# Existence Results for a Nonlocal Coupled System of Sequential Fractional Differential Equations Involvingψ-Hilfer Fractional Derivatives

Authors: Athasit Wongcharoen; Sotiris K. Ntouyas; Phollakrit Wongsantisuk; Jessada Tariboon
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, we discuss the existence and uniqueness of solutions for a new class of coupled system of sequential fractional differential equations involvingψ-Hilfer fractional derivatives, supplemented with multipoint boundary conditions. We make use of Banach’s fixed point theorem to obtain the uniqueness result and the Leray-Schauder alternative to obtain the existence result. Examples illustrating the main results are also constructed.


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## Body

## 1. Introduction

Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders. One of the most important advantages of fractional order models in comparison with integer order ones is that fractional integrals and derivatives are a powerful tool for the description of memory and hereditary properties of some materials. For details and applications, we refer the reader to the texts [1–6]. There are some different definitions of fractional derivatives, from the most popular of Riemann-Liouville and Caputo type fractional derivatives, to the other ones such as Hadamard fractional derivative and the Erdeyl-Kober fractional derivative. A generalization of both Riemann-Liouville and Caputo derivatives was given by Hilfer in [7], which is known as the Hilfer fractional derivativeDα,βxt of order α and a type β∈0,1. Some properties and applications of the Hilfer derivative can be found in [8, 9] and references cited therein.Initial value problems involving Hilfer fractional derivatives were studied by several authors (see, for example, [10–12]). Nonlocal boundary value problems for Hilfer fractional differential equation have been discussed in [13, 14]. Coupled systems for Hilfer fractional differential equations with nonlocal integral boundary conditions were studied in [15].The fractional derivative with another function, in the Hilfer sense, calledψ-Hilfer fractional derivative, has been introduced in [16], which unifies several different fractional operators. For some recent results on existence and uniqueness of initial value problems and results on Ulam-Hyers-Rassias stability, see [17–19] and references therein. Recently, in [20], the authors extended the results in [13] to ψ-Hilfer nonlocal implicit fractional boundary value problems. For recent results in ψ-Hilfer fractional derivative, we refer to [21–23] and references cited therein.In [24], the authors initiated the study of existence and uniqueness of solutions for a new class of boundary value problems of sequential ψ-Hilfer-type fractional differential equations with multipoint boundary conditions of the form(1)HDα,β;ψ+kHDα−1,β;ψxt=ft,xt,t∈a,b,xa=0,xb=∑i=1mλixθi,where HDα,β;ψ is the ψ-Hilfer fractional derivative of order α, 1<α≤2 and parameter β, 0≤β≤1, f:a,b×ℝ⟶ℝ is a continuous function, 0≤a<b,k,λi∈ℝ,i=1,2,⋯,m and a<θ1<θ2<⋯<θm<b. Existence and uniqueness results were proved by using classical fixed point theorems. The Banach’s fixed point theorem was used to obtain the uniqueness result, while nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem are applied to obtain the existence results for the problem (1).In this paper, we investigate the existence and uniqueness criteria for the solutions of the following nonlocal coupled system of sequentialψ-Hilfer fractional derivative of the form(2)HDα,β;ψ+kHDα−1,β;ψxt=ft,xt,yt,t∈a,b,HDp,q;ψ+vHDp−1,q;ψyt=gt,xt,yt,t∈a,b,xa=0,xb=∑i=1m−2λiyθi,ya=0,yb=∑j=1n−2μjxζj,where HDα,β;ψ, HDp,q;ψ are the ψ-Hilfer fractional derivatives of orders α and p, 1<α,p≤2, and two parameters β, q, 0≤β,q≤1, given constants k,ν,λi,μj∈ℝ, a≥0, and the points a<θ1<θ2<⋯<θm−2<b,a<ζ1<ζ2<⋯<ζn−2<b and f,g:a,b×ℝ×ℝ⟶ℝ are continuous functions.In order to study the problem (2), we convert it into an equivalent fixed point problem and then we use Banach’s fixed point theorem to prove the uniqueness of its solutions, while by applying the Leray-Schauder alternative [25], we obtain the existence result.The remaining part of the article is structured as follows: Section3 contains the main results for the problem (2). Examples illustrating the existence and uniqueness results are also included. We recall the related background material in Section 2, in which also we establish a lemma regarding a linear variant of the problem (2).


## 2. Preliminaries

Here, some notations and definitions of fractional calculus are reminded [1].Definition 1.
The Riemann-Liouville fractional integral of orderς>0 for a continuous function is defined by(3)Iςut=1Γς∫att−sς−1usds,provided the right-hand side exists on a,∞.Definition 2.
The Riemann-Liouville fractional derivative of orderς>0 of a continuous function is defined by(4)RLDςut=DnIn−ςut=1Γn−ςddtn∫att−sn−ς−1usds,n−1<ς<n,where n=ς+1 denotes the integer part of real number ς and D=d/dt, provided the right-hand side is point-wise defined on a,∞.Definition 3.
The Caputo fractional derivative of orderς>0 of a continuous function is defined by(5)CDςut=In−ςDnut=1Γn−ς∫att−sn−ς−1ddsnusds,n−1<ς<n,where the right-hand side is point-wise defined on a,∞.Definition 4 (Hilfer fractional derivative [7, 8]).
The Hilfer fractional derivative of orderα and parameter β of a function (also known as the generalized Riemann-Liouville and Caputo fractional derivatives) is defined by(6)HDα,βut=Iβn−αDnI1−βn−αut,where n−1<α<n, 0≤β≤1, and t>a.Remark 5.
Whenβ=0, the Hilfer fractional derivative corresponds to the Riemann-Liouville fractional derivative(7)HDα,0ut=DnIn−αut,while when β=1, the Hilfer fractional derivative corresponds to the Caputo fractional derivative(8)HDα,1ut=In−αDnut.
Letψ∈C1a,b,ℝ be an increasing function with ψ′t≠0 for all t∈a,b.Definition 6 ([1]).
Letα>0 and g∈L1a,b,ℝ. The ψ-Riemann-Liouville fractional integral of order α to a function g with respect to ψ is defined by(9)Iα;ψgt=1Γα∫atψ′sψt−ψsα−1gsds.Definition 7 ([16]).
Letn−1<α<n, n∈ℕ, and g,ψ∈Cna,b,ℝ such that ψ is increasing with ψ′t≠0 for all t∈a,b. The ψ-Hilfer fractional derivative HDα,β;ψ· of order α to a function g and type 0≤β≤1 is defined by(10)HDα,β;ψgt=Iβn−α;ψ1ψ′tddtnI1−βn−α;ψgt.Remark 8 ([1]).
Ifβ=0, then we have ψ-Riemann-Liouville fractional derivative as(11)HDα,0;ψgt:=RLDα;ψgt=1ψ′tddtnIn−α;ψgt,and if β=1, we obtain ψ-Caputo fractional derivative by(12)HDα,1;ψgt:=CDα;ψgt=In−α;ψ1ψ′tddtngt.Lemma 9 ([16]).
Letα,χ>0 and δ>0 be constants and ψ∈C1a,b,ℝ be an increasing function with ψ′t≠0 for all t∈a,b. Then, we have(13)Iα;ψIχ;ψht=Iα+χ;ψht,Iα;ψψt−ψaδ−1=ΓδΓα+δψt−ψaα+δ−1.The following lemma contains the compositional property of Riemann-Liouville fractional integral operator with theψ-Hilfer fractional derivative operator.Lemma 10 ([16]).
Letf∈La,b, n−1<α≤n, n∈ℕ,0≤β≤1,γ∗=α+nβ−αβ, and In−α1−βf∈ACka,b. Then,(14)Iα;ψHDα,β;ψft=ft−∑k=1nψt−ψaγ∗−kΓγ∗−k+1fψn−kI1−βn−α;ψfa,
wherefψn−k=1/ψ′td/dtn−k.The following lemma deals with a linear variant of the system (2).Lemma 11.
Letγ=α+2β−αβ, δ=p+2q−pq, and h,z∈Ca,b,ℝ be given functions. Then, the unique solution of ψ-Hilfer the fractional differential linear system(15)HDα,β;ψ+kHDα−1,β;ψxt=ht,t∈a,b,HDp,q;ψ+vHDp−1,q;ψyt=zt,t∈a,b,xa=0,xb=∑i=1m−2λiyθi,ya=0,yb=∑j=1n−2μjxζj,is given by(16)xt=Iα;ψht−kI1;ψxt+ψt−ψaγ−1ΛΓγΔ−ν∑i=1m−2λiI1;ψyθi+∑i=1m−2λiIp;ψzθi+kI1;ψxb−Iα;ψhb+B−k∑j=1n−2μjI1;ψxζj+∑j=1n−2μjIα;ψhζj+νI1;ψyb−Ip;ψzb,(17)yt=Ip;ψzt−νI1;ψyt+ψt−ψaδ−1ΛΓγA−k∑j=1n−2μjI1;ψxζj+∑j=1n−2μjIα;ψhζj+νI1;ψyb−Ip;ψzb+Ω−ν∑i=1m−2λiI1;ψyθi+∑i=1m−2λiIp;ψzθi+kI1;ψxb−Iα;ψhb,where(18)A=ψb−ψaγ−1Γγ,B=∑i=1m−2λiψθi−ψaδ−1Γδ,Ω=∑j=1n−2μjψζj−ψaγ−1Γγ,Δ=ψb−ψaδ−1Γδ,and it is assumed that(19)Λ≔AΔ−BΩ≠0.Proof.
Assume thatx is a solution of the nonlocal boundary value problem (15) on a,b. Operating fractional integral Iα;ψ on both sides of the first equation in (15) and using Lemma 10, we obtain for t∈a,b,(20)xt−∑k=12ψt−ψaγ−kΓγ−k+1fψ2−kI1−β2−α;ψxa+kI1;ψxt=Iα;ψht.Hence, using the fact that1−β2−α=2−γ, we have(21)xt=ψt−ψaγ−1Γγ1ψ′tddtI2−γ;ψxa+ψt−ψaγ−2Γγ−1I2−γ;ψxa−kI1;ψxt+Iα;ψht=ψt−ψaγ−1ΓγHDγ−1,β;ψxa+ψt−ψaγ−2Γγ−1I2−γ;ψxa−kI1;ψxt+Iα;ψht=c1ψt−ψaγ−1Γγ+c2ψt−ψaγ−2Γγ−1−kI1;ψxt+Iα;ψht,where c1=HDγ−1,β;ψxtt=a and c2=I2−γ;ψxtt=a.From the first boundary conditionxa=0, we can obtain c2=0, since limt⟶at−aγ−2=∞. Then, we get(22)xt=c1ψt−ψaγ−1Γγ−kI1;ψxt+Iα;ψht,t∈a,b.By a similar way, we obtain(23)yt=d1ψt−ψaδ−1Γδ−νI1;ψyt+Ip;ψzt,t∈a,b,where d1 is an arbitrary constant.From the second boundary conditionsxb=∑i=1m−2λiyθi and yb=∑j=1n−2μjxζj, we get the system(24)Ac1−Bd1=P,−Ωc1+Δd1=Q,where(25)P=−ν∑i=1m−2λiI1;ψyθi+∑i=1m−2λiIp;ψzθi+kI1;ψxb−Iα;ψhb,Q=−k∑j=1n−2μjI1;ψxζj+∑j=1n−2μjIα;ψhζj+νI1;ψyb−Ip;ψzb.Solving the system (24), we find that(26)c1=1ΛΔP+BQ,d1=1ΛAQ+ΩP.Substituting the value ofc1, d1 in (22) and (23) yields the solution (16) and (17). The converse follows by direct computation. This completes the proof.


## 3. Main Results

Let us introduce the spaceW=xt∣xt∈Ca,b,ℝ endowed with the norm ∥x∥=sup∣xt∣,t∈a,b. Obviously, W,· is a Banach space. Then, the product space W×W,∥x,y∥ is also a Banach space equipped with norm ∥x,y∥=∥x∥+∥y∥.In view of Lemma11, we define an operator S:W×W⟶W×W by(27)Sx,yt=S1x,ytS2x,yt,where(28)S1x,yt=Iα;ψfxyt−kI1;ψxt+ψt−ψaγ−1ΛΓγΔ−ν∑i=1m−2λiI1;ψyθi+∑i=1m−2λiIp;ψgxyθi+kI1;ψxb−Iα;ψfxyb+B−k∑j=1n−2μjI1;ψxζj+∑j=1n−2μjIα;ψfxyζj+νI1;ψyb−Ip;ψgxyb,S2x,yt=Ip;ψgxyt−νI1;ψyt+ψt−ψaδ−1ΛΓγA−k∑j=1n−2μjI1;ψxζj+∑j=1n−2μjIα;ψfxyζj+νI1;ψyb−Ip;ψgxyb+Ω−ν∑i=1m−2λiI1;ψyθi+∑i=1m−2λiIp;ψgxyθi+kI1;ψxb−Iα;ψfxyb,where(29)fxyt=ft,xt,yt,gxyt=gt,xt,yt,t∈a,b.For the sake of computational convenience, we put(30)X1=∣k∣b−a+∣A∣∣Λ∣Δkb−a+∣A∣∣Λ∣Bk∑j=1n−2∣μj∣ζj−a,(31)Y1=∣A∣∣Λ∣Δv∑i=1m−2∣λi∣θi−a+∣A∣∣Λ∣Bvb−a,(32)F1=ψb−ψaαΓα+11+∣A∣∣Λ∣Δ+∣A∣∣Λ∣B∑j=1n−2∣μj∣ψζj−ψaαΓα+1,(33)G1=∣A∣∣Λ∣Δ∑i=1m−2λiψθi−ψapΓp+1+∣A∣∣Λ∣Bψb−ψapΓp+1,(34)X2=∣Δ∣∣Λ∣Ak∑j=1n−2∣μj∣ζj−a+∣Δ∣∣Λ∣Ωkb−a,(35)Y2=vb−a+∣Δ∣∣Λ∣Avb−a+∣Δ∣∣Λ∣Ωv∑i=1m−2∣λi∣θi−a,(36)F2=∣Δ∣∣Λ∣∣A∣∑j=1n−2∣μj∣ψζj−ψaαΓα+1+∣Δ∣∣Λ∣∣Ω∣ψb−ψaαΓα+1,(37)G2=ψb−ψapΓp+11+∣Δ∣∣Λ∣∣A∣+∣Δ∣∣Λ∣∣Ω∣∑i=1m−2∣λi∣ψθi−ψapΓp+1.Our first result is based on Leray-Schauder alternative ([25] p. 4.).Lemma 12 (Leray-Schauder alternative).
LetF:E⟶E be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let(38)EF=x∈E:x=λFxforsome0<λ<1.
Then, either the setEF is unbounded, or F has at least one fixed point.Theorem 13.
Assume thatf,g:a,b×ℝ×ℝ⟶ℝ are continuous functions, and there exist real constants pi,qi≥0,i=1,2 and p0,q0>0 such that ∀xi,yi∈ℝ,i=1,2,(39)∣ft,x1,y1∣≤p0+p1∣x1∣+p2y1,∣gt,x2,y2∣≤q0+q1∣x2∣+q2y2.
If(40)M1=F1+F2p1+G1+G2q1+X1+X2<1,M2=F1+F2p2+G1+G2q2+Y1+Y2<1,where Xi,Yi,Fi,Gi,i=1,2 are given by (30)-(37); then, the system (2) has at least one solution on a,b.Proof.
The operatorS is continuous, by the continuity of functions f and g. We will show that the operator S:W×W⟶W×W is completely continuous. Let Zr=x,y∈W×W:∥x,y∥≤r be bounded set. Then, there exist positive constants Li,i=1,2 such that ∣ft,xt,yt∣≤L1,∣gt,xt,yt∣≤L2,∀x,y∈Zr. Then, for any x,y∈Zr, we have(41)∣S1x,yt∣≤Iα;ψ∣fxyt∣+∣k∣I1;ψ∣xt∣+ψt−ψaγ−1∣Λ∣Γγ∣Δ∣∣ν∣∑i=1m−2∣λi∣I1;ψyθi+∑i=1m−2∣λi∣Ip;ψ∣gxyθi∣+∣k∣I1;ψ∣xb∣+Iα;ψ∣fxyb∣+∣B∣∣k∣∑j=1n−2∣μj∣I1;ψxζj+∑j=1n−2∣μj∣Iα;ψ∣fxyζi∣+∣ν∣I1;ψ∣yb∣+Ip;ψ∣gxyb∣≤ψb−ψaαΓα+1L1+∣k∣b−a∥x∥+∣A∣∣Λ∣∣Δ∣∣ν∣∑i=1m−2∣λi∣θi−a∥y∥+∑i=1m−2∣λi∣ψθi−ψapΓp+1L2+∣k∣b−a∥x∥+ψb−ψaαΓα+1L1+∣B∣∣k∣∑j=1n−2∣μj∣ζj−a∥x∥+∑j=1n−2∣μj∣ψζj−ψaαΓα+1L1+∣ν∣b−a∥y∥+ψb−ψapΓp+1L2≤F1L1+G1L2+X1∥x∥+Y1∥y∥,which implies that(42)∥S1x,y∥≤F1L1+G1L2+X1∥x∥+Y1∥y∥.
Similarly, it can be shown that(43)∥S2x,y∥≤F2L1+G2L2+X2∥x∥+Y2∥y∥.
From the above inequalities, it follows that the operatorS is uniformly bounded, since(44)Sx,y≤F1+F2L1+F1+G2L2+X1+X2r+Y1+Y2r.
Next, we show thatS is equicontinuous. Let t1,t2∈a,b with t1<t2. Then, we have(45)∣S1xt2,yt2−S1xt1,yt1∣≤1Γα∣∫at1ψ′sψt2−ψsα−1−ψt1−ψsα−1fs,xs,ysds+∫t1t2ψ′sψt2−ψsα−1fs,xs,ysds∣+∣k∣∫t1t2∣xs∣ds+∣ψt2−ψaγ−1−ψt1−ψaγ−1∣∣Λ∣Γγ∣ΔP+BQ∣≤L12ψt2−ψt1α+∣ψt2α−ψt1α∣Γα+1+∣k∣r∣t2−t1∣+∣ψt2−ψaγ−1−ψt1−ψaγ−1∣∣Λ∣ΓγΔP+BQ.
Analogously, we can obtain(46)∣S2xt2,yt2−S2xt1,yt1∣≤L22ψt2−ψt1p+∣ψt2p−ψt1p∣Γp+1+∣ν∣r∣t2−t1∣+∣ψt2−ψaδ−1−ψt1−ψaδ−1∣∣Λ∣ΓδAQ+ΓP.
Therefore, the operatorSx,y is equicontinuous, and thus, the operator Sx,y is completely continuous.Finally, it will be verified that the setE=x,y∈W×W∣x,y=λSx,y,0≤λ≤1 is bounded. Let x,y∈E with x,y=λSx,y. For any t∈a,b, we have(47)xt=λS1x,yt,yt=λS2x,yt.Then,(48)∣xt∣≤ψb−ψaαΓα+1p0+p1∣x∣+p2∣y∣+∣k∣b−a∥x∥+∣A∣∣Λ∣∣Δ∣∣ν∣∑i=1m−2∣λi∣θi−a∥y∥+∑i=1m−2∣λi∣ψθi−ψapΓp+1q0+q1∣x∣+q2∣y∣+∣k∣b−a∥x∥+ψb−ψaαΓα+1p0+p1∣x∣+p2∣y∣+∣B∣∣k∣∑j=1n−2∣μj∣ζj−a∥x∥+∑j=1n−2∣μj∣ψζj−ψaαΓα+1p0+p1∣x∣+p2∣y∣+∣ν∣b−a∥y∥+ψb−ψapΓp+1q0+q1∣x∣+q2∣y∣≤F1p0+p1∣x∣+p2∣y∣+G1q0+q1∣x∣+q2∣y∣+X1∥x∥+Y1∥y∥=F1p0+G1q0+p1F1+q1G1+X1∥x∥+p2F1+q2G1+Y1∥y∥,∣yt∣≤F2p0+p1∣x∣+p2∣y∣+G2q0+q1∣x∣+q2∣y∣+X2∥x∥+Y2∥y∥=F2p0+G2q0+p1F2+q1G2+X2∥x∥+p2F2+q2G2+Y2∥y∥.Hence, we have(49)∥x∥≤F1p0+G1q0+p1F1+q1G1+X1∥x∥+p2F1+q2G1+Y1∥y∥,∥y∥≤F2p0+G2q0+p1F2+q1G2+X2∥x∥+p2F2+q2G2+Y2∥y∥,which imply that(50)∥x∥+∥y∥≤F1+F2p0+G1+G2q0+F1+F2p1+G1+G2q1+X1+X2∥x∥+F1+F2p2+G1+G2q2+Y1+Y2∥y∥.Consequently,(51)∥x,y∥≤F1+F2p0+G1+G2q0min1−M1,1−M2,which proves that E is bounded. Thus, the operator S, by Lemma 12, has at least one fixed point. Hence, the boundary value problem (2) has at least one solution. The proof is complete.The uniqueness of solutions of the system (2) is proved in the next theorem, via Banach’s contraction mapping principle.Theorem 14.
Assume thatf,g:a,b×ℝ×ℝ⟶ℝ are continuous functions, and there exist positive constants P,Q such that for all t∈a,b and ui,vi∈ℝ,i=1,2, we have(52)∣ft,u1,u2−ft,v1,v2∣≤P∣u1−v1∣+∣u2−v2∣,∣gt,u1,u2−gt,v1,v2∣≤Q∣u1−v1∣+∣u2−v2∣.
Then, the system (2) has a unique solution on a,b, provided that(53)F1+F2P+G1+G2Q+X1+X2+Y1+Y2<1,where Xi,Yi,Fi,Gi,i=1,2 are given by (30)-(37).Proof.
Definesupt∈a,bft,0,0=N1<∞,supt∈a,bgt,0,0=N2<∞ and r>0 such that(54)r>F1+F2N1+G1+G2N21−F1+F2P+G1+G2Q+X1+X2+Y1+Y2.
In the first step, we show thatSBr⊂Br, where Br=x,y∈W×W:∥x,y∥≤r. By the assumption H2, for x,y∈Br,t∈a,b, we have(55)∣ft,xt,yt∣≤∣ft,xt,yt−ft,0,0∣+∣ft,0,0∣≤P∣xt∣+∣yt∣+N1,≤P∥x∥+∥y∥+N1≤Pr+N1,∣gt,xt,yt∣≤Qr+N2.
Using the above estimates, we obtain(56)∣S1x,yt∣≤Iα;ψ∣fxyt∣+∣k∣I1;ψ∣xt∣+ψt−ψaγ−1∣Λ∣Γγ∣Δ∣∣ν∣∑i=1m−2∣λi∣I1;ψyθi+∑i=1m−2∣λi∣Ip;ψ∣gxyθi∣+∣k∣I1;ψ∣xb∣+Iα;ψ∣fxyb∣+∣B∣∣k∣∑j=1n−2∣μj∣I1;ψxζj+∑j=1n−2∣μj∣Iα;ψ∣fxyζj∣+∣ν∣I1;ψ∣yb∣+Ip;ψ∣gb,xb,yb∣≤ψb−ψaαΓα+1Pr+N1+∣k∣b−a∥x∥+∣A∣∣Λ∣∣Δ∣∣ν∣∑i=1m−2∣λi∣θi−a∥y∥+∑i=1m−2∣λi∣ψθi−ψapΓp+1Qr+N2+∣k∣b−a∥x∥+ψb−ψaαΓα+1Pr+N1+∣B∣∣k∣∑j=1n−2∣μj∣ζj−a∥x∥+∑j=1n−2∣μj∣ψζj−ψaαΓα+1Pr+N1+∣ν∣b−a∥y∥+ψb−ψapΓp+1Qr+N2≤F1P+G1Q+X1+Y1r+F1N1+G1N2.
Hence,(57)∥S1x,y∥≤F1P+G1Q+X1+Y1r+F1N1+G1N2.
In the same way, we can obtain that(58)∥S2x,y∥≤F2P+G2Q+X2+Y2r+F2N1+G2N2.
In consequence, it follows that(59)∥Sx,y∥≤F1+F2P+G1+G2Q+X1+X2+Y1+Y2r+F1+F2N1+G1+G2N2≤r,which shows that SBr⊂Br.We prove that the operatorS is a contraction. For x2,y2,x1,y1∈W×W and for any t∈a,b, we get(60)∣S1x2,y2t−S1x1,y1t∣≤ψb−ψaαΓα+1P∥x2−x1∥+∥y2−y1∥+∣k∣b−a∥x2−x1∥+∣A∣∣Λ∣∣Δ∣∣ν∣∑i=1m−2∣λi∣θi−a∥y2−y1∥+∑i=1m−2∣λi∣ψθi−ψapΓp+1Q∥x2−x1∥+∥y2−y1∥+∣k∣b−a∥x2−x1∥+ψb−ψaαΓα+1P∥x2−x1∥+∥y2−y1∥+∣B∣∣k∣∑j=1n−2∣μj∣ζj−a∥x∥+∑j=1n−2∣μj∣ψζj−ψaαΓα+1P∥x2−x1∥+∥y2−y1∥+∣ν∣b−a∥y2−y1∥+ψb−ψapΓp+1Q∥x2−x1∥+∥y2−y1∥≤F1P+G1Q+X1+Y1∥x2−x1∥+∥y2−y1∥,and consequently, we obtain(61)∥S1x2,y2−S1x1,y1∥≤F1P+G1Q+X1+Y1∥x2−x1∥+∥y2−y1∥.Similarly, we have(62)∥S2x2,y2t−S2x1,y1≤F2P+G2Q+X2+Y2∥x2−x1∥+∥y2−y1∥.It follows from above two equations (61) and (62) that(63)∥Sx2,y2−Sx1,y1∥≤F1+F2P+G1+G2Q+X1+X2+Y1+Y2∥x2−x1∥+∥y2−y1∥,which implies that the operator S is a contraction, by assumption (53). Consequently, the operator S has a unique fixed point, by Banach’s fixed point theorem, which is the unique solution of problem (2). This completes the proof.Example 15.
Consider the following system(64)HD32,13;t2+1+155HD12,13;t2+1xt=ft,xt,yt,t∈14,52,HD43,12;t2+1+158HD13,12;t2+1yt=gt,xt,yt,t∈14,52,x14=0,x52=13y34+16y32+19y2,14=0,y52=15x12+27x54+38x74+411x94,
Here,ψt=t2+1,α=3/2,β=1/3,p=4/3,q=1/2,k=1/55, ν=1/58,γ=5/3,δ=5/3,λ1=1/3,λ2=1/6,λ3=1/9,μ1=1/5,μ2=2/7,μ3=3/8,μ4=4/11,θ1=3/4,θ2=3/2,θ3=2,ζ1=1/2,ζ2=5/4,ζ3=7/4,ζ4=9/4, a=1/4, b=5/2, m=5, and n=6.From the given data, we can calculateA≈3.733460626, B≈0.8506267338, Ω≈2.529197097, Δ≈3.733460626, Λ≈11.78732558, X1≈0.09724746240, X2≈0.06772019882, Y1≈0.02206174386, Y2≈0.09253172371, F1≈26.59796061, F2≈15.10628827, G1≈3.858014303, and G2≈21.69467361.(i)
Let the nonlinear functionsf and g be defined on 1/4,5/2 by(65)ft,x,y=12e−∣xy∣+4339+4tx21+∣x∣+1320tysin6x,(66)gt,x,y=23cos2∣xy∣+4299+4txe−y+1360ty51+y4.It is obvious to check that the above functions satisfy(67)ft,x,y≤12+185x+180y,gt,x,y≤23+175x+190y,which can be set p0=1/2, p1=1/85, p2=1/80, q0=2/3, q1=1/75, and q2=1/90 as in the hypothesis H1 of Theorem 13. Then, we can find that(68)M1≈0.9963083888<1,M2≈0.9198153332<1.Thus, all assumptions of Theorem13 satisfy. The conclusion of Theorem 13 implies that problem (64) with (65) and (66) has at least one solution on 1/4,5/2.(ii)
Consider now the functionsf and g given by(69)ft,x,y=12+e−3t+4767+4tx2+2∣x∣1+∣x∣+1392tsiny,(70)gt,x,y=13+πlogt2+1364ttan−1x+4719+4ty2+2∣y∣1+∣y∣.Checking the Lipschitz condition forf and g, we obtain(71)∣ft,x1,y1−ft,x2,y2∣≤196∣x1−x2∣+198y1−y2,∣gt,x1,y1−gt,x2,y2∣≤191∣x1−x2∣+190y1−y2.Then, by settingP=1/96 and Q=1/90, the condition H2 of Theorem 14 is fulfilled. In addition, we find that(72)F1+F2P+G1+G2Q+X1+X2+Y1+Y2≈0.9978991426<1.Therefore, the system (64) with (69) and (70) has a unique solution on 1/4,5/2, by the benefit of Theorem 14.


## 4. Conclusion

We investigated the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations involving Hilfer fractional derivative with coupled nonlocal multipoint boundary conditions by applying the framework of fixed point theorems. The existence of a unique solution is obtained via Banach’s fixed point theorem, while the existence result is proved by using Leray-Schauder alternative. The results obtained in the present paper are new and significantly contribute to the existing literature on the topic.



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