The Existence of Solutions for a Fractional 2-Point Boundary Value Problems

Abstract

By using the coincidence degree theory, we consider the following 2-point boundary value problem for fractional differential equation , , where and are the standard Riemann-Liouville fractional derivative and fractional integral, respectively. A new result on the existence of solutions for above fractional boundary value problem is obtained.

1. Introduction

Fractional differential equations have been of great interest recently. This is because of the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of mathematics, fractional differential equations arise in a variety of different areas such as rheology, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science (see [1–4] and references cited therein). The research of fractional differential equations on boundary value problems, as one of the focal topics has attained a great deal of attention from many researchers (see [5–13]).

However, there are few papers which consider the boundary value problem at resonance for nonlinear ordinary differential equations of fractional order. In [14], Hu and Liu studied the following BVP of fractional equation at resonance: where , is the standard Caputo fractional derivative.

In [15], Zhang and Bai investigated the nonlinear nonlocal problem where , they consider the case , that is, the resonance case.

In [16], Bai investigated the boundary value problem at resonance is considered, where is a real number, and are the standard Riemann-Liouville fractional derivative and fractional integral, respectively, and is continuous and are given constants such that

In this paper, we study the 2-point boundary value problem where , , satisfies Carathéodory conditions, and are the standard Riemann-Liouville fractional derivative and fractional integral, respectively.

Setting

In this paper, we will always suppose that the following conditions hold:(): (): 

We say that boundary value problem (1.4) and (1.5) is at resonance, if BVP has , as a nontrivial solution.

The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions, and lemmas. In Section 3, we establish a theorem on existence of solutions for BVP (1.4)-(1.5) under nonlinear growth restriction of , basing on the coincidence degree theory due to Mawhin (see [17]).

Now, we will briefly recall some notation and an abstract existence result.

Let be real Banach spaces, a Fredholm map of index zero,s and continuous projectors such that It follows that is invertible. We denote the inverse of the map by . If is an open-bounded subset of such that , the map will be called -compact on if is bounded and is compact.

The lemma that we used is [17, Theorem 2.4].

Lemma 1.1. Let be a Fredholm operator of index zero and let be L-compact on . Assume that the following conditions are satisfied: (i), for all ; (ii), for all ; (iii),where is a projection as above with and is any isomorphism. Then the equation has at least one solution in .

2. Preliminaries

For the convenience of the reader, we present here some necessary basic knowledge and definitions about fractional calculus theory. These definitions can be found in the recent literature [1–16, 18].

Definition 2.1. The fractional integral of order of a function is given by provided the right side is pointwise defined on , where is the Gamma function.

Definition 2.2. The fractional derivative of order of a function is given by where , provided the right side is pointwise defined on .

Definition 2.3. We say that the map satisfies Carathéodory conditions with respect to if the following conditions are satisfied: (i)for each , the mapping is Lebesgue measurable; (ii)for almost every , the mapping is continuous on ; (iii) for each , there exists such that for a.e. and every , we have .

Lemma 2.4 (see [15]). Assume that with a fractional derivative of order that belongs to . Then for some , where is the smallest integer greater than or equal to .

We use the classical Banach space with the norm with the norm

Definition 2.5. For , we denote by the space of functions which have continuous derivatives up to order on such that is absolutely continuous: = and is absolutely continuous in .

Lemma 2.6 (see [15]). Given and we can define a linear space where . By means of the linear functional analysis theory, we can prove that with the is a Banach space.

Remark 2.7. If is a natural number, then is in accordance with the classical Banach space .

Lemma 2.8 (see [15]). is a sequentially compact set if and only if is uniformly bounded and equicontinuous. Here uniformly bounded means there exists , such that for every and equicontinuous means that , , such that

Lemma 2.9 (see [1]). Let . Assume that with a fractional integration of order that belongs to . Then the equality holds almost everywhere on .

Definition 2.10 (see [16]). Let denote the space of functions , represented by fractional integral of order of a summable function: .

Let , with the norm , defined by Lemma 2.6, with the norm , where is a Banach space.

Define to be the linear operator from to with we define by setting Then boundary value problem (1.4) and (1.5) can be written as .

3. Main Results

Lemma 3.1. Let L be defined by (2.12), then

Proof. In the following lemma, we use the unified notation of both for fractional integrals and fractional derivatives assuming that for .
Let , by Lemma 2.9, has solution Combine with (1.5), So, Let and let
Then a.e. and, if hold, then satisfies the boundary conditions (1.5). That is, and we have
Let . Then for , we have where which, due to the boundary value condition (1.5), implies that satisfies (3.5). In fact, from we have , from , , we have Hence, Therefore, The proof is complete.

Lemma 3.2. The mapping is a Fredholm operator of index zero, and where define by by and .

Proof. Consider the continuous linear mapping and defined by Using the above definitions, we construct the following auxiliary maps and : Since the condition (C2) holds, the mapping defined by is well defined.
Recall (C2) and note that and similarly we can derive that So, for , it follows from the four relations above that that is, the map is idempotent. In fact is a continuous linear projector.
Note that implies . Conversely, if , so but then we must have ; since the condition (C2) holds, this can only be the case if , that is, . In fact , take in the form so that , thus, , Let and assume that is not identically zero on . Then, since , from (3.5) and the condition (C2), we have So but we derive , which is a contradiction. Hence, ; thus .
Now, and so is a Fredholm operator of index zero.
Let be defined by Note that is a continuous linear projector and It is clear that .
Note that the projectors and are exact. Define by by Hence we have then and thus
In fact, if , then Also, if , then where and from the boundary value condition (1.5) and the fact that , , , we have , thus This shows that . The proof is complete. Using (3.16), we write

By Lemma 2.8 and a standard method, we obtain the following lemma.

Lemma 3.3 (see [16]). For every given is completely continuous.

Assume that the following conditions on the function are satisfied.(H1)There exist functions , and a constant such that for all , , one of the following inequalities is satisfied:(H2)There exists a constant , such that for satisfying for all , we have (H3)There exists a constant such that for every satisfying then either