Oscillations in Neutral Functional Differential Equations

摘要

A neutral functional differential equation as defined below includes the scalar differential-difference equation (1) d/dt(x(t)+ax(t-1)+ epsilon G(t,x(t-1))) = bx(t)+cx(t-1)+ epsilon F(t,x(t),x(t-1)) where epsilon is a parameter, a,b,c are constants and G(t,x), F(t,x,y) are continuous functions of t,x,y. For any continuous function phi defined on (-1,0), a solution of (1) is a continuous function x defined on some interval (-1, alpha), alpha > 0, which coincides with phi on (-1,0) and is such that the expression x(t) + ax(t-1) + G(x(t-1)) (not x(t)) is continuously differentiable and satisfies (1) on (0, alpha). The purpose of this paper is to prove for epsilon small the existence of bounded and periodic solutions of (1). (Author)