This paper studies the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations of order with antiperiodic boundary conditions. Our results are based on the nonlinear alternative of Leray-Schauder type and the contraction mapping principle. Two illustrative examples are also presented. 1. Introduction In this paper, we consider the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations: where , , and denotes the Caputo fractional derivative of order . Here our nonlinearity are given continuous functions. Fractional differential equations have recently been addressed by many researchers in various fields of science and engineering, such as rheology, porous media, fluid flows, chemical physics, and many other branches of science; see [1–4]. As a matter of fact, fractional-order models become more realistic and practical than the classical integer-order models; as a consequence, there are a large number of papers and books dealing with the existence and uniqueness of solutions to nonlinear fractional differential equations; see [5–14]. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications; see [15–17]. Antiperiodic boundary value problems arise in the mathematical modeling of a variety of physical process; many authors have paid much attention to such problems; for examples and details of Antiperiodic boundary conditions, see [5, 18–22]. In [5], Alsaedi et al. study an Antiperiodic boundary value problem of nonlinear fractional differential equations of order . It should be noted that, in [23], Ntouyas and Obaid have researched a coupled system of fractional differential equations with nonlocal integral boundary conditions, but this paper researches a coupled system of fractional differential equations with Antiperiodic boundary conditions. On the other hand, in [5, 19], the authors have discussed some existence results of solutions for Antiperiodic boundary value problems of fractional differential equation but not the coupled system. The rest of the papers above for the coupled systems have been devoted to the case of Riemann-Liouville fractional derivatives but not the Caputo fractional derivatives. This paper is organized as follows. In Section 2, we recall some basic definitions and preliminary results. In Section 3, we give the existence results of (1) by means of the Leray-Schauder alternative; then we obtain the uniqueness of solutions for system (1) by the
References
[1]
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
[2]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[3]
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
[4]
J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
[5]
A. Alsaedi, B. Ahmad, and A. Assolami, “On antiperiodic boundary value problems for higher-order fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 325984, 15 pages, 2012.
[6]
Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.
[7]
Z. Bai, W. Sun, and W. Zhang, “Positive solutions for boundary value problems of singular fractional differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 129640, 7 pages, 2013.
[8]
Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
[9]
R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
[10]
R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 918728, 47 pages, 2009.
[11]
X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions to singular positone and semipositone Dirichlet-type boundary value problems of nonlinear fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 16, pp. 5685–5696, 2011.
[12]
C.-Z. Bai and J.-X. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 150, no. 3, pp. 611–621, 2004.
[13]
X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4676–4688, 2009.
[14]
G. Wang, B. Ahmad, and L. Zhang, “Existence results for nonlinear fractional differential equations with closed boundary conditions and impulses,” Advances in Difference Equations, vol. 2012, article 169, 2012.
[15]
V. Gafiychuk, B. Datsko, V. Meleshko, and D. Blackmore, “Analysis of the solutions of coupled nonlinear fractional reaction-diffusion equations,” Chaos, Solitons & Fractals, vol. 41, no. 3, pp. 1095–1104, 2009.
[16]
Y. Chen, D. Chen, and Z. Lv, “The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions,” Bulletin of the Iranian Mathematical Society, vol. 38, no. 3, pp. 607–624, 2012.
[17]
B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers and Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009.
[18]
B. Ahmad and V. Otero-Espinar, “Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 625347, 2009.
[19]
R. P. Agarwal and B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1200–1214, 2011.
[20]
B. Ahmad, “Existence of solutions for fractional differential equations of order q∈ (2,3] with anti-periodic boundary conditions,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 385–391, 2010.
[21]
R. P. Agarwal and B. Ahmad, “Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems A, vol. 18, no. 4, pp. 457–470, 2011.
[22]
G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 792–804, 2011.
[23]
S. K. Ntouyas and M. Obaid, “A coupled system of fractional differential equations with nonlocal integral boundary conditions,” Advances in Difference Equations, vol. 2012, article 130, 2012.
[24]
A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003.