The State-dependent Delay Functional Differential Equations With Initial Value Conditions

In this paper, we consider the functional differential equation with state-dependent delay where assume h is a positive constant, C=C([-h,0],R) is a Banach space which normed with d∈C(R,(0,h)), r: Câ†'(0,h) for all φ∈C satisfy r(φ)=d(φ(-r(φ))). For all functional x∈C([-h,∞),R) and t∈[0,∞), xt∈C defined as xt(s)=x(t+s).∫:[0,∞)×Râ†'R, φ∈C. and extending the corresponding results in the literatures.In Chapter1, we introduce the research background and results of internal and external about equation (1), and supplying some essential knowledge.In Chapter2, we consider the existence and uniqueness of solutions about equation (1).establish the result to the question (1) in the small square [-h,â–³], use the proof by contradiction to prove the result is uniqueness.In Chapter3, we consider the continuous dependence on initial value conditions of equation (1).In Chapter4, we consider the slowly oscillating solutions of equation (1).This paper we extend the conclusion of related literature [J.Differential Equations,2003,195(1):46-65].