# Existence of Weak Solutions for Fractional Integrodifferential Equations with Multipoint Boundary Conditions

Authors: Haide Gou; Baolin Li
Journal: International Journal of Differential Equations
 (2018)

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## Abstract

By combining the techniques of fractional calculus with measure of weak noncompactness and fixed point theorem, we establish the existence of weak solutions of multipoint boundary value problem for fractional integrodifferential equations.


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## Body

## 1. Introduction

In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology. As long as the Banach space is reflexive, the weak compactness offers no problem since every bounded subset is relatively weakly compact and therefore the weak continuity suffices to prove nice existence results for differential and integral equations [1, 2]. De Blasi [3] introduced the concept of measure of weak noncompactness and proved the analogue of Sadovskiis fixed point theorem for the weak topology (see also [4]). As stressed in [5], in many applications, it is always not possible to show the weak continuity of the involved mappings, while the sequential weak continuity offers no problem. This is mainly due to the fact that Lebesgues dominated convergence theorem is valid for sequences but not for nets. Recall that a mapping between two Banach spaces is sequentially weakly continuous if it maps weakly convergent sequences into weakly convergent sequences.The theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored. There are many papers dealing with multipoint boundary value problems both on resonance case and on nonresonance case; for more details see [6–11]. However, as far as we know, few results can be found in the literature concerning multipoint boundary value problems for fractional differential equations in Banach spaces and weak topologies. Zhouet al. [12] discuss the existence of solutions for nonlinear multipoint boundary value problem of integrodifferential equations of fractional order as follows: (1)Dc0+αxt=ft,xt,Hxt,Kxt,t∈0,1,α∈1,2,a1x0-b1x′0=d1xξ1,a2x1+b2x′1=d2xξ2,with respect to strong topology, where Dc0+α denotes the fractional Caputo derivative and the operators given by (2)Hxt=∫0tgt,sxsds,Kxt=∫0tht,sxsds.Moreover, theory for boundary value problem of integrodifferential equations of fractional order in Banach spaces endowed with its weak topology has been few studied until now. In [13], we discussed the existence theorem of weak solutions nonlinear fractional integrodifferential equations in nonreflexive Banach spaces E: (3)Dc0+αxt=ft,xt,Txt,Sxt,t∈0,1,α∈1,2,a1x0-a2x′0=γ1,b1x1+b2x′1=γ2,and obtain a new result by using the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals, where Dc0+α denotes the fractional Caputo derivative and the operators given by (4)Txs=∫0sk1s,τgτ,xτdτ,Sxs=∫01k2s,τhτ,xτdτ.Our analysis relies on the Krasnoselskii fixed point theorem combined with the technique of measure of weak noncompactness.Motivated by the above works, in this paper, we use the techniques of measure of weak noncompactness combine with the fixed point theorem to discuss the existence theorem of weak solutions for a class of nonlinear fractional integrodifferential equations of the form(5)Dc0+αxt=ft,xt,Txt,Sxt,t∈0,1,α∈1,2,a1x0-b1x′0=d1xξ1,a2x1+b2x′1=d2xξ2,where T and S are two operators defined by (6)Tut=∫0tk1t,sgs,usds,Sut=∫0ak2t,shs,usds,E is a nonreflexive Banach space, Dc0+α denotes the fractional Caputo derivative, k1∈C(D,R+),k2∈C(D0,R+),D=t,s∈R2:0≤s≤t≤1, D0=t,s∈R2:0≤t,s≤1, a1,b1,d1,a2,b2,d2 are real numbers, 0<ξ1,ξ2<1, f:I×E3→E,g,h:I×E→E are given functions satisfying some assumptions that will be specified later, the integral is understood to be the Henstock-Kurzweil-Pettis, and solutions to (5) will be sought in E=C(I,Eω).The problems of our research are different between this paper and paper [13]. In paper [13], we studied two point boundary value problem by using the corresponding Green’s function and fixed point theorems; moreover, we get some good results. In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the multipoint boundary value problem of fractional integrodifferential equations equipped with the weak topology. Our results generalized some classical results and improve the assumptions conditions, so our results improve the results in [13].The paper is organized as follows: In Section2 we recall some basic known results. In Section 3 we discuss the existence theorem of weak solutions for problem (5).


## 2. Preliminaries

Throughout this paper, we introduce notations, definitions, and preliminary results which will be used.LetI=[0,1] be the real interval, let E be a real Banach space with norm ·, its dual space E∗ also B(E∗) denotes the closed unit ball in E∗, and Ew=(E,w)=(E,σ(E,E∗)) denotes the space E with its weak topology. Denote by C(I,Eω)=(C(I,E),ω) the space of all continuous functions from I to E endowed with the weak topology and the usual supremum norm x=supt∈I⁡xt.LetΩE be the collection of all nonempty bounded subsets of E, and let WE be the subset of ΩE consisting of all weakly compact subsets of E. Let Br denote the closed ball in E centered at 0 with radius r>0. The De Blasi [14] measure of weak noncompactness is the map β:ΩE→0,∞ defined by (7)βA=infr>0:thereexistsasetW∈WEsuchthatA⊆W+Brfor all A∈ΩE. The fundamental tool in this paper is the measure of weak noncompactness; for some properties of β(A) and more details see [3].Now, for the convenience of the reader, we recall some useful definitions of integrals.Definition 1 (see [15]).
A functionu:I→E is said to be Henstock-Kurzweil integrable on I if there exists an J∈E such that, for every ε>0, there exists δ(ξ):I⟶R+ such that, for every δ-fine partition D=Ii,ξii=1n, we have (8)∑i=1nuξiμIi-J<ε,and we denote the Henstock-Kurzweil integral J by (HK) ∫abu(s)ds.Definition 2 (see [15]).
A functionf:I→E is said to be Henstock-Kurzweil-Pettis integrable or simply HKP-integrable on I, if there exists a function g:I→E with the following properties:(i)
∀x∗∈E∗,x∗f is Henstock-Kurzweil integrable on I;(ii)
∀t∈I,∀x∗∈E∗,x∗g(t)=(HK)∫0tx∗f(s)ds.
This functiong will be called a primitive of f and be denote by g(t)=∫0tf(t)dt the Henstock-Kurzweil-Pettis integral of f on the interval I.Definition 3 (see [16]).
A familyM of functions f:S→E is called HK-equi-integrable if each f∈M is HK-integrable and for every ε>0 there exists a gauge δ on S such that, for every δ-fine HK-partition π of S, we have (9)∑I,s∈πfsλmI-HK∫S​≤ε,for all f∈M.Theorem 4 (see [16]).
Let(fn) be a pointwise bounded sequence of HKP integrable functions fn:S→E and let f:S→E be a function. Assume that,(i)
for everyx∗∈E∗,x∗(fn(t))→x∗(f(t))a.e.onS,(ii)
for every sequence(xk∗)⊂B(E∗), the sequence (xk∗(fn))k,n is HK-equi-integrable, then f is HKP-integrable and for every I∈I, and we have (10)limn→∞⁡FnI=FIin the weak topology σ(E,E∗), where F is the HKP-primitive of f and S is a fixed compact nondegenerate interval in Rn. Denote by I the family of all closed nondegenerate subintervals of S.Lemma 5 (see [17]).
IfB⊂C(I,E) is equicontinuous, u0∈C(I,E), then co¯B,u0 is also equicontinuous in C(I,E).Lemma 6 (see [17, 18]).
LetE be a Banach space, and let B⊂C(I,E) be bounded and equicontinuous. Then β(B(t)) is continuous on I, and β(B)=maxt∈I⁡β(B(t)).Lemma 7 (see [14, 19]).
LetE be a Banach space and let B⊂C(I,E) be bounded and equicontinuous. Then the map t→β(B(t)) is continuous on I and (11)βB=supt∈I⁡βBt=βBI,where B(t)=bt:b∈B and B(I)=⋃t∈Ibt:h∈B.Lemma 8 (see [17]).
LetB⊂C(I,E) be bounded and equicontinuous. Then β(B(t)) is continuous on I and (12)β∫IBsds≤∫IβBsds.We give the fixed point theorem, which play a key role in the proof of our main results.Lemma 9 (see [20]).
LetE be a Banach space and β a regular and set additive measure of weak noncompactness on E. Let C be a nonempty closed convex subset of E, x0∈C, and n0 a positive integer. Suppose F:C→C is β-convex power condensing about x0 and n0. If F is weakly sequentially continuous and F(C) is bounded, then F has a fixed point in C.The following we recall the definition of the Caputo derivative of fractional order.Definition 10.
Letx:I→E be a function. The fractional HKP-integral of the function x of order α∈R+ is defined by (13)I0+αxt≔∫0tt-sα-1Γαxsds.In the above definition the sign “∫​” denotes the HKP-integral integral.Definition 11.
The Riemann-Liouville derivative of orderα with the lower limit zero for a function f:0,∞→R can be written as (14)D0+αft=1Γn-αdndtn∫0tfst-sα+1-nds,t>0,n-1<α<n.Definition 12.
The Caputo fractional derivative of orderα for a function f:0,∞→E can be written as (15)Dc0+αft=D0+αft-∑k=0n-1tkk!fk0,t>0,n-1<α<n,where n=[α]+1 and [α] denotes the integer part of α.


## 3. Main Results

In this section, we present the existence of solutions to problem (5) in the space C(I,Eω).Definition 13.
A functionx∈C(I,Ew) is said to be a solution of problem (5) if x satisfies the equation Dc0+αx(t)=f(t,x(t),(Tx)(t),(Sx)(t)) on I and satisfies the conditions a1x(0)-b1x′(0)=d1x(ξ1),a2x(1)+b2x′(1)=d2x(ξ2).Lemma 14 (see [21]).
Letα>0. If one assumes u∈C(0,1)∩L(0,1), then the differential equation (16)Dc0+αut=0has solution u(t)=c0+c1t+c2t2+⋯+cntn-1,ci∈R,i=0,1,…,n,n=[α]+1.From the lemma above, we deduce the following statement.Lemma 15 (see [21]).
Assume thatu∈C(0,1)∩L(0,1) with a fractional derivative of order α>0 that belongs to C(0,1)∩L(0,1). Then (17)I0+αDc0+αut=ut+c0+c1t+c2t2+⋯+cntn-1for some ci∈R,i=0,1,…,n,n=[α]+1.The following we give the corresponding Greens function for problem (5).Lemma 16.
LetΔ≠0,ρ∈C(I,Ew) and α∈1,2, then the unique solution of (18)Dc0+αxt=ρt,t∈I,a1x0-b1x′0=d1xξ1,a2x1+b2x′1=d2xξ2is given by (19)xt=∫01Gt,sρsds,where the Green function G is given by (20)Gt,s=t-sα-1Γα+d1a21-t+b2+d2t-ξ2ξ1-sα-1ΔΓα-a2b1+d1ξ1+ta1-d11-sα-1ΔΓα-b2b1+d1ξ1+ta1-d11-sα-2ΔΓα-1+d2b1+d1ξ1+ta1-d1ξ2-sα-1ΔΓα,s≤ξ1,s≤t;d1a21-t+b2+d2t-ξ2ξ1-sα-1ΔΓα-a2b1+d1ξ1+ta1-d11-sα-1ΔΓα-b2b1+d1ξ1+ta1-d11-sα-2ΔΓα-1+d2b1+d1ξ1+ta1-d1ξ2-sα-1ΔΓα,s≤ξ1,t≤s;t-sα-1Γα-a2b1+d1ξ1+ta1-d11-sα-1ΔΓα-b2b1+d1ξ1+ta1-d11-sα-2ΔΓα-1+d2b1+d1ξ1+ta1-d1ξ2-sα-1ΔΓα,ξ1≤s≤ξ2,s≤t;-a2b1+d1ξ1+ta1-d11-sα-1ΔΓα-b2b1+d1ξ1+ta1-d11-sα-2ΔΓα-1+d2b1+d1ξ1+ta1-d1ξ2-sα-1ΔΓα,ξ1≤s≤ξ2,t≤s;t-sα-1Γα-a2b1+d1ξ1+ta1-d11-sα-1ΔΓα-b2b1+d1ξ1+ta1-d11-sα-2ΔΓα-1,ξ2≤s,s≤t;-a2b1+d1ξ1+ta1-d11-sα-1ΔΓα-b2b1+d1ξ1+ta1-d11-sα-2ΔΓα-1,ξ2≤s,t≤s.Proof.
Based on the idea of paper [7], assuming that x(t) satisfies (18), by Lemma 15, we formally put (21)xt=I0+αρt-c1-c2t=1Γα∫0tt-sα-1ρtds-c1-c2tfor some constants c1,c2∈R.
On the other hand, by the relationsD0+αI0+αx(t)=x(t) and I0+αI0+βx(t)=I0+α+βx(t), for α,β>0,x∈C(I,Ew), we get (22)x′t=1Γα-1∫0tt-sα-2ρsds-c2.By the boundary conditions of (18), we have (23)d1-a1c1+b1+d1ξ1c2=d1I0+αρξ1+b1I0+α-1ρ0-a1I0+αρ0,d2-a2c1+-a2-b2+d2ξ2c2=d2I0+αρξ2-b2I0+α-1ρ1-a2I0+αρ1,By the proof of paper [12], we get (24)c1=-d1a2+b2-d2ξ2ΔΓα∫0ξ1ξ1-sα-1ρsds+b1+d1ξ1Δa2∫011-sα-1Γαρsds+b2∫011-sα-2Γα-1ρsds-d2∫0ξ2ξ2-sα-1Γαρsds,c2=d1a2-d2ΔΓα∫0ξ1ξ1-sα-1ρsds+a1-d1Δa2∫011-sα-1Γαρsds+b2∫011-sα-2Γα-1ρsds-d2∫0ξ2ξ2-sα-1Γαρsds,where Δ=[(b1+d1ξ1)(a2-d2)+(a2+b2-d2ξ2)(a1-d1)]≠0. Substituting the values of c1 and c2 in (21), we get (25)xt=1Γα∫0tt-sα-1ρsds+d1a21-t+b2+d2t-ξ2ΔΓα∫0ξ1ξ1-sα-1ρsds-a2b1+d1ξ1+ta1-d1ΔΓα∫011-sα-1ρsds-b2b1+d1ξ1+ta1-d1ΔΓα-1∫011-sα-2ρsds+d2b1+d1ξ1+ta1-d1ΔΓα∫0ξ2ξ2-sα-1ρsds=∫01Gt,sρsds.This completes the proof.LetDr=z∈CI,Ew,z≤r, BV(I,R) denote the space of real bounded variation functions with its classical norm ·BV.Problem (5) will be studied under the following assumptions:(1)
For each weakly continuous functionx:I→E, the functions k1(t,·)g(·,x(·)),k2(t,·)h(·,x(·)), f(·,x(·),T(x)(·),S(x)(·)) are HKP-integrable, f:I×E3→E,g,h:I×E→E are weakly-weakly continuous function, and ∫0tg(s,x(s))ds,∫01h(s,x(s))ds are bounded.(2)
(i)
For anyr>0, there exist a HK-integrable function m:I→R+ and nondecreasing continuous functions ψ1:0,+∞→(0,∞),ψ2:0,+∞→[0,+∞),ψ3:0,+∞→0,+∞, ψ2,ψ3 satisfying ψ2(λx)≤λψ2(x),ψ3(λx)≤λψ3(x) for λ>0 such that (26)fs,x,y,z≤msψ1x+ψ2y+ψ3z,ψ2gs,x≤ψ2x,ψ3hs,x≤ψ3xfor alls∈I,(x,y,z)∈Dr×Dr×Dr with (27)∫01Msds<∫0∞dr∑i=13ψis.(ii)
For each bounded setX,Y,Z⊂Dr, and each for each closed interval J⊂I,t∈I, there exists positive constant l≥0 such that (28)β(k1J,JgJ,Y≤k1∗βYJ,β(k2J,JhJ,Z≤k2∗βZJβft,X,Y,Z≤lmax⁡βX,βY,βZ,whereM(s)=G∗msmax1,ak1∗,ak2∗,k1∗=supt∈I⁡k1(t,·)BV,k2∗=supt∈I⁡k2(t,·)BV.(3)
For eacht∈I,G(t,.), ki(t,·)∈BV(I,R),i=1,2 are continuous; i.e., the maps t↦G(t,.) and t↦ki(t,.) are .BV-continuous.(4)
The family{x∗f(·,x(·),T(x)(·),S(x)(·)):x∗∈E∗,x∗≤1} is uniformly HK-integrable over I for every x∈Dr.Remark 17.
From assumption(3) and the expression of function G(t,s), it is obvious that it is bounded and let G∗=supt∈I⁡G(t,·)BV.Now, we present the existence theorem for problem (5).Theorem 18.
Assume that conditions (5)-(20). Then problem (5) has a solution x∈C(I,Ew).Proof.
Letm=max⁡supt∈I⁡kit,·BV,i=1,2 and k0=max⁡supt∈I⁡∫0tgs,xsds,supt∈I⁡∫01hs,xsds. Let 0<k0<min⁡r0,r0/m, for x∈Dr0 and x∗∈E∗ such that x∗≤1; we have (29)x∗Txs=HK∫0tx∗k1t,sgs,xsds≤x∗supt∈I⁡k1t,·BV∫01gs,xsds≤m·k0≤r0,and also (30)sup⁡x∗Tx:x∈E∗,x∗≤1≤r0.So Tx∈Dr0. Similarly, we prove Sx∈Dr0.
Defining the set(31)Q≔x∈Dr0:x·≤r0,xt-xs≤r0G∗Gt2,·-Gt1,·BV,t1,t2∈I,it is clear that the convex closed and equicontinuous subset Q⊂Dr0⊂C(I,Ew), where (32)bt=I-1∫0tMsdsandIz=∫0zds∑i=13ψis.Clearly, (33)b′t=Mt∑i=13ψibt,andb0=0for all t∈I. Also notice that Q is a closed, convex, bounded, and equicontinuous subset of C(I,Ew). We define the operator F:C(I,Ew)→C(I,Ew) by (34)Fxt=∫01Gt,sfs,xs,Txs,Sxsds,t∈I,where G(·,·) is Green’s function defined by (20). Clearly the fixed points of the operator F are solutions of problem (5). Since for t∈I the function s↦G(t,s) is of bounded variation, then by the proof of Theorem 3.1 in [13] and assumption (4), the function G(t,·)f(·,x(·),T(x)(·),S(x)(·)) is HKP-integrable on I and thus the operator F makes sense.
We will show thatF satisfies the assumptions of Lemma 8; the proof will be given in three steps.
Step  1. We shall show that the operator F maps into itself. To see this, let x∈Q,t∈I. Without loss of generality, assume that Fx(t)≠0. By Hahn-Banach theorem, there exists x∗∈E∗ with x∗=1 and Fx(t)=x∗Fxt. Thus (35)Fxt=x∗Fxt≤x∗∫01Gt,sfs,xs,Txs,Sxsds≤supt∈I⁡Gt,·BV∫01msψ1bs+ak1∗ψ2bs+ak2∗ψ3bsds≤∫01b′sds≤I-1∫01Msds=r0.Then Fx=supt∈I⁡Fxt≤r0. Hence F:Q→Q.
Let0<t1<t2≤1, without loss of generality; assume that Fx(t2)-Fx(t1)≠0. By Hahn-Banach theorem, there exists x∗∈E∗ with x∗=1 and(36)Fxt2-Fxt1=x∗Fxt2-Fxt1≤∫01Gt2,s-Gt1,s·x∗fs,xs,Txs,Sxsds≤Gt2,·-Gt1,·BV∫01msψ1bs+ak1∗ψ2bs+ak2∗ψ3bsds≤1G∗Gt2,·-Gt1,·BV∫01b′sds≤1G∗Gt2,·-Gt1,·BVI-1∫01Msds=r0G∗Gt2,·-Gt1,·BV,and this estimation shows that F maps Q into itself.
Step  2. We will show that the operator F is weakly sequentially continuous. In order to be simple, we denote Tx(t)=ϕ(x)(t)=∫01k1(t,s)g(s,x(s))ds,Sx(t)=φ(x)(t)=∫01k2(t,s)h(s,x(s))ds. To see this, by Lemma 9 of [22], a sequence xn(·) weakly convergent to x(·)∈Q if and only if xn(·) tends weakly to x(t) for each t∈I. From Dinculeanu ([23, p. 380]) (C(I,E))∗=M(I,E∗), M(I,E∗) is the set of all bounded regular vector measures from I to E∗ which are of bounded variation). Let x∗∈E∗,t∈I. Put Pt=x∗δt, where δt is the Dirac measure concentrated at the point t. Then Pt∈M(I,E∗). Since xn converges weakly to x∈Q, then we have (37)limn→∞⁡pt,xn-x=0which means that (38)limn→∞⁡x∗,xn-x=0.Thus, for each t∈I, xn(t) converges weakly to x(t)∈E. Since g(s,·),h(s,·) are weakly-weakly sequentially continuous, then g(s,xn(s)) and h(s,xn(s)) converge weakly to g(s,x(s)) and h(s,x(s)), respectively. Hence, and by Theorem 4 and assumptions (1), we have (39)limn→∞⁡∫01∫0tk1t,sgs,xns-k1t,sgs,xsdsdms=0,∀m∈MI,E∗.This relation is equivalent to(40)limn→∞⁡m,ϕxn-ϕx=limn→∞⁡∫01ϕxnt-ϕxtdmt=0,∀m∈MI,E∗.
Similarly, we have(41)limn→∞⁡∫01∫01k2t,shs,xns-k2t,shs,xsdsdms=0,∀m∈MI,E∗.This relation is equivalent to (42)limn→∞⁡m,φxn-φx=limn→∞⁡∫01φxnt-φxtdmt=0,∀m∈MI,E∗.Therefore, the operators T,S are weakly sequentially continuous in Q.
Moreover, becausef is weakly-weakly sequentially continuous, we have that f(s,xn(s),(Txn)(s),(Sxn)(s)) converges weakly to f(s,x(s),(Tx)(s),(Sx)(s)) in E. By assumption (4), for every weakly convergent (xn)n⊂Dr0, the set (43)x∗f·,xn·,Txn·,Sxn·:n∈N,x∗∈BE∗is HK-equi-integrable. Since for t∈I the function s↦G(t,s) is of bounded variation, and by the proof of Theorem 3.1 in [13], the function G(t,·)f(·,xn(·),(Txn)(·),(Sxn)(·)) is HKP-integrable on I for every n≥1, and by Theorem 4, we have that ∫01G(t,s)f(s,xn(s),(Txn)(s),(Sxn)(s))ds converges weakly to ∫01G(t,s)f(s,x(s),(Tx)(s),(Sx)(s))ds in E which means that (44)limn→∞⁡∫01∫01(Gt,sfs,xns,Txns,Sxns-fs,xs,Txs,Sxsdsdms=0,for all m∈M(I,E∗). This relation is equivalent to (45)limn→∞⁡m,Fxn-Fx=limn→∞⁡∫01Fxnt-Fxtdmt=0,∀m∈MI,E∗.Therefore F is weakly-weakly sequentially continuous.
Step  3. We show that there is an integer n0 such that the operator F is β-power-convex condensing about 0 and n0. To see this, notice that, for each bounded set H⊆Q and for each t∈I, (46)βF1,0Ht=βFHt=β∫0tGt,sfs,xs,Txs,Sxsds:x∈H≤βG∗tco¯fs,xs,Txs,Sxs:x∈H,s∈I=G∗tβco¯fs,xs,Txs,Sxs:x∈H,s∈I≤G∗tβfI×HI×THI×SHI≤G∗t·max⁡1,k1∗,k2∗·lβHI.Let τ=G∗·max1,k1∗,k2∗·l>0. Lemma 7 implies (since H is equicontinuous) that (47)βF1,0Ht≤tτβH.Since F(1,0)(H) is equicontinuous, it follows from Lemma 5 that F(2,0)(H) is equicontinuous. Using (47), we get (48)βF2,0Ht=β∫0tGt,sfs,xs,Txs,Sxsds:x∈co¯F1,0H∪0≤β∫0tGt,sfs,xs,Txs,Sxsds:x∈V,where V=co¯(F1,0(H)∪0); it is clear that V is equicontinuous set. By Lemma 8, we get (49)βVs=βF1,0Hs≤sτβH,and therefore, (50)∫0tβVsds≤sτt22βH.Thus, (51)βF2,0Ht≤τt22βH.By induction, we get (52)βFn,0Ht≤τtnn!βH.And by Lemma 7, we have (53)βFn,0H≤τTnn!βH.Since limn→∞⁡τtn/n!=0, then there exist an n0 with τtn0/n0!<1, and we have (54)βFn0,0H≤βH.Consequently, F is β-power-convex condensing about 0 and n0, by Lemma 8, then problem (5) has a solution x∈C(I,Ew).


## 4. Conclusions

In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the multipoint boundary value problem of fractional integrodifferential equations equipped with the weak topology. Our results generalized some classical results.



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