Periodic solutions for a kind of higher-order neutral functional differential equation with variable parameter

Aijun Yang,Helin Wang

doi:10.1186/1687-1847-2014-187

Abstract

In this paper, we consider a kind of higher-order neutral equation with distributed delay and variable parameter: . By using the classical coincidence degree theory of Mawhin, sufficient conditions for the existence of periodic solutions are established. Recent results in the literature are generalized and significantly improved. Furthermore, two examples are given to illustrate that the results are almost sharp. MSC:34K10, 30D05, 34B45.

Highlights

This paper is devoted to the application of Mawhin’s continuation theorem to investigate the existence of periodic solutions for the following equation: x(t) – p(t)x(t – σ ) (n) + f x(t) x (t) + g x(t + s) dα(s) = q(t), ( . )–r where p, q are continuous periodic functions with period T >, p ∈ Cn(R, R) with |p(t)| =, f, g ∈ C(R, R), r >, n is a positive integer, σ ∈ R, α : [–r, ] → R+ is a bounded variation function, (α) and α( )α(–r), where is the total variation of α(s) over [–r, ].In recent years, the existence of periodic solution for functional differential equations has been studied extensively
In [ ], the authors studied the following equation with a deviating argument: x (t) + f x(t) x (t) + g x t – τ t, x(t) = e(t)
In [ ], the authors proved for the first time the lemma (Lemma . ) for the existence of A– with (Ax)(t) = x(t) – c(t)x(t – τ ) and some properties of A– when c(t) is not a constant

Summary

Introduction

We consider a kind of higher-order neutral equation with distributed delay and variable parameter: (x(t) – p(t)x(t – σ ))(n) + f (x(t))x (t) + g( By using the classical coincidence degree theory of Mawhin, sufficient conditions for the existence of periodic solutions are established. 1 Introduction This paper is devoted to the application of Mawhin’s continuation theorem to investigate the existence of periodic solutions for the following equation:

Objectives

Results

Conclusion