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

Original Research

## 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.

## Keywords

Fractional differential equations Cauchy problem Fixed point theorem Time scales

## Mathematics Subject Classification

34A08 34B15 34N05

## Notes

### Acknowledgements

I am indebted to the most respected Professor K. Rajendra Prasad and my heartfelt sincere thanks to the referees for their valuable suggestions and comments.

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