On Pantograph Problems Involving Weighted Caputo Fractional Operators with Respect to Another Function

In this investigation, weighted psi-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes into account the situation where ?(x)=x. With the aid of Banach's and Krasnoselskii's classic fixed point results, we have established a the qualitative study. Different values of ?(x) and w(x) are discussed as special cases that are relevant to our current results. Additionally, in light of our findings, we provide a more general fractional system with the weighted ?-Caputo-type that takes into account both the new problems and certain previously existing, related problems. Finally, we give two illustrations to support and validate the outcomes.