An exploration of global existence and uniqueness for neutral differential equations with state-dependent delay
摘要
This manuscript investigates abstract neutral differential equations with state-dependent delay in Banach spaces, in which the delay varies with the evolving state of the system. This article presents a unified abstract framework that enables a systematic analysis of existence, uniqueness, and qualitative properties of solutions. Within this framework, by using appropriate auxiliary lemmas together with the Banach fixed point theorem, we derive sufficient conditions that ensure the existence and uniqueness of mild solutions in suitable Lipschitz spaces. By imposing further regularity assumptions on the nonlinear terms and the delay function, we show that the mild solution attains the regularity of a strict solution. The analysis is carried out on both semi-infinite time intervals and the entire real line, enabling us to obtain results on global existence, uniqueness, including continuous dependence of solutions on the initial data. The developed framework unifies and extends several earlier results on neutral functional differential equations with state-dependent delays, and its applicability is illustrated through an example involving a neutral partial differential equation.