Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales

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.

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