Document Type: Research Paper

**Authors**

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China

**Abstract**

In this paper, we study the boundary-value problem of fractional order dynamic equations on time scales,

$$

^c{\Delta}^{\alpha}u(t)=f(t,u(t)),\;\;t\in

[0,1]_{\mathbb{T}^{\kappa^{2}}}:=J,\;\;1<\alpha<2,

$$ $$ u(0)+u^{\Delta}(0)=0,\;\;u(1)+u^{\Delta}(1)=0, $$

where $\mathbb{T}$ is a general time scale with $0,1\in \mathbb{T}$, $^c{\Delta}^{\alpha}$ is the Caputo $\Delta$-fractional derivative. We investigate the existence and uniqueness of solution for the problem by Banach's fixed point theorem and Schaefer's fixed point theorem. We also discuss the existence of positive solutions of the problem by using the Krasnoselskii theorem.

$$

^c{\Delta}^{\alpha}u(t)=f(t,u(t)),\;\;t\in

[0,1]_{\mathbb{T}^{\kappa^{2}}}:=J,\;\;1<\alpha<2,

$$ $$ u(0)+u^{\Delta}(0)=0,\;\;u(1)+u^{\Delta}(1)=0, $$

where $\mathbb{T}$ is a general time scale with $0,1\in \mathbb{T}$, $^c{\Delta}^{\alpha}$ is the Caputo $\Delta$-fractional derivative. We investigate the existence and uniqueness of solution for the problem by Banach's fixed point theorem and Schaefer's fixed point theorem. We also discuss the existence of positive solutions of the problem by using the Krasnoselskii theorem.

**Keywords**

**Main Subjects**

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Volume 42, Issue 2

March and April 2016

Pages 247-262

**Receive Date:**26 February 2014**Revise Date:**15 December 2014**Accept Date:**16 December 2014