Oscillation criteria for second order nonlinear neutral differential equations

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Abstract

By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form(r(t)(x(t)+p(t)x(t-τ)))+q(t)f(x(t),x(σ(t)))=0.

Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those in the recent paper by Shi and Wang [18] and related contributions to the subject. Examples are provided to illustrate the relevance of new theorems.

Introduction

In this paper, we are concerned with the oscillatory behavior of second order nonlinear neutral delay differential equations(r(t)(x(t)+p(t)x(t-τ)))+q(t)f(x(t),x(σ(t)))=0,where tt0>0,τ0 is a constant, r,σC1([t0,+),(0,+)),p,qC([t0,+),R), and fC(R2,R).

By a solution of Eq. (1) we mean a continuous function x(t), defined on [tx,+), such that r(t)(x(t)+p(t)x(t-τ)) is continuously differentiable and x(t) satisfies (1) for ttx. In the sequel, we assume that solutions of Eq. (1) exist and can be continued indefinitely to the right. Recall that a nontrivial solution x(t) of Eq. (1) is called oscillatory if there exists a sequence of real numbers {tk}k=1, diverging to +, such that x(tk)=0. Neutral differential Eq. (1) is said to be oscillatory if all its solutions oscillate.

Recently, an increasing interest in obtaining sufficient conditions for oscillatory or non-oscillatory behavior of different classes of differential and functional differential equations has been manifested. In particular, investigation of neutral differential equations is important since they are encountered in many applications in science and technology and are used, for instance, to describe distributed networks with lossless transmission lines, in the study of vibrating masses attached to an elastic bar, as well as in some variational problems, see [11]. It is well known [4] that the presence of a neutral term in a differential equation can cause oscillation, but it can also destroy oscillatory nature of a differential equation. In general, investigation of neutral differential equations is more difficult in comparison with ordinary differential equations, although certain similarities in the behavior of solutions of ordinary and neutral differential equations can be observed, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].

In what follows, we briefly review several important oscillation results obtained for second order neutral differential equations. Grammatikopoulos et al. [9] established that conditiont0+q(s)[1-p(s-σ)]ds=+,ensures oscillation of a linear neutral differential equation(x(t)+p(t)x(t-τ))+q(t)x(t-σ)=0.Sufficient conditions for the oscillation of solutions of a slightly more general neutral differential equation(x(t)+p(t)x(t-τ))+q(t)x(σ(t))=0,including the case when p=1, were obtained by Džurina and Mihalı´ková [2]. By using the integral averaging method, Ruan [17] derived a number of general oscillation criteria for a nonlinear neutral differential equation(r(t)(x(t)+p(t)x(t-τ)))+q(t)f(x(t-σ))=0,whereas Li [14] provided classification of non-oscillatory solutions of Eq. (2) and established necessary and/or sufficient conditions for the existence of eventually positive solutions. Ruan’s results for Eq. (2) have been further improved by Li and Liu [15] who exploited a generalized Riccati transformation. Interesting applications of the integral averaging technique to oscillation of several classes of nonlinear neutral differential equations can be found in the papers by Džurina and Lacková [1], Gai et al. [5], and Xu et al. [19]. In particular, the latter paper addresses the oscillation of a nonlinear neutral differential equation(r(t)(x(t)+p(t)x(t-τ)))+f(t,x(t),x(t-σ),x(t))=0,wheref(t,x(t),x(t-σ),x(t))q(t)f1(x(t))f2(x(t-σ))g(x(t)),f1(x)k1>0,f2(x)/xk2>0,g(x)k3>0. Recently, Shi and Wang [18] proved several oscillation criteria for Eq. (1), one of which we present below for the convenience of the reader. In what follows, we use the following notation:D0={(t,s):t0s<t<+}andD={(t,s):t0st<+}.

Theorem 1 [18, Theorem 2]

Let the following conditions hold:

  • (A1)

    for all tt0,0p(t)1,q(t)0, and q(t) is not identically zero for large t,

  • (A2)

    +r-1(s)ds=+,

  • (A3)

    for all tt0,σ(t)t,σ(t)>0, and limt+σ(t)=+,

  • (A4)

    f(x,y)yK>0, for y0, and f(x,y) has the sign of x and y if they have the same sign.

Suppose further that there exist functions HC1(D,R),hC(D0,R), and k,ρC1([t0,+),(0,+)) satisfying(i)H(t,t)=0,tt0;H(t,s)>0,t>st0;(ii)Hs(t,s)0,(t,s)D0;(iii)s[H(t,s)k(s)]+H(t,s)k(s)ρ(s)ρ(s)=-h(t,s)H(t,s)k(s).Assume also that0<infst0liminft+H(t,s)H(t,t0)+,andlimsupt+1H(t,t0)t0tr(σ(s))ρ(s)σ(s)h2(t,s)ds<+.If there exists a function BC([t0,+),R) such thatlimsupt+t0tσ(s)B+2(s)k(s)ρ(s)r(σ(s))ds=+,andlimsupt+1H(t,T)TtKH(t,s)k(s)ρ(s)q(s)(1-p(σ(s)))-r(σ(s))ρ(s)4σ(s)h2(t,s)dsB(T),for any Tt0, where B+(t)=max(B(t),0), then Eq. (1) is oscillatory.

Very recently, Rogovchenko and Tuncay [16] established new oscillation criteria for a second order nonlinear differential equation with a damping term(r(t)x(t))+p(t)x(t)+q(t)f(x(t))=0,without an assumption that has been required in related results reported in the literature over the last two decades. The purpose of this paper is to strengthen oscillation results obtained for Eq. (1) by Shi and Wang [18] using a generalized Riccati transformation and developing ideas exploited by Rogovchenko and Tuncay [16]. In order to illustrate the relevance of our theorems, two examples are provided.

Section snippets

Main results

We say that a continuous function H:D[0,+) belongs to the class I if:

  • (i)

    H(t,t)=0 and H(t,s)>0 for (t,s)D0;

  • (ii)

    H has a continuous partial derivative with respect to the second variable satisfying, for some locally integrable continuous function h,sH(t,s)=-h(t,s)H(t,s).

Theorem 2

Let conditions (A1)-(A3) of Theorem 1 hold with (A4) replaced by(A4)f(x,y)yκ>0, for y0, and xf(x,y)>0, for xy>0.

Suppose that there exits a function ρC1([t0,+),R) such that, for some β1 and for some HI,limsupt+1H(t,t0)t0tH(t,

References (20)

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Research of the second author has been supported in part through the grant from the Faculty of Sciences and Technology of the University of Kalmar.

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