Periodic boundary value problems for singular fractional differential equations with impulse effects

Yuji Liua,* and Shimin Lib

Periodic boundary value problems for singular fractional differential equations with impulse effects

Authors :

Yuji Liu^a,* and Shimin Li^b

Author Address :

^a,bDepartment of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou-510320, P R China.

*Corresponding author.

Abstract :

Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann-Liouville and Hadamard fractional derivatives with order $q\in (0, 1)$. Then we obtain exact expression of solutions of impulsive fractional differential equations, i.e., exact expression of piecewise continuous solutions. Finally, four classes of integral type periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearity $ p(t) f (t, x)$ in fractional differential equations to be singular at $ t = 0, 1$ and be involved a superlinear and sub-linear term. The analysis relies on Schaefer’s fixed point theorem.

Keywords :

singular fractional differential system, impulsive boundary value problem, Riemann-Liouville fractional derivative, Caputo fractional derivative, Hadanard fractional de

DOI :

Article Info :

Received : February 05, 2015; Accepted : August 23, 2015.