Nonlinear impulsive problems for fractional differential equations and Ulam stability

In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided.