Abstract
This paper studies the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations of orderα,β∈(4,5]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.
Highlights
In this paper, we consider the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations: cDαx (t) + f (t, x (t), y (t)) = 0, t ∈ [0, T], cDβy (t) + g (t, x (t), y (t)) = 0, t ∈ [0, T], (1)x(i) (0) = −x(i) (T), i = 0, 1, 2, 3, 4, y(i) (0) = −y(i) (T), i = 0, 1, 2, 3, 4, where 4 < α, β ≤ 5, and cDα denotes the Caputo fractional derivative of order α
The Caputo fractional derivative of order α > 0 of a continuous function y : (0, ∞) → R is given by cDαy (t)
It is obvious that a fixed point of the operator F is a solution of problem (1)
Summary
We consider the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations: cDαx (t) + f (t, x (t) , y (t)) = 0, t ∈ [0, T] , cDβy (t) + g (t, x (t) , y (t)) = 0, t ∈ [0, T] , (1). In [5], Alsaedi et al study an Antiperiodic boundary value problem of nonlinear fractional differential equations of order q ∈ 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.
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