New Existence Results for Nonlinear Fractional Differential Equations with Three-Point Integral Boundary Conditions
© Bashir Ahmad et al. 2011
Received: 30 October 2010
Accepted: 12 December 2010
Published: 20 December 2010
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.
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 . 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 .
provided the integral exists.
Lemma 2.4 (see ).
To prove the main results, we need the following assumptions:
3. Existence Results in a Banach Space
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 .
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 .
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
Thus, all the conditions of Theorem 4.1 are satisfied and consequently the problem (5.3) has at least one solution.
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.Google Scholar
- 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
- 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
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- 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, 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Krasnoselskii MA: Two remarks on the method of successive approximations. Uspekhi Matematicheskikh Nauk 1955, 10: 123-127.MathSciNetGoogle Scholar
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.