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

Original Research
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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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Copyright information

© Foundation for Scientific Research and Technological Innovation 2018

Authors and Affiliations

  1. 1.Department of MathematicsJazan UniversityJazanKingdom of Saudi Arabia

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