Solvability of Anti-periodic BVPs for Impulsive Fractional Differential Systems Involving Caputo and Riemann–Liouville Fractional Derivatives

Abstract

Abstract Sufficient conditions are given for the existence of solutions of anti-periodic value problems for impulsive fractional differential systems involving both Caputo and Riemann–Liouville fractional derivatives. We allow the nonlinearities p ( t ) f ( t , x , y , z , w ) $p(t)f(t,x,y,z,w)$ and q ( t ) g ( t , x , y , z , w ) $q(t)g(t,x,y,z,w)$ in fractional differential equations to be singular at t = 0 $t=0$ and t = 1 $t=1$ . Both f $f$ and g $g$ may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. Many unsuitable lemmas in recent published papers are pointed out in order not to mislead readers. A conclusion section is given at the end of the paper.