Periodic solutions for a kind of second order neutral functional differential equations

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Abstract

In this paper, criteria are established for the existence of periodic solutions to a second order sublinear neutral differential equation. Our method is based on careful a priori estimation and continuation theorem, and our sublinear condition is an improvement of the boundedness condition in some recent results.

Introduction

Forced second order differential equations such as Duffing and Rayleigh equations have been the subject of many investigations since they arise in many physical model. In particular, periodic solutions of such equations with delays were studied by using Mawhin’s continuation theorem by Iannacci and Nkashama [1], Wang and Cheng [2], and many others (see for example [3], [4], [5], [6], [7], [8]). But the works to study the existence of periodic solutions for neutral functional differential equations (NFDE) by using Mawhin’s continuation theorem rarely appeared [9], [10]. The reason for it lies in the following two respects. The first is that the criterion of L-compact of nonlinear operator N on the set Ω¯ is difficulty to establish; the second is that a priori bound of solutions is not easy to estimate. In paper [9], Enrice Serra studied a kind of first order NFDE in the following form:x(t)+ax(t-τ)=f(t,x(t))under the condition a<1 andα(t)liminf|s|f(t,s)slimsup|s|f(t,s)sβ(t),fora.e.tP,where α,βL,f:R×RR is Caraéodory function of 2π period in the first variable, P is a subset of [0,2π] with positive measure. The author obtained that the above equation has at least one periodic solution (see Theorem 3.1 in [9] ).

In this paper, we investigate the existence of periodic solutions to the second order NFDE[x(t)-cx(t-r)]+g(x(t-τ))=p(t),where c<1,r,τ are constants and the forcing function p:RR is a T periodic continuous function with zero mean over [0,T], i.e.,0Tp(s)ds=0.

In this paper, we will establish criteria for the existence of T periodic solutions of (1) under a sublinear condition on the function g. Our criteria, as will be explained in the last section, are compatible with the existent ones for the well known Duffing equations. The techniques for obtaining the existence criteria are based on a priori estimation and Mawhin’s continuation theorem.

Section snippets

Preparatory results

Let CT={xxC(R,R),x(t+T)x(t)} with the norm ϕ0=maxt[0,T]ϕ(t),ϕCT, and CT1={xxC1(R,R),x(t+T)x(t)} with the norm ϕ=max{ϕ0,ϕ0}. Clearly, CT and CT1 are two Banach spaces. We also define operator A in the following formA:CTCT,(Ax)(t)=x(t)-cx(t-r).

Lemma 1

LetB:CTCT,(Bx)(t)=cx(t-r).If c<1, then B satisfies the following conditions

  • (1)

    Bc.

  • (2)

    0T[Bx](t)dtc0Tx(t)dt,xCT.

The conclusion of case 1 is obviously true. The conclusion of case 2 follows from0T[Bx](t)dt0Tcx(t-r)dt=c

Main results

In order to use Lemma 3, throughout this paper, we assume that p is real T-periodic function with zero mean over [0,T], g is real continuous function and τ,r>0, c<1. We denote X=CT1,Y=CT.

Theorem 1

If there are constants D>0,β>0 and α[0,1) such thatxg(x)>0,xD,and eitherg(x)βxα,xD,org(x)-βxα,x-D,Then (1) has a T-periodic solution.

Proof

It is easy to see that Eq. (1) has a T-periodic solution if and only if the following operator equationLx=Nxhas a T-periodic solution, where L is defined in Section 2

Remarks and examples

Remark 1

It was proved in [3] that if there exist constants β,D>0 such that g(x)-β for x-D and xg(x)>0 for xD, then (1) has a 2π periodic solution when c = 0. In Theorem 1 we only assume sublinear g and hence is an improvement. Moreover, we extend the results in [11] to neutral differential equations.

Example 1

Consider the following equation:x(t)-12x(t-π)+x13[1+e-x(t-π)]=sint.Corresponding to Eq. (1), we have T=2π,c=12,p(t)=sint,g(x)=x13(1+e-x), andxg(x)>0,g(x)=x13(1+e-x)<2x13forx>π.So the conditions of

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Supported by the NNSF of China (No. 10571050) and Key Project of Hunan Province Department of Education of China.

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