- Research Article
- Open Access

# Existence of Solutions for Nonlinear Fractional Integro-Differential Equations with Three-Point Nonlocal Fractional Boundary Conditions

- Ahmed Alsaedi
^{1}and - Bashir Ahmad
^{1}Email author

**2010**:691721

https://doi.org/10.1155/2010/691721

© A. Alsaedi and B. Ahmad. 2010

**Received:**17 March 2010**Accepted:**11 June 2010**Published:**4 July 2010

## Abstract

We prove the existence and uniqueness of solutions for nonlinear integro-differential equations of fractional order with three-point nonlocal fractional boundary conditions by applying some standard fixed point theorems.

## Keywords

- Fractional Order
- Fractional Calculus
- Fractional Differential Equation
- Impedance Boundary Condition
- Surface Current Density

## 1. Introduction

Fractional calculus (differentiation and integration of arbitrary order) is proved to be an important tool in the modelling of dynamical systems associated with phenomena such as fractal and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, and fitting of experimental data [1–4].

Recently, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. For some recent work on fractional differential equations, see [5–11] and the references therein.

and satisfies the condition 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

We remark that fractional boundary conditions result in the existence of both electric and magnetic surface currents on the strip and are similar to the impedance boundary conditions with pure imaginary impedance, and in the physical optics approximation, the ratio of the surface currents is the same as for the impedance strip. For the comparison of the physical characteristics of the fractional and impedance strips such as radiation pattern, monostatic radar cross-section, and surface current densities, see [12]. The concept of nonlocal multipoint boundary conditions is quite important in various physical problems of applied nature when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the censors located at intermediate points. Some recent results on nonlocal fractional boundary value problems can be found in [13–15].

## 2. Preliminaries

Let us recall some basic definitions [1–3] on fractional calculus.

Definition 2.1.

provided the integral exists.

Definition 2.2.

provided the right-hand side is pointwise defined on

Lemma 2.3 (see [16]).

where ( is the smallest integer such that ).

Lemma 2.4 (see [2]).

Let . Then

Lemma 2.5.

Proof.

Substituting the values of and in (2.6), we obtain (2.5). This completes the proof.

## 3. Main Results

Theorem 3.1.

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

Proof.

where we have used the assumption . As therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

Now, we state Krasnoselskii's fixed point theorem [17] which is needed to prove the following result to prove the existence of at least one solution of (1.1).

Theorem 3.2.

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 there exists at least one solution of the boundary value problem (1.1) on

Proof.

Thus, It follows from the assumption that is a contraction mapping for

which is independent of So, is relatively compact on . Hence, By Arzela-Ascoli's Theorem, is compact on . Thus all the assumptions of Theorem 3.2 are satisfied and the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on

Example 3.

Thus, by Theorem 3.1, the boundary value problem (3.14) has a unique solution on

## 4. Conclusions

## Declarations

### Acknowledgment

The authors are grateful to the referees for their careful review of the manuscript.

## Authors’ Affiliations

## References

- Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives*. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar - Kilbas AA, Srivastava HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies*.*Volume 204*. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar - Sabatier J, Agrawal OP, Machad JAT (Eds):
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*. Springer, Dordrecht, The Netherlands; 2007:xiv+552.MATHGoogle Scholar - Daftardar-Gejji V, Bhalekar S:
**Boundary value problems for multi-term fractional differential equations.***Journal of Mathematical Analysis and Applications*2008,**345**(2):754-765. 10.1016/j.jmaa.2008.04.065MathSciNetView ArticleMATHGoogle Scholar - N'Guérékata GM:
**A Cauchy problem for some fractional abstract differential equation with non local conditions.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(5):1873-1876. 10.1016/j.na.2008.02.087MathSciNetView ArticleMATHGoogle Scholar - Mophou GM, N'Guérékata GM:
**On integral solutions of some nonlocal fractional differential equations with nondense domain.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4668-4675. 10.1016/j.na.2009.03.029MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Leela S, Devi JV:
*Theory of Fractional Dynamic Systems*. Cambridge Academic Publishers, Cambridge, UK; 2009.MATHGoogle Scholar - Gafiychuk V, Datsko B:
**Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems.***Computers & Mathematics with Applications*2010,**59**(3):1101-1107. 10.1016/j.camwa.2009.05.013MathSciNetView ArticleMATHGoogle Scholar - Mophou GM:
**Existence and uniqueness of mild solutions to impulsive fractional differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(3-4):1604-1615. 10.1016/j.na.2009.08.046MathSciNetView ArticleMATHGoogle Scholar - Ahmad B:
**Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations.***Applied Mathematics Letters*2010,**23**(4):390-394. 10.1016/j.aml.2009.11.004MathSciNetView ArticleMATHGoogle Scholar - Veliev EI, Ivakhnychenko MV, Ahmedov TM:
**scattering properties of the strip with fractional boundary conditions and comparison with the impedance strip.***Progress In Electromagnetics Research C*2008,**2:**189-205.Google Scholar - Ahmad B, Nieto JJ:
**Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations.***Abstract and Applied Analysis*2009,**2009:**-9.Google Scholar - Ahmad B, Graef JR:
**Coupled systems of nonlinear fractional differential equations with nonlocal boundary conditions.***PanAmerican Mathematical Journal*2009,**19**(3):29-39.MathSciNetMATHGoogle Scholar - Ahmad B, Nieto JJ:
**Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions.***Computers & Mathematics with Applicationsl*2009,**58**(9):1838-1843. 10.1016/j.camwa.2009.07.091MathSciNetView ArticleMATHGoogle Scholar - Bai Z, Lü H:
**Positive solutions for boundary value problem of nonlinear fractional differential equation.***Journal of Mathematical Analysis and Applications*2005,**311**(2):495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar - Smart DR:
*Fixed Point Theorems*. Cambridge University Press, London, UK; 1980:viii+93.MATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.