# Hilfer-Hadamard Nonlocal Integro-Multipoint Fractional Boundary Value Problems

Authors: Chanon Promsakon; Sotiris K. Ntouyas; Jessada Tariboon
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is concerned with the existence and uniqueness of solutions for a new class of boundary value problems, consisting by Hilfer-Hadamard fractional differential equations, supplemented with nonlocal integro-multipoint boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Examples illustrating the main results are also constructed.


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## Body

## 1. Introduction

The fractional calculus has always been an interesting research topic for many years. This is because fractional differential equations describe many real-world process related to memory and hereditary properties of various materials more accurately as compared to classical-order differential equations. Fractional differential equations arise in lots of engineering and clinical disciplines which include biology, physics, chemistry, economics, signal and image processing, and control theory (see the monographs and papers in [1–8]).Various types of fractional derivatives were introduced among which the following Riemann-Liouville and Caputo derivatives are the most widely used ones.(1)
Riemann-Liouville derivative. For n−1<α<n, the derivative of u is(1)RLDaαut≔DnIan−αut=1Γn−αddtn∫att−sn−α−1usds(2)
Caputo derivative. For n−1<α<n, the derivative of u is(2)CDaαut≔Ian−αDnut=1Γn−α∫att−sn−α−1ddsnusds,where Dn=d/dtn. Both Riemann-Liouville and Caputo derivatives are defined via fractional integral, the Riemann-Liouville fractional integral, which is defined by
(3)Iaαut=1Γα∫att−sα−1usds,n−1<α<nA generalization of derivatives of both Riemann-Liouville and Caputo was given by R. Hilfer in [9], known as the Hilfer fractional derivative of order α and a type β∈0,1, which interpolates between the Riemann-Liouville and Caputo derivatives, since it is reduced to the Riemann-Liouville and Caputo fractional derivatives when β=0 and β=1, respectively. The Hilfer fractional derivative of order α and parameter β of a function u is defined by
(4)HDaα,βut=Iaβn−αDnIa1−βn−αut,where n−1<α<n,0≤β≤1,t>a,D=d/dt. Some properties and applications of the Hilfer derivative are given in [10, 11] and references cited therein. Initial value problems involving Hilfer fractional derivatives were studied by several authors (see, for example, [12–15] and references therein). Nonlocal boundary value problems for Hilfer fractional derivative were studied in [16, 17].The Hadamard fractional calculus contains fractional derivative and integral with respect to the logarithmic function while someone say that it is generalized of the derivativetd/dtα, where α is an arbitrary order. The Hadamard calculus can be obtained by replacing as d/dt⟶td/dt, t−s·⟶loget−loges·, and ds⟶1/sds in (1)–(3). In the same way, the concept of Hilfer-Hadamard derivative is arrived by the modified definition in (4). Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional differential equations with two point boundary conditions were studied in [18].In this paper, we study existence and uniqueness of solutions for boundary value problems for Hilfer-Hadamard fractional differential equations with nonlocal integro-multipoint boundary conditions,(5)HHD1α,βxt=ft,xt,t∈1,T,x1=0,∑i=1mθixξi=λHI1δxη,where HHD1α,β is the Hilfer-Hadamard fractional derivative of order α∈1,2 and type β∈0,1, θi,λ∈ℝ, i=1,2,⋯,m, are given constants, and f:1,T×ℝ⟶ℝ is a given continuous function, HIδ is the Hadamard fractional integral of order δ>0, and η,ξi∈1,T, i=1,2,⋯,m, are given points.Existence and uniqueness results are established by using classical fixed point theorems. We make use of Banach’s fixed point theorem to obtain the uniqueness result, while Schaefer and Krasnoselskii’s fixed point theorem [19] as well as nonlinear alternative of Leray-Schauder type [20] is applied to obtain the existence results for the problem (5).The paper is constructed as follows: in Section2, we recall some basic facts needed in our study. The main results are proved in Section 3. Examples illustrating the main results are presented in Section 4.


## 2. Preliminaries

In this section, some basic definitions, lemmas, and theorems are mentioned.Definition 1 (Hadamard fractional integral [2]).
The Hadamard fractional integral of orderα>0 for a function f:a,∞⟶ℝ is defined as
(6)HIaαft=1Γα∫atlogtτα−1fττdτ,t>a,provided the integral exists, where log.=loge..Definition 2 (Hadamard fractional derivative [2]).
The Hadamard fractional derivative of orderα>0, applied to the fuction f:a,∞⟶ℝ, is defined as follows:
(7)HDaαft=δnHIan−αft,n=α+1,where δn=td/dtn and α denotes the integer part of the real number α.Definition 3 (Hilfer-Hadamard fractional derivative [11]).
Letn−1<α<n and 0≤β≤1, f∈L1a,b. The Hilfer-Hadamard fractional derivative of order α and tybe β of f is defined as
(8)HHDaα,βft=HIaβn−αδnHIan−α1−βft=HIaβn−αδnHIan−γft=HIaβn−αHDaγft,γ=α+nβ−αβ,where HIa. and HDa. is the Hadamard fractional integral and derivative defined by (6) and (7), respectively.The Hilfer-Hadamard fractional derivative may be viewed as interpolating between Hadamard and Caputo-Hadamard fractional derivatives. Indeed, forβ=0, this derivative reduces to the Hadamard fractional derivative while for β=1, it leads the Caputo-Hadamard derivative defined by
(9)HCDaαft=HIan−αδnft,n=α+1.We recall the following known theorem by Kilbas et al. [2] which will be used in the following.Theorem 4 ([2]).
Letα>0, 0≤β≤1, γ=α+nβ−αβ, n=α+1, and 0<a<b<∞. If f∈L1a,b and HIan−γft∈ACδna,b, then
(10)HIaαHHDaα,βft=HIaγHHDaγft=ft−∑j=0n−1δn−j−1HIan−γfaΓγ−jlogtaγ−j−1.
Sinceγ∈α,n, then the Γγ−j exists for all j=1,2,⋯,n−1.Finally, we will use the following well-known fixed point theorems on Banach space for proving the existence and uniqueness of the solutions to Hilfer-Hadamard fractional boundary value problem (5).Theorem 5 (Banach’s contraction principle [21]).
LetX be a Banach space, D⊂X be closed, and T:D⟶D be a contraction (i.e., there exists a constant L∈0,1 such that for any x,y∈X,∣Tx−Ty∣≤L∣x−y∣). Then, T has a unique fixed point on X.Theorem 6 (Krasnoselskii’s fixed point theorem)[19].
LetY be a bounded, closed, convex, and nonempty subset of a Banach space X. Let F1 and F2 be the operators satisfying the following conditions: (i) F1y1+F2y2∈Y whenever y1,y2∈Y; (ii) F1 is compact and continuous; and (iii) F2 is a contraction mapping. Then, there exists y∈Y such that y=F1y+F2y.Theorem 7 (Schaefer fixed point theorem [22]).
LetT:E⟶E be a completely continuous operator (i.e., a continuous map T restricted to any bounded set in E is compact). Let εT=x∈E:x=λTx,0≤λ≤1. Then, either the set εT is unbounded or T has at least one fixed point.Theorem 8 (Nonlinear alternative for single-valued maps) ([20]).
LetE be a Banach space, C a closed, convex subset of E, U an open subset of C, and 0∈U. Suppose that A:U¯⟶C is continuous and compact (that is, AU¯ is a relatively compact subset of C) map. Then, either
(i)
A has a fixed point in U¯, or(ii)
there isax∈∂U (the boundary of U in C) and λ∈0,1 with x=λAx


## 3. Main Results

In this section, we prove existence and uniqueness of solutions for nonlinear Hilfer-Hadamard fractional boundary value problem (5). Firstly, we start by proving a basic lemma concerning a linear variant of the boundary value problem (5), which will be used to transform the boundary value problem (5) into an equivalent integral equation. In this case, n=α+1=2; then, we have γ=α+2−αβ.Lemma 9.
Leth∈C1,T,ℝ and Λ≠0, where
(11)Λ=∑i=1mθilogξiγ−1−λΓγΓγ+δlogηγ+δ−1.
Then,x is a solution of the following linear Hilfer-Hadamard fractional differential equation:
(12)HHD1α,βxt=ht,1<α≤2,t∈1,T,supplemented with the boundary conditions in (5), if and only if
(13)xt=HI1αht+logtγ−1ΛλHI1α+δhη−∑i=1mθiHI1αhξi=1Γα∫1tlogtτα−1hττdτ+logtγ−1Λλ1Γα+δ∫1ηlogητα+δ−1hττdτ−∑i=1mθi1Γα∫1ξilogξiτα−1hττdτ,t∈1,T.Proof.
By taking the Hadamard fractional integral of orderα from 1 to t on both sides of (12) and using Theorem 4, it follows that
(14)xt−∑j=01δ2−j−1HI1+2−γx1Γγ−jlogtγ−j−1=HI1αht.
Then, we have(15)xt−δHI1+2−γx1Γγlogtγ−1−HI1+2−γx1Γγ−1logtγ−2=HI1αht.
Equation (15) can be rewritten by
(16)xt=c0logtγ−1+c1logtγ−2+1Γα∫1thsslogtsα−1ds,where c0,c1 are arbitrary constants. Now, the first boundary condition x1=0 together with (16) yields c1=0, since γ∈α,2. Putting c1=0 in (16), we get
(17)xt=c0logtγ−1+1Γα∫1tlogtsα−1hssds.
Next, the second boundary condition∑i=1mθixξi=λHI1δxη together with (17) yields
(18)c0=1ΛλHI1α+δhη−∑i=1mθiHI1αhξi.
Substituting the value ofc0 in (17), we get equation (13) as desired.
The converse follows by direct computation. The proof is completed.☐Let us introduce the Banach spaceX=C1,T,ℝ endowed with the norm defined by ∥x∥≔maxt∈1,T∣xt∣.In view of Lemma9, we define an operator F:X⟶X, where
(19)Fxt=1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ−∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ,t∈1,T.In the following, for convenience, we put(20)Ω=logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1.We need the following hypotheses in the sequel:(H1).
There exists a constantl>0 such that for all t∈1,T and ui∈ℝ,i=1,2,(21)ft,u1−ft,u2≤lu1−u2.(H2).
There exists a continuous nonnegative functionϕ∈C1,T,ℝ+ such that
(22)ft,u≤ϕt,foreacht,u∈1,T×ℝ.(H3).
There exists a real constantM>0 such that for all t∈1,T,u∈ℝ,(23)ft,u≤M.(H4).
There existp∈C1,T,ℝ+ and a continuous nondecreasing function ψ:ℝ+⟶ℝ+ such that
(24)ft,u≤ptψuforeacht,u∈1,T×ℝ.(H5).
There exists a constantK>0 such that
(25)KΩpψK>1.


### 3.1. Existence and Uniqueness Result via Banach’s Fixed Point Theorem

We prove an existence and uniqueness result based on Banach’s contraction mapping principle.Theorem 10.
Assume thatH1 holds. Then, boundary value problem (5) has a unique solution on 1,T, provided that lΩ<1, where Ω is defined by (20).Proof.
We will use Banach’s fixed point theorem to prove thatF, defined by (19), has a unique fixed point. Fixing N=maxt∈1,T∣ft,0∣<∞ and using hypothesis H1, we obtain
(26)ft,xt≤ft,xt−ft,0+ft,0≤lxt+ft,0≤lx+N.
Choose(27)r≥NΩ1−lΩ.
We divide the proof into two steps.
Step I. We show that FBr⊂Br, where Br=x∈X:x<r. Let x∈Br. Then, we have
(28)Fxt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤logTαΓα+1lx+N+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1lx+N=logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1lx+N≤logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1lr+N=Ωlr+N≤r.
Thus,(29)Fx=maxt∈1,TFxt≤r,which means that FBr⊂Br.
Step II. To show that the operator F is a contraction, let x1,x2∈X. Then, for any t∈1,T, we have
(30)Fx2t−Fx1t≤1Γα∫1tlogtτα−1fτ,x2τ−fs,x1ττdτ+logtγ−1ΛλΓα+δ∫1η‍logητα+δ−1fs,x2τ−fs,x1ττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,x2τ−fτ,x1ττdτ≤lx2−x1logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1.
Thus,(31)Fx2−Fx1=maxt∈1,TFx2t−Fx1t≤lΩx2−x1,which, in view of lΩ<1, shows that the operator F is a contraction. By Theorem 5, we get that the operator F has a unique fixed point. Therefore, the problem (5) has a unique solution on 1,T. The proof is completed.☐


### 3.2. Existence Result via Krasnoselskii’s Fixed Point Theorem

In this subsection, we prove an existence result based on Krasnoselskii’s fixed point theorem.Theorem 11.
Assume thatH1-H2 hold. Then, the problem (5) has at least one solution on 1,T, provided that
(32)logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1l<1.Proof.
By assumptionH2, we can fix ρ≥Ωϕ, where ϕ=supt∈1,Tϕt and consider Bρ=x∈C1,T,ℝ:x≤ρ. We split the operator F:C1,T,ℝ⟶C1,T,ℝ defined by (19) as F=F1+F2, where F1 and F2 are given by
(33)F1xt=1Γα∫1tlogtτα−1fτ,xττdτ,F2xt=logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ−∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ.
For anyx,y∈Bρ, we have
(34)F1xt+F2yt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,yττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,yττdτ≤logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1m∣θi∣logξiαΓα+1ϕ=Ωϕ≤r.
Hence,F1x+F2y≤ρ, which shows that F1x+F2y∈Bρ. It is easy to prove, using conditions H1 and (32), that the operator F2 is a contraction mapping.
Next, the operatorF1 is continuous by the continuity of f. Also, F1 is uniformly bounded on Bρ, since
(35)F1x≤logTαΓα+1ϕ.
Finally, we prove the compactness of the operatorF1. For t1,t2∈1,T,t1<t2, we have
(36)F1xt2−F1xt1≤1Γα∫1t1logt2τα−1−logt1τα−1fτ,xττdτ+1Γα∫t1t2logt2τα−1fτ,xττdτ≤ϕΓα+12logt2−logt1α+logt2α−logt1α,which tends to zero, independently of x∈Bρ, as t1⟶t2. Thus, F1 is equicontinuous. From the Arzelá-Ascoli theorem, we conclude that the operator F1 is compact on Bρ. Thus, the hypotheses of Krasnoselskii fixed point theorem are satisfied, and therefore, there exists at least one solution on 1,T. The proof is finished.☐


### 3.3. Existence Result via Schaefer’s Fixed Point Theorem

Our second existence result is based on Schaefer’s fixed point theorem.Theorem 12.
Assume thatH3 holds. Then, the boundary value problem (5) has at least one solution on 1,T.Proof.
We will prove that the operatorF, defined by (19), has a fixed point, by using Schaefer’s fixed point theorem. We divide the proof into two steps.
Step I. We show that the operator F:X⟶X is completely continuous.
We show first thatF is continuous. Let xn be a sequence such that xn⟶x in X. Then, for each t∈1,T, we have
(37)Fxnt−Fxt≤1Γα∫1tlogtτα−1fτ,xnτ−fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xnτ−fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xnτ−fτ,xττdτ.
Sincef is continuous, we get
(38)fs,xns−fs,xs⟶0asxn⟶x.
Then(39)Fxn−Fx⟶0asxn⟶x.
Hence,F is continuous.
Secondly, we show that the operatorF maps bounded sets into bounded sets in X. For a positive number R, let BR=x∈X:x≤R be a bounded ball in X. Then, for t∈1,T, we have
(40)Fxt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M,and
(41)Fx≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M.
Thirdly, we show thatF maps bounded sets into equicontinuous sets. Let t1,t2∈1,T with t1<t2 and u∈BR. Then, we have
(42)Fxt2−Fxt1≤1Γα∫1t1logt2τα−1−logt1τα−1fτ,xττdτ+1Γα∫t1t2logt2τα−1fτ,xττdτ+logt2γ−1−logt1γ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤MΓα+12logt2−logt1α+logt2α−logt1α+logt2γ−1−logt1γ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M,which tends to zero, independently of x∈BR, as t1⟶t2. Thus, the Arzelá-Ascoli theorem applies and hence, F:X⟶X is completely continuous.
Step II. We show that the set E=x∈X∣x=νFx,0≤ν≤1 is bounded. Let x∈E, then x=νFx. For any t∈1,T, we have xt=νFxt. Then, in view of the hypothesis H3, as in Step I, we obtain
(43)xt≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M.
Thus,(44)x≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M,which shows that the set E is bounded. By Theorem 7, we get that the operator F has at least one fixed point. Therefore, the boundary value problem (5) has at least one solution on 1,T. This completes the proof.☐


### 3.4. Existence Result via Leray-Schauder Nonlinear Alternative

Our final existence result is proved via Leray-Schauder nonlinear alternative.Theorem 13.
Assume thatH4 and H5 hold. Then, the boundary value problem (5) has at least one solution on 1,T.Proof.
As in Theorem12, we can prove that the operator F is completely continuous. We will prove that there exists an open set U⊆C1,T,ℝ with x≠μFx for μ∈0,1 and x∈∂U.
Letx∈C1,T,ℝ be such that x=μFx for some 0<μ<1. Then, for each t∈1,T, we have
(45)xt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1pψx.
Consequently,(46)U=x∈C1,T,ℝ:x<K.
In view ofH5, there is no solution x such that x≠K. Let us set
(47)U=x∈C1,T,ℝ:x<K.
The operatorF:U¯⟶C1,T,ℝ is continuous and completely continuous. From the choice of U, there is no u∈∂U such that x=μFx for some μ∈0,1. Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 8), we deduce that F has a fixed point x∈U¯ which is a solution of the boundary value problem (5). The proof is completed.☐


## 3.1. Existence and Uniqueness Result via Banach’s Fixed Point Theorem

We prove an existence and uniqueness result based on Banach’s contraction mapping principle.Theorem 10.
Assume thatH1 holds. Then, boundary value problem (5) has a unique solution on 1,T, provided that lΩ<1, where Ω is defined by (20).Proof.
We will use Banach’s fixed point theorem to prove thatF, defined by (19), has a unique fixed point. Fixing N=maxt∈1,T∣ft,0∣<∞ and using hypothesis H1, we obtain
(26)ft,xt≤ft,xt−ft,0+ft,0≤lxt+ft,0≤lx+N.
Choose(27)r≥NΩ1−lΩ.
We divide the proof into two steps.
Step I. We show that FBr⊂Br, where Br=x∈X:x<r. Let x∈Br. Then, we have
(28)Fxt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤logTαΓα+1lx+N+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1lx+N=logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1lx+N≤logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1lr+N=Ωlr+N≤r.
Thus,(29)Fx=maxt∈1,TFxt≤r,which means that FBr⊂Br.
Step II. To show that the operator F is a contraction, let x1,x2∈X. Then, for any t∈1,T, we have
(30)Fx2t−Fx1t≤1Γα∫1tlogtτα−1fτ,x2τ−fs,x1ττdτ+logtγ−1ΛλΓα+δ∫1η‍logητα+δ−1fs,x2τ−fs,x1ττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,x2τ−fτ,x1ττdτ≤lx2−x1logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1.
Thus,(31)Fx2−Fx1=maxt∈1,TFx2t−Fx1t≤lΩx2−x1,which, in view of lΩ<1, shows that the operator F is a contraction. By Theorem 5, we get that the operator F has a unique fixed point. Therefore, the problem (5) has a unique solution on 1,T. The proof is completed.☐


## 3.2. Existence Result via Krasnoselskii’s Fixed Point Theorem

In this subsection, we prove an existence result based on Krasnoselskii’s fixed point theorem.Theorem 11.
Assume thatH1-H2 hold. Then, the problem (5) has at least one solution on 1,T, provided that
(32)logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1l<1.Proof.
By assumptionH2, we can fix ρ≥Ωϕ, where ϕ=supt∈1,Tϕt and consider Bρ=x∈C1,T,ℝ:x≤ρ. We split the operator F:C1,T,ℝ⟶C1,T,ℝ defined by (19) as F=F1+F2, where F1 and F2 are given by
(33)F1xt=1Γα∫1tlogtτα−1fτ,xττdτ,F2xt=logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ−∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ.
For anyx,y∈Bρ, we have
(34)F1xt+F2yt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,yττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,yττdτ≤logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1m∣θi∣logξiαΓα+1ϕ=Ωϕ≤r.
Hence,F1x+F2y≤ρ, which shows that F1x+F2y∈Bρ. It is easy to prove, using conditions H1 and (32), that the operator F2 is a contraction mapping.
Next, the operatorF1 is continuous by the continuity of f. Also, F1 is uniformly bounded on Bρ, since
(35)F1x≤logTαΓα+1ϕ.
Finally, we prove the compactness of the operatorF1. For t1,t2∈1,T,t1<t2, we have
(36)F1xt2−F1xt1≤1Γα∫1t1logt2τα−1−logt1τα−1fτ,xττdτ+1Γα∫t1t2logt2τα−1fτ,xττdτ≤ϕΓα+12logt2−logt1α+logt2α−logt1α,which tends to zero, independently of x∈Bρ, as t1⟶t2. Thus, F1 is equicontinuous. From the Arzelá-Ascoli theorem, we conclude that the operator F1 is compact on Bρ. Thus, the hypotheses of Krasnoselskii fixed point theorem are satisfied, and therefore, there exists at least one solution on 1,T. The proof is finished.☐


## 3.3. Existence Result via Schaefer’s Fixed Point Theorem

Our second existence result is based on Schaefer’s fixed point theorem.Theorem 12.
Assume thatH3 holds. Then, the boundary value problem (5) has at least one solution on 1,T.Proof.
We will prove that the operatorF, defined by (19), has a fixed point, by using Schaefer’s fixed point theorem. We divide the proof into two steps.
Step I. We show that the operator F:X⟶X is completely continuous.
We show first thatF is continuous. Let xn be a sequence such that xn⟶x in X. Then, for each t∈1,T, we have
(37)Fxnt−Fxt≤1Γα∫1tlogtτα−1fτ,xnτ−fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xnτ−fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xnτ−fτ,xττdτ.
Sincef is continuous, we get
(38)fs,xns−fs,xs⟶0asxn⟶x.
Then(39)Fxn−Fx⟶0asxn⟶x.
Hence,F is continuous.
Secondly, we show that the operatorF maps bounded sets into bounded sets in X. For a positive number R, let BR=x∈X:x≤R be a bounded ball in X. Then, for t∈1,T, we have
(40)Fxt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M,and
(41)Fx≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M.
Thirdly, we show thatF maps bounded sets into equicontinuous sets. Let t1,t2∈1,T with t1<t2 and u∈BR. Then, we have
(42)Fxt2−Fxt1≤1Γα∫1t1logt2τα−1−logt1τα−1fτ,xττdτ+1Γα∫t1t2logt2τα−1fτ,xττdτ+logt2γ−1−logt1γ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤MΓα+12logt2−logt1α+logt2α−logt1α+logt2γ−1−logt1γ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M,which tends to zero, independently of x∈BR, as t1⟶t2. Thus, the Arzelá-Ascoli theorem applies and hence, F:X⟶X is completely continuous.
Step II. We show that the set E=x∈X∣x=νFx,0≤ν≤1 is bounded. Let x∈E, then x=νFx. For any t∈1,T, we have xt=νFxt. Then, in view of the hypothesis H3, as in Step I, we obtain
(43)xt≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M.
Thus,(44)x≤logTαΓα+1M+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1M,which shows that the set E is bounded. By Theorem 7, we get that the operator F has at least one fixed point. Therefore, the boundary value problem (5) has at least one solution on 1,T. This completes the proof.☐


## 3.4. Existence Result via Leray-Schauder Nonlinear Alternative

Our final existence result is proved via Leray-Schauder nonlinear alternative.Theorem 13.
Assume thatH4 and H5 hold. Then, the boundary value problem (5) has at least one solution on 1,T.Proof.
As in Theorem12, we can prove that the operator F is completely continuous. We will prove that there exists an open set U⊆C1,T,ℝ with x≠μFx for μ∈0,1 and x∈∂U.
Letx∈C1,T,ℝ be such that x=μFx for some 0<μ<1. Then, for each t∈1,T, we have
(45)xt≤1Γα∫1tlogtτα−1fτ,xττdτ+logtγ−1ΛλΓα+δ∫1ηlogητα+δ−1fτ,xττdτ+∑i=1mθi1Γα∫1ξilogξiτα−1fτ,xττdτ≤logTαΓα+1+logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1pψx.
Consequently,(46)U=x∈C1,T,ℝ:x<K.
In view ofH5, there is no solution x such that x≠K. Let us set
(47)U=x∈C1,T,ℝ:x<K.
The operatorF:U¯⟶C1,T,ℝ is continuous and completely continuous. From the choice of U, there is no u∈∂U such that x=μFx for some μ∈0,1. Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 8), we deduce that F has a fixed point x∈U¯ which is a solution of the boundary value problem (5). The proof is completed.☐


## 4. Examples

Example 1.
Consider the Hilfer-Hadamard nonlocal integro-multipoint fractional boundary value problem of the form(48)HHD13/2,1/2xt=ft,xt,t∈1,4.5,x1=0,629HI15/2x52=111x32+213x2+317x3+419x72+523x4.Set constantsα=3/2, β=1/2, T=9/2, m=5, θ1=1/11, θ2=2/13, θ3=3/17, θ4=4/19, θ5=5/23, ξ1=3/2, ξ2=2, ξ3=3, ξ4=7/2, ξ5=4, λ=6/29, δ=5/2, and η=5/2. Then, from these data, we compute that γ=7/4, Λ≈1.152315230, and Ω≈2.250815481,
(i)
Let a nonlinear unbounded functionf:1,4.5×ℝ⟶ℝ be defined by(49)ft,x=e1−tt+4x2+2x1+x+12t2+1Now, we can verify that functionft,x satisfies the Lipchitz condition H1 with l=2/5, that is, ft,x−ft,y≤2/5x−y, for all t∈1,4.5 and x,y∈ℝ. Then, we obtain lΩ≈0.9003261924<1. The conclusion can be gotten from Theorem 10 that the problem (47), with f given by (48), has a unique solution on 1,4.5.
(ii)
If the termx2 in (48) is replaced by ∣x∣, that is,(50)ft,x=3e1−tt+4x1+x+12t2+1,then the nonlinear function f is bounded by a function of t by
(51)ft,x≤3e1−tt+4+12t2+1≔ϕt,which is satisfied a condition H2. It is easy to check that function f in (49) is fulfilled the Lipchitz condition with constant l=3/5. Since
(52)logTγ−1Λλlogηα+δΓα+δ+1+∑i=1mθilogξiαΓα+1l≈0.5179201139<1,we can deduce that the Hilfer-Hardamard nonlocal integro-multipoint fractional boundary value problem (47), with f given by (49), has at least one solution on 1,4.5. In addition, we observe that Theorem 10 cannot be used in this case because lΩ≈1.350489289>1(iii)
Iff is a non-Lipschitzian function as(53)ft,x=e1−tt+4sin12x2+2x1+x+12t2+1,to claim the existence of boundary value problem (47), with f given by (52), we can find that
(54)ft,x≤15+12922+1=45340Hence,H3 is satisfied with M=453/40. Therefore, by the benefit of Theorem 12, problem (47) with f given by (52) has at least one solution on 1,4.5.
(iv)
Iff is a non-Lipschitzian function defined by(55)ft,x=e1−tt+4x261+x24+19t+12,we see that H4 holds as
(56)ft,x≤e1−tt+4x2+1,by setting pt=e1−t/t+4 and ψx=x2+1. Then, we obtain p=1/5 and there exists a constant K∈0.6273106970,1.594106403 satisfying condition H5. Thus, using Theorem 13, the boundary value problem (47), with f is given in (54), has at least one solution on 1,4.5


## 5. Conclusion

In this paper, we have presented the existence and uniqueness criteria for solutions for Hilfer-Hadamard fractional differential equation complemented with nonlocal integro-multipoint boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem. Once the fixed point operator is available, we make use of Banach contraction mapping principle to obtain the uniqueness result, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Our results are new in the given configuration and enrich the literature on boundary value problems involving Hilfer-Hadamard fractional differential equations.



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