Research Article | Open Access
Existence of Weak Solutions for Fractional Integrodifferential Equations with Multipoint Boundary Conditions
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.
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  introduced the concept of measure of weak noncompactness and proved the analogue of Sadovskiis fixed point theorem for the weak topology (see also ). As stressed in , 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. Zhou et al.  discuss the existence of solutions for nonlinear multipoint boundary value problem of integrodifferential equations of fractional order as follows: with respect to strong topology, where denotes the fractional Caputo derivative and the operators given by
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 , we discussed the existence theorem of weak solutions nonlinear fractional integrodifferential equations in nonreflexive Banach spaces : and obtain a new result by using the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals, where denotes the fractional Caputo derivative and the operators given by 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 where and are two operators defined by is a nonreflexive Banach space, denotes the fractional Caputo derivative, , , are real numbers, , 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 .
The problems of our research are different between this paper and paper . In paper , 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 .
Throughout this paper, we introduce notations, definitions, and preliminary results which will be used.
Let be the real interval, let be a real Banach space with norm , its dual space also denotes the closed unit ball in , and denotes the space with its weak topology. Denote by the space of all continuous functions from to endowed with the weak topology and the usual supremum norm .
Let be the collection of all nonempty bounded subsets of , and let be the subset of consisting of all weakly compact subsets of Let denote the closed ball in centered at with radius . The De Blasi  measure of weak noncompactness is the map defined by for all The fundamental tool in this paper is the measure of weak noncompactness; for some properties of and more details see .
Now, for the convenience of the reader, we recall some useful definitions of integrals.
Definition 1 (see ). A function is said to be Henstock-Kurzweil integrable on if there exists an such that, for every , there exists such that, for every -fine partition , we have and we denote the Henstock-Kurzweil integral by (HK)
Definition 2 (see ). A function is said to be Henstock-Kurzweil-Pettis integrable or simply HKP-integrable on , if there exists a function with the following properties:(i) is Henstock-Kurzweil integrable on ;(ii).This function will be called a primitive of and be denote by the Henstock-Kurzweil-Pettis integral of on the interval .
Definition 3 (see ). A family of functions is called -equi-integrable if each is HK-integrable and for every there exists a gauge on such that, for every -fine -partition of , we have for all .
Theorem 4 (see ). Let be a pointwise bounded sequence of integrable functions and let be a function. Assume that,(i)for every ,(ii)for every sequence , the sequence is -equi-integrable, then is -integrable and for every , and we have in the weak topology , where is the -primitive of and is a fixed compact nondegenerate interval in . Denote by the family of all closed nondegenerate subintervals of .
Lemma 5 (see ). If is equicontinuous, , then is also equicontinuous in .
Lemma 8 (see ). Let be bounded and equicontinuous. Then is continuous on and
We give the fixed point theorem, which play a key role in the proof of our main results.
Lemma 9 (see ). Let be a Banach space and a regular and set additive measure of weak noncompactness on . Let be a nonempty closed convex subset of , , and a positive integer. Suppose is -convex power condensing about and . If is weakly sequentially continuous and is bounded, then has a fixed point in .
The following we recall the definition of the Caputo derivative of fractional order.
Definition 10. Let be a function. The fractional HKP-integral of the function of order is defined by
In the above definition the sign “” denotes the HKP-integral integral.
Definition 11. The Riemann-Liouville derivative of order with the lower limit zero for a function can be written as
Definition 12. The Caputo fractional derivative of order for a function can be written as where and denotes the integer part of .
3. Main Results
In this section, we present the existence of solutions to problem (5) in the space .
Definition 13. A function is said to be a solution of problem (5) if x satisfies the equation on and satisfies the conditions
Lemma 14 (see ). Let . If one assumes , then the differential equation has solution
From the lemma above, we deduce the following statement.
Lemma 15 (see ). Assume that with a fractional derivative of order that belongs to . Then for some .
The following we give the corresponding Greens function for problem (5).
Lemma 16. Let and , then the unique solution of is given by where the Green function is given by
Proof. Based on the idea of paper , assuming that satisfies (18), by Lemma 15, we formally put for some constants
On the other hand, by the relations and , for , we get By the boundary conditions of (18), we have By the proof of paper , we get where . Substituting the values of and in (21), we get This completes the proof.
Let , denote the space of real bounded variation functions with its classical norm
Problem (5) will be studied under the following assumptions:(1)For each weakly continuous function , the functions , are HKP-integrable, are weakly-weakly continuous function, and are bounded.(2)(i)For any , there exist a HK-integrable function and nondecreasing continuous functions , satisfying for such that for all with (ii)For each bounded set , and each for each closed interval , there exists positive constant such that where (3)For each , are continuous; i.e., the maps and are -continuous.(4)The family is uniformly HK-integrable over for every .
Remark 17. From assumption and the expression of function , it is obvious that it is bounded and let .
Now, we present the existence theorem for problem (5).
Proof. Let and . Let , for and such that ; we have and also So . Similarly, we prove .
Defining the set it is clear that the convex closed and equicontinuous subset , where Clearly, for all . Also notice that is a closed, convex, bounded, and equicontinuous subset of . We define the operator by where is Green’s function defined by (20). Clearly the fixed points of the operator are solutions of problem (5). Since for the function is of bounded variation, then by the proof of Theorem 3.1 in  and assumption , the function is HKP-integrable on and thus the operator makes sense.
We will show that satisfies the assumptions of Lemma 8; the proof will be given in three steps.
Step 1. We shall show that the operator maps into itself. To see this, let Without loss of generality, assume that . By Hahn-Banach theorem, there exists with and Thus Then Hence
Let , without loss of generality; assume that . By Hahn-Banach theorem, there exists with andand this estimation shows that maps into itself.
Step 2. We will show that the operator is weakly sequentially continuous. In order to be simple, we denote . To see this, by Lemma 9 of , a sequence weakly convergent to if and only if tends weakly to for each . From Dinculeanu ([23, p. 380]) , is the set of all bounded regular vector measures from to which are of bounded variation). Let Put , where is the Dirac measure concentrated at the point . Then . Since converges weakly to , then we have which means that Thus, for each , converges weakly to Since are weakly-weakly sequentially continuous, then and converge weakly to and , respectively. Hence, and by Theorem 4 and assumptions , we have This relation is equivalent toSimilarly, we have This relation is equivalent to Therefore, the operators are weakly sequentially continuous in .
Moreover, because is weakly-weakly sequentially continuous, we have that converges weakly to in . By assumption , for every weakly convergent , the set is HK-equi-integrable. Since for the function is of bounded variation, and by the proof of Theorem 3.1 in , the function is HKP-integrable on for every , and by Theorem 4, we have that converges weakly to in which means that for all This relation is equivalent to Therefore is weakly-weakly sequentially continuous.
Step 3. We show that there is an integer such that the operator is -power-convex condensing about and . To see this, notice that, for each bounded set and for each , Let . Lemma 7 implies (since is equicontinuous) that Since is equicontinuous, it follows from Lemma 5 that is equicontinuous. Using (47), we get where ; it is clear that is equicontinuous set. By Lemma 8, we get and therefore,