

We consider the multivalued operator $\mathfrak{L}$: $C(J,X)\to \mathcal{P}\left(C\right(J,X\left)\right)$,

$$\begin{array}{c}\hfill \mathfrak{L}\left(u\right)=\left\{\phi \in C(J,X):\phi \left(t\right)=\left({\Phi }_{1}u\right)\left(t\right)+\left({\Phi }_{2}u\right)\left(t\right),g\in S_{G,u},t\in (0,T]\right\}.\end{array}$$

Clearly, the solvability of problem (1) follows the multivalued operator $\mathfrak{L}$ satisfying all conditions of Lemma 2. We divide the proof of the theorem into five steps.

Step 1. $\mathfrak{L}$ is convex for any $u\in C(J,X)$.

Let ${\phi }_1,{\phi }_2$ belong to $\mathfrak{L}\left(u\right)$. Then, there exist $g_1,g_2\in S_{G,u}$ such that, for each $t\in J$,

$$\begin{array}{c}\hfill {\phi }_i\left(t\right)=t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0+t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g_i\left(s\right)ds,i=1,2.\end{array}$$

Set $\varpi \in [0,1]$. We have, for each $t\in J$,

$$\begin{array}{c}\varpi {\phi }_1\left(t\right)+(1-\varpi ){\phi }_2\left(t\right)=t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0\hfill \\ +t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)[\varpi g_1\left(s\right)+(1-\varpi )g_2\left(s\right)]ds.\hfill \end{array}$$

Clearly, $S_{G,u}$ is convex since G has convex values. Thus, $\varpi g_1+(1-\varpi )g_2\in S_{G,u}$. Hence,

$$\varpi {\phi }_1+(1-\varpi ){\phi }_2\in \mathfrak{L}\left(u\right).$$

Step 2. $\mathfrak{L}$ maps bounded sets into equicontinuous sets of $C(J,X)$.

For each $u\in {\Omega }_r$, $\phi \in \mathfrak{L}\left(u\right)$, there exists a $g\in S_{G,u}$ such that

$$\begin{array}{c}\phi \left(t\right):=\left({\Phi }_{1}u\right)\left(t\right)+\left({\Phi }_{2}u\right)\left(t\right)\hfill \\ =t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0+t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g\left(s\right)ds.\hfill \end{array}$$

Firstly, we show that $\left\{{\Phi }_{1}u,u\in {\Omega }_r\right\}$ is equicontinuous.

By (3), for $\tau =0$, $\sigma \in (0,T]$, we obtain

$$\begin{array}{c}\hfill |\left({\Phi }_{1}u\right)\left(\sigma \right)-\left({\Phi }_{1}u\right)\left(0\right)|\le |{\sigma }^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(\sigma \right)y_0-0|\to 0,\mathrm{as}\sigma \to 0.\end{array}$$

For any $\tau ,\sigma \in (0,T]$ and $\tau <\sigma$, we have

$$\begin{array}{cc}\hfill & |\left({\Phi }_{1}u\right)\left(\sigma \right)-\left({\Phi }_{1}u\right)\left(\tau \right)|\hfill \\ \hfill & \le |{\sigma }^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(\sigma \right)y_0-{\tau }^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(\tau \right)y_0|\hfill \\ \hfill & \le |{\sigma }^{1-q+pq-pk}\left|\right|{\mathcal{H}}_{p,q}\left(\sigma \right)y_0-{\mathcal{H}}_{p,q}\left(\tau \right)y_0|\hfill \\ \hfill & +|{\sigma }^{1-q+pq-pk}-{\tau }^{1-q+pq-pk}\left|\right|{\mathcal{H}}_{p,q}\left(\tau \right)y_0|\hfill \\ \hfill & \to 0,\mathrm{as}\sigma \to \tau .\hfill \end{array}$$

(5)

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

Next, we show that $\left\{{\Phi }_{2}u,u\in {\Omega }_r\right\}$ is also equicontinuous.

• For $\tau =0$, $0<\sigma <T$, by (4), we have

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

For $0<\tau <\sigma \le T$, we obtain

$$\begin{array}{cc}\hfill & |\left({\Phi }_{2}u\right)\left(\sigma \right)-\left({\Phi }_{2}u\right)\left(\tau \right)|\hfill \\ \hfill & \le \left|{\tau }^{1-q+pq-pk}{\int }_{\tau }^{\sigma }{(\sigma -s)}^{p-1}{\mathcal{K}}_p(\sigma -s)g\left(s\right)ds\right|\hfill \\ \hfill & +\left|{\tau }^{1-q+pq-pk}{\int }_0^{\tau }\left({(\sigma -s)}^{p-1}-{(\tau -s)}^{p-1}\right){\mathcal{K}}_p(\sigma -s)g\left(s\right)ds\right|\hfill \\ \hfill & +\left|{\tau }^{1-q+pq-pk}{\int }_0^{\tau }{(\tau -s)}^{p-1}\left({\mathcal{K}}_p(\sigma -s)-{\mathcal{K}}_p(\tau -s)\right)g\left(s\right)ds\right|\hfill \\ \hfill & +|{\sigma }^{1-q+pq-pk}-{\tau }^{1-q+pq-pk}|\left|{\int }_0^{\sigma }{(\sigma -s)}^{p-1}{\mathcal{K}}_p(\sigma -s)g\left(s\right)ds\right|\hfill \\ \hfill & \le I_1+I_2+I_3+I_4,\hfill \end{array}$$

(6)

where

$$\begin{array}{c}{I}_1=L_1{\tau }^{1-q+pq-pk}|{\int }_0^{\sigma }{(\sigma -s)}^{pk-1}m\left(s\right)ds-{\int }_0^{\tau }{(\tau -s)}^{pk-1}m\left(s\right)ds|,\hfill \\ I_2=2L_1{\tau }^{1-q+pq-pk}{\int }_0^{\tau }\left({(\tau -s)}^{p-1}-{(\sigma -s)}^{p-1}\right){(\sigma -s)}^{p(k-1)}m\left(s\right)ds,\hfill \\ I_3={\tau }^{1-q+pq-pk}\left|{\int }_0^{\tau }{(\tau -s)}^{p-1}\left({\mathcal{K}}_p(\sigma -s)-{\mathcal{K}}_p(\tau -s)\right)g\left(s\right)ds\right|,\hfill \\ I_4=|{\sigma }^{1-q+pq-pk}-{\tau }^{1-q+pq-pk}|\left|L_1{\int }_0^{\sigma }{(\sigma -s)}^{pk-1}m\left(s\right)ds\right|.\hfill \end{array}$$

Since $I_{0+}^{pk}m\in C((0,T],{\mathbb{R}}_{+})$, we find that ${lim}_{\sigma \to \tau }{I}_1=0$. It is also noticed that

$$({(\tau -s)}^{p-1}-{(\sigma -s)}^{p-1}){(\sigma -s)}^{p(k-1)}m\left(s\right)\le {(\tau -s)}^{pk-1}m\left(s\right),\mathrm{for}s\in [0,\tau ).$$

Therefore,

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

which means $I_2\to 0$ as $\sigma \to \tau$.

From (H2), for $\epsilon >0$, we obtain

$$\begin{array}{cc}\hfill & I_3\le {\tau }^{1-q+pq-pk}{\int }_0^{\tau -\epsilon }{(\tau -s)}^{p-1}\parallel {\mathcal{K}}_p(\sigma -s)-{\mathcal{K}}_p(\tau -s){\parallel }_{\mathcal{L}\left(X\right)}\left|g\left(s\right)\right|ds\hfill \\ \hfill & +{\tau }^{1-q+pq-pk}{\int }_{\tau -\epsilon }^{\tau }{(\tau -s)}^{p-1}\parallel {\mathcal{K}}_p(\sigma -s)-{\mathcal{K}}_p(\tau -s){\parallel }_{\mathcal{L}\left(X\right)}\left|g\left(s\right)\right|ds\hfill \\ \hfill & \le {\tau }^{1-q+pq-pk}{\int }_0^{\tau }{(\tau -s)}^{p-1}m\left(s\right)ds\underset{s\in [0,\tau -\epsilon ]}{sup}{\parallel {\mathcal{K}}_p(\sigma -s)-{\mathcal{K}}_p(\tau -s)\parallel }_{\mathcal{L}\left(X\right)}\hfill \\ \hfill & +2L_1{\tau }^{1-q+pq-pk}{\int }_{\tau -\epsilon }^{\tau }{(\tau -s)}^{pk-1}m\left(s\right)ds\hfill \\ \hfill & \le I_{31}+I_{32}+I_{33},\hfill \end{array}$$

where

$$\begin{array}{c}{I}_{31}={\tau }^{1-q+pq-pk}{\int }_0^{\tau }{(\tau -s)}^{p-1}m\left(s\right)ds\underset{s\in [0,\tau -\epsilon ]}{sup}{\parallel {\mathcal{K}}_p(\sigma -s)-{\mathcal{K}}_p(\tau -s)\parallel }_{\mathcal{L}\left(X\right)},\hfill \\ I_{32}=2L_{1}L{\tau }^{1-q+pq-pk}|{\int }_0^{\tau }{(\tau -s)}^{pk-1}m\left(s\right)ds-{\int }_0^{\tau -\epsilon }{(\tau -\epsilon -s)}^{pk-1}m\left(s\right)ds|,\hfill \\ I_{33}=2L_{1}L{\tau }^{1-q+pq-pk}{\int }_0^{\tau -\epsilon }({(\tau -\epsilon -s)}^{pk-1}-{(\tau -s)}^{pk-1})m\left(s\right)ds.\hfill \end{array}$$

From (H1) and Lemma 7, we easily obtain that $I_{31}\to 0$ as $\sigma \to \tau$. Similarly to the process of proving that $I_1\to 0$ and $I_2\to 0$, we see that $I_{32}\to 0$ and $I_{33}\to 0$ as $\epsilon \to 0$. Thus, $I_3$ tends to zero as $\sigma \to \tau$. Clearly, $I_4\to 0$ as $\sigma \to \tau$.

Hence, $\left\{{\Phi }_{2}u)\left(t\right),u\in {\Omega }_r\right\}$ is equicontinuous. Therefore, $\mathfrak{L}\left({\Omega }_r\right)\subset C(J,X)$ is equicontinuous.

Step 3. $\mathfrak{L}{\Omega }_r\subseteq {\Omega }_r$.

From Step 2, $\Phi \Omega \subset C(J,X)$. By (H4), for $t>0$ and any $u\in {\Omega }_r$,

$$\begin{array}{cc}\hfill & |\phi (t)|\le \left|t^{1-q+\lambda q-\lambda k}{\mathcal{H}}_{\lambda ,q}\left(t\right)y_0\right|+\left|t^{1-q+\lambda q-\lambda k}{\int }_0^t{\mathcal{J}}_{\lambda }(t-s)g\left(s\right)ds\right|\hfill \\ \hfill & \le L_2\left|y_0\right|+L_1{t}^{1-q+\lambda q-\lambda k}{\int }_0^t{(t-s)}^{\lambda k-1}m\left(s\right)ds<r.\hfill \end{array}$$

(7)

In the case where $t=0$, we know that $\left|\phi \right(0\left)\right|=0<r$. Hence,

$$\parallel \mathfrak{L}\left(u\right)\parallel \equiv sup\left\{\parallel \phi \parallel :\phi \in \mathfrak{L}\left(u\right)\right\}<r,$$

Therefore, $\mathfrak{L}{\Omega }_r\subseteq {\Omega }_r$.

Step 4. $\Lambda \left(t\right)=\{\phi \left(t\right):\phi \in \mathfrak{L}\left({\Omega }_r\right)\}$ is relatively compact in X.

Obviously, $\Lambda \left(0\right)=\{\left(\mathfrak{L}\phi \right)\left(0\right):\phi \in {\Omega }_r\}$ is compact, and we only need to show the case where $t>0$. For $t\in (0,T]$, $x\in {\Omega }_r$, and any $\varsigma >0$, we define

$$\begin{array}{c}\hfill {\Lambda }_{\epsilon ,\varsigma }\left(t\right)=\left\{\left(\mathfrak{L}{\phi }_{\epsilon ,\varsigma }\right)\left(t\right)\mid \phi \in {\Omega }_r\right\}\end{array}$$

where

$$\begin{array}{cc}\hfill & {\phi }_{\epsilon ,\varsigma }\left(t\right)\hfill \\ \hfill & :=t^{1-q+pq-pk}\left({\mathcal{H}}_{p,q}\left(t\right)y_0+{\int }_0^{t-\epsilon }{\int }_{\varsigma }^{\infty }p\theta {(t-s)}^{p-1}{M}_p\left(\theta \right)Q\left({(t-s)}^p\theta \right)g\left(s\right)d\theta ds\right)\hfill \\ \hfill & =t^{1-q+pq-pk}({\mathcal{H}}_{p,q}\left(t\right)y_0\hfill \\ & +Q\left({\epsilon }^p\varsigma \right){\int }_0^{t-\epsilon }{\int }_{\varsigma }^{\infty }p\theta {(t-s)}^{p-1}{M}_p\left(\theta \right)Q({(t-s)}^p\theta -{\epsilon }^p\varsigma )g\left(s\right)d\theta ds).\hfill \end{array}$$

Since the operator $Q\left({\epsilon }^p\varsigma \right)$, ${\epsilon }^p\varsigma >0$ is compact in X, the sets $\{\left({\mathfrak{L}}_{\epsilon ,\varsigma }\phi \right)\left(t\right)\mid \phi \in {\Omega }_r\}$ are relatively compact in X. On the other hand, for every $u\in {\Omega }_r$, we have

$$\begin{array}{cc}\hfill & |\phi \left(t\right)-{\phi }_{\epsilon ,\varsigma }\left(t\right)|\hfill \\ \hfill & \le t^{1-q+pq-pk}\left|{\int }_0^t{\int }_0^{\varsigma }p\theta {(t-s)}^{p-1}{M}_p\left(\theta \right)Q\left({(t-s)}^p\theta \right)g\left(s\right)d\theta ds\right|\hfill \\ \hfill & +t^{1-q+pq-pk}\left|{\int }_{t-\epsilon }^t{\int }_{\varsigma }^{\infty }p\theta {(t-s)}^{p-1}{M}_p\left(\theta \right)Q\left({(t-s)}^p\theta \right)g\left(s\right)d\theta ds\right|\hfill \\ \hfill & \le p{C}_0{t}^{1-q+pq-pk}{\int }_0^t{\int }_0^{\varsigma }{(t-s)}^{pk-1}\left|g\left(s\right)\right|{\theta }^k{M}_p\left(\theta \right)d\theta ds\hfill \\ \hfill & +p{C}_0{t}^{1-q+pq-pk}{\int }_{t-\epsilon }^t{\int }_0^{\infty }{(t-s)}^{pk-1}\left|g\left(s\right)\right|{\theta }^k{M}_p\left(\theta \right)d\theta ds\hfill \\ \hfill & \le p{C}_0{t}^{1-q+pq-pk}{\int }_0^t{\int }_0^{\varsigma }{(t-s)}^{pk-1}m\left(s\right){\theta }^k{M}_p\left(\theta \right)d\theta ds\hfill \\ \hfill & +p{C}_0{t}^{1-q+pq-pk}{\int }_{t-\epsilon }^t{(t-s)}^{pk-1}m\left(s\right)ds{\int }_0^{\infty }{\theta }^k{M}_p\left(\theta \right)d\theta \hfill \\ \hfill & \to 0,\mathrm{as}\varsigma \to 0,\epsilon \to 0.\hfill \end{array}$$

This means that the set $\Lambda \left(t\right)$ can be arbitrarily approximated by relatively compact sets. Therefore, $\Lambda \left(t\right)$ is relatively compact in X for all $t\in (0,T]$. Due to its compactness at t = 0, $\Lambda \left(t\right)=\{\phi \left(t\right):\phi \in \mathfrak{L}\left({\Omega }_r\right)\}$ is relatively compact in X for all $t\in J$.

Step 5. $\mathfrak{L}$ has a closed graph.

Set $u_n\to u_{\ast }$ as $n\to \infty$, ${\phi }_n\in \mathfrak{L}\left(u_n\right)$, and ${\phi }_n\to {\phi }_{\ast }$ as $n\to \infty$. We will prove that ${\phi }_{\ast }\in \mathfrak{L}\left(u_{\ast }\right)$. Since ${\phi }_n\in \mathfrak{L}\left(u_n\right)$, there exists a $g_n\in S_{G,u_n}$ such that

$$\begin{array}{c}\hfill {\phi }_n\left(t\right)=t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0+t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g_n\left(s\right)ds,i=1,2.\end{array}$$

We need show that there exists a $g_{\ast }\in S_{G,u_{\ast }}$ such that

$$\begin{array}{c}\hfill {\phi }_{\ast }\left(t\right)=t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0+t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g_{\ast }\left(s\right)ds,i=1,2.\end{array}$$

Define the mapping $\mathcal{F}:L^1(J,X)\to C(J,X)$,

$$\begin{array}{c}\hfill \left(\mathcal{F}g\right)\left(t\right)=t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g\left(s\right)ds.\end{array}$$

It is easy to see that $\mathcal{F}$ is linear and continuous.

For any $t\in (0,T]$, ${(t-s)}^{pk-1}|g_n\left(s\right)-g_{\ast }\left(s\right)|\le 2{(t-s)}^{pk-1}m\left(s\right)$. The Lebesgue-dominated convergence theorem implies

$${\int }_0^t{(t-s)}^{pk-1}|g_n\left(s\right)-g_{\ast }\left(s\right)|ds\to 0,\mathrm{as} n\to \infty .$$

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

$$\begin{array}{cc}\hfill & |({\phi }_n\left(t\right)-t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0)-({\phi }_{\ast }\left(t\right)-t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0)|\hfill \\ \hfill & \le t^{1-q+pq-pk}{\int }_0^t\left|{\mathcal{J}}_p(t-s)(g_n\left(s\right)-g_{\ast }\left(s\right))\right|ds\hfill \\ \hfill & \le L_1{t}^{1-q+pq-pk}{\int }_0^t{(t-s)}^{pk-1}|g_n\left(s\right)-g_{\ast }\left(s\right)|ds\to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

By Lemma 1, we find that $\mathcal{F}\circ S_{G,u}$ is a closed graph operator. In addition, from the definition of $\mathcal{F}$, we can obtain

$${\phi }_n-t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0\in \mathcal{F}\left(S_{G,u_n}\right).$$

Noting that $u_n\to u_{\ast }$ as $n\to \infty$, it follows again by Lemma 1 that

$$\begin{array}{c}\hfill {\phi }_{\ast }\left(t\right)-t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0=t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g_{\ast }\left(s\right)ds,i=1,2,\end{array}$$

for some $g_{\ast }\in S_{G,u_{\ast }}$. Thus,

$$\begin{array}{c}\hfill {\phi }_{\ast }\left(t\right)=t^{1-q+pq-pk}{\mathcal{H}}_{p,q}\left(t\right)y_0+t^{1-q+pq-pk}{\int }_0^t{\mathcal{J}}_p(t-s)g_{\ast }\left(s\right)ds,i=1,2.\end{array}$$

This proves that ${\phi }_{\ast }\in \mathfrak{L}\left(u_{\ast }\right)$.

Based on Steps 1–5 and the Arzela–Ascoli theorem, we can summarize that $\mathfrak{L}$ is a compact multivalued map that is u.s.c. with convex closed values. By Lemma 2, we can conclude that $\mathfrak{L}$ has a fixed point $u\in \mathfrak{L}\left({\Omega }_r\right)$. Set $y\left(t\right)=t^{-1+q-pq+pk}u\left(t\right)$ for $t\in (0,T]$. Then, there is y belonging to $\mathfrak{L}\left({\tilde{\Omega }}_r\right)$, defined as a subset of $\mathfrak{L}\left(y\right)=\left\{\varphi \in C_p(J,X):\varphi \left(t\right)=(\Psi y)\left(t\right),g\in S_{G,u},t\in (0,T]\right\}.$ Therefore, y is a mild solution of (1) in ${\tilde{\Omega }}_r$. The proof is completed. □

3.2. Noncompact Operator Case

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

• (H4) For any bounded $E\subset X$, there exists a constant $\eta >0$ such that

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

Theorem 2.

Suppose that (H1), (H2), (H3), and (H4) hold. Then, (1) is solvable on J.

Proof. 

By the proof of Theorem 1, it remains to prove that there exists a $\widehat{\Delta }\subset {\Omega }_r$ such that the set $\{\left(\mathfrak{L}\phi \right)\left(t\right)\mid \phi \in \widehat{\Delta }\}$ is relatively compact in X.

For any bounded subset $E_0\subset {\Omega }_r$, let

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

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

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

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

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

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

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

Through mathematical induction, it can be shown that

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

Because

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

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

$${\left(4L_1\eta \right)}^{\widehat{n}}{t}^{\widehat{n}pk}{\displaystyle \frac{{\Gamma }^{\widehat{n}}\left(pk\right)\Gamma (q-pq+pk))}{\Gamma ((\widehat{n}+1)pk+q-pq)}}\le {\left(4L_1\eta \right)}^{\widehat{n}}{T}^{\widehat{n}pk}{\displaystyle \frac{{\Gamma }^{\widehat{n}}\left(pk\right)\Gamma (q-pq+pk))}{\Gamma ((\widehat{n}+1)pk+q-pq)}}=\mathsf{\ell }<1$$

Thus,

$$\begin{array}{c}\hfill \chi ({\mathfrak{L}}^{\widehat{n}}\left(E_0\left(t\right)\right))\le \mathsf{\ell }\chi \left(E_0\right).\end{array}$$

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

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

So,

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

Set

$${\Delta }_0={\Omega }_r,{\Delta }_1=\overline{co}({\mathfrak{L}}^{\widehat{n}}\left({\Delta }_0\right)),\cdots ,{\Delta }_n=\overline{co}({\mathfrak{L}}^{\widehat{n}}\left({\Delta }_{n-1}\right)),n=2,3,\cdots .$$

In view of [30], we find

(i) ${\Delta }_0\supset {\Delta }_1\supset {\Delta }_2\supset \cdots \supset {\Delta }_{n-1}\supset {\Delta }_n\supset \cdots ;$

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

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

Thanks to the method in [30], we can show that $\mathfrak{L}(\widehat{\Delta })\subset \widehat{\Delta }.$ Due to ${\mathfrak{L}}^1\left({\Delta }_0\right)=\mathfrak{L}\left({\Delta }_0\right)\subset {\Delta }_0,$ we have $\overline{co}({\mathfrak{L}}^1\left({\Delta }_0\right))\subset {\Delta }_0.$ Thus,

$$\begin{array}{ccc}\hfill {\mathfrak{L}}^2\left({\Delta }_0\right)=\mathfrak{L}(\overline{co}({\mathfrak{L}}^1\left({\Delta }_0\right)))& \subset & \mathfrak{L}\left({\Delta }_0\right)={\mathfrak{L}}^1\left({\Delta }_0\right),\hfill \\ \hfill {\mathfrak{L}}^3\left({\Delta }_0\right)=\mathfrak{L}(\overline{co}({\mathfrak{L}}^2\left({\Delta }_0\right)))& \subset & \mathfrak{L}(\overline{co}({\mathfrak{L}}^1\left({\Delta }_0\right)))={\mathfrak{L}}^2\left({\Delta }_0\right),\hfill \\ & \cdots & \\ \hfill {\mathfrak{L}}^{\widehat{n}}\left({\Delta }_0\right)=\mathfrak{L}(\overline{co}({\mathfrak{L}}^{\widehat{n}-1}\left({\Delta }_0\right)))& \subset & \mathfrak{L}(\overline{co}({\mathfrak{L}}^{\widehat{n}-2}\left({\Delta }_0\right)))={\mathfrak{L}}^{\widehat{n}-1}\left({\Delta }_0\right).\hfill \end{array}$$

Therefore, ${\Delta }_1=\overline{co}({\mathfrak{L}}^{\widehat{n}}\left({\Delta }_0\right))\subset \overline{co}({\mathfrak{L}}^{\widehat{n}-1}\left({\Delta }_0\right)),$ so $\mathfrak{L}\left({\Delta }_1\right)\subset \mathfrak{L}(\overline{co}({\mathfrak{L}}^{\widehat{n}-1}\left({\Delta }_0\right)))={\mathfrak{L}}^{\widehat{n}}\left({\Delta }_0\right)\subset \overline{co}({\mathfrak{L}}^{\widehat{n}}\left({\Delta }_0\right))={\Delta }_1.$ By using the same method, we can show that $\mathfrak{L}\left({\Delta }_n\right)\subset {\Delta }_n(n=0,1,2,\cdots ).$ From [30], we obtain $\mathfrak{L}(\widehat{\Delta })\subset {\bigcap }_{n=0}^{\infty }\mathfrak{L}\left({\Delta }_n\right)\subset {\bigcap }_{n=0}^{\infty }{\Delta }_n=\widehat{\Delta }.$ Hence, $\mathfrak{L}(\widehat{\Delta })$ is compact. As a consequence of Lemma 2, we can conclude that $\mathfrak{L}$ has a fixed point $u\in \mathfrak{L}(\widehat{\Delta })$. Let $y\left(t\right)=t^{-1+q-pq+pk}u\left(t\right)$ for $t\in (0,T]$. Then, y is a mild solution of (1) in ${\tilde{\Omega }}_r$. The proof is completed. □

Remark 2.

In the recent paper [18], the authors studied the existence of (1). Unfortunately, the results and proofs in [18] are incorrect.

An operator Φ in Theorem 3.1 of [18] is defined by

$$\Phi \left(x\left(t\right)\right)=\left\{\begin{array}{ccc}0,\hfill & \hfill & t=0,\hfill \\ t^{(1-pk)(1-q)}\left[{\mathcal{H}}_{p,q}\left(t\right)y_0+{\int }_0^t{(t-s)}^{p-1}{\mathcal{K}}_p(t-s)g\left(s\right)ds\right],\hfill & \hfill & t\in (0,T],\hfill \end{array}\right.$$

(8)

where

$${\mathcal{H}}_{p,q}\left(t\right)=I_{0+}^{q(1-p)}\left[t^{p-1}{\mathcal{K}}_p\left(t\right)\right],{\mathcal{K}}_p\left(t\right)={\int }_0^{\infty }p\theta M_p\left(\theta \right)Q\left(t^p\theta \right)d\theta ,$$

${\left\{Q\left(t\right)\right\}}_{t\ge 0}$ is the semigroup associated with A, $0<k<1$, $g\in S_{G,y}$.

However, we notice that

$$\underset{t\to 0+}{lim}{t}^{(1-pk)(1-q)}{\mathcal{H}}_{p,q}\left(t\right)y_0\ne 0,\mathrm{for} y_0\ne 0.$$

For example, if $q=1$, by Proposition 2.3 in [9], we find that ${lim}_{t\to 0+}{\mathcal{K}}_p\left(t\right)y_0=y_0/\Gamma \left(p\right)$. So,

$$\begin{array}{cc}\hfill \underset{t\to 0+}{lim}{t}^{(1-pk)(1-q)}{\mathcal{H}}_{p,1}\left(t\right)y_0=& {\displaystyle \frac{1}{\Gamma (1-p)}}\underset{t\to 0+}{lim}{\int }_0^t{(t-s)}^{-p}{r}^{p-1}{\mathcal{K}}_p\left(s\right)y_{0}ds\hfill \\ \hfill =& {\displaystyle \frac{1}{\Gamma (1-p)}}\underset{t\to 0+}{lim}{\int }_0^1{(1-v)}^{-p}{v}^{p-1}{\mathcal{K}}_p\left(tv\right)y_{0}dv\hfill \\ \hfill =& y_0\ne 0,\mathrm{for} y_0\ne 0.\hfill \end{array}$$

Therefore, the proof of Theorem 3.1 in [18] is incorrect.

Example 1.

Let $X=L^2([0,\pi ],\mathbb{R})$. Consider the following fractional partial differential inclusions on $J:=(0,T]$

$$\left\{\begin{array}{c}{}^H{D}_{0+}^{p,q}y(t,z)\in {\partial }_z^{2}y(t,z)+t^{-\alpha },z\in [0,\pi ],t\in J,\hfill \\ y(t,0)=y(t,\pi )=0,t\in J,\hfill \\ I_{0+}^{(1-p)(1-q)}y(0,z)=y_0\left(z\right),z\in [0,\pi ],\hfill \end{array}\right.$$

(9)

where $\alpha \in (q,1-p+pq)$ We define an operator A by $Av=v^{\prime \prime }$ with the domain

$$D\left(A\right)=\{v\in X:v,v^{\prime \prime } \mathrm{are} \mathrm{absolutely} \mathrm{continuous}, v^{\prime \prime }\in X,v\left(0\right)=v\left(\pi \right)=0\}.$$

Then, A generates a compact, analytic, self-adjoint semigroup ${\left\{Q\left(t\right)\right\}}_{t>0}$. Then, problem (9) can be rewritten as follows:

$$\left\{\begin{array}{c}{}^H{D}_{0+}^{p,q}y\left(t\right)\in Ay\left(t\right)+G(t,y\left(t\right)),t\in J,\hfill \\ I_{0+}^{(1-p)(1-q)}y\left(0\right)=y_0,\hfill \end{array}\right.$$

(10)

where $G(t,y):=t^{-\alpha }$ and $\left|G(t,y\left(t\right))\right|\le t^{-\alpha }$, $t\in (0,\infty )$. Let $m\left(t\right)=t^{-\alpha }$, for $t>0$. Then,

$$I_{0+}^{q}m\left(t\right)={\displaystyle \frac{\Gamma (1-\alpha )}{\Gamma (1+q-\alpha )}}{t}^{q-\alpha },t^{(1-q)(1-p)}{I}_{0+}^{q}m\left(t\right)={\displaystyle \frac{\Gamma (1-\alpha )}{\Gamma (1+q-\alpha )}}{t}^{\kappa },$$

where $\kappa =1-p+qp-\alpha \in (0,1)$. This means that the condition (H3) is satisfied. By Theorem 1, (9) is solvable on J.

4. Conclusions

This paper studies the existence of mild solutions for Hilfer fractional evolution inclusions with almost sectorial operators. We address two cases, i.e., when the almost sectorial operator A generates either a compact or noncompact semigroup. Key results rely on fixed-point theorems, fractional calculus, and Kuratowski’s measure of noncompactness. We extend prior work on Riemann–Liouville and Caputo derivatives to the Hilfer derivative, which interpolates between the two derivatives ($0<p<1$, $q=0$ and $q=1$). This allows the modeling of systems with intermediate memory properties. We identify and resolve a flaw in prior research (Theorem 3.1 of [20]), where incorrect operator definitions led to invalid conclusions. We rigorously redefine the solution operators to ensure continuity at $t=0$. Our results can be extended to stochastic inclusions, impulsive systems, and functional evolution inclusions with infinite delays.