Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρ k , ϕ k)-Hilfer Fractional Integro-Differential Equations.

In this paper, we establish the existence and stability results for the (ρ k , ϕ k) -Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρ k , ϕ k) -Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach's fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O'Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results. [ABSTRACT FROM AUTHOR]