Impulsive nonlocal boundary value problems for $(k,\psi )$-Hilfer proportional fractional differential equations: Existence, stability, and application to pantograph equations

This paper examines a class of impulsive boundary value problems related to fractional pantograph differential equations governed by the $(k,\psi )$-Hilfer proportional fractional derivative. The problem is reformulated into an equivalent integral equation, which provides a convenient framework for further analysis. The existence and uniqueness of solutions are derived using Banach's fixed-point theorem. Moreover, several forms of Ulam stability are examined. Illustrative examples and graphical computations are included to support the applicability of the theoretical results.