Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy. (5th May 2014) - Xia, Yong-Hui
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 ) | ≤ μ e − ε | t | . However, this condition is very restricted. There are few functions satisfying | f ( t, x ) | ≤ μ e − ε | t | . In some sense, this assumption is not reasonable enough. More suitable assumption should be | f ( t, x ) | ≤ μ . 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 ) | ≤ μ . 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.