Some Results for a Four-Point Boundary Value Problems for a Coupled System Involving Caputo Derivatives

Abstract

Motivated by the problem (1.1) in [5], in this paper, we prove the existence and uniqueness of solutions for the following system of fractional differential equations with four point boundary conditions:
$$
\left\{\begin{array}{l}
D^\alpha x(t)+f\left(t, y(t), D^\delta y(t)\right)=0, t \in J \\
D^\beta y(t)+g\left(t, x(t), D^\sigma x(t)\right)=0, t \in J \\
x(0)=y(0)=0, x(1)-\lambda_1 x(\eta)=0, y(1)-\lambda_1 y(\eta)=0 \\
x^{\prime \prime}(0)=y^{\prime \prime}(0)=0, x^{\prime \prime}(1)-\lambda_2 x^{\prime \prime}(\xi)=0, y^{\prime \prime}(1)-\lambda_2 y^{\prime \prime}(\xi)=0
\end{array}\right.
$$
where \(3<\alpha, \beta \leq 4, \alpha-2<\sigma \leq \alpha-1, \beta-2<\delta \leq \beta-1,0<\xi, \eta<1\), and \(D^\alpha, D^\beta\), \(D^\delta\) and \(D^\sigma\), are the Caputo fractional derivatives, \(J=[0,1], \lambda_1, \lambda_2\) are real constants with \(\lambda_1 \eta \neq 1, \lambda_2 \xi \neq 1\) and \(f, g\) continuous functions on \([0,1] \times \mathbb{R}^2\).