Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives
© Alsaedi et al.; licensee Springer. 2014
Received: 20 February 2014
Accepted: 2 April 2014
Published: 7 May 2014
We investigate the uniqueness of solutions for fully anti-periodic fractional boundary value problems of order with nonlinearity depending on lower-order fractional derivatives. Our results are based on some standard fixed point theorems. The paper concludes with illustrative examples.
Keywordsdifferential equations of fractional order anti-periodic fractional boundary conditions uniqueness fixed point
where denotes the Caputo fractional derivative of order q and f is a given continuous function.
In the last few decades, fractional calculus has evolved as an attractive field of research in view of its extensive applications in basic and technical sciences. Examples can be found in physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. [2–5].
The subject of boundary value problems of differential equations, having an enriched history, has been progressing at the same pace as before. In the context of fractional boundary value problems, there has been a much development in the last ten years; for instance, see [6–25] and the references cited therein.
In view of the importance of anti-periodic boundary conditions in the mathematical modeling of a variety of physical processes [26–28], the study of anti-periodic boundary value problems has received considerable attention. Some recent work on anti-periodic boundary value problems of fractional order can be found in a series of papers [29–34] and the references therein.
provided the integral exists.
where denotes the integer part of the real number q.
Lemma 2.1 
3 Uniqueness of solutions
This section is devoted to the uniqueness of solutions for the problems at hand by means of Banach’s contraction principle.
3.1 Uniqueness result for the problem (1.1)-(1.2)
with , where N is given by (3.2). Then the anti-periodic boundary value problem (1.1)-(1.2) has a unique solution.
3.2 Uniqueness result for the problem (1.3)-(1.2)
Here, we study the uniqueness of solutions for the problem of (1.3)-(1.2). For that, let be a Banach space endowed with the norm , .
Then the anti-periodic boundary value problem (1.3)-(1.2) has a unique solution on .
Since , therefore, the operator is a contraction. Thus, it follows by the contraction mapping principle that the problem (1.3)-(1.2) has a unique solution on . □
- (a)Consider the anti-periodic fractional boundary value problem given by(4.1)
- (b)Consider the following anti-periodic fractional boundary value problem:(4.2)
Clearly all the assumptions of Theorem 3.2 are satisfied with (). Hence, the problem (4.2) has a unique solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
- Ahmad B, Nieto JJ, Alsaedi A, Mohamad N: On a new class of anti-periodic fractional boundary value problems. Abstr. Appl. Anal. 2013., 2013: Article ID 606454Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.Google Scholar
- Agarwal RP, Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 2010, 11: 3532–3554. 10.1016/j.nonrwa.2010.01.002MathSciNetView ArticleGoogle Scholar
- Agarwal RP, Ntouyas SK, Ahmad B, Alhothuali MS: Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions. Adv. Differ. Equ. 2013., 2013: Article ID 128Google Scholar
- Ahmad B: On nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order. Results Math. 2013, 63: 183–194. 10.1007/s00025-011-0187-9MathSciNetView ArticleGoogle Scholar
- Bai ZB: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916–924. 10.1016/j.na.2009.07.033MathSciNetView ArticleGoogle Scholar
- Bai C: Existence of solutions for a nonlinear fractional boundary value problem via a local minimum theorem. Electron. J. Differ. Equ. 2012., 2012: Article ID 176Google Scholar
- Caballero J, Harjani J, Sadarangani K: On existence and uniqueness of positive solutions to a class of fractional boundary value problems. Bound. Value Probl. 2011., 2011: Article ID 25Google Scholar
- Chen F: Coincidence degree and fractional boundary value problems with impulses. Comput. Math. Appl. 2012, 64: 3444–3455. 10.1016/j.camwa.2012.02.022MathSciNetView ArticleGoogle Scholar
- Ding X-L, Jiang Y-L: Analytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions. Nonlinear Anal., Real World Appl. 2013, 14: 1026–1033. 10.1016/j.nonrwa.2012.08.014MathSciNetView ArticleGoogle Scholar
- Ford NJ, Morgado ML: Fractional boundary value problems: analysis and numerical methods. Fract. Calc. Appl. Anal. 2011, 14(4):554–567. 10.2478/s13540-011-0034-4MathSciNetView ArticleGoogle Scholar
- Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl. 2010, 59: 1101–1107. 10.1016/j.camwa.2009.05.013MathSciNetView ArticleGoogle Scholar
- Graef JR, Kong L, Kong Q, Wang M: Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 2012, 15: 509–528. 10.2478/s13540-012-0036-xMathSciNetGoogle Scholar
- Graef JR, Kong L, Yang B: Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 2012, 15: 8–24. 10.2478/s13540-012-0002-7MathSciNetGoogle Scholar
- Wang G, Liu S, Baleanu D, Zhang L: Existence results for nonlinear fractional differential equations involving different Riemann-Liouville fractional derivatives. Adv. Differ. Equ. 2013., 2013: Article ID 280Google Scholar
- Hernández E, O’Regan D, Balachandran K: Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators. Indag. Math. 2013, 24: 68–82. 10.1016/j.indag.2012.06.007MathSciNetView ArticleGoogle Scholar
- Jarad F, Abdeljawad T, Baleanu D: Stability of q -fractional non-autonomous systems. Nonlinear Anal., Real World Appl. 2013, 14: 780–784. 10.1016/j.nonrwa.2012.08.001MathSciNetView ArticleGoogle Scholar
- Lizama C: Solutions of two-term time fractional order differential equations with nonlocal initial conditions. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: Article ID 82Google Scholar
- Machado JT, Galhano AM, Trujillo JJ: Science metrics on fractional calculus development since 1966. Fract. Calc. Appl. Anal. 2013, 16: 479–500. 10.2478/s13540-013-0030-yMathSciNetGoogle Scholar
- Sun H-R, Zhang Q-G: Existence of solutions for a fractional boundary value problem via the mountain pass method and an iterative technique. Comput. Math. Appl. 2012, 64: 3436–3443. 10.1016/j.camwa.2012.02.023MathSciNetView ArticleGoogle Scholar
- Wang Y, Liu L, Wu Y: Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Anal. 2011, 74: 6434–6441. 10.1016/j.na.2011.06.026MathSciNetView ArticleGoogle Scholar
- Zhao X, Chai C, Ge W: Existence and nonexistence results for a class of fractional boundary value problems. J. Appl. Math. Comput. 2013, 41: 17–31. 10.1007/s12190-012-0590-8MathSciNetView ArticleGoogle Scholar
- Aubertin M, Henneron T, Piriou F, Guerin P, Mipo J-C: Periodic and anti-periodic boundary conditions with the Lagrange multipliers in the FEM. IEEE Trans. Magn. 2010, 46: 3417–3420.View ArticleGoogle Scholar
- Cardy JL: Finite-size scaling in strips: antiperiodic boundary conditions. J. Phys. A, Math. Gen. 1984, 17: L961-L964. 10.1088/0305-4470/17/18/005MathSciNetView ArticleGoogle Scholar
- Chen Y, Nieto JJ, O’Regan D: Anti-periodic solutions for evolution equations associated with maximal monotone mappings. Appl. Math. Lett. 2011, 24: 302–307. 10.1016/j.aml.2010.10.010MathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 2010, 35: 295–304.MathSciNetGoogle Scholar
- Ahmad B, Nieto JJ: Anti-periodic fractional boundary value problem with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 2012, 15: 451–462. 10.2478/s13540-012-0032-1MathSciNetGoogle Scholar
- Benchohra M, Hamidi N, Henderson J: Fractional differential equations with anti-periodic boundary conditions. Numer. Funct. Anal. Optim. 2013, 34: 404–414. 10.1080/01630563.2012.763140MathSciNetView ArticleGoogle Scholar
- Fang W, Zhenhai L: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv. Differ. Equ. 2012., 2012: Article ID 116Google Scholar
- Wang G, Ahmad B, Zhang L: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 2011, 74: 792–804. 10.1016/j.na.2010.09.030MathSciNetView ArticleGoogle Scholar
- Wang X, Guo X, Tang G: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. J. Appl. Math. Comput. 2013, 41: 367–375. 10.1007/s12190-012-0613-5MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.