Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses Caputo fractional derivative function with respect to another function, supplemented Riemann–Stieltjes integral conditions. The unique solution is obtained via Banach’s contraction mapping principle, while an result established using Leray–Schauder nonlinear alternative. Examples illustrating main are also constructed.