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
If u is a solution of (1.1) and (1.2)(1.3), then
where
Define an operator
Observe that the problem (1.1) and (1.2)(1.3) has solutions if and only if the operator
equation
Lemma 3.1 The operator
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
Hence,
(ii) For any t_{1}, t_{2 }∈ [0, T], u ∈ B, we have
Thus,
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
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
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
In view of (A_{1}), for every t ∈ [0, T], we have
By assumption (A_{2}), Λ < 1, therefore, the operator
Theorem 3.4 (Krasnoselskii's fixed point theorem [22]). Let
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
For
Thus,
Also,
Now we prove the compactness of the operator
In view of (H_{1}), we define
which is independent of u and tends to zero as t_{2 }→ t_{1}. So,
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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