Existence and Uniqueness of Solutions for a Nonlinear Coupled System of Fractional Differential Equations on Time Scales

Abstract

In this paper, we establish the criteria for the existence and uniqueness of solutions of a two-point BVP for a system of nonlinear fractional differential equations on time scales.

$$\begin{aligned} \begin{aligned} \Delta _{a^{\star }}^{\alpha _{1}-1}x(t)&=f_{1}(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb {T},\\ \Delta _{a^{\star }}^{\alpha _{2}-1}y(t)&=f_{2}(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb {T},\\ \end{aligned} \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} \begin{aligned} x(a)=0,&\quad x^{\Delta }(b)=0,\quad x^{\Delta \Delta }(b)=0,\\ y(a)=0,&\quad y^{\Delta }(b)=0,\quad y^{\Delta \Delta }(b)=0. \end{aligned} \end{aligned}$$

where $\mathbb {T}$ is any time scale (nonempty closed subsets of the reals), $2<\alpha _{i}<3$ and $f_{i}\in C_{rd}([a,b]\times \mathbb {R}\times \mathbb {R}, \mathbb {R})$ and $\Delta _{a^{\star }}^{\alpha _{i}-1}$ denotes the delta fractional derivative on time scales $\mathbb {T}$ of order $\alpha _{i}-1$ for $i=1, 2$. By using the Banach contraction principle. Finally, an example is given to illustrate the main result.