Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations

In this paper, we consider an initial. value problem for a coupled system of multi-term nonlinear fractional differential equations {D(alpha)u(t) = f(t, v(t), D(beta 1)v(t), ... , D-beta N v(t)), D(alpha-i)u(0) = 0, i = 1, 2, ... , n(1), D(sigma)v(t) = g(t, u(t), D(rho 1)u(t), ... , D-rho N u(t)), D(sigma-j)v(0) = 0, j = 1, 2, ... , n(2,) where t is an element of (0, 1], alpha > beta(1) > beta(2) > ... beta(N) > 0, sigma > rho(1) > rho(1) > ... rho(N) > 0, n(1) = [alpha] + 1, n(2) = [sigma] + 1 for alpha, sigma is not an element of N and n(1) = alpha, n(2) = sigma for alpha, sigma is an element of N, beta(q,) rho(q) < 1 for any q is an element of {1, 2, ... , N}, D is the standard Riemann-Liouville differentiation and f, g : [0, 1] x RN+1 -> R are given functions. By means of Schauder fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution are obtained, respectively. As an application, some examples are presented to illustrate the main results. (C) 2012 Elsevier Ltd. All rights reserved.