Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra System with Distributed Delays and Impulses

Abstract

By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays: $x_i^{\prime }(t)=x_i(t)[r_i(t)-{\sum }_{j=\mathrm{1}}^n{a}_{ij}(t)x_j(t)-{\sum }_{j=\mathrm{1}}^n\mathrm{‍}{b}_{ij}(t){\int }_{-{\tau }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(\xi )x_j(t+\xi )d\xi -{\sum }_{j=\mathrm{1}}^n\mathrm{‍}{c}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{g}_{ij}(\xi )x_j^{\prime }(t+\xi )d\xi ],\hspace{0.17em}\hspace{0.17em}\mathrm{\Delta }{x}_i(t_k)=-I_{ik}(x_i(t_k)),\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2},\dots ,n,\hspace{0.17em}\hspace{0.17em}k=\mathrm{1,2},\dots$$.\mathrm{}$ Some verifiable criteria are established easily.