This article investigates a boundary value problem of Riemann-Liouville fractional integro-differential 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.

In this article, we study the existence and uniqueness of solutions for the following nonlinear fractional integro-differential equation:

subject to the boundary conditions of fractional order given by

where D^α denotes the Riemann-Liouville 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 fractional-order models more realistic and practical than the classical integer-order 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 [3-19] and references therein).

Let us recall some basic definitions (see [20,21]).

Definition 2.1 The Riemann-Liouville 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 Riemann-Liouville 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^αu = I^-αu. Also for β ∈ [0, α), it is valid that D^β I^au = I^α-βu.

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^α1 = 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^αI^αu = 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 Riemann-Liouville 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 c₀ = 0 and

where A is defined by (2.2).

Substituting the values of c₀ and c₁ in (2.3), the unique solution of (2.1) subject to the boundary conditions (1.2)-(1.3) is given by

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

Define an operator as

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

Hence, is uniformly bounded.

(ii) For any t₁, t₂ ∈ [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

(A₁) there exist positive functions L₁(t), L₂(t), L₃(t) such that

(A₂) Λ = (ξ₁ + |ν₁|T^α-1ξ₂)(1 + γ₀ + δ₀) < 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 (A₁), for every t ∈ [0, T], we have

By assumption (A₂), 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:

(H₁)

(H₂) |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

For , we find that

Thus, . It follows from the assumption (H₁) 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 (H₁), we define , and consequently we have

which is independent of u and tends to zero as t₂ → t₁. So, is relatively compact on . Hence, by the Arzelá-Ascoli 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.   □

The authors declare that they have no competing interests.

Both authors, BA and JJN, contributed to each part of this work equally and read and approved the final version of the manuscript.

This study was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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