Ulam stability for nonlinear boundary value problems for impulsive Caputo type fractional delay differential equations

Abstract In this paper we consider impulsive delay differential equations with the Caputo fractional derivative with respect to another function on a finite interval. We set up and study a problem that consists of an initial condition on the initial time interval and a nonlinear boundary condition. This mixed boundary value problem depends on a parameter. We obtain an integral presentation of the solution and establish the existence and uniqueness of the solution for any value of the real valued parameter. We discuss the Ulam type stability for boundary value problems and, based on the application of an appropriate modification of the classical definition, we suggest a way to study this type of stability. We keep the idea that the solution of the studied problem will depend significantly on the chosen solution of the corresponding differential inequality, and this is based on an appropriate choice of the parameter involved in the boundary condition. Some examples are given to illustrate the theoretical results.