Open Access

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

Advances in Difference Equations20102011:107384

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

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.

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, [14]. For some recent development on the topic, see [521] and the references therein.

We discuss the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations of order with three-point integral boundary conditions given by
(1.1)

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.

For a continuous function , the Caputo derivative of fractional order is defined as
(2.1)

where denotes the integer part of the real number .

Definition 2.2.

The Riemann-Liouville fractional integral of order is defined as
(2.2)

provided the integral exists.

Definition 2.3.

The Riemann-Liouville fractional derivative of order for a continuous function is defined by
(2.3)

provided the right-hand side is pointwise defined on .

Lemma 2.4 (see [2]).

For , the general solution of the fractional differential equation is given by
(2.4)

where , ( ).

In view of Lemma 2.4, it follows that
(2.5)

for some , ( ).

Lemma 2.5.

A unique solution of the boundary value problem (1.1) is given by
(2.6)

Proof.

For some constants , we have
(2.7)
From , we have . Applying the second boundary condition for (1.1), we find that
(2.8)
which imply that
(2.9)

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

In view of Lemma 2.5, we define an operator by
(2.10)

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

, for all , , ;

, for all , and .

For convenience, let us set
(2.11)

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.

Setting and choosing , we show that , where . For , we have
(3.1)
Now, for and for each , we obtain
(3.2)

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.

Let be a jointly continuous function mapping bounded subsets of into relatively compact subsets of , and the assumptions and hold with
(3.3)

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

Proof.

Letting , we fix
(3.4)
and consider . We define the operators and on as
(3.5)
For , we find that
(3.6)
Thus, . It follows from the assumption together with (3.3) that is a contraction mapping. Continuity of implies that the operator is continuous. Also, is uniformly bounded on as
(3.7)

Now we prove the compactness of the operator .

In view of , we define , and consequently we have
(3.8)

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.

Let us define an operator as
(4.1)
where
(4.2)
In view of the fixed point problem (4.1), we just need to prove the existence of at least one solution satisfying (4.1). Define a suitable ball with radius as
(4.3)
where will be fixed later. Then, it is sufficient to show that satisfies
(4.4)
Let us set
(4.5)
Then, by the Arzelá-Ascoli Theorem, is completely continuous. If (4.4) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that
(4.6)
where denotes the unit operator. By the nonzero property of Leray-Schauder degree, for at least one . In order to prove (4.4), we assume that for some and for all so that
(4.7)
which, on taking norm ( ) and solving for , yields
(4.8)

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

5. Examples

Example 5.1.

Consider the following three-point integral fractional boundary value problem:
(5.1)
Here, , , , and . As , therefore, is satisfied with . Further,
(5.2)

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

Example 5.2.

Consider the following boundary value problem:
(5.3)
Here, , , , and
(5.4)
Clearly and
(5.5)

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

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, University of Ioannina

References

  1. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar
  2. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
  3. Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, The Netherlands; 2007:xiv+552.Google Scholar
  4. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
  5. Agarwal RP, de Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Analysis: Real World Applications 2010,11(5):3532-3554. 10.1016/j.nonrwa.2010.01.002MathSciNetView ArticleMATHGoogle Scholar
  6. Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods & Applications 2010,72(6):2859-2862. 10.1016/j.na.2009.11.029MathSciNetView ArticleMATHGoogle Scholar
  7. Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Computers & Mathematics with Applications 2010,59(3):1095-1100.MathSciNetView ArticleMATHGoogle Scholar
  8. Ahmad B, Sivasundaram S: On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Applied Mathematics and Computation 2010,217(2):480-487. 10.1016/j.amc.2010.05.080MathSciNetView ArticleMATHGoogle Scholar
  9. 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
  10. Ahmad B:Existence of solutions for fractional differential equations of order with anti-periodic boundary conditions. Journal of Applied Mathematics and Computing 2010,34(1-2):385-391. 10.1007/s12190-009-0328-4MathSciNetView ArticleMATHGoogle Scholar
  11. 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
  12. Ahmad B, Alsaedi A: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory and Applications 2010, 2010:-17.Google Scholar
  13. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
  14. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computers & Mathematics with Applications 2009,58(9):1838-1843. 10.1016/j.camwa.2009.07.091MathSciNetView ArticleMATHGoogle Scholar
  15. Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2010,72(2):916-924. 10.1016/j.na.2009.07.033MathSciNetView ArticleMATHGoogle Scholar
  16. Balachandran K, Trujillo JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2010,72(12):4587-4593. 10.1016/j.na.2010.02.035MathSciNetView ArticleMATHGoogle Scholar
  17. Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7-8):2391-2396. 10.1016/j.na.2009.01.073MathSciNetView ArticleMATHGoogle Scholar
  18. Lazarević MP, Spasić AM: Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach. Mathematical and Computer Modelling 2009,49(3-4):475-481. 10.1016/j.mcm.2008.09.011MathSciNetView ArticleMATHGoogle Scholar
  19. Nieto JJ: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Applied Mathematics Letters 2010,23(10):1248-1251. 10.1016/j.aml.2010.06.007MathSciNetView ArticleMATHGoogle Scholar
  20. Wei Z, Li Q, Che J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. Journal of Mathematical Analysis and Applications 2010,367(1):260-272. 10.1016/j.jmaa.2010.01.023MathSciNetView ArticleMATHGoogle Scholar
  21. Zhang S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Computers & Mathematics with Applications 2010,59(3):1300-1309.MathSciNetView ArticleMATHGoogle Scholar
  22. Krasnoselskii MA: Two remarks on the method of successive approximations. Uspekhi Matematicheskikh Nauk 1955, 10: 123-127.MathSciNetGoogle Scholar

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© Bashir Ahmad et al. 2011

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.