Abstract

In this work we study an approximation of a mild solution y of a semilinear first order abstract differential problem with delay, which depends of an initial history condition and an unbounded closed linear operator Agenerating a C-0-semigroup on a Banach space X. The approximation considers the mild solutions (z(delta)) delta>0 of the corresponding family of differential equations with piecewise constant argument, varying the semilinear term with a parameter d. Our main results is about the obtaining of the solution z(delta) in terms of a difference equation on X and conditions to ensure uniform convergence of z(delta) to y as delta -> 0, on compact and unbounded intervals. We obtain explicit exponential decay estimates for the error function using the stability of the semigroup and the Halanay's inequality. Also with a new idea and method we prove that the approximation is stable and there exists a preservation of asymptotic stability between the solution of delayed differential equation and its corresponding difference equation, obtained by piecewise constant argument. (C) 2021 Elsevier Inc. All rights reserved.