Existence and uniqueness of solutions for fractional differential system with four-point coupled boundary conditions

Abstract

The goal of this paper is to study the existence and uniqueness of solutions for fractional differential system with four-point coupled boundary conditions of the type:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle D^{\alpha _{1}}_{0+}u_{1}(t)+f_1(t,u_{1}(t),u_{2}(t))=0,\\ \displaystyle u_{1}(0)=u_{1}'(0)=0,\ u_{1}(1)=a_{1}u_{2}(\xi _{1}),\\ \displaystyle D^{\alpha _{2}}_{0+}u_{2}(t)+f_2(t,u_{1}(t),u_{2}(t))=0,\\ \displaystyle u_{2}(0)=u_{2}'(0)=0,\ u_{2}(1)=a_{2}u_{1}(\xi _{2}).\\ \end{array} \right. \end{aligned}$$

Our hypotheses on the nonlinearities $f_1$ and $f_2$ are formulated with a mild Lipschitz assumption. The main tools used are spectral analysis of matrices and Perov’s fixed point theorem. An example is also given to illustrate the applicability of the results.