Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy

Abstract

Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies $|f(t,x)|\le \mu e^{-\epsilon |t|}$. However, this condition is very restricted. There are few functions satisfying $|f(t,x)|\le \mu e^{-\epsilon |t|}$. In some sense, this assumption is not reasonable enough. More suitable assumption should be $|f(t,x)|\le \mu$. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption $|f(t,x)|\le \mu$. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.