Some Results On Second-order Neutral Functional Differential Equations With Infinite Distributed Delay

In this paper, we consider two types of second-order neutral functional differential equationswith infinite distributed delay. By choosing available operators and applying Krasnoselskii's fixedpoint theorem, we obtain sufficient conditions for the existence of positive periodic solutions tosuch equations. Consider the following two types of second-order neutral functional differentialequations with infinite distributed delayandhere A is a positive parameter; w and c are two constants with w＞0,|c|＜1;a(t)∈C(R,(0,∞)),b(t)∈C(R,(0,∞)), a(t) and b(t) are w-periodic functions; f(x)∈C(R,[0,∞)),K(r)∈C((-∞,0],[0,∞))There are four sections in the paper. The first section is an introduction to the history ofthe natual delay differential equation. In the second section, we introduce Krasnoselskii's fixedpoint theorem, Arzela-Ascoli theorem and so on. In the third section, the following equationswill be discussed, and we give some Lemmas, which are useful for our next study. At last, wefocus on the existence of positive w-periodic solutions for (1.1) and (1.2), and the proof will bedevided into two part: first consider a special case whenλ= 1, next consider the general one.