Nonlinear Perturbations and Fixed-Point Telescoping: A Unified Framework for Semilinear Evolution Equations

We develop a comprehensive framework for solving semilinear evolution equations by combiningfixed-point iteration with high-order telescoping semigroup approximations. For theabstract problem u′(t) = Au(t) + N(u(t)), where A generates a strongly continuous semigroupand N is locally Lipschitz, we introduce a telescoping Picard iteration that replaces the exactsemigroup etA with rational approximants Mn(t) of order k ≥ 1. We prove O(n−k) convergenceof the telescoping iterates to the mild solution uniformly on compact time intervals, and—under an additional operator asymptotic expansion hypothesis onMn(t)—an enhanced adjacentdifferencerate ∥un+1 − un∥ = O(n−(k+1)). The framework is validated on seven benchmarkproblems: semilinear heat, nonlinear Schr¨odinger, reaction-diffusion, Burgers, Navier-Stokes,Gross-Pitaevskii, and Cahn-Hilliard equations. We extend the theory to adaptive refinementstrategies and stochastic perturbations. Numerical experiments confirm the theoretical rates(R2 > 0.999) with typical errors of O(10−9) at n = 100. Complete proofs, verification boxes foreach application, Python implementations, and comprehensive numerical data are provided.