Multiple Positive Periodic Solutions for Two Kinds of Higher-Dimension Impulsive Differential Equations with Multiple Delays and Two Parameters

Abstract

By applying the fixed point theorem, we derive some new criteria for the existence of multiple positive periodic solutions for two kinds of  n-dimension periodic impulsive functional differential equations with multiple delays and two parameters:xi′(t)=ai(t)xi(t)-λbi(t)fi(t,x(t),x(t-τ1(t)),…,x(t-τn(t)))), a.e.,t>0,t≠tk,k∈Z+,xi(tk+)-xi(tk-)=μcikxi(tk),i=1,2,…,n,k∈Z+,andxi′(t)=-ai(t)xi(t)+λbi(t)fi(t,x(t),x(t-τ1(t)),…,x(t-τn(t)))), a.e.,t>0,t≠tk,k∈Z+,xi(tk+)-xi(tk-)=μcikxi(tk),i=1,2,…,n,k∈Z+.As an application, we study some special cases of the previous systems, which have been studied extensively in the literature.