Solvability of Multi-Order $\psi$-Hilfer Fractional Langevin Equations with Infinite Delay in Weighted Phase Spaces

This work introduces a new class of fractional Langevin equations with infinite delay by incorporating generalized Hilfer derivatives within a weighted Banach space framework. Unlike existing studies that rely on standard Caputo or Hilfer operators without incorporating delay-weight interactions, the proposed formulation simultaneously accounts for nonlocal memory and non-uniform decay effects. This dual mechanism enables the modeling of long-term hereditary responses in systems where past states contribute with varying intensity over time, a behavior unattainable in traditional fractional models. Existence and uniqueness of solutions are established using Banach's fixed point theorem and the Leray--Schauder alternative, both adapted to the weighted setting. Two illustrative examples are provided to demonstrate how generalized weighted Hilfer derivatives alter the dynamical behavior compared to classical formulations. These results offer a new perspective on delayed fractional systems, expanding the analytical toolkit available for modeling complex memory-driven processes.