Asymptotic behavior of nonoscillatory solutions to n-th order nonlinear neutral differential equations

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Abstract

For a class of n-th order nonlinear neutral differential equations, sufficient conditions for all nonoscillatory solutions to satisfy limt+t1nx(t)=a are established. For another class of equations, necessary and sufficient conditions for nonoscillatory solutions to satisfy the above condition are obtained. Our theorems complement and extend recent results reported in the literature.

Introduction

In this paper, we are concerned with the asymptotic behavior of nonoscillatory solutions to the n-th order nonlinear neutral differential equation of the form (x(t)+p(t)x(tτ))(n)+f(t,x(t),x(ρ(t)),x(t),x(σ(t)))=0. 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 equations have many important applications in science and technology and are used, for instance, to describe distributed networks with lossless transmission lines. It is known [6] that the presence of a neutral term in a differential equation can cause oscillation, and it also can destroy the oscillatory nature of the original differential equation. In general, the study of neutral differential equations brings more complications compared to that of ordinary differential equations, although in some cases certain similarities in the behavior of solutions of ordinary and neutral differential equations can be observed.

Recently, several authors discussed the existence of solutions to second order nonlinear differential equations u+f(t,u,u)=0,tt01 and u+f(t,u)=0,tt01 which behave like nontrivial linear functions at infinity; see, for instance, Constantin [1], [2], Lipovan [16], Mustafa and Rogovchenko [17], [18], [19], [20], Philos et al. [23], Rogovchenko and Rogovchenko [24], [25], Rogovchenko [26], Rogovchenko and Villari [27], Seifert [28], Serrin and Zou [29], Yin [31], Zhao [32].

Many interesting results regarding asymptotic properties of solutions of different classes of functional differential equations have been obtained by Dahiya and Singh [3], Dahiya and Zafer [4], Džurina [5], Graef and Spikes [8], Grammatikopoulos et al. [9], Kong et al. [12], Kulcsár [13], Lacková [14], Ladas [15], Naito [21], [22], Tanaka [30], and other authors.

In particular, Kong et al. [12] gave a classification of nonoscillatory solutions of an odd order linear neutral differential equation (x(t)x(tτ))(n)+p(t)x(tσ)=0 and established conditions for the existence of each type of nonoscillatory solution. Naito [21] proved that an n-th order nonlinear neutral differential equation dndtn[x(t)+λx(tτ)]+σF(t,x(g(t)))=0 has a solution satisfying limt+tkx(t)=c>0 if and only if t0+tnk1F(t,c[g(t)]k)dt<+for some c>0, whereas Naito [22] derived a necessary and sufficient condition for a neutral differential equation dndtn[x(t)h(t)x(τ(t))]+f(t,x(g(t)))=0 to have a positive solution satisfying limt+x(t)h(t)x(τ(t))tk=c>0.

Recently, Džurina [5] extended results of the second author [26] to a class of neutral differential equations (x(t)+px(tτ))+f(t,x(t),x(t))=0, establishing conditions under which all nonoscillatory solutions behave like linear functions at+b as t+ for some a,bR.

The purpose of this paper is to find sufficient conditions which guarantee that all nonoscillatory solutions of the general n-th order nonlinear neutral differential equation (1) satisfy, for some real a, limt+x(t)tn1=a. In addition, for a particular case of Eq. (1) of the form (x(t)+p(t)x(tτ))(n)+f(t,x(t),x(ρ(t)))=0, we establish necessary and sufficient conditions for the existence of nonoscillatory solutions satisfying condition (3). Our theorems extend to Eq. (1) recent results on asymptotic behavior of nonoscillatory solutions of second order nonlinear neutral differential equations reported by Džurina [5] and complement research on asymptotic behavior of nonoscillatory solutions of functional differential equations reported by Dahiya and Singh [3], Dahiya and Zafer [4], Graef et al. [7] Graef and Spikes [8], Grammatikopoulos et al. [9], Kong et al. [12], Kulcsár [13], Ladas [15], Naito [21], [22], and Tanaka [30]. Examples are provided to illustrate the main results obtained in the paper.

Section snippets

Auxiliary lemma

Let R+=[0,+). In what follows, we suppose that

(A1) fC[R+×R4,R], and there exist functions ϕk,ωlC[R+,R+],k=1,,5,l=1,4, such that either |f(t,u1,u2,v1,v2)|ϕ1(t)+ϕ2(t)ω1(|u1|tn1)+ϕ3(t)ω2(|u2|[ρ(t)]n1), or |f(t,u1,u2,v1,v2)|ϕ4(t)+ϕ5(t)ω3(|u1|tn1)ω4(|u2|[ρ(t)]n1), where, for s>0, the functions ωl(s) are positive, nondecreasing and t0+ϕk(s)ds=Ak<+,k=1,,5; (A2) ρ,σC[R+,R+],ρ(t)t,σ(t)t,limt+ρ(t)=+, and limt+σ(t)=+;

(A3) pC[R+,R+],0p(t)p<1, and limt+p(t)=p0.

For tt0, let Ψ1(t

Continuation of solutions

The following result has independent interest and is used to assure that any nonoscillatory solution of Eq. (1) can be indefinitely continued to the right.

Theorem 3

Suppose that there exists a nonoscillatory solution x(t) of Eq.(1)defined on [t0,T),t0<T<+ , which cannot be continued to the right beyond T .

(i) If f(t,u1,u2,v1,v2) satisfies(5), then G1(+)<+ .

(ii) If f(t,u1,u2,v1,v2) satisfies(6), then G2(+)<+ .

Proof

(i) Let x(t) be a nonoscillatory solution of Eq. (1) defined on [t0,T),t0<T<+, which cannot

Asymptotic behavior of solutions of Eq. (1)

Theorem 5

Suppose that(5)holds andG1(+)=+.Then any nonoscillatory solution x(t) of Eq.(1)satisfies(3), and there exist nonoscillatory solutions for which a0 .

Proof

Let x(t) be a nonoscillatory solution of Eq. (1) and z(t) be defined by (15). Then (20) holds, and Φ1(t)M+A1+t0t[ϕ2(s)ω1(Φ1(s))+ϕ3(s)ω2(Φ1(s))]ds. Observing that ϕ2(t)ω1(Φ1(t))+ϕ3(t)ω2(Φ1(t))[ϕ2(t)+ϕ3(t)]Ω1(Φ1(t)), one has Φ1(t)M+A1+t0t[ϕ2(s)+ϕ3(s)]Ω1(Φ1(s))ds. An application of the Bihari inequality yields Φ1(t)G11(G1(M+A1)+t0t[ϕ2(s)+ϕ3(s

Asymptotic behavior of solutions of Eq. (4)

In this section, we study asymptotic behavior of solutions of Eq. (4). In what follows, we suppose that

(B1) fC[R+×R2,R], and there exist functions ϕk,ηlC[R+,R+],k=1,,5,l=1,,4, such that, for s>0,ηj(s) are nondecreasing, and either |f(t,u1,u2)|ϕ1(t)+ϕ2(t)η1(|u1|tn1)+ϕ3(t)η2(|u2|[ρ(t)]n1), or |f(t,u1,u2)|ϕ4(t)+ϕ5(t)η3(|u1|tn1)η4(|u2|[ρ(t)]n1); (B2) ρC[R+,R+],ρ(t)t, and limt+ρ(t)=+;

(B3) pC[R+,R+],0p(t)p<1, and limt+p(t)=p0;

(B4) if u1 and u2 have the same sign, then f(t,u1,u2)

Acknowledgements

The authors express their sincere gratitude to the referee for careful reading of the manuscript and valuable suggestions that helped to improve the paper. The research of the second author has been supported in part by the Abdus Salam International Centre for Theoretical Physics through Associate Membership and by the Faculty of Science and Technology of the University of Kalmar through a research grant.

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