Regularization of Second-Order Functional Differential Equations in Weighted Spaces

Abstract

This paper develops a regularization framework for second-order functional differential equations in weighted Banach spaces. The main object is the delayed equation u′′(t) + a(t)u′(t) + b(t)u(t) = F(t, u(t), u(t − τ )), and its regularized counterpart u′′ ε (t) + a(t)u′ε(t) + b(t)uε(t) + εLuε(t) = F(t, uε(t), uε(t − τ )).

The analysis is motivated by previous work of Manuel J. Alves and Elena V. Alves on functional differential equations, weighted functional settings, operator methods and delayed or memory-dependent combustion models. The local assertions are reorganized into four principal results: existence and uniqueness, uniform a priori estimates, compactness and convergence, and spectral stability. Complete proofs are supplied. Additional computational examples and professional PGFPlots figures illustrate weighted decay, delay amplification, error rates, stability regions and non-stationary combustion profiles. The revised version also defines a concrete regularizing operator, includes reproducible Python and MATLAB simulations, demonstrates the explicit O(ε) convergence rate, makes the regularizing operator explicit in every theorem, gives a deeper numerical stability analysis, and compares the approach with five recent papers published in 2023–2026