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

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.