# Solvability of Fractional Differential Inclusion with a Generalized Caputo Derivative

Authors: Tamer Nabil
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.


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## Body

## 1. Introduction

The history of the theory of fractional calculus goes back to 1695 when Leibniz sent a question to L-Hôpital [1]. Although in the starter fractional calculus had an efflorescence as a mathematical analysis idea, nowadays, its use has also sawing into many other subjects of engineering and science such as biology, physics, mechanics, chemistry, and bioengineering [2–5].It is known that differential inclusions are more general than differential equations and various phenomena of science, control, and engineering are successfully modeled as fractional differential inclusions [6, 7].Recently, fractional differential inclusions with nonlocal conditions have attracted the attention of many researchers. In 2011, El-Sayed et al. [8] established the solvability of the ordinary differential inclusion with deviated-advanced nonlocal condition.
(1)η′t∈Ft,ηt,a.e.t∈0,1∑i=1l1ciηθti=β∑j=1l2djηϕτi,ci,dj>0,ti,τj∈0,1,∀i=1,⋯,l1,j=1,⋯,l2.In the few past years, there has been important works in fractional differential inclusions with other types of nonlocal conditions. Detailedly, in 2015, Wang et al. [9] established the existence of solutions for the Caputo fractional differential inclusions involving nonlocal conditions. In the second year, Lian et al. [10] established the solvability of the fractional differential inclusions with nonlocal conditions by using the measure of noncompactness and several-valued fixed-point approach. In 2019, Castaing et al. [11] studied the solvability of a new class of the Riemann–Liouville fractional differential inclusion with nonlocal integral conditions in a separable Banach space.In the above-cited monographs, the Caputo and Riemann–Liouville derivatives were utilized. In 2017, Almeida [12] obtained the new generalized Caputo fractional derivative, that is, a Caputo-kind operator of a function with respect to another function. Indeed, this fractional operator is more general than Riemann–Liouville, Hadamard, Erdely Kober, and Caputo operator kinds. More details about the generalized Caputo fractional operator are found in [13, 14]. Since then, generalized fractional operators draw increasing attention due to their advantages, because the generalized fractional operators will give us new opportunities to improve the theoretical results and to model a lot of real-life events. In 2019, Promsakon et al. [15] established the solvability of a new class of impulsive fractional boundary value problems involving the generalized Caputo fractional derivative. In 2020, Belmor et al. [16] investigated the solvability of fractional differential inclusion including the generalized Caputo derivative with integral nonlocal conditions. There are other works that showed interest in the generalized Caputo operators; we mention for example [17–20].Nowadays, Herzallah and Radwan [21] studied the fractional version of the system (1) with the classical Caputo operator, namely
(2)D∗0+αηt∈Atηt+Ft,ηt,a.e.t∈0,T,T>0,∑i=1l1ciηθti=β∑j=1l2djηϕτi,ci,dj>0,ti,τj∈0,T,∀i=1,⋯,l1,j=1,⋯,l2.Motivated by the above-cited contributions, in particular systems (1) and (2), we propose a new fractional differential inclusion involving generalized Caputo operator, given by
(3)D∗0+,Qαηt∈Atηt+Ft,ηt,a.e.t∈I=0,T,T>0,∑i=1l1ciηθti=β∑j=1l2djηϕτi,ci,dj>0,ti,τj∈I,∀i=1,⋯,l1,j=1,⋯,l2.where ∗D0+,Qα is the generalized Caputo derivative w.r.t. the function Q such that α∈0,1, At:DA⊆ℝ⟶ℝ is linear bounded operator and F:I×ℝ⟶Pℝ+. We show the existence of solution for the proposed system (3). The proposed system (3) is more flexible since it allows us to choose fractional derivative depending on the particular established phenomenon. Therefore, the tools of generalized fractional differential inclusions facilitate the investigation of optimal controls and stochastic processing, in particular, modeling of control processes that are considered by selecting a trial function [7]. Moreover, nonlocal conditions give more accurate measurements, precise results, and efficient effect than the classical boundary conditions.An outline of this paper is as follows. In Section2, some bases and results are given needed in the sequel. In Section 3, we study the solvability of the generalized system (3). In Section 4, we apply the abstract results in order to establish the existence of solution for some illustrative examples.


## 2. Preliminaries

In this part, we recall some definitions and theorems that will be used later. LetE,.E be a Banach space and PE=Z⊂E:Z≠∅. Now, throughout this paper, let
(4)PbdE=Z∈PE:Zisbounded,PcpE=Z∈PE:Ziscompact,PcvE=Z∈PE:Zisconvex,PcvpE=PcpE∩PcvE.LetW⊆E. The fixed point of set-valued map Ψ:W⟶PE is a point ω∈W such that ω∈Ψω. The graph of Ψ is defined as
(5)GΨ=ω1,ω2∈W×E:ω2∈Ψw1.A selection ofΨ is a single-valued map ψ:W⟶E such that Gψ⊆GΨ.Ψ:E⟶PE is closed (convex) valued if Ψω is closed (convex) for each ω∈E, and Ψ is bounded on bounded sets if ΨW=⋃ω∈WΨω is bounded for each W∈PbdE, that is, supω∈Wsupz:z∈Ψω<∞.Therefore,Ψ is completely continuous if Ψ (W) is relatively compact for each W∈PbdW. In fact, if Ψ is completely continuous with nonempty compact values, then Ψ is upper semicontinuous (u.s.c., for short) if and only if G (Ψ) is closed.LetL1I,E=η:I⟶E:ηE:I⟶R+beLebesgueintegrable; then L1I,E is Banach with the norm ηL1=∫IηtEdt.Definition 1 [22].
A multivalued functionΨ:I×E⟶PEis called L1−Carathéodory if(i)
t⟶Ψt,ω is measurable for each ω ∈ E,(ii)
ω⟶Ψt,ω is u.s.c. for almost all t ∈ I,(iii)
for eachr∈0,∞, there exists pr∈L1I,R+ such that
Ψt,ω=supνE:ν∈Ψt,ω≤prt,∀ωE≤rand almost every t∈I.Definition 2 [23].
LetCI,E be the Banach space of all continuous functions η:I⟶E with the norm η∞=supt∈IηtE. Therefore, let CmI,E be the Banach space of all m−differentiable maps η:I⟶E with ηmt∈CI,E,m∈ℕ.Definition 3 [24].
For everyη∈CI,E, define the family of L1−selection of Ψ:I×E⟶PcvpE as
(6)SΨ,η1=ψ∈L1I,ℝ:ψt∈Ψt,ηtfora.e.t∈I.
Therefore,SΨ,η1 is a nonempty set.Lemma 4 [24].
LetΨ:I×E⟶PcvpE be a L1−Carathéodory-multivalued function and Φ:L1I,E⟶CI,E be a continuous linear mapping. Then
(7)Φ∘SΨ1:CI,E⟶PcvpCI,E,η↦ΦSΨ,η1,is a closed graph operator in CI,E×CI,E.An important role is played by the fixed-point principle to obtain the solvability of various types of operator equations (see, for example, [25–29]). We will apply the following fixed-point theorem to obtain the main results.Theorem 5 [30].
LetZ be a convex closed subset of E, Ω be an open subset of Z, 0∈Ω and Ψ:Ω¯⟶PcvpZ is u.s.c. and compact operator. Then, either
(1)
Ψ has a fixed point in Ω¯, or(2)
there existsω∈∂Ω and λ∈0,1 with ω∈λΨω.Next, we outline some definitions of the generalized fractional operators [12, 13]. For more details about fractional operators, the readers are also referred to [1, 31].Definition 6.
LetQ:a,b⟶ℝ+ be an increasing function having a derivative Q′t∈Ca,b,ℝ such that Q′t≠0 for all t∈a,b. The left generalized Riemann–Liouville fractional integral of order α∈n,n+1 for some n∈ℕ of an integrable function η:a,b⟶ℝ w.r.t. the function Q is given by [12]
(8)Ja+,Qαηt=1Γα∫atQ′ζQt−Qζα−1ηζdζ,t>a.ChoosingQt=lnt and replacing in (8), we have the Hadamard fractional integral, given by [32]
(9)JHa+αηt=1Γα∫atlntζα−1ηζdζζ,a>0,t>a.ChoosingQt=t and replacing in (8), we get the classical Riemann–Liouville integral, given by [1]
(10)Ja+αηt=1Γα∫att−ζα−1ηζdζ,t>a.The left generalized Riemann–Liouville fractional derivative of orderα∈n,n+1 for some n∈ℕ of an integrable function η:a,b⟶ℝ w.r.t. the function Q is given by [12]
(11)Da+,Qαηt=1Γn−α1Q′tddtn∫atQ′ζQt−Qζn−α−1ηζdζt>a.ChoosingQt=lnt and replacing in (11), we have the Hadamard fractional derivative, given by [32]
(12)DHa+αηt=1Γn−αtddtn∫atlntζn−α−1ηζdζ,a>0,t>a.ChoosingQt=t and replacing in (11), we get classical Riemann–Liouville derivative, given by [31]
(13)Da+αηt=1Γn−αddtn∫att−ζn−α−1ηζdζ,t>a.Definition 7.
Letα∈n,n+1 for some n∈ℕ and Q∈Cna,b,ℝ be an increasing mapping such that Q′t≠0 for all t∈a,b. Consider η:a,b⟶ℝ be an integrable function. The left generalized Caputo fractional derivative of order α>0,α∈ℝ+, w.r.t. the function Q is given by [12]
(14)D∗a+,Qαηt=Ja+,Qn−α1Q′tddtnηt,t>a.ChoosingQt=t and replacing in (14), we obtain the classical Caputo fractional derivative. Choosing Qt=lnt and replacing in (14), we have the Caputo-Hadamard fractional derivative, given by [32]
(15)DC−Ha+αηt=1Γn−α∫atlnζtn−α−1tddtnηtdtt=JHa+n−αtddtnηt,a>0,t>a.Further, the generalized Caputo derivative can be defined via the generalized Riemann–Liouville fractional derivative as [13]
(16)D∗a+,Qαηt=Da+,Qαηt−∑k=0n−1ηQkak!Qt−Qak,where ηQkt=1/Q′td/dtkηt.The following lemma, which concerns some properties of generalized fractional operators, plays a key role in the sequel.Lemma 8.
[13]. Suppose that η:I⟶ℝ, then
(1)
ifη∈CI,ℝ, then D∗0+,QαJ0+,Qαηt=ηt,(2)
ifη∈CnI,ℝ, then
(17)J0+,Qα∗D0+,Qαηt=ηt−∑k=0n−1ηQk0k!Qt−Q0k,(3)
ifη∈CnI,ℝ and α∈0,1, then
(18)J0+,QαD∗0+,Qαηt=ηt−η0.


## 3. Main Results

The differential inclusions using fractional derivatives have been proven to be of major interest to the academic community, not only mathematicians but also researchers in other fields. There is a motivating way to obtain the solvability of the differential inclusions; this way is representing the solution by integral equation.The solvability of system (3) will be established under the following hypotheses:(H1) For allηt∈I, there exists LA∈0,∞ such that LA=maxt∈IAt.(H2)∑i=1l1ci≠β∑j=1l2dj.(H3) The functionF:I×ℝ⟶ℝ is L1−Carathéodory and has nonempty, convex, and compact values.(H4) The functionsθ,ϕ:I⟶I are continuous such that θt≤t and ϕt≥t for all t∈I.(H5) There exists a functionp∈L1I,R+ and K>0 such that Ft,ηt≤Kpt, and there exists M∗>0 such that
(19)Γα+1M∗LAM∗QT−Q0α+KMΓα+1σβ∑j=1l2dj+σ∑i=1l1ci+1>1,where
(20)M=1Γα∫0TQ′sQT−Qsα−1psds.The integral representation of the system (3) will be given in the following lemma.Lemma 9.
Let the hypotheses (H1)-(H2) hold. Suppose thatψ:I⟶CI,ℝ, then the solution η(t) of the following problem
(21)D∗0+,Qαηt=Atηt+ψt,a.e.t∈I,∑i=1l1ciηθti=β∑j=1l2djηϕτi,ci,dj>0,ti,τj∈I,∀i=1,⋯,l1,j=1,⋯l2,is given by
(22)ηt=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds,where σ=1/∑i=1l1ci−β∑j=1l2dj.Proof.
Applying the operatorJ0+,Qα on both sides of equation (21). Then, by Lemma 8, we get
(23)J0+,QαD∗0+,Qαηt=J0+,QαAtηt+ψt.Therefore, we obtain(24)ηt=η0+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds.Puttingt=θti in equation (24), we get
(25)ηθti=η0+1Γα∫0θtiQ′sQθti−Qsα−1Asηsds+1Γα∫0θtiQ′sQθti−Qsα−1ψsds.Thus, we have(26)∑i=1l1ciηθti=∑i=1liciη0+∑i=1l1ci1Γα∫0θtiQ′sQθti−Qsα−1Asηsds+∑i=1l1ci1Γα∫0θtiQ′sQθti−Qsα−1ψsds.Puttingt=ϕτj in equation (24), we get
(27)ηϕτj=η0+1Γα∫0ϕτjQ′sQϕτj−Qsα−1Asηsds+1Γα∫0ϕτjQ′sQϕτj−Qsα−1ψsds.Thus, we have(28)∑j=1l2djηϕτj=∑j=1l2djη0+∑j=1l2dj1Γα∫0ϕτjQ′sQϕτj−Qsα−1Asηsds+∑j=1l2dj1Γα∫0ϕτjQ′sQϕτj−Qsα−1ψsds.Hence, we obtain(29)η0=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψsds.Substituting equation (29) into equation (24), we obtain the result.We note that a functionη∈CI,ℝ is called a solution for system (3) if there exists a map ψ∈L1I,ℝ such that ψ∈Ft,ηt a.e. on I and η(t) is given by
(30)ηt=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds, where σ=1/∑i=1l1ci−β∑j=1l2dj.Now, we establish the solvability of problem (3).Theorem 10.
Suppose that the hypotheses (H1)-(H5) are satisfied; then the system (3) has at least one solution.Proof.
By hypothesis (H3) and Lemma4, there exists a single-value map ψ∈SF,η1. Define the multivalued operator ϒ:CI,ℝ⟶PCI,ℝ as
(31)ϒηt=h∈CI,ℝ:ht=σβ∑j=1L2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1L2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1cj∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1cj∫0θtiQ′sQθti−Qsα−1Γαψsds+∫0tQ′sQt−Qsα−1ΓαAsηsds+∫0tQ′sQt−Qsα−1Γαψsds,ψ∈SF,η1.
Then for everyh∈ϒη and η∈CI,ℝ, there exists ψ∈SF,η1 such that
(32)ht=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds.
Now, we can obtain the proof in 4 steps.
Step 1. ϒη is convex for all η∈CI,ℝ. Let h1,h2∈ϒη; then there exists ψ1,ψ2∈SF,η1 such that for every t∈I, we get
(33)h1t=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψ1sds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψ1sds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψ1sds,h2t=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψ2sds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψ2sds+1Γα∫0tQ′sQt−Q−sα−1Asηsds+1Γα∫0tQ′sQt−Q−sα−1ψ2sds.
Let0≤ρ≤1, then
(34)ρh1+1−ρh2t=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαρψ1s+1−ρψ2sds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαρψ1s+1−ρψ2sds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ρψ1s+1−ρψ2sds.
From (H3), we haveF takes convex values. Hence, ρh1+1−ρh2∈ϒη. Thus, SF,η1 is convex.
Step 2.ϒ is completely continuous. First, we will prove that ϒ is bounded. Let r>0. Define Br=η∈CI,ℝ:η∞≤r and let η∈Br. From (H4), we have that
(35)ht≤σβ∑j=1l2dj∫0τjQ′sQτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0τjQ′sQτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0tiQ′sQti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0tiQ′sQti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds,for all t∈I. Therefore, we get
(36)h∞≤σβ∑j=1l2dj∫0τjQ′sQτj−Qsα−1ΓαAsη∞ds+σβK∑j=1l2dj∫0τjQ′sQτj−Qsα−1Γαpsds+σ∑i=1l1ci∫0tiQ′sQti−Qsα−1ΓαAsη∞ds+σK∑i=1l1ci∫0tiQ′sQti−Qsα−1Γαpsds+1Γα∫0tQ′sQt−Qsα−1Asη∞ds+KΓα∫0tQ′sQt−Qsα−1psds≤LArQT−Q0αΓα+1+KMσβ∑j=1l2dj+σ∑i=1l1ci+1.
Hence,ϒ sends bounded sets to bounded sets in CI,ℝ.
Secondly, we will proveϒ sends Br into equicontinuous sets of CI,ℝ. Let t1∗,t2∗∈I such that t1∗<t2∗.
Then, for allh∈ϒη and η∈Br, we have
(37)ht2∗−ht1∗=1Γα∫0t2∗Q′sQt2∗−Qsα−1Asηsds−1Γα∫0t1∗Q′sQt1∗−Qsα−1Asηsds+1Γα∫0t2∗Q′sQt2∗−Qsα−1ψsds−1Γα∫0t1∗Q′sQt1∗−Qsα−1ψsds.
Hence, we get(38)ht2∗−ht1∗≤1Γα∫0t1∗Q′sQt2∗−Qsα−1−Qt1∗−Qsα−1Asη∞+ψ∞ds+1Γα∫t1∗t2∗Q′sQt2∗−Qsα−1Asη∞+ψ∞ds.
Next, by the Lagrange mean-value classical theorem, we obtain(39)ht2∗−ht1∗≤α−1t2∗−t1∗Q′t^Γα∫0t1∗Q′sQt^−Qsα−2LAr+ψ∞ds+1Γα∫t1∗t2∗Q′sQt2∗−Qsα−1LAr+ψ∞ds,where t1∗<t^<t2∗. As t1∗⟶t2∗, ht2∗−ht1∗⟶0. Thus, ϒBris equicontinuous. From the Arzela-Ascoli theorem, we get ϒ is completely continuous.
Step 3. ϒ is u.s.c. We only need to show that ϒ has a closed graph to be u.s.c. Let ηn⟶η^ and hn⟶h^ where hn∈ϒηn.We need to show that h^∈ϒη^.Associated with hn∈ϒηn,there exists ψn∈SF,ηn1 such that for all t∈I, we have
(40)hnt=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηnsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψnsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηnsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψnsds+1Γα∫0tQ′sQt−Qsα−1Asηnsds+1Γα∫0tQ′sQt−Qsα−1ψnsds.
We want to show that there existsψ^∈SF,η^1 such that for each t∈I, we get
(41)h^t=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsη^sds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψ^sds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsη^sds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψ^sds+1Γα∫0tQ′sQt−Qsα−1Asη^sds+1Γα∫0tQ′sQt−Qsα−1ψ^sds.
Define the linear continuous operatorΘ:L1I,ℝ⟶CI,ℝ by
(42)Θψt=σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds.
Hence, we get(43)hnt−h^t≤σβLA∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαηns−η^sds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψns−ψ^sds+σLA∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαηns−η^sds+σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψns−ψ^sds+LAΓα∫0tQ′sQt−Qsα−1ηns−η^sds+1Γα∫0tQ′sQt−Qsα−1ψns−ψ^sds.
Thus,hnt⟶h^t as n⟶∞. From Lemma 4, we can see Θ○SF,η1 is a closed graph in CI,ℝ×CI,ℝ and hn∈ΘSF,ηn1. Since ηn⟶η^, then h^t satisfies equation (41) for some ψ∈SF,η^1. Thus, ϒ is an u.s.c.
Step 4. There exists an open set Ω∈CI,ℝ such that η∈δϒη for some δ∈0,1 and η∈∂Ω.
Letδ∈0,1 and η∈δϒη. Then, there exists ψ∈L1I,ℝ with ψ∈SF,η1 such that for all t∈I, we have
(44)ηt=δσβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0ϕτjQ′sQϕτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0θtiQ′sQθti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds.
As in the proof of Step 2, we get that(45)η∞≤δLAη∞QT−Q0αΓα+1+KMσβ∑j=1l2dj+σ∑i=1l1ci+1≤LAη∞QT−Q0αΓα+1+KMσβ∑j=1l2dj+σ∑i=1l1ci+1.
Hence, we get(46)Γα+1η∞LAη∞QT−Q0α+KMΓα+1σβ∑j=1l2dj+σ∑i=1l1cj+1≤1.
From (H5), there existsM∗ such that M∗≠η∞. Let Ω=η∈CI,ℝ:η∞<M∗+1. Thus, there is no η∈∂Ω such that η∈δϒη for δ∈0,1. Hence, ϒ:Ω¯⟶PcvpBr is u.s.c. From Theorem 5, we deduce that ϒ has a fixed point η∈Ω¯ which is a solution of the system (3).Theorem 11.
Assume that (H1)-(H4) hold. In addition, suppose
(H6) there exists a nondecreasing continuous functionΞ:0,∞⟶0,∞ and q∈CI,ℝ+ such that
(47)Ft,ηt=supρ:ρ∈Wt,η≤qtΞρ,for all t,η∈I×ℝ, and there exists κ>0 such that
(48)Γα+1κLAκ+q∞ΞκQT−Q0ασβ∑j=1l2dj+σ∑1=1l1ci+1≤1.
Then, the system (3) has at least one solution on I.Proof.
Define the multivalued operatorϒ:CI,ℝ⟶PCI,ℝ as in equation (31) of Theorem 10.
Step 1. ϒη is convex for all η∈CI,ℝ. By doing the same steps as in the proof of Theorem 10, we can get that ϒη is convex for all η∈CI,ℝ.
Step 2. ϒ is completely continuous. First, we will prove that ϒ is bounded. Let r0>0. Define Br0=η∈CI,ℝ:η∞≤r0 and let η∈Br0. From (H4), we have that
(49)ht≤σβ∑j=1l2dj∫0τjQ′sQτj−Qsα−1ΓαAsηsds+σβ∑j=1l2dj∫0τjQ′sQτj−Qsα−1Γαψsds−σ∑i=1l1ci∫0tiQ′sQti−Qsα−1ΓαAsηsds−σ∑i=1l1ci∫0tiQ′sQti−Qsα−1Γαψsds+1Γα∫0tQ′sQt−Qsα−1Asηsds+1Γα∫0tQ′sQt−Qsα−1ψsds,for all t∈I. Therefore, we get
(50)h∞≤σβ∑j=1l2dj∫0τjQ′sQτj−Qsα−1ΓαAsη∞ds+σβq∞Ξr0∑j=1l2dj∫0τjQ′sQτj−Qsα−1Γαds+σ∑i=1l1ci∫0tiQ′sQti−Qsα−1ΓαAsη∞ds+σq∞Ξr0∑i=1l1ci∫0tiQ′sQτj−Qsα−1Γαds+1Γα∫0tQ′sQt−Qsα−1Asη∞ds+q∞Ξr0Γα∫0tQ′sQt−Qsα−1ds≤LAr0+q∞Ξr0Γα+1QT−Q0ασβ∑j=1l2dj+σ∑i=1l1ci+1.
Hence,ϒ sends bounded sets to bounded sets in CI,R.
Secondly, we will proveϒ sends Br0 into equicontinuous sets of CI,ℝ. Let t1∗,t2∗∈I such that t1∗<t2∗.
Then, for allh∈ϒη and η∈Br0, we have
(51)ht2∗−ht1∗=1Γα∫0t2∗Q′sQt2∗−Qsα−1Asηsds−1Γα∫0t1∗Q′sQt1∗−Qsα−1Asηsds+1Γα∫0t2∗Q′sQt2∗−Qsα−1ψsds−1Γα∫0t1∗Q′sQt1∗−Qsα−1ψsds.
Hence, we get(52)ht2∗−ht1∗≤LAro+q∞Ξr0Γα∫0t1∗Q′sQt2∗−Qsα−1−Qt1∗−Qsα−1ds+LAro+q∞Ξr0Γα∫t1∗t2∗Q′sQt2∗−Qsα−1ds≤LAro+q∞Ξr0Γα∫0t1∗Q′sQt2∗−Qsα−1−Qt1∗−Qsα−1ds+LAro+q∞Ξr0Γα∫t1∗t2∗Q′sQt2∗−Qsα−1ds≤LAro+q∞Ξr0Γα+1Qt2∗−Q0α−Qt1∗−Q0α.
In view of continuity ofQ, we have ht2∗−ht1∗⟶0 as t1∗⟶t2∗. Thus, ϒ is completely continuous.
Step 3. ϒ is u.s.c. As in the proof of Theorem 10, we have that ϒ is an u.s.c.
Step 4. There exists an open set Ω∈CI,ℝ such that η/δ∈ϒη for some δ∈0,1 and η∈∂Ω.
Letδ∈0,1 and η∈δϒη. Then, there exists ψ∈L1I,ℝ with ψ∈SF,η1 such that for all t∈I, we have η∈δYη satisfies (44). As in the proof of Step 2, we have that
(53)η∞≤δLAη∞+q∞Ξη∞Γα+1QT−Q0ασβ∑j=1l2dj+σ∑i=1l1ci+1≤LAη∞+q∞Ξη∞Γα+1QT−Q0ασβ∑j=1l2dj+σ∑i=1l1ci+1.
Hence, we get(54)Γα+1η∞LAη∞+q∞Ξη∞QT−Q0ασβ∑j=1l2dj+σ∑1=1l1ci+1≤1.
From (H6), there existsκ such that κ≠η∞. Let Λ=η∈CI,ℝ:η∞<κ+1. Thus, there is no η∈∂Λ such that η∈δϒη for δ∈0,1. Hence, ϒ:Λ¯⟶PcvpBr0 is u.s.c. From Theorem 5, we deduce that ϒ has a fixed point η∈Λ¯ which is a solution of the system (3).PuttingQt=t in (3), we have the following result.Corollary 12.
Assume that (H1)-(H4) hold. In addition, suppose
(H7) there exists a nondecreasing continuous functionΞ:0,∞⟶0,∞ and q∈CI,ℝ+ such that
(55)Ft,ηt=supρ:ρ∈Wt,η≤qtΞρ,for all t,η∈I×ℝ, and there exists κ∗>0 such that
(56)Γα+1κ∗LAκ∗+q∞Ξκ∗Tασβ∑j=1l2dj+σ∑i=1l1ci+1≤1.
Then the system (2) has at least one solution on I.ChoosingQt=lnt in system (3) and changing the interval I with the interval Ie=1,e in (3) and in all conditions (H1)-(H4), we obtain the following Caputo–Hadamard fractional system
(57)DC−H1+αηt∈Atηt+Ft,ηt,a.e.t∈Ie=1,e,∑i=1l1ciηθti=β∑j=1l2djηϕτi,ci,dj>0,ti,τj∈Ie,∀i=1,⋯,l1,j=1,⋯,l2.The following result is a direct consequence of Theorem11.Corollary 13.
Assume that (H1)-(H4) hold onIe. In addition, suppose
(H8) there exists a nondecreasing continuous functionΞ:0,∞⟶0,∞ and q∈CIe,ℝ+ such that
(58)Ft,ηt=supρ:ρ∈Wt,η≤qtΞ∣ρ∣,for all t,η∈Ie×ℝ, and there exists κ^>0 such that
(59)Γα+1κ^LAκ^+q∞Ξκ^σβ∑j=1l2dj+σ∑i=1l1ci+1≤1.Then, the system (57) has at least one solution on Ie.


## 4. Applications

In the following examples, we point to how applied the abstract results in particular systems.Example 14.
Consider the following generalized fractional differential inclusion(60)D∗0+,Q25ηt−cosηtt2+100∈0,ηtηt+t2+1+3t2+6t4,a.e.t∈0,1,∑i=124iηti4=94∑j=133−jητj0.25,ti,τj∈0,1,∀i=1,2;j=1,2,3.LetQt=t2+t/2. Here, we get α=2/5,(61)Atηt=cosηtt2+100,Ft,η=0,ηtηt+t2+1+3t2+6t4,θ:0,1⟶0,1,θt=t4,ϕ:0,1⟶0,1,ϕt=t0.25From given information, we obtain thatLA=0.01,T=1,l1=2,l2=3,∑i=124i=5/16,∑j=133−j=7/8,β=2.25. Hence, σ=−0.6037735849. In addition, it has Ft,η≤10. Thus, pt=1,K=10 and M=1.1273957159. Then, there exists a constant M∗ such that M∗∈25.53,∞ satisfying the inequality of (H5).By Theorem10, we know the system (60) has at least one solution.Example 15.
Consider the following generalized fractional differential inclusion(62)D∗0+,Q23ηt−cosηtt2+100∈0,tηtt+1,a.e.t∈0,1,∑i=424iηti4=94∑j=133−jητj0.25,ti,τj∈0,1,∀i=1,2;j=1,2,3.LetQt=t+1. Here, we get α=2/3,
(63)Atηt=cosntt2+100Ft,η=0,tηtt+1,θ:0,1⟶0,1,θt=t4,ϕ:0,1⟶0,1,ϕt=t0.25.Thus,LA=0.01, T=1, l1=2, l2=3, ∑i=124i=5/16, ∑j=133−j=7/8, and β=2.25. Therefore, σ=−0.6037735849. In addition, it has Ft,η≤t/t+1ηt Further, qt=t/t+1,q∞=1, and Ξη∞=η∞. The condition (H6) of Theorem 11 is satisfied with κ∈0.5558151022,∞. Consequently, all the hypotheses of Theorem 11 are satisfied. Thus, the problem (62) has at least one solution.


## 5. Conclusions

In this paper, we established the solvability of fractional differential inclusions involving the generalized Caputo operator by applying Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. The proposed system studied in the present work is more practical and more generalized. The results given in this paper extended and developed some previous works. We presented some examples to illustrate the solvability results.



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