Existence and uniqueness of solution for perturbed nonautonomous systems with nonuniform exponential dichotomy (Q1724826)

Summary: Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies \(| f(t, x) | \leq \mu e^{- \varepsilon | t |}\). However, this condition is very restricted. There are few functions satisfying \(| f(t, x) | \leq \mu e^{- \varepsilon | t |}\). In some sense, this assumption is not reasonable enough. More suitable assumption should be \(| f(t, x) | \leq \mu\). 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 \(| f(t, x) | \leq \mu\). 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.