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

摘要

In this paper, we consider an initial value problem for a coupled system of multi-term nonlinear fractional differential equations {D~α u(t)=f(t,v(t),D~(β1) v(t),...,D~(βN) v(t)), D~(α-i) u(0) = 0, i= 1,2, ...,n_1, D~σ u(t)=f(t,v(t),D~(ρ1) v(t),...,D~(ρN) v(t)), D~(σ-j) u(0) = 0, j= 1,2, ...,n_2, where t ∈ (0,1], α > β_1 > β_2 > ··· β_n > 0, σ > ρ_1 > ρ_2 >··· ρ_N > 0, n_1 = [α] + 1, n_2 = [σ] + 1 for α, σ ∈ N and n_1 = α, n_2 = σ for α, σ ∈ N, β_q, p_q < 1 for any q ∈ {1,2.....N), D is the standard Riemann-Liouville differentiation and f, g : [0, 1] × R~(N+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.