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^@au(t)=f(t,v(t),D^@b^"^1v(t),...,D^@b^"^Nv(t)),D^@a^-^iu(0)=0,i=1,2,...,n"1,D^@sv(t)=g(t,u(t),D^@r^"^1u(t),...,D^@r^"^Nu(t)),D^@s^-^jv(0)=0,j=1,2,...,n"2, where [email protected]?(0,1], @a>@b"1>@b"2>[email protected]"N>0, @s>@r"1>@r"2>[email protected]"N>0, n"1=[@a]+1, n"2=[@s]+1 for @a,@[email protected]?N and n"[email protected], n"[email protected] for @a,@[email protected]?N, @b"q,@r"q 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.

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