# BVP for Hadamard Sequential Fractional Hybrid Differential Inclusions

Authors: Ning Li; Haibo Gu; Yiru Chen
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The study is concerned with the Hadamard sequential fractional hybrid differential inclusions with two-point hybrid integral boundary conditions. With the help of the Dhage fixed-point theorem for the product of two operators and the Covitz-Nadler fixed-point theorem in the case of fractional inclusions, we obtain the existence results of solutions for Hadamard sequential fractional hybrid differential inclusions. Finally, two examples are presented to illustrate the main results.


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## Body

## 1. Introduction

Nowadays, with the increasing demand of researchers for the study of natural phenomena, the use of fractional differential operators and fractional differential equations become an effective means to achieve this goal. Compared with integer order operators, fractional operators, which can simulate natural phenomena better, are a class of operators developed in recent years. This kind of operator has been expanded and widely used in modeling real-world phenomena such as biomathematics, electrical circuits, medicine, disease transmission, and control [1–6]. Also, some studies in the biological models with fractional-order derivative have been conducted in recent years [7–9]. In the past year, fractional differential operators and fractional differential equations have been used in modeling the spread of some viruses, such as Zika virus and mumps virus [10, 11]. All of these have enabled researchers to discover the structure of fractional boundary value problems (BVP) and the hereditary nature of their solutions from various aspects. In this regard, many researchers investigated advanced fractional-order modelings and related theoretical results and qualitative behaviors of such fractional-order boundary value problems, see [12–20] and the references therein.There have been appeared different versions of fractional operators during these years. Much of the work on fractional differential equations only involves either Riemann-Liouville derivative or Caputo derivative [21–29]. Guo et al. ([30, 31]) discussed the existence and Hyers–Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1<β<2 and the existence and Hyers–Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal condition. Ma et al. [32] investigated the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space.However, there is another concept of fractional derivative in the literature which was introduced by Hadamard in 1892 [33]. This derivative is known as Hadamard fractional derivative and differs from aforementioned derivatives in the sense that the kernel of the integral in its definition contains logarithmic function of arbitrary exponent. Many researchers have studied and obtained some results on the existence of solutions of Hadamard fractional differential equations in recent years. Yang ([34, 35]) studied the extremal solutions for a coupled system of nonlinear Hadamard fractional differential equations with Cauchy initial value conditions and the existence and nonexistence of positive solutions for the eigenvalue problems of nonlinear Hadamard fractional differential equations with p-Laplacian operator. Tomar et al. [36] established certain generalized Hermite-Hadamard inequalities for generalized convex functions via local fractional integral.In 1993, Miller and Ross also defined another type of fractional derivative called sequential derivative, which is a combination of the existing derivative operators. From then on, the attention of some researchers was attracted to finding a connection between the Hadamard fractional derivative and the sequential fractional derivative [37–40]. In [41], by using the topological degree theory and Leray–Schauder fixed-point theory, Rezapour and Etemad studied the existence of solutions for the following Caputo–Hadamard fractional boundary value problem via mixed multiorder integroderivative conditions:(1)DCH1+α+λDCH1+ςut=ft,ut,t∈1,M,u1=0,μ1DCH1+γ1uM+DCH1+γ2uη=δ1,μ2IH1+q1uM+IH1+q2uη=δ2,where λ,μ1μ2,∈0,1,γ1,γ2∈0,ς−α with 2<α<ς<3,q1,q2>0,δ1,δ2∈R. The symbol DCH1+ξ points out the Caputo–Hadamard fractional derivative of order ξ∈α,ς,γ1,γ2, with the notation IH1+q standing for the Hadamard fractional integrals of order q∈q1,q2. The function f formulated by f:1,M×R⟶R is assumed to be continuous on 1,M×R with respect to its both components.As a generalization of fractional boundary value problems, hybrid differential problems with different kinds of boundary conditions have received a lot of attention in recent years [42–44]. The research in this field started from Dhage and Lakshmikantham in 2010 [45]. There is a new concept of differential equation in the literature which was introduced by Dhage and Lakshmikantham. They described this novel differential equation as a hybrid differential equation and investigated the extremal solutions of this new BVP by using some useful fundamental differential inequalities [45]. So far, there are few studies about the existence and various properties of solutions for hybrid boundary value problems of fractional order. In [46], by using a fixed-point theorem due to Dhage, the authors developed some existence theorem for Hadamard-type fractional hybrid differential inclusions problem:(2)DHαxtft,xt∈Ft,xt,t∈1,e,α∈1,2,x1=xe=0,where DHα is the Hadamard fractional derivative, f∈C1,e×R,R∖0, F: 1,e×R⟶PR is a multivalued map, and PR is the family of all nonempty subsets of R. In [47], by using a hybrid fixed-point theorem of Schaefer type for a sum of three operators due to Dhage, the authors investigated the existence of solutions for the nonlocal fractional BVP of hybrid inclusion problem given by(3)DC0+ϱws−∑j=1mIR0+βjhjs,wsfs,ws∈Fs,ws,s∈J≔0,1,w0=μw,w1=α,where μ:CJ,R⟶R, α∈R,ϱ∈1,2, DC0+ϱ is the Caputo derivative, and IR0+βj is the Riemann-Liouville integral of order φ>0, such that φ∈β1,β2,…,βm. In [48], by using the well-known Dhage fixed-point theorems for single-valued and set-valued maps, Baleanu and Etemad studied a new fractional hybrid model of thermostat in which the thermostat controls an amount of heat based on the temperature detected by sensors. This hybrid differential inclusions of Caputo type are illustrated by(4)−DC0+ϱxsφs,ws∈Φs,xs,ϱ∈1,2,s∈0,1,Dxsφs,xs|s=0=0,λDC0+ϱ−1xsφs,xs|s=1+xsφs,xs|s=η=0,where λ>0,η∈0,1,ϱ−1∈0,1,D=d/ds,DC0+α is the Caputo derivative of fractional order α∈ϱ,ϱ−1, the function Φ:0,1×R⟶PR is a multivalued map, and φ∈C0,1×R,R∖0. In [49], the authors investigated the following fractional three-point hybrid problem:(5)DC0+ϱwsgs,ws=Gs,ws,s∈0,1,w0=0,wsgs,ws|s=0+IR0+ϱ∗wsgs,ws|s=η=0,wsgs,ws|s=0+IR0+ϱ∗wsgs,ws|s=1=0,where ϱ∈2,3ϱ∗>0,η∈0,1. The function G: 0,1×R⟶R is continuous, and g∈C0,1×R,R∖0. In [50], by using various novel analytical techniques based on α−ψ−contractive mappings, endpoints, and the fixed points of the product operators, the authors investigated a new category of the sequential hybrid inclusion problem with three-point integroderivative boundary conditions:(6)p1DC0+ϱ+p2DC0+ϱ−1wsζs,ws,IR0+γws∈Ss,ws,s∈0,1,wsζs,ws,IR0+γws|s=0=0,DC0+1wsζs,ws,IR0+γws|s=0+DC0+2wsζs,ws,IR0+γws|s=0=0,wsζs,ws,IR0+γws|s=1+IR0+ξwsζs,ws,IR0+γws|s=p=0,where ϱ∈2,3,p∈0,1,p1,p2,γ,ξ>0,DC0+⋅ and IR0+⋅ denote the Caputo-fractional derivative and the Riemann-Liouville fractional integral, respectively. Note that DC0+1=d/ds and DC0+2=d2/ds2. The nonzero continuous real-valued function ζ is supposed to be defined on 0,1×R×R. S:0,1×R⟶PR is a set-valued map equipped via some properties.Motivated by these problems, in this study, we will study the following Hadamard sequential fractional hybrid differential inclusion with two-point hybrid Hadamard integroboundary conditions:(7)DHα+λDHα−1xtρt,xt,IHpxt∈Gt,xt,IHpxt,t∈1,e,α1xξρξ,xξ,IHpxξ=α2IHrxeρe,xe,IHpxe,β1xeρe,xe,IHpxe=β2IHrxξρξ,xξ,IHpxξ,where α∈1,2,ξ∈1,e,p∈0,1,λ,r>0,αi,βi∈R,i=1,2,DH⋅ and IH⋅ denote the Hadamard fractional derivative and the Hadamard fractional integral, respectively. The nonzero continuous real-valued function ρ is supposed to be defined 1,e×R×R, and G: 1,e×R×R⟶PR is a set-valued map equipped via some properties.The Hadamard sequential fractional hybrid differential inclusion BVP (7) is modeled with respect to the generalized operators with kernels, including logarithmic functions. In other words, the presented formulation for the given Hadamard sequential fractional hybrid differential inclusion BVP (7) involves two different derivatives in the format of the Hadamard. The supposed abstract fractional hybrid differential inclusion problem (7) with given hybrid integral boundary conditions can describe some mathematical models of real and physical processes in which some parameters are often adjusted to suitable situations. The value of these parameters can change the effects of fractional derivatives and integrals. Moreover, we express that such a Hadamard sequential fractional hybrid differential inclusion BVP is new and enriches the literature on boundary value problems for nonlinear Hadamard fractional differential inclusions. In this way, with the help of Dhage fixed-point theorem and Covitz-Nadler fixed-point theorem in the case of multivalued mapping, we try to find the existence criteria of solutions for the proposed problem (7).The rest of this study is organized as follows. In Section2, some preliminary facts that we need in the sequel are given. In Section 3, the existence results of solution for system (7) are discussed. In Section 4, two examples are given to prove validity of the results we obtained.


## 2. Preliminaries

Definition 1 (see [2]).
The Hadamard derivative of fractional-orderα for a function f:1,∞⟶R is defined as(8)DHαft=1Γn−αtddtn∫1tlogtsn−α−1fssds,n−1<α<n,n=α+1,provided the right side is pointwise defined on 1,∞, where Γ⋅ is the gamma function and log⋅=loge⋅.Definition 2 (see [2]).
The Hadamard fractional integral of orderβ for a function g is defined as(9)IHβgt=1Γβ∫1tlogtsβ−1gssds,β>0,provided the integral exists.Definition 3 (see [15]).
Letw:1,+∞⟶R is a sufficiently smooth function; then, the sequential fractional derivative is defined by(10)DHϱws=DHϱ1DHϱ2…DHϱnws,where ϱ=ϱ1,ϱ2,…,ϱn is a multiindex.Lemma 1.
For anyh∈C1,e,R. A function x∈AC1,e,R is a solution of the Hadamard sequential fractional hybrid differential equations:(11)DHα+λDHα−1xtρt,xt,IHpxt=ht,t∈1,e,α∈1,2,supplemented with the boundary conditions in (7) if and only if it satisfies the following integral equation:(12)xt=ρt,xt,IHpxtt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1hτdτds−α1ξ−λ∫1ξsλ−1IHα−1hsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1hτdτds−β1e−λ∫1esλ−1IHα−1hsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1hτdτds−β1e−λ∫1esλ−1IHα−1hsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1hτdτds−α1ξ−λ∫1ξsλ−1IHα−1hsds+t−λ∫1tsλ−1IHα−1hsds,where(13)Δ=A1B2−A2B1≠0,A1=α1ξ−λ−α2Γr∫1elogesr−1s−λ−1ds,(14)A2=α1ξ−λ∫1ξsλ−1logsα−2ds−α2Γr∫1elogesr−1s−λ−1∫1sτλ−1logτα−2dτds,B1=β1e−λ−β2Γr∫1ξlogξsr−1s−λ−1ds,B2=β1e−λ∫1esλ−1logsα−2ds−β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1logτα−2dτds.Proof.
As argued in [2], the solution of Hadamard differential equation in (11) can be written as(15)xt=ρt,xt,IHpxtc0t−λ+c1t−λ∫1tsλ−1logsα−2ds+t−λ∫1tsλ−1IHα−1hsds,where ci,i=0,1 are the unknown arbitrary constants. Making use of the integral boundary conditions given by (7) in (15), we obtain(16)A1A2B1B2c0c1=J1J2,where Ai and Bii=1,2 are, respectively, given by (13) and (14), and(17)J1=α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1hτdτds−α1ξ−λ∫1ξsλ−1IHα−1hsds,(18)J2=β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1hτdτds−β1e−λ∫1esλ−1IHα−1hsds.
Solving (16) for c0 and c1 and using notation (13), we find that(19)c0=1ΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1hτdτds−α1ξ−λ∫1ξsλ−1IHα−1hsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1hτdτds−β1e−λ∫1esλ−1IHα−1hsds,(20)c1=1ΔA1β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1hτdτds−β1e−λ∫1esλ−1IHα−1hsds−B1α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1hτdτds−α1ξ−λ∫1ξsλ−1IHα−1hsds.
Substituting the values ofc0 and c1 in (15), we get the desired solution (12). This completes the proof.
For a normed spaceX,⋅, let PbX=Y∈PX:Yis bounded, PcpX=Y∈PX:Yis compact, PclX=Y∈PX:Yis closed, and Pcp,cvX=Y∈PX:Yis compact and convex.Definition 4 (see [51]).
A multivalued mapG:X⟶PX is convex (closed) valued if Gx is convex (closed) for all x∈X.Definition 5 (see [51]).
The multivalued mapG is bounded on bounded sets if GB=∪x∈BGx is bounded in X for all B∈PbXi.e.,supx∈Bsupy:y∈Gx<∞.Definition 6 (see [51]).
A multivalued mapG is called upper semicontinuous (u.s.c.) on X if for each x0∈X, the set Gx0 is a nonempty closed subset of X, and if for each open set N of X containing Gx0, there exists an open neighborhood N0 of x0, such that GN0⊆N.Definition 7 (see [51]).
A multivalued mapG is said to be completely continuous if GB is relatively compact for every B∈PbX.Definition 8 (see [51]).
A multivalued mapG has a fixed point if there is x∈X, such that x∈Gx. The fixed point set of the multivalued operator G will be denoted by FixG.Definition 9 (see [51]).
A multivalued mapG:0,1⟶PclR is said to be measurable if for every y∈R, the function(21)t⟼dy,Gt=infy−z:z∈Gt,is measurable.Lemma 2 (see [51]).
If the multivalued mapG is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, that is, xn⟶x∗,yn⟶y∗, and yn∈Gxn imply y∗∈Gx∗.LetC1,e,R denote a Banach space of continuous functions from 1,e into R with the norm x=supt∈1,ext. Let Lp1,e,R be the Banach space of measurable functions x:1,e⟶R which are p-th Lebesgue integrable and normed by xLp=∫1extpdt1/p.Definition 10 (see [51]).
A collection of selections of multivalued mapG at point x∈C1,e is defined by(22)SG,x=vs∈L11,e:vs∈Gt,xt,IHpxtfor a.e.t∈1,e.Definition 11 (see [51]).
A multivalued mapG:1,e×R2⟶PR is said to be Caratheodory if(i)
t⟼Gt,x,y is measurable for each x,y∈R(ii)
x,y⟼Gt,x,y is upper semicontinuous for almost all t∈1,eDefinition 12 (see [37]).
A functionx∈AC1,e,R is called a solution of problem (7) if there exists a function v∈L11,e,R with vt∈Gt,xt,IHpxt, a.e. on 1,e, such that(23)DHα+λDHα−1xtρt,xt,IHpxt=vt,a.e.t∈1,e,α1xξρξ,xξ,IHpxξ=α2IHrxeρe,xe,IHpxe,β1xeρe,xe,IHpxe=β2IHrxξρξ,xξ,IHpxξ.Lemma 3 (see [52]).
LetX be a Banach space. Let G:1,e×R2⟶Pcp,cvX be an L1-Caratheodory multivalued map, and let Θ be a linear continuous mapping from L11,e,X to C1,e,X. Then, the operator(24)Θ°SG:C1,e,X⟶Pcp,cvC1,e,X,x⟼θ°SGx=ΘSG,x,is a closed graph operator in C1,e,X×C[1,e,X.Lemma 4 (see [53]).
LetX be a Banach algebra and A:X⟶X be a single-valued and B:X⟶Pcp,cvX be a multivalued operator satisfying the following:(i)
A is single-valued Lipschitz with a Lipschitz constant k(ii)
B is compact and upper semicontinuous operator(iii)
2MK<1, where M=BX
Then, either(i)
The operator inclusionx∈AxBx has a solution or(ii)
The setE=u∈X:μu∈AuBu,μ>1 is unbounded.Definition 13 (see [51]).
A multivalued mapG:X⟶PX is said to be a contraction mapping if there is a constant 0<λ<1, such that(25)HdGx,Gy≤λx−yX,for every x,y∈X, where Hd is the Hausdorff metric.Lemma 5 (see [54]).
LetX,d be a complete metric space. If N:X⟶PclX is a contraction, then FixN≠∅.


## 3. Main Results

In this section, we will study the existence results of solutions for problem (7). First of all, we fix our terminology.LetX=C1,e,R denote the space equipped with the norm x=supt∈1,ext. Observe that X,⋅ is a Banach space, and X,⋅ with multiplication given by x⋅x′s=xsx′s is a Banach algebra.Now, we enlist the assumptions that we need in the sequel.H1 The function ρ:1,e×R×R⟶R∖0 is continuous, and there exists a bounded function Ψ, with bound Ψ, such that Ψt>0,a.e.t∈1,e, and(26)ρt,x1,y1−ρt,x2,y2≤Ψtx1−x2+y1−y2,a.e.t∈1,e,∀x1,y1,x2,y2∈R.H2G:1,e×R×R⟶ℛ is Caratheodory and has nonempty compact and convex valuesH3 There exists a constant p1∈0,p and a function g∈L1/p11,e,R+, such that(27)Gt,x,y=supv:v∈Gt,x,y≤gt,For allx,y∈R and for a.e. t∈[1,e.H4 There exists a positive real number ℛ, such that(28)ℛ>ρ0M1gL1/p11−M0M1gL1/p1,whereM0M1gL1/p1<1/2,ρ0=supt∈1,eρt,0,0.H5 There exists a continuous nondecreasing, subhomogeneous function Φ:R+⟶R+ (that is, Φμx≤μΦx for all μ≥1 and x∈R+) and a function ς∈L1/p11,e,R+, such that(29)Gt,x,yP≔supy:y∈Gt,x,y≤ςtΦx+y,For eacht,x,y∈1,e×R2.H6 There exists a constant r>0, such that(30)r>ρ0M11+1/Γp+1ςL1/p1Φr1−M0M11+1/Γp+1ςL1/p1Φr,where(31)M0M11+1Γp+1ςL1/p1Φr<12,ρ0=supt∈1,eρt,0,0.H7G:1,e×R2⟶PcpR satisfying the condition G⋅,x,y:1,e⟶PcpR is measurable for each x,y∈R2H8HdGt,x,y,Gt,x¯,y¯≤ζtx−x¯+y−y¯ for a.e. t∈1,e and for all x,y,x¯,y¯∈R with ζ∈L1/p11,e,R+ and d0,Gt,0,0≤ζt for a.e. t∈1,e.H9 The function ρ:1,e×R×R⟶R∖0 is continuous, and there exists a function η∈C1,e,R+, such that(32)ρt,x,y≤ηt,∀t,x,y∈1,e×R2.In assumptionH8, Hd is the Hausdorff metric, where d is the Euclidean metric in R defined by dx,y=x−y for x,y∈R.Furthermore, we set the notations:(33)M1=1Δα−1α2α−1B2+B1Γα−1Γr+11+a2−p11−p1+1+a1−p1+β2α−1A2+A1logξ1+a1−p1+r+1Γα−1Γr+11+a2−p11−p1+1+a1−p1+β1α−1A2+A1Γα−11+a2−p11−p1+1+a1−p1+α1α−1B2+B1logξ1+a1−p1+1Γα−11+a2−p11−p1+1+a1−p1+Δα−1Γα−11+a2−p11−p1+1+a1−p1,M2=1Δα2B2Γα−1Γr+11+a2−p11−p1+1+a1−p1+β2A2logξ1+a1−p1+r+1Γα−1Γr+11+a2−p11−p1+1+a1−p1+α1B2logξ1+a1−p1+1Γα−11+a2−p11−p1+1+a1−p1+β1A2Γα−11+a2−p11−p1+1+a1−p1,M3=1Δα−1α2B1Γα−1Γr+11+a2−p11−p1+1+a1−p1+β2A1logξ1+a1−p1+r+1Γα−1Γr+11+a2−p11−p1+1+a1−p1+α1B1logξ1+a1−p1+1Γα−11+a2−p11−p1+1+a1−p1+β1A1Γα−11+a2−p11−p1+1+a1−p1,Ψ=supt∈1,eΨt,η=supt∈1,eηt,a=α−21−p1,ζL1/p1=∫1eζt1/p1dtp1.Theorem 1.
Let the hypothesesH1–H4 be satisfied. Then, inclusion problem (7) has at least one mild solution on C1,e,R.Proof.
Consider the operatorN:X⟶PX defined by(34)Nxt=w∈C1,e,R:wt=ρt,xt,IHpxtt−λΔB2α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds,v∈SG,x,and we define two operators A:X⟶X by(35)Axt=ρt,xt,IHpxt,t∈1,e,and ℬ:X⟶PX by(36)ℬxt=w∈C1,e,R:wt=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds,v∈SG,x.
Observe thatNx=Axℬx. We will show that the operators A and ℬ satisfy all the conditions of Lemma 4. For the sake of convenience, we split the proof into several steps.Step 1.
A is a Lipschitz on X, that is, (i) of Lemma 4 holds.
Letx,y∈X. By H1, we have(37)Axt−Ayt=ρt,xt,IHpxt−ρt,yt,IHpyt≤Ψtxt−yt+IHpxt−IHpyt≤Ψtxt−yt+1Γp∫1tlogtsp−11sxs−ysds≤Ψx−y+x−yΓp∫1tlogt−logsp−1dlogs≤Ψx−y+x−yΓp⋅logtpp≤Ψx−y+1Γ1+px−y≤Ψ1+1Γ1+px−y.
Therefore,(38)Ax−Ay=supt∈1,eAxt−Ayt≤Ψ1+1Γ1+px−y,for all x,y∈X. So, A is a Lipschitz on X with Lipschitz constant M0=Ψ1+1/Γ1+p.Step 2.
The multivalued operatorℬ is compact and upper semicontinuous on X, that is, (ii) of Lemma 4 holds.
First, we show thatℬ has convex values. Let w1,w2∈ℬx, and then, there are v1,v2∈SG,x, such that(39)wi=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1viτdτds−α1ξ−λ∫1ξsλ−1IHα−1visds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1viτdτds−β1e−λ∫1esλ−1IHα−1visds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1viτdτds−β1e−λ∫1esλ−1IHα−1visds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1viτdτds−α1ξ−λ∫1ξsλ−1IHα−1visds+t−λ∫1tsλ−1IHα−1visds,i=1,2,t∈1,e. For any θ∈0,1, we have(40)θw1t+1−θw2t=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1θv1τ+1−θv2τdτds−α1ξ−λ∫1ξsλ−1IHα−1θv1s+1−θv2sds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1θv1τ+1−θv2τdτds−β1e−λ∫1esλ−1IHα−1θv1s+1−θv2sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1θv1τ+1−θv2τdτds−β1e−λ∫1esλ−1IHα−1θv1s+1−θv2sds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1θv1τ+1−θv2τdτds−α1ξ−λ∫1ξsλ−1IHα−1θv1s+(1−θv2sds+t−λ∫1tsλ−1IHα−1θv1s+1−θv2sds,where v¯t=θv1t+1−θv2t∈Gt,xt,IHqxt for all t∈1,e. Hence, θu1t+1−θu2t∈ℬx, and consequently, ℬx is convex for each x∈X. As a result, ℬ defines a multivalued operator ℬ:X⟶PcvX.
Next, we show thatℬ maps bounded sets into bounded sets in X. To see this, let Q be a bounded set in X, and then, there exists a real number r>0, such that x≤r,∀x∈Q. Now, for each h∈ℬx, there exist v∈SG,x, such that(41)ht=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds.
Then, for eacht∈1,e, using H2, we have(42)ht≤t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds+α1ξ−λ∫1ξsλ−1×IHα−1vsds+A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1vτdτds+β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2ds×A1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds+β1e−λ∫1e∫1esλ−1IHα−1vsds+B1α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1vτdτds+α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds≤t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1gτdτds+α1ξ−λ∫1ξsλ−1×IHα−1gsds+A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1gτdτds+β1e−λ∫1esλ−1IHα−1gsds+1Δt−λ∫1tsλ−1logsα−2ds×A1β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1gτdτds+β1e−λ∫1esλ−1IHα−1gsds+B1α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1gτdτds+α1ξ−λ∫1ξsλ−1IHα−1gsds+t−λ∫1tsλ−1IHα−1gsds≤gLp1/1ΔB2α2Γα−1Γr∫1elogesr−1s−1∫1sτ−1∫1τlogτσα−2/1−p1σ−1/1−p1dσ1−p1dτds+α1Γα−1∫1ξs−1∫1slogsτα−2/1−p1τ−1/1−p1dτ1−p1ds+A2β2Γα−1Γr∫1ξlogξsr−1s−1∫1sτ−1∫1τlogτσα−2/1−p1σ−1/1−p1dσ1−p1dτds+β1Γα−1∫1es−1∫1slogsτα−2/1−p1τ−1/1−p1dτ1−p1ds+gL1/p1Δ∫1ts−1logsα−2dsA1β2Γα−1Γr∫1ξlogξsr−1s−1×∫1sτ−1∫1τlogτσα−2/1−p1σ−1/1−p1dσ1−p1dτds+β1Γα−1∫1es−1∫1slogsτα−2/1−p1τ−1/1−p1dτ1−p1ds+B1α2Γα−1Γr∫1elogesr−1s−1∫1sτ−1∫1τlogτσα−2/1−p1σ−1/1−p1dσ1−p1dτds+α1Γα−1∫1ξs−1∫1slogsτα−2/1−p1τ−1/1−p1dτ1−p1ds+gL1/p1Γα−1∫1ts−1∫1slogsτα−2/1−p1τ−1/1−p1dτ1−p1ds≤gL1/p1ΔB2α2Γα−1Γr∫1elogesr−1s−1∫1sτ−1∫1τlogτσα−2/1−p11σdσ1−p1dτds+α1Γα−1∫1eξs−1∫1slogsτα−2/1−p11τdτ1−p1ds+A2β2Γα−1Γr∫1ξlogξsr−1s−1×∫1sτ−1∫1e∫1τlogτσα−2/1−p11σdσ1−p1dτds+β1Γα−1∫1es−1∫1e∫1slogsτα−2/1−p11τdτ1−p1ds+gL1/p1∣Δ∣∫1ts−1logsα−2dsA1β2Γα−1Γr∫1ξlogξsr−1s−1×∫1sτ−1∫1τlogτσα−2/1−p11σdσ1−p1dτds+β1Γα−1∫1es−1∫1slogsτα−2/1−p11τdτ1−p1ds+B1α2Γα−1Γr∫1elogesr−1s−1∫1sτ−1∫1τlogτσα−2/1−p11σdσ1−p1dτds+α1Γα−1∫1ξs−1∫1slogsτα−2/1−p11τdτ1−p1ds+gL1/p1Γα−1∫1ts−1∫1slogsτα−2/1−p11τdτ1−p1ds≤gL1/p1ΔB2α2Γα−1Γr1+a2−p11−p1+1+a1−p1∫1elogesr−1s−1logs1+a1−p1+1ds+α1logξ1+a1−p1+1Γα−11+a2−p11−p1+1+a1−p1+A2β2Γα−1Γr1+a2−p11−p1+1+a1−p1×∫1ξlogξsr−1s−1logs1+a1−p1+1ds+β1Γα−11+a2−p11−p1+1+a1−p1+gL1/p1Δα−1A1β2Γα−1Γr1+a2−p11−p1+1+a1−p1×∫1ξlogξsr−1s−1logs1+a1−p1+1ds+β1Γα−11+a2−p11−p1+1+a1−p1+B1α2Γα−1Γr1+a2−p11−p1+1+a1−p1×∫1elogesr−1s−1logs1+a1−p1+1ds+α1logξ1+a1−p1+1Γα−11+a2−p11−p1+1+a1−p1+gL1/p1Γα−11+a2−p11−p1+1+a1−p1≤M1gL1/p1,hence,(43)h≤M1gL1/p1.
Therefore,ℬQ is uniformly bounded.
Next, we show thatℬ maps bounded sets into equicontinuous sets. For this purpose, we assume that Q be, as above, a bounded set and h∈ℬx for some x∈Q, and then, there exists a v∈SG,x, such that(44)ht=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds.
Thus, for anyt1,t2∈1,e, t2>t1, we have(45)ht2−ht1≤M2gL1/p1Δt2−λ−t1−λ+M3gL1/p1Δt2−λ∫1t2sλ−1logsα−2ds−t1−λ∫1t1sλ−1logsα−2ds+gL1/p11+a1−p1Γα−1t2−λ∫1t2sλ−1logs1+a1−p1ds−t1−λ∫1t1sλ−1logs1+a1−p1ds≤M2gL1/p1Δt2−λ−t1−λ+M3gL1/p1Δt2−λ−t1−λ⋅∫1t2logsα−2s1−λds+t1−λ∫t1t2logsα−2s1−λds|+gL1/p11+a1−p1Γα−1t2−λ−t1−λ⋅∫1t2logs1+a1−p1s1−λds+t1−λ∫t1t2logs1+a1−p1s1−λds|⟶0,independent of x∈Q as t1−t2⟶0.
Therefore,ℬQ is an equicontinuous set in X. Now, an application of the Arzela-Ascoli theorem yields that ℬQ is relatively compact.
In our next step, we show thatℬ is upper semicontinuous. By Lemma 2, ℬ will be upper semicontinuous if we prove that it has a closed graph. Let xn⟶x∗,hn∈ℬxn, and hn⟶h∗. Then, we need to show that h∗∈ℬx∗. Associated with hn∈ℬxn, there exists vn∈SG,xn, such that for each t∈1,e,(46)hnt=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vnτdτds−α1ξ−λ∫1ξsλ−1IHα−1vnsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vnτdτds−β1e−λ∫1esλ−1IHα−1vnsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vnτdτds−β1e−λ∫1esλ−1IHα−1vnsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vnτdτds−α1ξ−λ∫1ξsλ−1IHα−1vnsds+t−λ∫1tsλ−1IHα−1vnsds.
Thus, it suffices to show that there existsv∗∈SG,x∗, such that for each t∈1,e,(47)h∗t=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v∗τdτds−α1ξ−λ∫1ξsλ−1IHα−1v∗sds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v∗τdτds−β1e−λ∫1esλ−1IHα−1v∗sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v∗τdτds−β1e−λ∫1esλ−1IHα−1v∗sds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v∗τdτds−α1ξ−λ∫1ξsλ−1IHα−1v∗sds+t−λ∫1tsλ−1IHα−1v∗sds.
Let us consider the linear operatorΘ:L11,e,R⟶C1,e,R given by(48)vt⟼Θvt=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds.
Notice that the operatorΘ is continuous. Indeed, for vn,v∗∈L11,e,R with vn⟶v∗ in L11,e,R, we obtain(49)‖Θvnt−Θv∗t‖≤1Δ‖B2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vnτ−v∗τdτds−α1ξ−λ∫1ξsλ−1IHα−1vns−v∗sds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vnτ−v∗τdτds−β1e−λ∫1esλ−1IHα−1vns−v∗sds‖+1Δα−1‖A1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vnτ−v∗τdτds−β1e−λ∫1esλ−1IHα−1vns−v∗sds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vnτ−v∗τdτds−α1ξ−λ∫1ξsλ−1IHα−1vns−v∗sds‖+∫1ts−1IHα−1vns−v∗sds,∀t∈1,e,which implies that Θvn⟶Θv∗ in C1,e,R.
Thus, it follows by Lemma3 that Θ°SG is a closed graph operator. Furthermore, we have hnt∈ΘSG,xn. Since xn⟶x∗, therefore, we have(50)h∗t=t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v∗τdτds−α1ξ−λ∫1ξsλ−1IHα−1v∗sds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v∗τdτds−β1e−λ∫1esλ−1IHα−1v∗sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v∗τdτds−β1e−λ∫1esλ−1IHα−1v∗sds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v∗τdτds−α1ξ−λ∫1ξsλ−1IHα−1v∗sds+t−λ∫1tsλ−1IHα−1v∗sds,for some v∗∈SG,x∗.
As a result, we have that the operatorℬ is compact and upper semicontinuous.Step 3.
Now, we show that2MK<1, that is, (iii) of Lemma 4 holds.
This is obvious byH4 since we have(51)M=BX=supℬx:x∈X≤M1gL1/p1,and K=M0.
Thus, all the conditions of Lemma4 are satisfied, and a direct application of it yields that either conclusion (i) or conclusion (ii) holds. We show that conclusion (ii) is not possible.
Supposed the conclusion (ii) is true. Letu∈E be arbitrary. Then, we have, for λ>1,λu∈Auℬu, and then, there exists v∈SG,x such that(52)ut=λ−1ρt,xt,IHpxtt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1pvτdτds−α1ξ−λ∫1ξsλ−1IHα−1pvsds−A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1pvτdτds−β1e−λ∫1esλ−1IHα−1pvsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1pvτdτds−β1e−λ∫1esλ−1IHα−1pvsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1pvτdτds−α1ξ−λ∫1ξsλ−1IHα−1pvsds+t−λ∫1tsλ−1IHα−1pvsds,and so, for all t∈1,e, we have(53)ut≤λ−1ρt,ut,IHput×t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds+α1ξ−λ∫1ξsλ−1IHα−1vsds+A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds+β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds+β1e−λ∫1esλ−1IHα−1vsds+B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds+α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds≤ρt,ut,IHput−ρt,0,0+ρt,0,0×t−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds+α1ξ−λ∫1ξsλ−1IHα−1vsds+A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds+β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1vτdτds+β1e−λ∫1esλ−1IHα−1vsds+B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds+α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds≤M1gL1/p1Ψ1+1Γ1+pu+ρ0.
Therefore,(54)u=supt∈1,eut≤M1gL1/p1Ψ1+1Γ1+pu+ρ0,where we have put ρ0=supt∈1,eρt,0,0. Then, with u=ℛ, we have(55)ℛ≤ρ0M1gL1/p11−M0M1gL1/p1.
This is contradictory. Thus, conclusion (ii) of Lemma4 does not hold by (28). Therefore, the operator equation Axℬx and consequent problem (7) have a solution on 1,e. This completes the proof.Theorem 2.
Suppose that the conditionsH1,H2,H5, and H6 hold. Then, inclusion problem (7) has at least one mild solution on C1,e,R.Proof.
The proof is similar to that of Theorem1 and is omitted.Theorem 3.
Suppose that the conditionsH1,H3, and H7–H9 hold. If(56)λ0≔M11+1Γ1+pgL1/p1Ψ+ηζL1/p1<1,where M1,Ψ,η,ζL1/p1 are given by (33); then, inclusion problem (7) has at least one mild solution on C1,e,R.Proof.
Observe that the setSG,x is nonempty for each x∈C1,e by assumption H7, and thus, G has a measurable selection. We now show that the operator N:C1,e⟶PC1,e satisfies the assumptions of Lemma 5. To establish that Nx∈PclC1,e, for each x∈C1,e, let wnn≥1⊂Nx be such that wn⟶w as n⟶∞ in C1,e. Then, w∈C1,e, and there exists vn∈SG,x, such that for each t∈1,e, we have(57)wnt=ρt,xt,IHpxtt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vnτdτds−α1ξ−λ∫1ξsλ−1IHα−1vnsds−A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1vnτdτds−β1e−λ∫1esλ−1IHα−1vnsds+1Δt−λ∫1tsλ−1logsα−2ds×A1β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1vnτdτds−β1e−λ∫1esλ−1IHα−1vnsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vnτdτds−α1ξ−λ∫1ξsλ−1IHα−1vnsds+t−λ∫1tsλ−1IHα−1vnsds,with vnt∈Gt,xt,IHqxt,t∈1,e.
SinceG has compact values, therefore, we can pass onto a subsequence (denoted in a same way) to obtain that vn converges to v in L11,e. Thus, v∈SG,x, and for each t∈1,e, we have wnt⟶wt, where(58)wt=ρt,xt,IHpxtt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds−A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds+1Δt−λ∫1tsλ−1logsα−2ds×A1β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1vτdτds−β1e−λ∫1esλ−1IHα−1vsds−B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1vτdτds−α1ξ−λ∫1ξsλ−1IHα−1vsds+t−λ∫1tsλ−1IHα−1vsds.
Hence,w∈Nx.
Next, we show thatN is a contraction, that is,(59)Hd1Nx,Nx¯≤λ0x−x¯,∀x,x¯∈X,where λ0 is defined in (56), and d1 is the metric induced by the norm ⋅ in C1,e.
For this, letx,x¯∈1,e and w1∈Nx. Then, there exists v1∈SG,x, such that for all t∈1,e, we obtain(60)w1t=ρt,xt,IHpxtt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v1τdτds−α1ξ−λ∫1ξsλ−1IHα−1v1sds−A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1v1τdτds−β1e−λ∫1esλ−1IHα−1v1sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v1τdτds−β1e−λ∫1esλ−1IHα−1v1sds−B1α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1v1τdτds−α1ξ−λ∫1ξsλ−1IHα−1v1sds+t−λ∫1tsλ−1IHα−1v1sds.
ByH7, we have(61)Hd1Gt,xt,IHpxt,Gt,x¯t,IHpx¯t≤ζtxt−x¯t+IHpxt−IHpx¯t,for a.e. t∈1,e, and then, there exists ψ∈Gt,x¯t,DHqx¯t, such that(62)v1t−ψ≤ζtxt−x¯t+IHpxt−IHpx¯t,a.e.t∈1,e.
We defineU˜:1,e⟶PR by U˜t=ψ∈R:v1t−ψ≤ζtxt−x¯t+IHpxt−IHpx¯t. As the multivalued operator Vt=U˜t∩Gt,x¯t,IHqx¯t is measurable (proposition III.4, [55]), there exists a function v2t which is a measurable selection for Vt. Hence, v2t∈Gt,x¯t,IHqx¯t for a.e. t∈1,e and(63)v1t−v2t≤ζtxt−x¯t+IHpxt−IHpx¯t,a.e.t∈1,e.
Let us define the functionw2t,t∈1,e by(64)w2t=ρt,x¯t,IHpx¯tt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v2τdτds−α1ξ−λ∫1ξsλ−1IHα−1v2sds−A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1v2τdτds−β1e−λ∫1esλ−1IHα−1v2sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v2τdτds−β1e−λ∫1esλ−1IHα−1v2sds−B1α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1v2τdτds−α1ξ−λ∫1ξsλ−1IHα−1v2sds+t−λ∫1tsλ−1IHα−1v2sds.
Then, we conclude that(65)w1t−w2t≤ρt,xt,IHpxt−ρt,x¯t,IHpx¯tt−λΔB2α2Γr∫1elogesr−1s−λ−1×∫1sτλ−1IHα−1v1τdτds+α1ξ−λ∫1ξsλ−1IHα−1v1sds+A2β2Γr∫1ξlogξsr−1s−λ−1∫1sτλ−1IHα−1v1τdτds+β1e−λ∫1esλ−1IHα−1v1sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v1τdτds+β1e−λ∫1esλ−1IHα−1v1sds+B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v1τdτds+α1ξ−λ∫1ξsλ−1IHα−1v1sds+t−λ∫1tsλ−1IHα−1v1sds+ρt,x¯t,IHpx¯tt−λΔB2α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v1τ−v2τdτds+α1ξ−λ∫1ξsλ−1IHα−1v1s−v2sds+A2β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v1τ−v2τdτds+β1e−λ∫1esλ−1IHα−1v1s−v2sds+1Δt−λ∫1tsλ−1logsα−2dsA1β2Γr∫1ξlogξsr−1s−λ−1×∫1sτλ−1IHα−1v1τ−v2τdτds+β1e−λ∫1esλ−1IHα−1v1s−v2sds+B1α2Γr∫1elogesr−1s−λ−1∫1sτλ−1IHα−1v1τ−v2τdτds+α1ξ−λ∫1ξsλ−1IHα−1v1s−v2sds+t−λ∫1tsλ−1IHα−1v1s−v2sds≤M1gL1/p1ρt,xt,IHpxt−ρt,x¯t,IHpx¯t+M1ζL1/p11+1Γ1+px−x¯×ρt,x¯t,IHpx¯t≤M11+1Γ1+pgL1/p1Ψ+ηζL1/p1x−x¯,∀t∈1,e,which yield(66)w1−w2=supt∈1,ew1t−w2t≤M11+1Γ1+pgL1/p1Ψ+ηζL1/p1x−x¯=λ0x−x¯.
By interchanging the roles ofx and x¯, we obtain a similar relation, and thus, we get(67)Hd1Nx,Nx¯≤λ0x−x¯.
In view of the conditionλ0<1 (given by (56)), it follows that N is a contraction, and therefore, by Lemma 5, N has a fixed point x, which is a solution of problem (7). This completes the proof.


## 4. Examples

(a) Consider the following equation:(68)DH3/2+DH1/2xte1−t/460tan−1xt+IH1/2xt+π/4+2∈Gt,xt,IH1/2xt,t∈1,e,3xe1/2e1−e1/2/460tan−1xe1/2+IH1/2xe1/2+π/4+2=IH1/2xee1−e/460tan−1xe+IH1/2xe+π/4+2,3xee1−e/460tan−1xe+IH1/2xe+π/4+2=IH1/2xe1/2e1−e1/2/460tan−1xe1/2+IH1/2xe1/2+π/4+2,where G:1,e×R×R⟶PR is a multivalued map given by(69)t⟶Gt,xt,IH1/2xt=x3+IH1/2x320x3+IH1/2x3+4,sinx+IH1/2x9sinx+IH1/2x+1+89.By conditionH1, Ψt=e1−t/460 with Ψ=1/460. For g˜∈G, we have(70)g˜≤maxx3+IH1/2x320x3+IH1/2x3+4,sinx+IH1/2x9sinx+IH1/2x+1+89≤1,∀x∈R,Gt,x,y=supy:y∈Gt,x,y≤1=gt,∀x,y∈R.Letp1=1/4; then, gt∈L41,e,R+. Using the given data, we find that A1≤1.8196,A2≤4.2428,B1≤1.1041,B2≤6,Δ≤15.6018,M0≤0.0048,M1≤84.6585,ρ0=1/460+2. Furthermore,(71)M0M1gL4≤0.4485<12,and ℛ>160e−11/41/460+2≥ρ0M1gL4/1−M0M1gL4. Hence, all the conditions of Theorem 1 are satisfied, and accordingly, problem (68) has a solution on 1,e.(b) Let us consider the following inclusion problem:(72)DH3/2+DH1/2xt3/800et−1+20sinxt+IH1/2xt/1+IH1/2xt+1/10∈Gt,xt,IH1/2xt,t∈1,e,3xe1/23/800ee1/2−1+20sinxe1/2+IH1/2xe1/2/1+IH1/2xe1/2+1/10=IH1/2xe3/800ee−1+20sinxe+IH1/2xe/1+IH1/2xe+1/10,3xe3/800ee−1+20sinxe+IH1/2xe/1+IH1/2xe+1/10=IH1/2xe1/23/800ee−1+20sinxe1/2+IH1/2xe1/2/1+IH1/2xe1/2+1/10.In order to demonstrate the application of Theorem3, we consider(73)t⟶Gt,xt,IH1/2xt=0,15124txt128+xt+tan−1IH1/2xt1+tan−1IH1/2xt+1300+t.Clearly,(74)Hd1Gt,x,IH1/2x,Gt,x¯,IH1/2x¯≤3512t4x−x¯,Gt,x,y=supv:v∈Gt,x,y≤1=gt,∀x,y∈R.Lettingζt=3/512t4, it is easy to check that d0,Gt,0,0≤ζt holds for almost t∈1,e and that ζt∈L41,e,R+p1=1/4, ζL4=3/512. From the following inequalities, we get ηt=6/800et−1+20+1/10 and η=88/820:(75)ρt,x,y≤6800et−1+20+110,t,x,y∈1,e×R2.In addition, by conditionH1, we obtain Ψt=3/800et−1+20 with Ψ=3/820. Furthermore, using the given data, we find that A1≤1.8196,A2≤4.2428,B1≤1.1040,B2≤6,Δ≤15.6018,M1≤84.6585. Furthermore,(76)M11+1Γ1+pgL4Ψ+ηζL4≤0.8682<1.Thus, all the conditions of Theorem3 are satisfied. Hence, it follows by the conclusion of Theorem 3 that there exists a solution for problem (72) on 1,e.


## 5. Conclusion

Nowadays, we need to study more natural phenomena to gain more abilities for modeling. Therefore, fractional calculus came into being, and today, their importance has become more and more apparent to researchers. In this way, it is necessary to design different and complicated modelings by utilizing the fractional differential problems. This is useful in making modern software which helps us to allow for more cost-free testing and less material consumption. In this work, we have developed the existence theory for a class of Hadamard sequential fractional hybrid differential inclusions equipped with two-point hybrid Hadamard integral boundary value conditions. The nonlinearities in the given problems implicitly depend on the unknown function together with its Hadamard fractional integral of orderp∈0,1. We apply fixed-point theorem due to Dhage and Covitz-Nadler fixed-point theorem to establish the desired results. Eventually, we give two numerical examples to support the applicability of our findings.The work accomplished in this study is new and enriches the literature on boundary value problems for nonlinear Hadamard fractional differential inclusions. For future works, one can extend the given fractional boundary value problem to more general structures, such as finitely point multistrip integral boundary value conditions given by newly introduced generalized fractional operators with nonsingular kernels.



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