- Research Article
- Open Access

# New Existence Results for Nonlinear Fractional Differential Equations with Three-Point Integral Boundary Conditions

- Bashir Ahmad
^{1}Email author, - Sotiris K. Ntouyas
^{2}and - Ahmed Alsaedi
^{1}

**2011**:107384

https://doi.org/10.1155/2011/107384

© Bashir Ahmad et al. 2011

**Received:**30 October 2010**Accepted:**12 December 2010**Published:**20 December 2010

## Abstract

This paper studies a boundary value problem of nonlinear fractional differential equations of order with three-point integral boundary conditions. Some new existence and uniqueness results are obtained by using standard fixed point theorems and Leray-Schauder degree theory. Our results are new in the sense that the nonlocal parameter in three-point integral boundary conditions appears in the integral part of the conditions in contrast to the available literature on three-point boundary value problems which deals with the three-point boundary conditions restrictions on the solution or gradient of the solution of the problem. Some illustrative examples are also discussed.

## Keywords

- Fractional Order
- Fractional Derivative
- Fractional Calculus
- Fractional Differential Equation
- Topological Degree

## 1. Introduction

In recent years, boundary value problems for nonlinear fractional differential equations have been addressed by several researchers. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes; see [1]. These characteristics of the fractional derivatives make the fractional-order models more realistic and practical than the classical integer-order models. As a matter of fact, 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, and fitting of experimental data, [1–4]. For some recent development on the topic, see [5–21] and the references therein.

where denotes the Caputo fractional derivative of order , is continuous, and is such that . Here, is a Banach space and denotes the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by .

Note that the three-point boundary condition in (1.1) corresponds to the area under the curve of solutions from to .

## 2. Preliminaries

Let us recall some basic definitions of fractional calculus [2, 4].

Definition 2.1.

where denotes the integer part of the real number .

Definition 2.2.

provided the integral exists.

Definition 2.3.

provided the right-hand side is pointwise defined on .

Lemma 2.4 (see [2]).

where , ( ).

for some , ( ).

Lemma 2.5.

Proof.

Substituting the values of and in (2.7), we obtain the solution (2.6).

To prove the main results, we need the following assumptions:

, for all , , ;

, for all , and .

## 3. Existence Results in a Banach Space

Theorem 3.1.

Assume that is a jointly continuous function and satisfies the assumption with , where is given by (2.11). Then the boundary value problem (1.1) has a unique solution.

Proof.

where is given by (2.11). Observe that depends only on the parameters involved in the problem. As , therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii's fixed point theorem [22].

Theorem 3.2 (Krasnoselskii's fixed point theorem).

Let be a closed convex and nonempty subset of a Banach space . Let be the operators such that (i) whenever ; (ii) is compact and continuous; (iii) is a contraction mapping. Then there exists such that .

Theorem 3.3.

Then the boundary value problem (1.1) has at least one solution on .

Proof.

Now we prove the compactness of the operator .

which is independent of . Thus, is equicontinuous. Using the fact that maps bounded subsets into relatively compact subsets, we have that is relatively compact in for every , where is a bounded subset of . So is relatively compact on . Hence, by the Arzelá-Ascoli Theorem, is compact on . Thus all the assumptions of Theorem 3.2 are satisfied. So the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on .

## 4. Existence of Solution via Leray-Schauder Degree Theory

Theorem 4.1.

Let . Assume that there exist constants , where is given by (2.11) and such that for all . Then the boundary value problem (1.1) has at least one solution.

Proof.

Letting , (4.4) holds. This completes the proof.

## 5. Examples

Example 5.1.

Thus, by the conclusion of Theorem 3.1, the boundary value problem (5.1) has a unique solution on .

Example 5.2.

Thus, all the conditions of Theorem 4.1 are satisfied and consequently the problem (5.3) has at least one solution.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.