We use the continuation theorem of coincidence degree theory and Liapunov functions to study the existence and stability of periodic solutions for the Cohen-Grossberg neural network with multiple delays.

The existence and the global attractivity of a positive periodic solution of the delay differential equation y(t)=y(t)F[t,y(t-τ1(t)),…, y(t-τn(t))] are studied by using some techniques of the Mawhin coincidence degree theory and the constructing suitable Liapunov functionals. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved.

We apply the generalized form of Leggett-Williams fixed point theorem to prove that the following nonlinear functional differential equation X'(t)=-a(t)x(t)+f(t,xt) has at least two positive T-periodic solutions, where a (t) is a T-periodic function satisfying exp (∫0a(u)du)>1, f (t,xt) is a nonnegative function defined on R×BC, where BC denotes the Banach space of bounded continuous functions.

The existence of a positive periodic solution for {dt dP(t) dH(t)=r(t)H(t)[1-K(t) H(t-τ(t)))]-a(t)H(t)P(t), dt dP(t)=-b(t)P(t)+β(t)P(t)H(t)-σ(t)) is established, wherer, K, a, b, βare positive periodic continuous functions with period ω>0, and τ, σare periodic continuous functions with period ω.

In this paper, we employ Avery-Henderson fixed point theorem to study the existence of positive periodic solutions to the following nonlinear nonautonomous functional differential system with feedback control: dx dt =−r(t)x(t)+ F(t,xt,u(t−δ(t))), du dt =−h(t)u(t)+ g(t)x(t−σ(t)). We show that the system above has at least two positive periodic solutions under certain growth condition imposed on F.