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 {Dαu(t)=f(t,v(t),Dβ1v(t),…,DβNv(t)),Dα−iu(0)=0,i=1,2,…,n1,Dσv(t)=g(t,u(t),Dρ1u(t),…,DρNu(t)),Dσ−jv(0)=0,j=1,2,…,n2, where t∈(0,1]t∈(0,1], α>β1>β2>⋯βN>0α>β1>β2>⋯βN>0, σ>ρ1>ρ2>⋯ρN>0σ>ρ1>ρ2>⋯ρN>0, n1=[α]+1n1=[α]+1, n2=[σ]+1n2=[σ]+1 for α,σ∉N and n1=αn1=α, n2=σn2=σ for α,σ∈N, βq,ρq<1 for any q∈{1,2,…,N}q∈{1,2,…,N}, DD is the standard Riemann–Liouville differentiation and f,g:[0,1]×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.