Abstract
This article investigates a boundary value problem of RiemannLiouville fractional integrodifferential equations with fractional nonlocal integral boundary conditions. Some new existence results are obtained by applying standard fixed point theorems.
2010 Mathematics Subject Classification: 26A33; 34A34; 34B15.
Keywords:
RiemannLiouville calculus; fractional integrodifferential equations; fractional boundary conditions; fixed point theorems1 Introduction
In this article, we study the existence and uniqueness of solutions for the following nonlinear fractional integrodifferential equation:
subject to the boundary conditions of fractional order given by
where D^{α }denotes the RiemannLiouville fractional derivative of order α, f: [0, T] × ℝ × ℝ × ℝ → ℝ is continuous, and
with γ and δ being continuous functions on [0, T] × [0, T].
Boundary value problems for nonlinear fractional differential equations have recently been investigated by several researchers. As a matter of fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes (see [1]) and make the fractionalorder models more realistic and practical than the classical integerorder models. Fractional differential equations arise in many engineering and scientific disciplines, such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. (see [1,2]). For some recent development on the topic, (see [319] and references therein).
2 Preliminaries
Let us recall some basic definitions (see [20,21]).
Definition 2.1 The RiemannLiouville fractional integral of order α > 0 for a continuous function u: (0, ∞) → ℝ is defined as
provided the integral exists.
Definition 2.2 For a continuous function u: (0, ∞) → ℝ, the RiemannLiouville derivative of fractional order α > 0, n = [α] + 1 ([α] denotes the integer part of the real number α) is defined as
provided it exists.
For α < 0, we use the convention that D^{α}u = I^{α}u. Also for β ∈ [0, α), it is valid that D^{β }I^{a}u = I^{αβ}u.
Note that for λ >1, λ ≠ α  1, α  2,..., α  n, we have
and
In particular, for the constant function u(t) = 1, we obtain
For α ∈ ℕ, we get, of course, D^{α}1 = 0 because of the poles of the gamma function at the points 0, 1, 2,....
For α > 0, the general solution of the homogeneous equation
in C(0, T) ∩ L(0, T) is
where c_{i}, i = 1, 2,..., n  1, are arbitrary real constants.
We always have D^{α}I^{α}u = u, and
To define the solution for the nonlinear problem (1.1) and (1.2)(1.3), we consider the following linear equation
where σ ∈ C[0, T].
We define
such that A ≠ Γ(α).
The general solution of (2.1) is given by
with I^{α }the usual RiemannLiouville fractional integral of order α.
From (2.3), we have
Using the conditions (1.2) and (1.3) in (2.4) and (2.5), we find that c_{0 }= 0 and
where A is defined by (2.2).
Substituting the values of c_{0 }and c_{1 }in (2.3), the unique solution of (2.1) subject to the boundary conditions (1.2)(1.3) is given by
3 Main results
Let denotes the Banach space of all continuous functions from [0, T] → ℝ endowed with the norm defined by ║u║ = sup{u(t), t ∈ [0, T]}.
If u is a solution of (1.1) and (1.2)(1.3), then
where
Observe that the problem (1.1) and (1.2)(1.3) has solutions if and only if the operator equation has fixed points.
Lemma 3.1 The operator is compact.
Proof
(i) Let B be a bounded set in C[0, T]. Then, there exists a constant M such that f(t,u(t), (φu)(t), (ψu)(t)) ≤ M, ∀u ∈ B, t∈[0, T]. Thus
which implies that
(ii) For any t_{1}, t_{2 }∈ [0, T], u ∈ B, we have
Thus, is equicontinuous. Consequently, the operator is compact. This completes the proof. □
We need the following known fixed point theorem to prove the existence of solutions for the problem at hand.
Theorem 3.1 ([22]) Let E be a Banach space. Assume that T: E → E be a completely continuous operator and the set V = {x ∈ E  x = μTx, 0 < μ < 1} be bounded.
Then, T has a fixed point in E.
Theorem 3.2 Assume that there exists a constant M > 0 such that
Then, the problem (1.1) and (1.2)(1.3) has at least one solution on [0,T].
Proof We consider the set
and show that the set V is bounded. Let u ∈ V, then , 0 < μ < 1. For any t ∈ [0, T], we have
As in part (i) of Lemma 3.1, we have
This implies that the set V is bounded independently of μ ∈ (0,1). Using Lemma 3.1 and Theorem 3.1, we obtain that the operator has at least a fixed point, which implies that the problem (1.1) and (1.2)(1.3) has at least one solution. This completes the proof.
Theorem 3.3 Assume that
(A_{1}) there exist positive functions L_{1}(t), L_{2}(t), L_{3}(t) such that
(A_{2}) Λ = (ξ_{1 }+ ν_{1}T^{α1}ξ_{2})(1 + γ_{0 }+ δ_{0}) < 1, where
Then the problem (1.1) and (1.2)(1.3) has a unique solution on C[0, T].
Proof Let us set sup_{t∈[0, T] }f(t,0,0,0) = M, and choose
Then we show that , where . For x ∈ B_{r}, we have
In view of (A_{1}), for every t ∈ [0, T], we have
By assumption (A_{2}), Λ < 1, therefore, the operator is a contraction. Hence, by Banach fixed point theorem, we deduce that has a unique fixed point which in fact is a unique solution of problem (1.1) and (1.2)(1.3). This completes the proof. □
Theorem 3.4 (Krasnoselskii's fixed point theorem [22]). Let be a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i) whenever ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then, there exists such that z = Az + Bz.
Theorem 3.5 Assume that f: [0, T] × ℝ × ℝ × ℝ → ℝ is a continuous function and the following assumptions hold:
(H_{1})
(H_{2}) f (t,u) ≤ μ(t), ∀(t,u)∈[0, T] × ℝ, and μ ∈ C([0, T],ℝ^{+}).
If
then the boundary value problem (1.1) and (1.2)(1.3) has at least one solution on [0, T].
Proof Letting sup_{t∈[0, T] }μ(t) = μ, we fix
and consider . We define the operators and on as
Thus, . It follows from the assumption (H_{1}) together with (3.1) that is a contraction mapping. Continuity of f implies that the operator is continuous.
Also, is uniformly bounded on as
Now we prove the compactness of the operator .
In view of (H_{1}), we define , and consequently we have
which is independent of u and tends to zero as t_{2 }→ t_{1}. So, is relatively compact on . Hence, by the ArzeláAscoli Theorem, is compact on . Thus, all the assumptions of Theorem 3.4 are satisfied. So the conclusion of Theorem 3.4 implies that the boundary value problem (1.1) and (1.2)(1.3) has at least one solution on [0, T]. This completes the proof. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors, BA and JJN, contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
This study was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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