Quasi-periodic solutions for differential equations with an elliptic equilibrium under delayed perturbation

We employ the Craig-Wayne-Bourgain (CWB) method to construct quasi-periodic solutions for the nonlinear delayed perturbation equations. The linearized equation at each step of the iterations is a differentialdifference equation on the torus. We shall combine the techniques of Green's function estimate and the reducibility method in KAM theory to solve the linear equation, which generalizes the applicability of the CWB method. As an application, we study the positive quasi-periodic solutions for a class of Lotka-Volterra equations with quasi-periodic coefficients and time delay. (c) 2023 Elsevier Inc. All rights reserved.