# Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator

Authors: Adel Lachouri; Abdelouaheb Ardjouni; Fahd Jarad; Mohammed S. Abdo
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the existence of solutions to initial value problems for a nonlinear generalized Caputo fractional differential inclusion with Lipschitz set-valued functions. The applied fractional operator is given by the kernelkρ,s=ξρ−ξs and the derivative operator 1/ξ′ρd/dρ. The existence result is obtained via fixed point theorems due to Covitz and Nadler. Moreover, we also characterize the topological properties of the set of solutions for such inclusions. The obtained results generalize previous works in the literature, where the classical Caputo fractional derivative is considered. In the end, an example demonstrating the effectiveness of the theoretical results is presented.


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## Body

## 1. Introduction

Fractional calculus is a field of mathematical analysis that embraces the integrals and derivatives of functions of any real or complex order. For the past few decades, this field has been one of the hand-over-fist sprawling fields of mathematics by virtue of the amazing findings obtained when researchers enrolled the fractional operators in their attempts to construe some problems that arise in nature (see [1–6]). As a matter of fact, the classical fractional calculus consisted of one main integral operator, namely, the Riemann-Liouville fractional integral, and two fractional derivatives, namely, the Riemann-Liouville and Caputo derivatives. Because of the penurious number of operators, researchers were compelled to discover and develop new fractional operators that allow them to better comprehend the world around them. In the last 10 years or so, new fractional operators have been introduced and wielded on many occasions. One can touch upon the operators in [7–13]. It should be noted that some of these operators are an extension or generalization of the Riemann-Liouville integrals and Riemann-Liouville and Caputo derivatives that are nonlocal but singular kerneled. The others are brand-new ones, and they are nonlocal and contain nonsingular kernels.One of the main applications of advanced fractional calculus is the theory of fractional evolution inclusions since they are abstract formulations for numerous problems arising in physics and engineering. Evolution equations and inclusions are commonly applied to describe the systems and models that change or evolve over time. Many studies have investigated the existence and uniqueness of solutions for fractional evolution problems based on semigroup and fixed point theories; e.g., Bedi et al. [14] studied the existence of mild solutions for Hilfer-type fractional neutral impulsive evolution problems in a Banach space. The same author, with others in [15], discussed the stability and controllability results for Hilfer-type fractional evolution equations in a Banach space. The existence of saturated mild (and global) solutions of Caputo-type fractional semilinear evolution problems with noncompact semigroups has been obtained in Banach spaces by Chen et al. [16]. The authors in [17] studied the existence of local (and global) solutions and the uniqueness of a mild solution to fractional semilinear evolution problems with compact (and noncompact) semigroups in Banach spaces. Zhou and Jiao [18] considered a class of nonlocal Cauchy problems for a Caputo-type fractional neutral evolution problem to investigate the existence and uniqueness of mild solutions. For differential equations-inclusions governed by Cauchy conditions or boundary conditions, the issue of importance is to tackle the existence, uniqueness, and stability of their solutions. These properties constitute the most essential parts of the analysis of these equations. There are heaps of papers that went about the existence, uniqueness, and stability results of differential equations-inclusions within the scope of a variety of fractional derivatives based on different types of fixed point techniques. For instance, Abdo et al. [19] have proven the existence and different types of stability of solutions for ψ-Hilfer-type fractional integro-problems. The authors in [20] investigated some existence results of Caputo-type fractional neutral inclusions by using weak topology. The existence of solutions for a generalized Caputo-type fractional differential inclusion problem with integral boundary conditions has been studied by Belmor et al. [21]. In [22], the authors discussed the existence of solutions for a Caputo-type fractional for higher-order fractional inclusion problems. Lachouri et al. [23] considered a nonlinear Hilfer-type nonlocal fractional inclusion problem to prove the existence results, taking into account the convex and nonconvex values of the nonlinear term.Recently, Chen et al. [16] proved the existence of saturated mild solutions and global mild solutions for fractional evolution equations of the type(1)CDr1σρ+Aσρ=hρ,σρ,ρ≥0,σ0=σ0∈X, where CDr1 is the Caputo fractional derivative of order r1∈0,1, A:DA⊂X⟶X is a closed linear operator and −A generates a uniformly bounded C0-semigroup Tρρ≥0 in X, and h:0,∞×X⟶X is the given function.In [17], Suechoei and Ngiamsunthorn studied the following fractional evolution equations:(2)DCa+r1;ξσρ=Aσρ+hρ,σρ,ρ∈0,T,σ0=σ0,σ0∈X, where CDa+r1;ξ is the ξ-Caputo FD of order r1∈0,1, A is the infinitesimal generator of a C0-semigroup of uniformly bounded linear operators Tρρ≥0 on X, and h:0,∞×X⟶X is the given function.Motivated by the aforementioned works and inspired by [17], we consider the following fractional evolution inclusion involving ξ-Caputo FD:(3)DCa+r1;ξσρ∈Aσρ+fρ,σρ,ρ∈0,T,σ0=σ0,σ0∈X,where f:0,T×X⟶OX is a set-valued map (svm) from 0,T×X to the family of OX⊂X and X is a Banach space with the norm ..The novelty of the present work is choosing a more general operator than the classical fractional operators. More precisely, problem (3) is reduced to a Caputo-type problem, for ξρ=ρ; Hadamard-Caputo-type problem, for ξρ=logρ; and Katugampola-Caputo-type problem, for ξρ=ρδ, δ>0. In addition, we describe the topological properties of the considered solution in the present work.In this paper, we aim to prove the existence of mild solutions for problem (3) involving the generalized Caputo derivative using the fixed point theorem (FPT) of Nadler and Covitz and to characterize the topological properties of the set of solutions for such inclusions.The acquired results are more general and cover many of the parallel problems that contain special cases of functionξ, such as [16, 18].


## 2. Preliminary Notions

### 2.1. Fractional Calculus (FC)

In this portion, we introduce several basic notions of FC and necessary lemmas that will be needed in such a study.LetI∗=0,T. Denoted by E=CI∗,X, we have the Banach space of continuous functions σ:I∗⟶X with(4)σE=supσρ:ρ∈I∗.Letσ,ξ∈CnI∗,ℝ such that ξ is increasing and ξ′ρ≠0, ∀ρ∈I∗, and(5)M=supρ∈0,∞Tρ<∞.Definition 1 (see [24]).
Theξ-Riemann-Liouville FI of a function σ of order r1 is defined by(6)Ia+r1;ξσρ=1Γr1∫aρξ′υξρ−ξυr1−1συdυ.Definition 2 (see [24]).
Theξ-Riemann-Liouville FD of order r1 for a function σ is given by(7)Da+r1;ξσρ=1ξ′ρddρnIa+n−r1;ξσρ,where n=r1+1, n∈ℕ.Definition 3 (see [13]).
Theξ-Caputo FD of a function σ∈CnI∗,ℝ of order r1 is described by(8)CDa+r1;ξσρ=Ia+n−r1;ξσnρ,where σnρ=1/ξ′ρd/dρnσρ and n=r1+1, n∈ℕ.Lemma 4 (see [11, 25]).
Letr1,r2,μ>0. Then,(9)Ia+r1;ξξρ−ξar2−1ρ=Γr2Γr1+r2ξρ−ξar1+r2−1,CDa+r1;ξξρ−ξar2−1ρ=Γr2Γr2−r1ξρ−ξar1+r2−1.Lemma 5 (see [13]).
Ifσ∈CnI∗,ℝ, r1∈n−1,n, and r2∈0,1, then(10)Ia+r1;ξCDa+r1;ξσρ=σρ−∑k=0n−1σna+k!ξρ−ξak.
In particular, givenr1∈0,1, we have(11)Ia+r1;ξCDa+r1;ξσρ=σρ−σa.Definition 6 (see [11]).
Letσ,ξ:a,∞⟶ℝ be real-valued functions. The generalized Laplace transform of σ is defined by(12)Lσρυ=∫aρe−υξρ−ξaσρξ′ρdρ,forallυ.Theorem 7 (see [11]).
Letr1>0 and ϰ be a piecewise continuous function on each interval a,ρ and ξρ-exponential order. Then,(13)LIa+r1;ξσρυ=Ia+r1;ξσρυr1.Definition 8 (see [26, 27]).
Letr1∈0,1 and ρ∈ℂ. The Wright-type function is defined by(14)ϑr1ρ=∑m=0∞−ρmm!Γ−r1m+1−r1=∑m=0∞−ρmΓr1m+1sinπr1m+1m!.Proposition 9 (see [26, 27]).
The Wright functionϑr1 is an entire function and has the properties listed below:(1)
ϑr1≥0 for ρ≥0 and ∫0∞ϑr1ρdρ=1(2)
∫0∞ϑr1ρραdρ=Γ1+α/Γ1+αr1, for α>−1(3)
∫0∞ϑr1ρe−kρdρ=Er1−k, k∈ℂ(4)
α∫0∞ρϑr1ρe−kρdρ=Er1,r1−k, k∈ℂNext, we recall some conceptions concerning the semigroups of linear operators. For more details, see [28, 29].For strongly continuous semigroups (i.e., C0-semigroup Tρρ≥0 on X), we define the generator(15)Aσ=limρ⟶0+Tρσ−σσ,σ∈X.ByDA, we denote the domain of A, that is,(16)DA=σ∈X:limρ⟶0+Tρσ−σσexists.Lemma 10 (see [28, 29]).
LetTρρ≥0 be a C0-semigroup; then, there exist constants C>0 and a≥0 such that(17)Tρ≤Ceaτ,∀ρ≥0.Lemma 11 (see [28, 29]).
A linear operatorA is the infinitesimal generator of a C0-semigroup iff(1)
A is closed, and DA¯=X(2)
The resolvent setRA of A contains ℝ+, and for every λ∈ℝ+, we have(18)Rλ,A≤1λ,where Rλ,A=λr1I−A−1σ=∫0∞e−λτTρσdρ.In relation to problem (3), we need the following lemma which was proven in [17].Lemma 12.
The mild solution of IVPs(19)DCa+r1;ξσρ=Aσρ+qρ,σρ,ρ∈0,T,σ0=σ0,is obtained as(20)σρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υqυ,συdυ,where(21)Sr1;ξρ,υσ=∫0∞ϑr1θTξρ−ξυr1θσdθ,Tr1;ξρ,υσ=r1∫0∞θϑr1θTξρ−ξυr1θσdθ,for 0≤υ≤ρ≤T.Lemma 13 (see [18]).
The operatorsTr1;ξ and Sr1;ξ have the following properties:(1)
For any fixedρ≥υ≥0, Tr1;ξ and Sr1;ξ are bounded linear operators with(22)Sr1;ξρ,υσ≤Mσ,Tr1;ξρ,υσ≤r1MΓ1+r1=MΓr1σ,∀σ∈X.(2)
Tr1;ξ and Sr1;ξ are strongly continuous ∀ρ≥υ≥0; that is, for every ϖ∈X and 0≤υ≤ρ1≤ρ2≤T, we have(23)Sr1;ξρ2,υσ−Sr1;ξρ1,υσ⟶0,Tr1;ξρ2,υσ−Tr1;ξρ1,υσ⟶0,as ρ1⟶ρ2.(3)
IfTρ is a compact operator ∀ρ>0, then Tr1;ξρ,υ and Sr1;ξρ,υ are compact for all ρ,υ>0.(4)
The operatorsTr1;ξρ,υ and Sr1;ξρ,υ are continuous in the uniform operator topology.


## 2.1. Fractional Calculus (FC)

In this portion, we introduce several basic notions of FC and necessary lemmas that will be needed in such a study.LetI∗=0,T. Denoted by E=CI∗,X, we have the Banach space of continuous functions σ:I∗⟶X with(4)σE=supσρ:ρ∈I∗.Letσ,ξ∈CnI∗,ℝ such that ξ is increasing and ξ′ρ≠0, ∀ρ∈I∗, and(5)M=supρ∈0,∞Tρ<∞.Definition 1 (see [24]).
Theξ-Riemann-Liouville FI of a function σ of order r1 is defined by(6)Ia+r1;ξσρ=1Γr1∫aρξ′υξρ−ξυr1−1συdυ.Definition 2 (see [24]).
Theξ-Riemann-Liouville FD of order r1 for a function σ is given by(7)Da+r1;ξσρ=1ξ′ρddρnIa+n−r1;ξσρ,where n=r1+1, n∈ℕ.Definition 3 (see [13]).
Theξ-Caputo FD of a function σ∈CnI∗,ℝ of order r1 is described by(8)CDa+r1;ξσρ=Ia+n−r1;ξσnρ,where σnρ=1/ξ′ρd/dρnσρ and n=r1+1, n∈ℕ.Lemma 4 (see [11, 25]).
Letr1,r2,μ>0. Then,(9)Ia+r1;ξξρ−ξar2−1ρ=Γr2Γr1+r2ξρ−ξar1+r2−1,CDa+r1;ξξρ−ξar2−1ρ=Γr2Γr2−r1ξρ−ξar1+r2−1.Lemma 5 (see [13]).
Ifσ∈CnI∗,ℝ, r1∈n−1,n, and r2∈0,1, then(10)Ia+r1;ξCDa+r1;ξσρ=σρ−∑k=0n−1σna+k!ξρ−ξak.
In particular, givenr1∈0,1, we have(11)Ia+r1;ξCDa+r1;ξσρ=σρ−σa.Definition 6 (see [11]).
Letσ,ξ:a,∞⟶ℝ be real-valued functions. The generalized Laplace transform of σ is defined by(12)Lσρυ=∫aρe−υξρ−ξaσρξ′ρdρ,forallυ.Theorem 7 (see [11]).
Letr1>0 and ϰ be a piecewise continuous function on each interval a,ρ and ξρ-exponential order. Then,(13)LIa+r1;ξσρυ=Ia+r1;ξσρυr1.Definition 8 (see [26, 27]).
Letr1∈0,1 and ρ∈ℂ. The Wright-type function is defined by(14)ϑr1ρ=∑m=0∞−ρmm!Γ−r1m+1−r1=∑m=0∞−ρmΓr1m+1sinπr1m+1m!.Proposition 9 (see [26, 27]).
The Wright functionϑr1 is an entire function and has the properties listed below:(1)
ϑr1≥0 for ρ≥0 and ∫0∞ϑr1ρdρ=1(2)
∫0∞ϑr1ρραdρ=Γ1+α/Γ1+αr1, for α>−1(3)
∫0∞ϑr1ρe−kρdρ=Er1−k, k∈ℂ(4)
α∫0∞ρϑr1ρe−kρdρ=Er1,r1−k, k∈ℂNext, we recall some conceptions concerning the semigroups of linear operators. For more details, see [28, 29].For strongly continuous semigroups (i.e., C0-semigroup Tρρ≥0 on X), we define the generator(15)Aσ=limρ⟶0+Tρσ−σσ,σ∈X.ByDA, we denote the domain of A, that is,(16)DA=σ∈X:limρ⟶0+Tρσ−σσexists.Lemma 10 (see [28, 29]).
LetTρρ≥0 be a C0-semigroup; then, there exist constants C>0 and a≥0 such that(17)Tρ≤Ceaτ,∀ρ≥0.Lemma 11 (see [28, 29]).
A linear operatorA is the infinitesimal generator of a C0-semigroup iff(1)
A is closed, and DA¯=X(2)
The resolvent setRA of A contains ℝ+, and for every λ∈ℝ+, we have(18)Rλ,A≤1λ,where Rλ,A=λr1I−A−1σ=∫0∞e−λτTρσdρ.In relation to problem (3), we need the following lemma which was proven in [17].Lemma 12.
The mild solution of IVPs(19)DCa+r1;ξσρ=Aσρ+qρ,σρ,ρ∈0,T,σ0=σ0,is obtained as(20)σρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υqυ,συdυ,where(21)Sr1;ξρ,υσ=∫0∞ϑr1θTξρ−ξυr1θσdθ,Tr1;ξρ,υσ=r1∫0∞θϑr1θTξρ−ξυr1θσdθ,for 0≤υ≤ρ≤T.Lemma 13 (see [18]).
The operatorsTr1;ξ and Sr1;ξ have the following properties:(1)
For any fixedρ≥υ≥0, Tr1;ξ and Sr1;ξ are bounded linear operators with(22)Sr1;ξρ,υσ≤Mσ,Tr1;ξρ,υσ≤r1MΓ1+r1=MΓr1σ,∀σ∈X.(2)
Tr1;ξ and Sr1;ξ are strongly continuous ∀ρ≥υ≥0; that is, for every ϖ∈X and 0≤υ≤ρ1≤ρ2≤T, we have(23)Sr1;ξρ2,υσ−Sr1;ξρ1,υσ⟶0,Tr1;ξρ2,υσ−Tr1;ξρ1,υσ⟶0,as ρ1⟶ρ2.(3)
IfTρ is a compact operator ∀ρ>0, then Tr1;ξρ,υ and Sr1;ξρ,υ are compact for all ρ,υ>0.(4)
The operatorsTr1;ξρ,υ and Sr1;ξρ,υ are continuous in the uniform operator topology.


## 3. Main Results

In what follows, we will utilize the notationOcp,cX=N∈Ov:Niscompactandconvex for a normed space X. For more information about the svm, we refer the reader to [30, 31].Definition 14.
A functionσ∈C1I∗,X is a solution of (3), if ∃℘~∈L1I∗,X with ℘~ρ∈fρ,σ∀ρ∈I∗ fulfilling the initial condition(24)σ0=σ0,σρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~υdυ.
The first outcome treats the nonconvexf based on the theorem of Covitz and Nadler [32].Letσ,d be a metric space induced from the normed space σ,.. Consider Hd:Oσ×Oσ⟶ℝ+∪∞ defined by(25)HdC~,D~=maxsupc~∈C~dc~,D~,supd~∈D~dC~,d~,where dC~,d~=infc~∈C~dc~,d~ and dc~,D~=infd~∈D~dc~,d~. Then, Ob,clσ,Hd is a metric space (see [33]).Definition 15.
A svmF:X⟶OclX is ν-Lipschitz iff ∃ν>0 such that(26)HdFσ,Fρ≤νdσ,ρ,∀σ,ρ∈X.
Particularly, ifν<1, then F is a contraction.Theorem 16.
Let(27)ρ=MΓr1+1ξT−ξ0r1,and assume that
(As1)f:I∗×X⟶OcpX such that f.,σ:I∗⟶OcpX is measurable for any σ∈X.
(As2)Hdfρ,σ,fρ,σ¯≤zρσ−σ¯ for (a.e.) all ρ∈I∗ and σ,σ¯∈ℝ with z∈CI∗,ℝ+ and d0,fρ,0≤zρ for (a.e.) all ρ∈I∗.
Then, (3) possesses at least one solution on I∗ if(28)ρz∞<1.Proof.
At first, to switch problem (3) into a FP, we formulate F:E⟶OE as(29)Fσ=ϕ∈E:ϕρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~υdυ,for ℘~∈Rf,σ. The solution of (3) is obviously an FP of F. The following are the steps in the proofing process.
By virtue of assumption (As1) and [34] (Theorem III.6), f has a measurable selection, and thus, Rf,σ≠∅. In the sequel, we demonstrate that F:E⟶OE defined in (29) fulfills the assumptions of FPT of Covitz and Nadler. First, we show that Fσ is closed for every σ∈E. Let unn=0∞∈Fσ such that un⟶un⟶∞ in E. Then, u∈E and there is ℘~n∈Rf,σn such that(30)unρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~nυdυ,ρ∈I∗.
Accordingly, there is a subsequence℘~nthat converges to ℘~ in L1I∗,X because f has compact values. As an outcome, ℘~∈Rf,σ, and we get(31)unρ⟶uρ=Sr1;ξρ,0v0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~υdυ,∀ρ∈I∗.
Hence,u∈Fσ.
It remains to demonstrate that there is aθ∈0,1θ=ρz∞ such that(32)HdFσ,Fσ¯≤θσ−σ¯,∀σ,σ¯∈E.
Letσ,σ¯∈E and ϕ1∈Fσ. Then, there exists ℘~1ρ∈fρ,σρ such that(33)ϕ1ρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~1υdυ.
By (As2), we have(34)Hdfρ,σ,fρ,σ¯≤zρσρ−σ¯ρ.
So, there existsς^ρ∈fρ,σ¯ such that(35)℘~1ρ−ς^≤zρσρ−σ¯ρ,ρ∈I∗.
As follows, we build a svmχ:I∗⟶OX, where(36)χρ=ς^∈X:℘~1ρ−ς^≤zρσρ−σ¯ρ.
From [34] (Proposition III.4), the svm χρ∩fρ,σ¯ is measurable. We now select the function ℘~2ρ∈fρ,σ¯ such that(37)℘~1ρ−℘~2ρ≤zρσρ−σ¯ρ,∀ρ∈I∗.
We define(38)ϕ2ρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~2υdυ,∀ρ∈I∗.
As a consequence, we get(39)ϕ1ρ−ϕ2ρ≤∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~1υ−℘~2υdυ≤Mz∞σ−σ¯EΓr1+1ξT−ξ0r1.
Therefore,(40)ϕ1−ϕ2E≤ρz∞σ−σ¯E.
By the analogous relation, obtained by interchanging the roles ofσ and σ¯, we get(41)HdFσ,Fσ¯≤ρz∞σ−σ¯E.
BecauseF is a contraction, we conclude that it has a FP, which is a solution of (3) based on the Covitz and Nadler theorem.Next, we study the topological properties of the set of solutions of problem (3).Theorem 17.
Assume that
(As3)Tρ is a compact operator for each ρ>0.
(As4)f:I∗×X⟶Ocp,cX is a L1-Carathéodory set-valued map.
(As5)∃ℓ~1∈CI∗,ℝ+ and a nondecreasing ℓ~2∈Cℝ+,ℝ+ such that(42)fρ,σO=supγ:γ∈fρ,σ≤ℓ~1ρℓ~2σE,∀ρ,σ∈I∗×X.
Then, the set of solutions of (3) is convex and relatively compact.Proof.
We consider the operatorF defined in (29) and demonstrate in the following steps that the set of solutions of (3) is convex and relatively compact in E.(Step 1)
The svmFσ is convex for every σ∈E.
Letϕ1,ϕ2∈Fσ. Then, there exist ℘~1,℘~2∈Rf,σ such that(43)ϕiρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~iυdυ,i=1,2,∀ρ∈I∗.
Letϰ∈0,1. Then, for every ρ∈I∗,(44)ϰϕ1+1−ϰϕ2ρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υϰ℘~1υ+1−ϰ℘~2υdυ.
Sincef has convex values, Rf,σ is convex and ϰ℘~1ρ+1−ϰ℘~2ρ∈Rf,σ. Thus, ϰϕ1+1−ϰϕ2∈Fσ.(Step 2)
FBr is bounded on bounded sets of S.
For a constantr∈ℝ+, let Br=σ∈S:σE≤r be a bounded set in S. Then, for each ϕ∈Fσ and υ∈Br, there exists ℘~∈Rf,σ such that(45)ϕρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~υdυ.
Under hypothesis (As4) and Lemma13 (1), for any ρ∈I∗, we attain(46)ϕρ≤Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~υdυ≤Mσ0+ℓ~1∞ℓ~2rMΓr1+1ξT−ξ0r1.
Thus,(47)ϕE≤Mσ0+ρℓ~1∞ℓ~2r,where ρ is defined in (27).(Step 3)
F sends bounded sets of E into equicontinuous sets.
Letσ∈Br and ϕ∈Fσ. Then, there is a function ℘~∈Rf,σ such that(48)ϕρ=Sr1;ξρ,0σ0+∫0ρξ′υξρ−ξυr1−1Tr1;ξρ,υ℘~υdυ.
Letρ1,ρ2∈I∗, ρ1<ρ2. Then,(49)ϕρ2−ϕρ1≤Sr1;ξρ2,0σ0−Sr1;ξρ1,0σ0+∫0ρ1ξ′υξρ1−ξυr1−1Tr1;ξρ2,υ℘~υdυ+∫ρ1ρ2ξ′υξρ2−ξυr1−1Tr1;ξρ2,υ℘~υdυ+∫0ρ1ξ′υξρ1−ξυr1−1Tr1;ξρ2,υ℘~υdυ−∫0ρ1ξ′υξρ1−ξυr1−1Tr1;ξρ2,υ℘~υdυ−∫0ρ1ξ′υξρ1−ξυr1−1Tr1;ξρ1,υ℘~υdυ≤Sr1;ξρ2,0σ0−Sr1;ξρ1,0σ0+∫ρ1ρ2ξ′υξρ2−ξυr1−1Tr1;ξρ2,υ℘~υdυ+∫0ρ1ξ′υξρ2−ξυr1−1−ξρ1−ξυr1−1Tr1;ξρ2,υ℘~υdυ+∫0ρ1ξ′υξρ1−ξυr1−1Tr1;ξρ2,υ−Tr1;ξρ1,υ℘~υdυ≕I1+I2+I3+I4.
From Lemma13, it is obvious that I1⟶0 as ρ1⟶ρ2 and we get(50)I2≤Mℓ~1∞ℓ~2rΓr1+1ξρ2−ξρ1r1,I3≤Mℓ~1∞ℓ~2rΓr1+1ξρ2r1−ξρ1r1−ξρ2−ξρ1r1,as ρ1⟶ρ2, which yields to I2⟶0 and I3⟶0. For ρ1=0 and ρ2∈0,T, it is clear that I4=0. Then, for any 0<ε<ρ1, we get(51)I4≤∫0ρ1−εξ′υξρ1−ξυr1−1Tr1;ξρ2,υ−Tr1;ξρ1,υ℘~υdυ+∫ρ1−ερ1ξ′υξρ1−ξυr1−1Tr1;ξρ2,υ−Tr1;ξρ1,υ℘~υdυ≤ℓ~1ℓ~2rr1ξρ1−ξ0r1−ξρ1−ξρ1−εr1×sup0≤ε<ρ1Tr1;ξρ2,υ−Tr1;ξρ1,υ+2Mℓ~1∞ℓ~2rΓr1+1ξρ1−ξρ1−εr1,as ρ1⟶ρ2 and ε⟶0, which yields to I4⟶0; thus,(52)ϕρ2−ϕρ1⟶0.
Hence,FBr is equicontinuous.(Step 4)
We show that for eachρ∈I∗, Hρ=Fε,δσρ:σ∈Br is relatively compact in X.
Clearly,H0 is relatively compact in X. Let ρ∈I∗ be fixed, and for any ε,δ>0 and σ∈Br, define(53)Fσ=∫0∞ϑr1θTξρ−ξυr1θσ0dθ+r1∫0ρ−ε∫δ∞ξ′υξρ−ξυr1−1θϑr1θ×Tξρ−ξυr1θ℘~υdθdυ=∫0∞ϑr1θTξρ−ξυr1θσ0dθ+r1∫0ρ−ε∫δ∞ξ′υξρ−ξυr1−1θϑr1θ×Tξρ−ξυr1θ+εr1δ−εr1δ℘~υdθdυ=∫0∞ϑr1θTξρ−ξυr1θσ0dθ+r1∫0ρ−ε∫δ∞ξ′υξρ−ξυr1−1θϑr1θ×Tεr1δTξρ−ξυr1θ−εr1δ℘~υdθdυ=∫0∞ϑr1θTξρ−ξυr1θσ0dθ+r1Tεr1δ∫0ρ−ε∫δ∞ξ′υξρ−ξυr1−1θϑr1θ×Tξρ−ξυr1θ−εr1δ℘~υdθdυ.
Then, by the compactness ofTεr1δ for εr1δ>0, we see that the set Hε,δρ=Fε,δσρ:σ∈Br is relatively compact in X for all ε,δ>0. Moreover, for each σ∈Br, we have(54)Fσρ−Fε,δσρ=r1∫0ρ∫0δξ′υξρ−ξυr1−1θϑr1θTξρ−ξυr1θ℘~υdθdυ+∫0ρ∫δ∞ξ′υξρ−ξυr1−1θϑr1θTξρ−ξυr1θ℘~υdθdυ−∫0ρ−ε∫δ∞ξ′υξρ−ξυr1−1θϑr1θTξρ−ξυr1θ℘~υdθdυ≤r1∫0ρ∫0δξ′υξρ−ξυr1−1θϑr1θTξρ−ξυr1θ℘~υdθdυ+r1∫ρ−ερ∫δ∞ξ′υξρ−ξυr1−1θϑr1θTξρ−ξυr1θ℘~υdθdυ≤r1Mℓ~1∞ℓ~2r∫0ρξ′υξρ−ξυr1−1dυ∫0δθϑr1θdθ+r1Mℓ~1∞ℓ~2r∫ρ−ερξ′υξρ−ξυr1−1dυ∫δ∞θϑr1θdθ≤r1Mℓ~1∞ℓ~2r∫0ρξ′υξρ−ξυr1−1dυ∫0δθϑr1θdθ+r1Mℓ~1∞ℓ~2r∫ρ−ερξ′υξρ−ξυr1−1dυ∫0∞θϑr1θdθ≤Mℓ~1∞ℓ~2rξρ−ξ0r1∫0δθϑr1θdθ+Mℓ~1∞ℓ~2rΓr+1ξρ−ξρ−εr1⟶0,asε,δ⟶0+.
Hence, there are relatively compact sets arbitrarily close toHρ for ρ>0, Therefore, Hρ is relatively compact in X.
From the Arzela-Ascoli theorem,FBr is relatively compact in E. Consequently, Fσ is relatively compact in ∀σ∈E.


## 4. An Example

LetX=L20,π. Consider the following fractional partial differential inclusions with Caputo derivative ξρ=ρ:(55)DCa+1/2;ρσρ,v∈∂2∂v2σρ,v+fρ,σρ,v,ρ∈0,1,σ∈0,π,σρ,0=σρ,π=0,σ0,v=σ0,where r1=1/2 and T=1, and we define an operator A by Aσ=σ′′ for σ∈DA, where DA=σ∈X:σ,σ′areabsolutelycontinuous,andσ0=σπ=0. Then, A generates a strongly continuous semigroup Tρρ≥0 which is compact. Furthermore, Tρ≤e−ρ≤1=M, ρ≥0. Let f:0,1×ℝ⟶Oℝ defined by(56)σρ,v⟶fρ,σρ,v=0,sinσρ,vexpρ2+2+115.Clearly,Hdfρ,σ,fρ,σ¯≤zρσ−σ¯, where zρ=1/expρ2+2 and d0,fρ,0=1/15≤zρ for (a.e.) all ρ∈0,1. Besides, we obtain z∞=1/3 which implies ρz≃0.38<1. Consequently, all items of Theorem 16 are satisfied. Then, there exists at least one solution of (10) on 0,1.Remark 18.
Our current results for problem (3) remain true for the following cases:(i)
Caputo-type inclusion problem:ξρ=ρ.(ii)
Caputo-Hadamard-type inclusion problem:ξρ=logρ.(iii)
Generalized Caputo-type inclusion problem:ξρ=ρρ,ρ>0.Remark 19.
The obtained results for problem (3) include the results of Suechoei and Ngiamsunthorn [17]; i.e., problem (3) is reduced to the considered problem in [17] when fρ,σρ=hρ,σρ.


## 5. Concluding Remarks

We have studied the class of IVPs for generalized Caputo FDIs with Lipschitz set-valued functions. We obtained the existence result by using the FPT of Covitz and Nadler. Also, we characterized the topological properties of the set of solutions. We confirm that our acquired outcomes are new in the frame of generalized Caputo FDIs with initial conditions and it greatly contributes to enriching the existing literature on this theme.In future works, many cases can be established when one takes a more generalized operator, for example,ψ-Hilfer fractional operator [13]. Further, it will be of interest to study the existing problem in this article for the Mittag-Leffler power law [8] and for fractal fractional operators [35].



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