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.

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. Zhou et al. [12] 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 [13], we discussed the existence theorem of weak solutions nonlinear fractional integrodifferential equations in nonreflexive Banach spaces E:

()

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 T and S are two operators defined by

()

E is a nonreflexive Banach space,

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, E_ω).

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 Section 2 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.

Let I = [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 E_w = (E, w) = (E, σ(E, E^∗)) denotes the space E with its weak topology. Denote by C(I, E_ω) = (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‖ = sup_t∈I⁡|x(t)|.

Let Ω_E be the collection of all nonempty bounded subsets of E, and let

be the subset of Ω_E consisting of all weakly compact subsets of E. Let B_r 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 : Ω_E → [0, ∞ defined by

()

for all A ∈ Ω_E. 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 function u : 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 such that, for every δ-fine partition , we have

()

and we denote the Henstock-Kurzweil integral J by (HK)

Definition 2 (see [15].)A function f : 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)
.

This function g will be called a primitive of f and be denote by the Henstock-Kurzweil-Pettis integral of f on the interval I.

Definition 3 (see [16].)A family of functions f : S → E is called HK-equi-integrable if each 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

()

for all

Theorem 4 (see [16].)Let (f_n) be a pointwise bounded sequence of HKP integrable functions f_n : S → E and let f : S → E be a function. Assume that,

(i)
for every x^∗ ∈ E^∗, x^∗(f_n(t)) → x^∗(f(t)) a.e. on S,
(ii)
for every sequence , the sequence is HK-equi-integrable, then f is HKP-integrable and for every , and we have
()

in the weak topology σ(E, E^∗), where F is the HKP-primitive of f and S is a fixed compact nondegenerate interval in

. Denote by

the family of all closed nondegenerate subintervals of S.

Lemma 5 (see [17].)If B ⊂ C(I, E) is equicontinuous, u₀ ∈ C(I, E), then is also equicontinuous in C(I, E).

Lemma 6 (see [17], [18].)Let E 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) = max_t∈I⁡β(B(t)).

Lemma 7 (see [14], [19].)Let E 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

()

where B(t) = {b(t) : b ∈ B} and

Lemma 8 (see [17].)Let B ⊂ C(I, E) be bounded and equicontinuous. Then β(B(t)) is continuous on I and

()

We give the fixed point theorem, which play a key role in the proof of our main results.

Lemma 9 (see [20].)Let E 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. Let x : I → E be a function. The fractional HKP-integral of the function x of order is defined by

()

In the above definition the sign “∫” denotes the HKP-integral integral.

Definition 11. The Riemann-Liouville derivative of order α with the lower limit zero for a function f : [0, ∞ → R can be written as

()

Definition 12. The Caputo fractional derivative of order α for a function f : [0, ∞ → E can be written as

()

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, E_ω).

Definition 13. A function x ∈ C(I, E_w) is said to be a solution of problem (5) if x satisfies the equation on I and satisfies the conditions a₁x(0) − b₁x^′(0) = d₁x(ξ₁), a₂x(1) + b₂x^′(1) = d₂x(ξ₂).

Lemma 14 (see [21].)Let α > 0. If one assumes u ∈ C(0,1)∩L(0,1), then the differential equation

()

has solution

From the lemma above, we deduce the following statement.

Lemma 15 (see [21].)Assume that u ∈ C(0,1)∩L(0,1) with a fractional derivative of order α > 0 that belongs to C(0,1)∩L(0,1). Then

()

for some

The following we give the corresponding Greens function for problem (5).

Lemma 16. Let Δ ≠ 0, ρ ∈ C(I, E_w) and α ∈ 1,2], then the unique solution of

()

is given by

()

where the Green function G is given by

()

Proof. Based on the idea of paper [7], assuming that x(t) satisfies (18), by Lemma 15, we formally put

()

for some constants

On the other hand, by the relations and , for α, β > 0, x ∈ C(I, E_w), we get

()

By the boundary conditions of (18), we have

()

By the proof of paper [12], we get

()

where Δ = [(b₁ + d₁ξ₁)(a₂ − d₂)+(a₂ + b₂ − d₂ξ₂)(a₁ − d₁)] ≠ 0. Substituting the values of c₁ and c₂ in (21), we get

()

This completes the proof.

Let D_r = {z ∈ C(I, E_w), ‖z‖ ≤ 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 function x : 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 are bounded.
(2)
- (i)
  For any r > 0, there exist a HK-integrable function 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
  ()
- for all s ∈ I, (x, y, z) ∈ D_r × D_r × D_r with
  ()
- (ii)
  For each bounded set X, Y, Z ⊂ D_r, and each for each closed interval J ⊂ I, t ∈ I, there exists positive constant l ≥ 0 such that
  ()
- where
(3)
For each t ∈ I, G(t.), are continuous; i.e., the maps t ↦ G(t.) and t ↦ k_i(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 ∈ D_r.

Remark 17. From assumption (3) and the expression of function G(t, s), it is obvious that it is bounded and let G^∗ = sup_t∈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, E_w).

Proof. Let and . Let 0 < k₀ < min⁡(r₀, r₀/m), for and x^∗ ∈ E^∗ such that ‖x‖^∗ ≤ 1; we have

()

and also

()

. Similarly, we prove

Defining the set

()

it is clear that the convex closed and equicontinuous subset

, where

()

Clearly,

()

for all t ∈ I. Also notice that Q is a closed, convex, bounded, and equicontinuous subset of C(I, E_w). We define the operator F : C(I, E_w) → C(I, E_w) by

()

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 that F 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^∗(Fx(t))|. Thus

()

Then ‖Fx‖ = sup_t∈I⁡|Fx(t)| ≤ r₀. Hence F : Q → Q.

Let 0 < 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

()

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 . To see this, by Lemma 9 of [22], a sequence x_n(·) weakly convergent to x(·) ∈ Q if and only if x_n(·) 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 P_t = x^∗δ_t, where δ_t is the Dirac measure concentrated at the point t. Then P_t ∈ M(I, E^∗). Since x_n converges weakly to x ∈ Q, then we have

()

which means that

()

Thus, for each t ∈ I, x_n(t) converges weakly to x(t) ∈ E. Since g(s, ·), h(s, ·) are weakly-weakly sequentially continuous, then g(s, x_n(s)) and h(s, x_n(s)) converge weakly to g(s, x(s)) and h(s, x(s)), respectively. Hence, and by Theorem 4 and assumptions (1), we have

()

This relation is equivalent to

()

Similarly, we have

()

This relation is equivalent to

()

Therefore, the operators T, S are weakly sequentially continuous in Q.

Moreover, because f is weakly-weakly sequentially continuous, we have that f(s, x_n(s), (Tx_n)(s), (Sx_n)(s)) converges weakly to f(s, x(s), (Tx)(s), (Sx)(s)) in E. By assumption (4), for every weakly convergent , the set

()

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(·, x_n(·), (Tx_n)(·), (Sx_n)(·)) is HKP-integrable on I for every n ≥ 1, and by Theorem 4, we have that

converges weakly to

in E which means that

()

for all m ∈ M(I, E^∗). This relation is equivalent to

()

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,

()

Let

. Lemma 7 implies (since H is equicontinuous) that

()

Since F^(1,0)(H) is equicontinuous, it follows from Lemma 5 that F^(2,0)(H) is equicontinuous. Using (47), we get

()

where

; it is clear that V is equicontinuous set. By Lemma 8, we get

()

and therefore,

()

Thus,

()

By induction, we get

()

And by Lemma 7, we have

()

Since lim_n→∞⁡((τt)ⁿ/n!) = 0, then there exist an n₀ with

, and we have

()

Consequently, F is β-power-convex condensing about 0 and n₀, by Lemma 8, then problem (5) has a solution x ∈ C(I, E_w).

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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11061031).

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

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Conclusions

Conflicts of Interest

Authors’ Contributions

Acknowledgments

References

References

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

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Conclusions

Conflicts of Interest

Authors’ Contributions

Acknowledgments

References

References

Related

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