Approximation of mild solutions of delay differential equations on Banach spaces

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 A generating a C0-semigroup on a Banach space X. The approximation considers the mild solutions (zδ)δ>0 of the corresponding family of differential equations with piecewise constant argument, varying the semilinear term with a parameter δ. Our main results is about the obtaining of the solution zδ in terms of a difference equation on X and conditions to ensure uniform convergence of zδ to y as δ→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.