Existence, Uniqueness, and Ulam–Hyers–Rassias Stability of a Nonlinear ψ-Hilfer Variable-Order Fractional Integrodifferential System with Nonlocal Integral Boundary Conditions

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 approximate 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 integral 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 variableorder framework.