# Existence Results for Nonlinear Boundary Value Problems of Fractional Integrodifferential Equations with Integral Boundary Conditions

- Bashir Ahmad
^{1}Email author and - Juan J Nieto
^{2}

**2009**:708576

**DOI: **10.1155/2009/708576

© B. Ahmad and J.J. Nieto 2009

**Received: **9 December 2008

**Accepted: **23 January 2009

**Published: **12 February 2009

## Abstract

This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem.

## 1. Introduction

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For examples and details, see [1–22] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.

Integrodifferential equations arise in many engineering and scientific disciplines, often as approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. Some of the applications are unsteady aerodynamics and aero elastic phenomena, visco elasticity, visco elastic panel in super sonic gas flow, fluid dynamics, electrodynamics of complex medium, many models of population growth, polymer rheology, neural network modeling, sandwich system identification, materials with fading memory, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, heat conduction in materials with memory, theory of lossless transmission lines, theory of population dynamics, compartmental systems, nuclear reactors, and mathematical modeling of a hereditary phenomena. For details, see [23–29] and the references therein.

Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper [30]. For more details of nonlocal and integral boundary conditions, see [31–37] and references therein.

and are real numbers. 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

## 2. Preliminaries

First of all, we recall some basic definitions [15, 18, 20].

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

In passing, we remark that the definition of Riemann-Liouville fractional derivative, which did certainly play an important role in the development of theory of fractional derivatives and integrals, could hardly produce the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. The same applies to the boundary value problems of fractional differential equations. It was Caputo definition of fractional derivative which solved this problem. In fact, the Caputo derivative becomes the conventional th derivative of the function as and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer derivatives. Another difference is that the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see [20].

Lemma 2.4 (see [22]).

where ( ).

for some ( ).

Now, we state a known result due to Krasnosel'skiĭ [38] which is needed to prove the existence of at least one solution of (1.1).

Theorem 2.5.

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

Lemma 2.6.

Proof.

where is given by (2.8). This completes the proof.

## 3. Main Results

Theorem 3.1.

Assume that is jointly continuous and maps bounded subsets of into relatively compact subsets of is continuous with and are continuous functions. Further, there exist positive constants such that

(A_{1})
for all

(A_{2})
with

Proof.

which 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.

Theorem 3.2.

_{1})-(A

_{2}) hold with where and

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

Proof.

_{1}), (A

_{2}) that is a contraction mapping for

_{1}), we define and consequently we have

which is independent of So is relatively compact on Hence, By Arzela Ascoli Theorem, is compact on Thus all the assumptions of Theorem 2.5 are satisfied and the conclusion of Theorem 2.5 implies that the boundary value problem (1.1) has at least one solution on

Example 3.3.

_{1}) and (A

_{2}) are satisfied with Further,

Thus, by Theorem 3.1, the boundary value problem (3.15) has a unique solution on

## Declarations

### Acknowledgments

The authors are grateful to the anonymous referee for his/her valuable suggestions that led to the improvement of the original manuscript. The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

## Authors’ Affiliations

## References

- Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. preprint
- Ahmad B, Sivasundaram S: Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions. to appear in
*Dynamic Systems and Applications* - Araya D, Lizama C: Almost automorphic mild solutions to fractional differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(11):3692–3705. 10.1016/j.na.2007.10.004MATHMathSciNetView ArticleGoogle 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.2005.02.052MATHMathSciNetView ArticleGoogle Scholar - Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solution for a nonlinear fractional differential equation. preprint
- Benchohra M, Hamani S, Nieto JJ, Slimani BA: Existence results for differential inclusions with fractional order and impulses. preprint
- Bonilla B, Rivero M, Rodríguez-Germá L, Trujillo JJ: Fractional differential equations as alternative models to nonlinear differential equations.
*Applied Mathematics and Computation*2007, 187(1):79–88. 10.1016/j.amc.2006.08.105MATHMathSciNetView ArticleGoogle Scholar - Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions.
*Mathematical and Computer Modelling*2009, 49(3–4):605–609. 10.1016/j.mcm.2008.03.014MATHMathSciNetView ArticleGoogle Scholar - Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of time fractional reaction-diffusion systems.
*Journal of Computational and Applied Mathematics*2008, 220(1–2):215–225. 10.1016/j.cam.2007.08.011MATHMathSciNetView ArticleGoogle Scholar - Daftardar-Gejji V: Positive solutions of a system of non-autonomous fractional differential equations.
*Journal of Mathematical Analysis and Applications*2005, 302(1):56–64. 10.1016/j.jmaa.2004.08.007MATHMathSciNetView ArticleGoogle 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.2008.04.065MATHMathSciNetView ArticleGoogle Scholar - El-Shahed M: Positive solutions for boundary value problem of nonlinear fractional differential equation.
*Abstract and Applied Analysis*2007, 2007:-8.Google Scholar - Ibrahim RW, Darus M: Subordination and superordination for univalent solutions for fractional differential equations.
*Journal of Mathematical Analysis and Applications*2008, 345(2):871–879. 10.1016/j.jmaa.2008.05.017MATHMathSciNetView ArticleGoogle Scholar - Jafari H, Seifi S: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation.
*Communications in Nonlinear Science and Numerical Simulation*2009, 14(5):2006–2012. 10.1016/j.cnsns.2008.05.008MATHMathSciNetView ArticleGoogle 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 - Ladaci S, Loiseau JJ, Charef A: Fractional order adaptive high-gain controllers for a class of linear systems.
*Communications in Nonlinear Science and Numerical Simulation*2008, 13(4):707–714. 10.1016/j.cnsns.2006.06.009MATHMathSciNetView ArticleGoogle Scholar - Lazarević MP: Finite time stability analysis of fractional control of robotic time-delay systems.
*Mechanics Research Communications*2006, 33(2):269–279. 10.1016/j.mechrescom.2005.08.010MATHMathSciNetView ArticleGoogle Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar - Rida SZ, El-Sherbiny HM, Arafa AAM: On the solution of the fractional nonlinear Schrödinger equation.
*Physics Letters A*2008, 372(5):553–558. 10.1016/j.physleta.2007.06.071MATHMathSciNetView ArticleGoogle Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives: Theory and Applications*. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar - Varlamov V: Differential and integral relations involving fractional derivatives of Airy functions and applications.
*Journal of Mathematical Analysis and Applications*2008, 348(1):101–115. 10.1016/j.jmaa.2008.06.052MATHMathSciNetView ArticleGoogle Scholar - Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations.
*Electronic Journal of Differential Equations*2006, 2006(36):1–12.Google Scholar - Ahmad B, Sivasundaram S: Some existence results for fractional integrodifferential equations with nonlinear conditions.
*Communications in Applied Analysis*2008, 12: 107–112.MATHMathSciNetGoogle Scholar - Ahmad B, Alghamdi BS: Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions.
*Computer Physics Communications*2008, 179(6):409–416. 10.1016/j.cpc.2008.04.008MATHMathSciNetView ArticleGoogle Scholar - Ahmad B: On the existence of
-periodic solutions for Duffing type integro-differential equations with -Laplacian.
*Lobachevskii Journal of Mathematics*2008, 29(1):1–4.MATHMathSciNetView ArticleGoogle Scholar - Chang YK, Nieto JJ: Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. to appear in
*Numerical Functional Analysis and Optimization* - Luo Z, Nieto JJ: New results for the periodic boundary value problem for impulsive integro-differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(6):2248–2260. 10.1016/j.na.2008.03.004MATHMathSciNetView ArticleGoogle Scholar - Mesloub S: On a mixed nonlinear one point boundary value problem for an integrodifferential equation.
*Boundary Value Problems*2008, 2008:-8.Google Scholar - Nieto JJ, Rodríguez-López R: New comparison results for impulsive integro-differential equations and applications.
*Journal of Mathematical Analysis and Applications*2007, 328(2):1343–1368. 10.1016/j.jmaa.2006.06.029MATHMathSciNetView ArticleGoogle Scholar - Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions.
*Nonlinear Analysis: Real World Applications*2008, 9(4):1727–1740. 10.1016/j.nonrwa.2007.05.005MATHMathSciNetView ArticleGoogle Scholar - Ahmad B, Alsaedi A: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions.
*Nonlinear Analysis: Real World Applications*2009, 10(1):358–367. 10.1016/j.nonrwa.2007.09.004MATHMathSciNetView ArticleGoogle Scholar - Benchohra M, Hamani S, Nieto JJ: The method of upper and lower solutions for second order differential inclusions with integral boundary conditions.
*Rocky Mountain Journal of Mathematics*. In press - Boucherif A: Second-order boundary value problems with integral boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(1):364–371. 10.1016/j.na.2007.12.007MATHMathSciNetView ArticleGoogle Scholar - Chang Y-K, Nieto JJ, Li W-S: On impulsive hyperbolic differential inclusions with nonlocal initial conditions.
*Journal of Optimization Theory and Applications*2009, 140(3):431–442. 10.1007/s10957-008-9468-1MATHMathSciNetView ArticleGoogle Scholar - Chang YK, Nieto JJ, Li WS: Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces. to appear in
*Journal of Optimization Theory and Applications* - Feng M, Du B, Ge W: Impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian.
*Nonlinear Analysis: Theory, Methods & Applications*. In press - Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(1):216–225. 10.1016/j.na.2006.10.044MATHMathSciNetView ArticleGoogle Scholar - Krasnosel'skiĭ MA: Two remarks on the method of successive approximations.
*Uspekhi Matematicheskikh Nauk*1955, 10(1(63)):123–127.Google 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.