Generalized anti-periodic boundary value problems of impulsive fractional differential equations

In this paper, a generalized anti-periodic boundary value problem for an impulsive fractional differential equation is studied. A natural formula of solutions is derived and new existence theorems of the solutions are established under the mixed nonlinear D -contraction condition, comparison condition, sublinear growth condition and nonlinear growth condition via fixed point methods. In particular, a new special hybrid singular type Gronwall inequality is established to obtain a prior bounds of the solutions.