Well-Posedness of the Cauchy Problem for Nonlinear Differential Equations with Variable Delays

Abstract

In the present paper, we prove a theorem on the continuous dependence of the solution on perturbations of the initial data (the initial time, the initial value of the trajectory, the initial function) and the right-hand side; the perturbation of the right-hand side of the differential equation is assumed to be small in the integral sense. Such theorems play an important role in the investigation of optimal control problems [1–3]. The theorem proved here is a straightforward generalization of theorems in [1–4]. The well-posedness of the Cauchy problem for various classes of differential equations for the case in which the perturbation of the right-hand side is assumed to be small in the integral sense was considered in [5–9].