The Positive Periodic Solutions Of A Delay Differential Equation With Impulse

In recent years, the researches on mathematical biology which base on the biological dy-namical system have been developing rapidly. The researches on continuous biological dynamical systems are gradually completed and the discussions on impulsive dynamical systems also obtain huge achievements.In this paper, based on impulsive differential equation’s theorem and fixed index theorem in topological degree, we establish a mathematical models for the periodic delay differential equation with impulse, and the existence of positive periodic solutions are studied in the last. We mainly consider the existence,nonexistence of positive ω-periodic solutions for the periodic equation x’(t)=a{t)g(x(t))x{t)-λb{t)f{x(t-Ï„(t))), t≠tk,x{tk+)=x(tk)+δ,t=tk, where a,b e C(R,[0,∞))are w-periodic,f0w a(t)dt>0,f0wb(t)>0, f,g∈C([0,∞),[0,∞). and f[u)> Of oru>0,g(x) is bounded,τ∈C(R,R) is a continuous w-periodic function. Defiuef0=limuâ†'0+f(u)/u,f∞=limuâ†'∞f(u)/u,i0=number of zeros in the set{f0,f∞},i∞=number of inifinities in the set {f0,f∞}.We show that the equation has i0or i∞positive ω periodic solution(s) for sufficiently large or small λ>0,respectively.