Positive Periodic Solution for the Generalized Neutral Differential Equation with Multiple Delays and Impulse

Abstract

By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for $k$-set contraction, we established some new criteria for the existence of positive periodic solution of the following generalized neutral delay functional differential equation with impulse: $x\mathrm{\text{'}}(t)=x(t)[a(t)-f(t,x(t),x(t-{\tau }_{\mathrm{1}}(t,x(t))),\dots ,x(t-{\tau }_n(t,x(t))),x\mathrm{\text{'}}(t-{\gamma }_{\mathrm{1}}(t,x(t))),\dots ,x\mathrm{\text{'}}(t-{\gamma }_m(t,x(t))))],t\ne t_k,k\in Z_{+};x(t_k^{+})=x(t_k^{-})+{\theta }_k(x(t_k)),k\in Z_{+}$. As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.