Regularized Urysohn Operators for Nonlinear Systems with Memory: Theory, Numerical Methods, and Applications to Artificial Intelligence and Combustion

Abstract

Nonlinear systems with memory arise in learning algorithms, hereditary mechanics, biological regulation, combustion, and other processes whose present state depends on a distributed history of the unknown. This paper develops a compact mathematical and computational framework for regularized Urysohn operators acting on Banach spaces of continuous trajectories. We consider nonlinear integralmemory models of the form u(t) = g(t) + T ∫0 K(t, s, u(s), u(t − τ (s))) ds,  and their regularized counterparts uε(t) + εLuε(t) = g(t) + T∫ 0 K(t, s, uε(s), uε(t − τ (s))) ds, ε > 0,where L is a stabilizing positive operator and K is a nonlinear Urysohn kernel. Under natural continuity, growth, Lipschitz, and compactness assumptions we prove existence, uniqueness, a priori estimates, compactness of approximate solutions, stability with respect to data, and convergence as the regularization parameter tends to zero. We also present Nyström and collocation schemes, derive practical error estimates, and give algorithms for Picard and damped Newton iterations. Two application modules are discussed: memory-augmented artificial intelligence models, where Urysohn terms represent nonlinear temporal attention or reservoir-type memory, and solid-propellant combustion, where the kernel captures thermal memory and pressure-dependent burning laws. The numerical illustrations emphasize regularization convergence, stability regions, parameter sensitivity, and comparison of discretization methods. The results show that regularized Urysohn operators provide a flexible bridge between rigorous nonlinear analysis and computational models with memory