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# Existence and Uniqueness Results for a Coupled System of Nonlinear Fractional Differential Equations with Antiperiodic Boundary Conditions

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Abstract:

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

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