Authors: Dr. M. K. Vediappan, Dr. K. Srinivasan
Abstract: This paper establishes a comprehensive well-posedness and stability theory for a class of nonlinear ψ-Hilfer variable-order fractional integrodifferential equations (VO-FIDEs) of the form ᴙ^{α(⋅),β}_{ψ} x(t) = f(t, x(t), ∫₀ᵗ κ(t,s,x(s))ds) subject to nonlocal integral boundary conditions on a finite interval [a, b]. The fractional derivative is taken in the ψ-Hilfer sense with a continuous variable order α : [a,b] → (0,1] and type β ∈ [0,1], which simultaneously unifies the Riemann–Liouville, Caputo, Hilfer, and Hadamard operators as special cases. Three principal results are established: (i) existence of at least one solution via the Schauder fixed-point theorem in a suitably weighted Banach space; (ii) uniqueness of the solution via the Banach contraction principle under a generalized Lipschitz condition; and (iii) Ulam–Hyers–Rassias (UHR) stability, providing quantitative bounds on the deviation of approxi-mate solutions from exact ones. The variable-order framework captures systems whose memory depth evolves dynamically, a feature relevant to viscoelastic materials, anomalous diffusion with space-dependent porosity, and variable-memory epidemic models. New inte-gral inequalities for ψ-Hilfer variable-order operators are derived as auxiliary results. Two illustrative examples confirm the theoretical findings, and a comparison with constant-order results reveals the strictly broader applicability of the variable-order framework.
DOI: http://doi.org/10.5281/zenodo.20347750
