A note on periodic solutions for delay-differential systems

Abstract:

Let $f(t,x,y):[0,\infty ) \times {R^n} \times {R^n} \to {R^n}$ be continuous and 1-periodic in t, $\tau (t):[0,\infty ) \to [0,h](0 < h \leqq 1)$ continuous and 1-periodic. A simple geometric condition (Theorem 1) is given for the existence of 1-periodic solutions $x(t)$ of the nonlinear delay-differential system $x’(t) = f(t,x(t),x(t - \tau (t)))$, with $x(t)$ in a given bounded convex open set G in ${R^n}$. The addition of a Lipschitz condition in x and monotonicity in y allows one to calculate $x(t)$ by a monotone sequence of successive approximations (Theorem 2). Extensions to a more general functional differential equation $x’(t) = g(t,x(t),{x_t})$ are given.