# Existence Results forψ-Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions

Authors: Chanakarn Kiataramkul; Sotiris K. Ntouyas; Jessada Tariboon
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this research work, we study a new class ofψ-Hilfer hybrid fractional integro-differential boundary value problems with nonlocal boundary conditions. Existence results are established for single and multivalued cases, by using suitable fixed-point theorems for the product of two single or multivalued operators. Examples illustrating the main results are also constructed.


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## Body

## 1. Introduction

Some real-world problems in physics, mechanics, and other fields can be described better with the help of fractional differential equations. So, differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous monographs have appeared devoted to fractional differential equations, for example, see [1–8]. Recently, differential equations and inclusions equipped with various boundary conditions have been widely investigated by many researchers, see [9–14] and the references cited therein.Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers [15–18].We will give a brief history on the subject of hybrid differential and fractional differential equations. In 2010, Dhage and Lakshmikantham [19] initiated the study of the first-order hybrid differential equation
(1)ddtxtft,xt=gt,xt,t∈J≔0,T,x0=x0∈ℝ,where f∈CJ×ℝ,ℝ\0 and g∈CJ×ℝ,ℝ. They established the existence, uniqueness results, and some fundamental differential inequalities.In 2011, Zhao et al. [15] discussed the following hybrid fractional initial value problem
(2)Dqxtft,xt=gt,xt,t∈J,x0=x0∈ℝ,where Dq is the Riemann-Liouville fractional derivative of order 0<q<1,f∈CJ×ℝ,ℝ\0, and g∈CJ×ℝ,ℝ.Sun et al. [16] studied the following hybrid fractional boundary value problem
(3)Dqxtft,xt=gt,xt,t∈0,1,x0=x1=0,where Dq is the Riemann-Liouville fractional derivative of order 1<q<2,f∈C0,1×ℝ,ℝ\0, and g∈C0,1×ℝ,ℝ.In [20], the authors studied the existence of solutions for a nonlocal boundary value problem of hybrid fractional integro-differential equations given by
(4)Dαxt−∑i=1nIβihit,xtgt,xt=ft,xt,t∈0,1,x0=mx,x1=A,where Dα is the Caputo fractional derivative of order α with 1<α≤2, Iβi is the Riemann-Liouville fractional integral of order βi>0, hi∈C0,1×ℝ,ℝ for i=1,2,⋯,n, g∈C0,1×ℝ,ℝ\0, f∈C0,1×ℝ,ℝ, a functional m:C0,1,ℝ⟶ℝ, and A∈ℝ. The main result was obtained by means of a hybrid fixed-point theorem for three operators in a Banach algebra due to Dhage [21].The existence of solutions for an initial value problem of hybrid fractional integro-differential equations, given by(5)DαDωxt−∑i=1nIβihit,xtgt,xt=ft,xt,t∈0,1,x0=0,Dωx0=0,was studied in [22]. Here, Dθ is the Caputo fractional derivative of order θ∈α,ω with 0<α,ω≤1; Iβi is the Riemann-Liouville fractional integral of order βi>0, hi∈C0,1×ℝ,ℝ for i=1,2,⋯,n, g∈C0,1×ℝ,ℝ\0, f∈C0,1×ℝ,ℝ. A generalization of Krasnoselskii fixed-point theorem due to Dhage [21] was used in the proof of the existence result.The problem (5) was extended in [23] to boundary value problems of the form
(6)DαDωxt−∑i=1nIβihit,xtgt,xt=ft,xt,t∈0,1,x0=0,Dωx0=0,x1=δxη,0<δ<1,0<η<1,where Dθ is the Caputo fractional derivative of order θ∈α,ω with 0<α≤1, 1<ω≤2; Iβi is the Riemann-Liouville fractional integral of order βi>0, hi∈C0,1×ℝ,ℝ for i=1,2,⋯,n, g∈C0,1×ℝ,ℝ\0, f∈C0,1×ℝ,ℝ. An existence result is proved via Dhage’s [21] fixed-point theorem.For recent results on hybrid boundary value problems of fractional differential equations and inclusions, we refer to [24–26] and references cited therein. In the literature, there do exist several definitions of fractional integrals and derivatives. One of them is the Hilfer fractional derivative, which composites both Riemann-Liouville and Caputo fractional derivatives [27]. Fractional differential equations involving Hilfer derivative have many applications, and we refer to [28] and the references cited therein. There are actual world occurrences with uncharacteristic dynamics such as atmospheric diffusion of pollution, signal transmissions through strong magnetic fields, the effect of the theory of the profitability of stocks in economic markets, the theoretical simulation of dielectric relaxation in glass forming materials, and network traffic. See [29, 30] and references cited therein.In [31], an initial value problem was discussed for hybrid fractional differential equations involving ψ-Hilfer fractional derivative of the form
(7)Dα,β;ψxtft,xt=gt,xt,t∈J,HI1−γ,ψx0f0,x0=x0∈ℝwhere HDα,β;ψ is the ψ-Hilfer fractional derivative with 0<α<1,0≤β≤1,α≤γ=α+β−αβ<1,f∈C1−γ;ψJ×ℝ,ℝ\0, and g∈C1−γ;ψJ×ℝ,ℝ. For some recent results on ψ-Hilfer fractional initial value problems, see [32–37] and references cited therein.In the present work, we study aψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem of the form
(8)HDa+α,ρ;ψxtgt,xt−∑i=1nIa+βi;ψhit,xt=ft,xt,t∈a,b,xa=0,xb=mx,where HDa+α,ρ;ψ is the ψ-Hilfer fractional derivative operator of order α, with 0<α≤2, 0≤ρ≤1; Ia+βi;ψ is ψ-Riemann-Liouville fractional integral of order βi>0, for i=1,2,⋯,n, g∈Ca,b×ℝ,ℝ\0, f∈Ca,b×ℝ,ℝ, m:Ca,b,ℝ⟶ℝ,hi∈Ca,b×ℝ,ℝ with hia,0=0 for i=1,2,⋯,n. An existence result is established via a fixed-point theorem for the product of two operators due to Dhage [21].As a second problem, we investigate the existence of solutions for the following inclusionψ-Hilfer fractional hybrid integro-differential equations with nonlocal boundary conditions of the form
(9)DHα,β;ψxtgt,xt−∑i=1nIa+βi;ψhit,xt∈Ft,xt,t∈a,b,xa=0,xb=mx,where F:a,b×ℝ⟶Pℝ is a multivalued map, Pℝ is the family of all subsets of ℝ, and the other quantities are the same as in boundary value problem (8). Here, the existence result is based on a multivalued fixed-point theorem for the product of two operators due to Dhage [38].The rest of the paper is arranged as follows: in Section2, we recall some notations, definitions, and lemmas from fractional calculus needed in our study. Also, we prove an auxiliary lemma helping us to transform the hybrid boundary value problem (8) into an equivalent integral equation. The main existence result for the ψ-Hilfer hybrid boundary value problem (8) is contained in Section 3. The obtained result is illustrated by a numerical example. Section 4 is devoted in the study of the inclusion case of the hybrid boundary value problem (8) by considering the multivalued hybrid boundary value problem (9). Some special cases are discussed in Section 5.


## 2. Preliminaries

This section is assigned to recall some notation in relation to fractional calculus. We denote byACna,b,ℝ the n-times absolutely continuous functions given by
(10)ACna,b,ℝ=f:a,b⟶ℝ;fn−1∈ACa,b,ℝ.Definition 1 (see [2]).
Leta,b, −∞≤a<b≤∞, be a finite or infinite interval of the half-axis 0,∞ and α>0. Also, let ψ be an increasing and positive monotone function on a,b, having a continuous derivative ψ′x on a,b. The ψ-Riemann-Liouville fractional integral of a function f with respect to another function ψ on a,b is defined by
(11)Ia+α;ψft=1Γα∫atψ′sψt−ψsα−1fsds,t>a>0,where Γ· is the Euler Gamma function.Definition 2 (see [2]).
Letψ′t≠0 and α>0, n∈ℕ. The Riemann-Liouville derivatives of a function f with respect to another function ψ of order α is defined by
(12)Da+α;ψft=1ψ′tddtnIa+n−α;ψft=1Γn−α1ψ′tddtn∫atψ′sψt−ψsn−α−1fsds,where n=α+1, α is represent the integer part of the real number α.Definition 3 (see [32]).
Letn−1<α<n with n∈ℕ, a,b is the interval such that −∞≤a<b≤∞ and f,ψ∈Cna,b,ℝ two functions such that ψ is increasing and ψ′t≠0, for all t∈a,b. The ψ-Hilfer fractional derivative of a function f of order α and type 0≤ρ≤1 is defined by
(13)HDa+α,ρ;ψft=Ia+ρn−α;ψ1ψ′tddtnIa+1−ρn−α;ψft=Ia+γ−α;ψDa+γ;ψft,where n=α+1; α represents the integer part of the real number α with γ=α+ρn−α.Lemma 4 (see [2]).
Letα,β>0, then we have the following semigroup property given by
(14)Ia+α;ψIa+β;ψft=Ia+α+β;ψft,t>a.
Next, we present theψ-fractional integral and derivatives of a power function.Proposition 5 (see [2, 32]).
Letα≥0, υ>0, and t>a, then, ψ-fractional integral and derivative of a power function are given by
(15)Ia+α;ψψs−ψaυ−1t=ΓυΓυ+αψt−ψaυ+α−1,HDa+α,ρ;ψψs−ψaυ−1t=ΓυΓυ−αψt−ψaυ−α−1,n−1<α<n,υ>n.Lemma 6 (see [33]).
Letm−1<α<m, n−1<β<n, n,m∈ℕ, n≤m, 0≤ρ≤1, and α≥β+ρn−β. If f∈Cna,b,ℝ, then
(16)HDa+β,ρ;ψIa+α;ψft=Ia+α−β;ψft.Lemma 7 (see [32]).
Iff∈Cna,b,ℝ, n−1<α<n, 0≤ρ≤1, and γ=α+ρn−α, then
(17)Ia+α;ψHDa+α,ρ;ψft=ft−∑k=1nψt−ψaγ−kΓγ−k+1∇ψn−kIa+1−ρn−α;ψfa,for all t∈a,b, where ∇ψnft≔1/ψ′td/dtnft.Lemma 8.
Let1<α≤2, 0≤ρ≤1, γ=α+ρ2−α, and z∈Ca,b,ℝ. Then, x is a solution of the ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem of the form
(18)Da+α,ρ;ψxtgt,xt−∑i=1nIa+βi;ψhit,xt=zt,t∈a,b,xa=0,xb=mx,if and only if x satisfies the equation
(19)xt=gt,xt∑i=1nIa+βi;ψhit,xt+Ia+α;ψzt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψzb.Proof.
Letx∈Ca,b,ℝ be a solution of the problem (18). Applying the operator Ia+α;ψ to both sides of (18) and using Lemma 7, we obtain
(20)xtgt,xt−∑i=1nIa+βi;ψhit,xt=Ia+α;ψzt+ψt−ψaγ−1Γγc1+ψt−ψaγ−2Γγ−1c2,where c1, c2∈ℝ. By using the first boundary condition, xa=0, we get the constant c2=0. From the second boundary condition, xb=mx, we find that
(21)c1=Γγψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψzb.
Substituting the value ofc1 and c2 in (20), we obtain (19).
Conversely, by a direct computation, it is easy to show that the solutionx given by (19) satisfies the problem (18). The proof of Lemma 8 is completed.☐LetX=CJ,ℝ be the Banach space of continuous real-valued functions defined on a,b, equipped with the norm x=supt∈a,bxt and a multiplication xyt=xtyt,∀t∈a,b. Then, clearly, X is a Banach algebra with above-defined supremum norm and multiplication in it.Lemma 9 (see [21]).
LetS be a nonempty, closed convex, and bounded subset of the Banach algebra X and A:X⟶X,B:S⟶X two operators such that
(a)
A is Lipschitzian with a Lipschitz constant k(b)
B is completely continuous(c)
x=AxBy⟹x∈S for all y∈S(d)
Mk<1, where M=BS=supBx:x∈S
Then, the operator equationx=AxBx has a solution.


## 3. Existence Result for the Problem (8)

In view of Lemma8, we define an operator Q:X⟶X by
(22)Qxt=gt,xt∑i=1nIa+βi;ψhit,xt+Ia+α;ψft,xt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψfb,mx.Notice that the problem (8) has solutions if and only if the operator Q has fixed points.Theorem 10.
Assume that:
(A1) The function g:a,b×ℝ⟶ℝ\0 is continuous and there exists a positive function ϕ, with bound ∥ϕ∥, such that
(23)gt,x−gt,y≤ϕtx−y,for t∈a,b and x, y∈ℝ
(A2) hit,x≤λit,λi∈Ca,b,ℝ,∀t,x∈a,b×ℝ,i=1,2,⋯,n,ft,x≤μt,gt,x≤νt,∀t,x∈a,b×ℝ,μ,ν∈Ca,b,ℝ
(A3) There exists a positive constant K>0 such that
(24)mxgb,mx≤K,∀x∈Ca,b,ℝ,
(A4) There exists a positive real number r>0 such that
(25)ν2∑i=1nψb−ψaβiΓβi+1λi+2ψb−ψaαΓα+1μ+K≤r.
Then, theψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem (8) has at least one solution on a,b, provided that
(26)ϕ2∑i=1nψb−ψaβiΓβi+1λi+2ψb−ψaαΓα+1μ+K<1.Proof.
We consider a subsetS of X defined by S=x∈X:x≤r, where r satisfies the inequality (25). Observe that S is a closed, convex, and bounded subset of the Banach space X. We set supt∈a,bλit=λi,i=1,2,⋯,m,supt∈a,bμt=μ, and supt∈a,bνt=ν.
Next, we define two more operatorsA:X⟶X and B:S⟶X as follows:
(27)Axt=gt,xt,t∈a,b,Bxt=∑i=1nIa+βi;ψhit,xt+Ia+α;ψft,xt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψfb,mx.
Clearly,Qx=AxBx. In the next steps, we show that the operators A and B fulfil all the assumptions of Lemma 9. The proof is divided into three steps.☐Step 1.
We show that the operatorA is Lipschitzian with Lipschitz constant k, i.e., condition a of Lemma 9 is fulfilled. Let x,y∈S. Then we have
(28)Axt−Ayt=gt,xt−gt,yt≤ϕtxt−yt≤ϕx−y.
Consequently,(29)Ax−Ay≤ϕx−y.
Hence, the operatorA is Lipschitzian with Lipschitz constant k=ϕ.Step 2.
We show that the conditionb of Lemma 9 is satisfied, i.e., the operator B is completely continuous on S. First, we will prove its continuity. Let xjt be a sequence of functions in S converging to a function xt∈S. Then, by the Lebesgue dominant theorem, for each t∈a,b, we have
(30)limj⟶∞Bxjt=limj⟶∞∑i=1nIa+βi;ψhit,xjt+Ia+α;ψft,xjt+ψt−ψaγ−1ψb−ψaγ−1mxjgb,mxj−∑i=1nIa+βi;ψhib,mxj−Ia+α;ψfb,mxj=∑i=1nIa+βi;ψlimj⟶∞hit,xjt+Ia+α;ψlimj⟶∞ft,xjt+ψt−ψaγ−1ψb−ψaγ−1limj⟶∞mxjgb,mxj−∑i=1nIa+βi;ψlimj⟶∞hib,mxj−Ia+α;ψlimj⟶∞fb,mxj=∑i=1nIa+βi;ψhit,xt+Ia+α;ψft,xt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψfb,mx=Bxt.
Therefore,B is a continuous operator on S. Next, we show that the operator B is uniformly bounded on S. For any x∈S, we have
(31)Bxt≤∑i=1nIa+βi;ψhit,xt+Ia+α;ψft,xt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx+∑i=1nIa+βi;ψhib,mx+Ia+α;ψfb,mx≤2∑i=1nψb−ψaβiΓβi+1λi+2ψb−ψaαΓα+1μ+K≔M.
Hence,Bx≤M,∀t∈a,b, which shows that the operator B is uniformly bounded on S. Now, we show that the operator B is equicontinuous. Let t1<t2 and x∈S. Then, we have
(32)Bxt2−Bxt1≤∑i=1n1Γβi∫at1ψ′sψt2−ψsβi−1−ψt1−ψsβi−1his,xsds+1Γβi∫t1t2ψ′sψt2−ψsβi−1his,xsds+1Γα∫at1ψ′sψt2−ψsα−1−ψt1−ψsα−1fs,xsds+1Γα∫t1t2ψ′sψt2−ψsα−1fs,xsds+ψt2−ψaγ1−1−ψt1−ψaγ1−1ψb−ψaγ−1mxgb,mx+∑i=1nIa+βi;ψhib,mx+Ia+α;ψfb,mx≤∑i=1nλiΓβi+1ψt2−ψaβi−ψt1−ψaβi+∥μ∥Γα+1ψt2−ψaα−ψt1−ψaα+ψt2−ψaγ1−1−ψt1−ψaγ1−1ψb−ψaγ−1K+∑i=1nψb−ψaβiΓβi+1λi+ψb−ψaαΓα+1μ.
Ast2−t1⟶0, the right-hand side tends to zero, independently of x. Thus, B is equicontinuous. Therefore, it follows by Aezelá-Ascoli theorem that B is a completely continuous operator on S.Step 3.
We show that the third conditionc of Lemma 9 is fulfilled. For any y∈S, we have
(33)xt=AxtByt=AxtByt≤gt,xt∑i=1nIa+βi;ψhit,yt+Ia+α;ψft,yt+ψt−ψaγ−1ψb−ψaγ−1mygb,my+∑i=1nIa+βi;ψhib,my+Ia+α;ψfb,my≤ν2∑i=1nψb−ψaβiΓβi+1λi+2ψb−ψaαΓα+1μ+K≤r,which implies x≤r, and so, x∈S.Moreover, by (26), it holds kM<1 which is fulfilled condition d of Lemma 9. Hence, all the conditions of Lemma 9 are satisfied, and consequently, the operator equation xt=AxtBxt has at least one solution in S. Therefore, there exists a solution of the ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem (8) in a,b. The proof is finished.Now, we present an example ofψ-Hilfer hybrid fractional integro-differential boundary value problem to illustrate our main result.Example 11.
Consider the boundary value problem of the form(34)DH1/23/2,1/4;1−e−2txtgt,xt−I1/23/4;1−e−2th1t,xt−I1/25/4;1−e−2th2t,xt=e−x2t4t+6x4t1+x4t+cos8xt26t+3+116,t∈12,72,x12=0,x72=2sinx32+3cosx74+4tan−1x52,where
(35)gt,x=tx71+x+2t+5,h1t,x=1−e−x4/1+x42t+1,h2t,x=sin4x3+cos2πt.
Here,α=3/2, ρ=1/4, ψt=1−e−2t, a=1/2, b=7/2, β1=3/4, β2=5/4, and mx=2sinx+3cosx+4tan−1x. Then, we can find that
(36)gt,x−gt,y≤t7x−y,and we choose ϕt=t/7. In addition, h1t,x≤1/2t+1≔λ1t, h2t,x≤1/3+cos2πt≔λ2t, ft,x≤1/4t+6+1/26t+3+1/16≔μt, gt,x≤t/7+2t+5≔νt. Then, we have ϕ=1/2, λ1=1/2, λ2=1/3, μ=1/4, ν=25/2 and
(37)mxg7/2,mx≤5+2π12≔K,which lead to
(38)ϕ2∑i=1nψb−ψaβiΓβi+1λi+2ψb−ψaαΓα+1μ+K≈0.8524738760<1.
Therefore, by Theorem10, the ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem (34)-(35) has at least one solution xt on 1/2,7/2, such that ∥x∥≤r, where r is satisfies
(39)ν2∑i=1nψb−ψaβiΓβi+1λi+2ψb−ψaαΓα+1μ+K≈21.31184690≤r.


## 4. Existence Result for the Problem (9)

First of all, we recall some basic concepts for multivalued maps [39–41]. For a normed space X,∥·∥, let Pcp,cvX=Y∈PX:Yiscompactandconvex. For each x∈X, define the set of selections of F by
(40)SF,x≔v∈L1a,b,ℝ:vt∈Ft,xt,fora.e.t∈a,b.Lemma 12 (see [42]).
LetF:a,b×ℝ⟶Pcp,cvℝ be an L1−Carathéodory multivalued map and let Θ be a linear continuous mapping from L1a,b,ℝ to Ca,b,ℝ. Then, the operator
(41)Θ∘SF:Ca,b,ℝ⟶Pcp,cvCa,b,ℝ,x⟼Θ∘SFx=ΘSF,x,is a closed graph operator in Ca,b,ℝ×Ca,b,ℝ.Remark 13.
We recall that a multivalued mapF:a,b×ℝ⟶Pℝ is said to be L1−Carathéodory if it⟼Ft,x is measurable for each x∈ℝ; (ii) x⟼Ft,x is upper semicontinuous for almost all t∈a,b; (iii) for each α>0, there exists φα∈L1a,b,ℝ+ such that Ft,x=supv:v∈Ft,x≤φαt for all x∈ℝ with x≤α and for a.e.t∈a,b.The following multivalued fixed-point theorem for the product of two operators in a Banach algebra, due to Dhage [38, Theorem 4.13], plays a key role in proving the existence result for the nonlocal boundary value problem (9).Lemma 14 (see [38]).
LetX be a Banach algebra and let A:X⟶X be a single valued and B:X⟶Pcp,cvX be a multivalued operator satisfying the conditions:
(a)
A is single-valued Lipschitz with a Lipschitz constant k(b)
B is compact and upper semicontinuous(c)
2Mk<1, where M=BS=supBx:x∈S
Then, either (i) the operator inclusionx∈AxBx has a solution or (ii) the set E=u∈X∣λu∈AuBu,λ>1 is unbounded.Definition 15.
A functionx∈ACa,b,ℝ is a solution of the problem (9) if xa=0,xb=mx, and there exists function v∈L1a,b,ℝ such that vt∈Ft,xta.e. on a,b and
(42)xt=gt,xt∑i=1nIa+βi;ψhit,xt+Ia+α;ψvt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvb.Theorem 16.
Assume that (A1) and (A3) hold. In addition, we suppose that
(B1) F:a,b×ℝ⟶Pcp,cvℝ+ is L1-Carathéodory multivalued map;
(B2) The functions g and hi,i=1,2,⋯,n satisfy condition A2;
(B3) There exists a continuous function q∈Ca,b,ℝ+ such that
(43)∥Ft,x∥P≔supy:y∈Ft,x≤qt,foreacht,x∈a,b×ℝ.
Then, the nonlocalψ-Hilfer hybrid inclusion boundary value problem (9) has at least one solution on a,b, provided that
(44)Ω≔q2qψb−ψaαΓα+1+2∑i=1nψb−ψaβiΓβi+1λi+K<12.Proof.
To transform the boundary value problem (9) into a fixed-point problem, by using Lemma 8, we define a multivalued operator Q1:X⟶PX as
(45)Q1x=h∈X:ht=gt,xt∑i=1nIa+βi;ψhit,xt+Ia+α;ψvtψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvb,v∈SF,x,t∈a,b..
Next, we introduce the operatorA:X⟶X as in (27) and the multivalued operator B1:X⟶PX by
(46)B1x=h∈X:ht=gt,xt∑i=1nIa+a^i;ψhit,xt+Ia+α;ψvtψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvb,v∈SF,x,t∈a,b..
We will show that the operatorsA and B1 satisfy the hypotheses of Lemma 14. The proof is given in a series of steps.
Step 1.B1 is convex valued. Let z1,z2∈B1. Then, there exist v1,v2∈SF,x such that
(47)zjt=∑i=1nIa+βi;ψhit,xt+Ia+α;ψvjt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvjb,j=1,2.
For anyθ∈0,1, we have
(48)θz1t+1−θz2t=∑i=1nIa+βi;ψhit,xt+Ia+α;ψθv1t+1−θv2t+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψθv1b+1−θv2b.
SinceFt,xt is convex, θv1t+1−θv2t∈Ft,xt for all t∈a,b, and so, θz1+1−θz2∈SF,x. Thus, θz1+1−θz2∈Ft,xt, which means that B1 is convex valued on X.
Step 2.A1 is single-valued Lipschitz operator on X. It is proved in Step 1 of Theorem 10.
Step 3. The operatorB1 is completely continuous and upper semicontinuous on X. Let S be a bounded set of X. Then, there exists a constant r such that x≤r, for all x∈S. We prove first that the operator B1 is completely continuous. Let x∈B1S. Then, there exists v∈SF,x such that
(49)ht=∑i=1nIa+βi;ψhit,xt+Ia+α;ψvt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvb,for any x∈S. Then, we have
(50)ht≤2Ia+α;ψvb+mxgb,mx+2∑i=1nIa+βi;ψhib,mx≤2qψb−ψaαΓα+1+2∑i=1nψb−ψaβiΓβi+1λi+K=M1,for all t∈a,b, which implies that x≤M1. Therefore, the operator B1 is uniformly bounded on X.
Next, we show thatB1S is an equicontinuous set in X. Let t1, t2∈a,b with t1<t2 and x∈B1S. Then, we have
(51)B1xt2−B1xt1≤∑i=1n1Γβi∫at1ψ′sψt2−ψsβi−1−ψt1−ψsβi−1his,xsds+1Γβi∫t1t2ψ′sψt2−ψsβi−1his,xsds+1Γα∫at1ψ′sψt2−ψsα−1−ψt1−ψsα−1vsds+1Γα∫t1t2ψ′sψt2−ψsα−1vsds+ψt2−ψaγ−1−ψt1−ψaγ−1ψb−ψaγ−1mxgb,mx+∑i=1nIa+βi;ψhib,mx+Ia+α;ψvb≤∑i=1nλiΓβi+1ψt2−ψaβi−ψt1−ψaβi+qΓα+1ψt2−ψaα−ψt1−ψaα+ψt2−ψaγ−1−ψt1−ψaγ−1ψb−ψaγ−1∑i=1nψb−ψaβiΓβi+1λi+qψb−ψaαΓα+1+K.
Ast2−t1⟶0, the right-hand side of the above inequality tends to zero independently of x, and thus, the operator B1 is equicontinuous. In consequence, the operator B1 is completely continuous by the Arzelá-Ascoli theorem.
Next, we show thatB1 is an upper semicontinuous multivalued mapping on X. It is known, by [[39], Proposition 1.2], that B1 will be upper semicontinuous if we establish that it has a closed graph, since already shown to be completely continuous. Thus, we will prove that B1 has a closed graph.
Letxn be a sequence in X such that xn⟶x∗. Let yn be a sequence such that yn∈B1xn and yn⟶y∗. We shall show that y∗∈B1x∗. Since yn∈B1xn, there exists a vn∈SF,xn such that
(52)xnt=∑i=1nIa+βi;ψhit,xt+Ia+α;ψvnt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvnb,t∈a,b.
We must prove that there is av∗∈SF,x∗ such that
(53)x∗t=∑i=1nIa+βi;ψhit,xt+Ia+α;ψv∗t+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψv∗b,t∈a,b.
Consider the continuous linear operatorL:L1a,b,ℝ⟶Ca,b,ℝ defined by
(54)Lvt=∑i=1nIa+βi;ψhit,xt+Ia+α;ψvt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvb,t∈a,b.
Observe that(55)xnt−x∗t=2Ia+α;ψvns−v∗s⟶0,as n⟶∞. From Lemma 12, it follows that L∘SF,x is a closed graph operator. Further, we have ynt∈LSF,x. Since yn⟶y∗, therefore, we have
(56)x∗t=∑i=1nIa+βi;ψhit,xt+Ia+α;ψv∗t+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψv∗b,t∈a,b.
As a result, we have thatB1 is a compact and upper semicontinuous operator on X.
Step 4. We show that the condition (c) of Lemma14 holds, that is, kM<1/2. This is obvious by (44).
Step 5. Finally, we show that the conclusion (ii) of Lemma14 does not hold.
Letx be any solution of the boundary value problem (9) such that λx∈AxB1x for some λ>1. Then, there is a v∈SF,x such that
(57)xt=1λgt,xt∑i=1nIa+βi;ψhit,xt+Ia+α;ψvt+ψt−ψaγ−1ψb−ψaγ−1mxgb,mx−∑i=1nIa+βi;ψhib,mx−Ia+α;ψvb,t∈a,b.
Then, we have(58)xt≤gt,xt∑i=1nIa+βi;ψhit,xt+Ia+α;ψvt+mxgb,mx+∑i=1nIa+βi;ψhib,mx+Ia+α;ψvb≤ν2qψb−ψaαΓα+1+2∑i=1nψb−ψaβiΓβi+1λi+K.
Taking the supremum fort∈a,b of the above inequality, we obtain a constant M>0 such that
(59)x≤ν2∥q∥ψb−ψaαΓα+1+2∑i=1nψb−ψaβiΓβi+1λi+K≔M,which means that the set E=x∈X:λx∈AxB1x,λ>1 is bounded.
As a result, the conclusion (ii) of Lemma14 does not hold. Hence, the conclusion (i) holds, and consequently, the boundary value problem (9) has at least one solution on a,b. This completes the proof.☐Example 17.
Consider the boundary value problem of the form(60)DH1/47/4,3/5;loget2+1xtgt,xt−I1/41/4;loget2+1h1t,xt−I1/47/4;loget2+1h2t,xt−I1/411/4;loget2+1h3t,xt∈0,=e−x2t4t+15x121+x12+14t+32,t∈14,114,x14=0,x114=5e−x234+2x25/4x25/4+1+7sin2x74+3cos6x94.where
(61)gt,x=8t4t+1x2+2x1+x+4t+7,h1t,x=x8t+1x+1,h2t,x=sin2x12t+2,h3t,x=1−e−∣x∣16t+3.
Here,α=7/4, ρ=3/5, ψt=loget2+1, a=1/4, b=11/4, β1=1/4, β2=7/4, β3=11/4, mx=5e−x2+2x2/x2+1+7sin2x+3cos6x. Now, we find that h1t,x≤1/8t+1≔λ1t, h2t,x≤1/12t+2≔λ2t, and h3t,x≤1/16t+3≔λ3t which yield λ1=1/3, λ2=1/5, λ3=1/7. In addition, we have
(62)mxg11/4,mx≤1718≔K,Ft,xP≤14t+15+14t+32≔qt,which implies q=1/8. Hence,
(63)Ω≈0.4726347379<12.
Therefore, by applying Theorem16, the ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem (60)-(61) has at least one solution on 1/4,11/4.


## 5. Special Cases

The problem (8) considered in the present work is general in the sense that it includes the following classes of new boundary value problems of ψ-Hilfer fractional differential equations.
(I)
Letgt,x=1 and hit,x=0,i=1,2,⋯,n for all t∈a,b and x∈ℝ. Then, the problem (8) reduces to the following ψ-Hilfer fractional boundary value problem
(64)DHa+α,ρ;ψxt=ft,xt,t∈a,b,xa=0,xb=mx.In the case whenμx=∑i=1mλixθi, where a<θ1<θ2<⋯<θn<b, the corresponding ψ-Hilfer fractional boundary value problem was studied in [34] for k=0.
(II)
Lethit,x=0,i=1,2,⋯,n for all t∈a,b and x∈ℝ. Then, the problem (8) reduces to the following ψ-Hilfer fractional boundary value problem
(65)DHa+α,ρ;ψxtgt,xt=ft,xt,t∈a,b,xa=0,xb=mx.(III)
Letgt,x=1 for all t∈a,b and x∈ℝ. Then, the problem (8) reduces to the following ψ-Hilfer fractional boundary value problem
(66)DHa+α,ρ;ψxt−∑i=1nIa+βi;ψhit,xt=ft,xt,t∈a,b,xa=0,xb=mx.Therefore, the main result of this paper also includes the existence results for the solutions of the above-mentionedψ-Hilfer boundary value problems of fractional differential equations as special cases.


## 6. Conclusions

In this paper, we studied a new class ofψ-Hilfer hybrid fractional integro-differential boundary value problems with nonlocal boundary conditions. After proving a basic lemma, helping us to transform the considered system into a fixed-point problem, an existence result is proved via a fixed-point theorem for the product of two operators due to Dhage [21]. The multivalued analogue is also studied, and an existence theorem was established with the help of a multivalued fixed-point theorem for the product of two operators due to Dhage [38]. Numerical examples illustrating the obtained results are also presented. Some special cases are also discussed. The obtained results are new and enrich the existing literature on hybrid fractional differential equations and inclusions.



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