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

Yuji Liu; Shimin Li

doi:10.26637/mjm304/006

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 derivative, Caputo type Hadamard fractional derivative, fixed point theorem

Mathematics Subject Classification:

92D25, 34A37, 34K15

Authors Imprint References Funding Related Articles Downloads

Yuji Liu Department of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou-510320, P R China.
Shimin Li Department of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou-510320, P R China.

Pages: 423-490
Date Published: 01-10-2015
Vol. 3 No. 04 (2015): Malaya Journal of Matematik (MJM)

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The research was supported by the National Natural Science Foundation of China (No: 11401111), the Natural Science Foundation of Guangdong province (No:S2011010001900), the Natural Science Foundation of institution of higher education of Guangdong province (No:2014KTSCX126) and the Foundation for High- level talents in Guangdong Higher Education Project.