AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation27896210.1155/2010/278962278962Research ArticleOscillation for Third-Order Nonlinear Differential Equations with Deviating ArgumentBartušekMiroslav1CecchiMariella2DošláZuzana1MariniMauro2EloePaul1Department of Mathematics and StatisticsMasaryk UniversityCZ-61137 BrnoCzech Republic2Department of Electronic and TelecommunicationsUniversity of FlorenceI-50139 FlorenceItaly201017012010201021102009050120102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argument x(t)+q(t)x(t)+r(t)f(x(φ(t)))=0. Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operator x=x+q(t)x is oscillatory.

1. Introduction

The aim of this paper is to investigate the third order nonlinear functional differential equation with deviating argument

x′′′(t)+q(t)x(t)+r(t)f(x(φ(t)))=0.

The following assumptions will be made. q,r,φ are continuous functions for t0, r(t)>0, φ:[0,)[0,), φ(0)=0, limtφ(t)=, and f is a continuous function, f:, such that

f(u)u>0foru0. In this paper we will restrict our attention to solutions x of (1.1) which are defined in a neighborhood of infinity and sup{|x(s)|:s>t}>0 for any t of this neighborhood. As usual, a solution of (1.1) is said to be oscillatory if it has a sequence of zeros converging to infinity; otherwise it is said to be nonoscillatory.

Throughout the paper we assume that the operator x=x′′′+q(t)x is oscillatory, that is, the second-order equation

h′′(t)+q(t)h(t)=0 is oscillatory.

It is well known, see, for example, , that if (1.3) is nonoscillatory, then (1.1) can be written as a two-term equation of the form

(a1(t)(a2(t)x))+r(t)f(x(φ(t)))=0, where ai,i=1,2, are continuous positive functions for t0.

Asymptotic properties of equations of type (1.4) have been widely investigated in the literature. We refer to  in case when 01/ai(t)dt= (i=1,2), that is, the disconjugate differential operator 1x=(a1(t)(a2(t)x)) is in the so-called canonical form [5, 8, 9] when this property does not occur. Some of these results extend the pioneering works [10, 11], devoted to the equation

x′′′(t)+q(t)x(t)+r(t)x  μ(t)=0, where μ is the quotient of odd positive integers. Other contributions deal with the solvability of certain boundary value problems associated to equations of type (1.1) on compact or noncompact intervals see, for example, [12, 13] or [14, 15], respectively, and references therein.

Recently, oscillation criteria for (1.4) with damping term, that is, for

(a1(t)(a2(t)x))+q(t)x+r(t)f(x(φ(t)))=0

have been presented in  by using a generalized Riccati transformation and an integral averaging technique. Here oscillation means that any solution x of this equation is oscillatory or satisfies limtx(t)=0. Several examples [9, Examples  1–5] concern the case when the second-order equation (1.3) is nonoscillatory and so such an equation can be reduced to a two-term equation of the form (1.4).

If the differential operator x=x′′′+q(t)x is oscillatory, then very little is known. According to Kiguradze , we say that (1.1) has property A if each of its solutions either is oscillatory, or satisfies the condition

limtx(i)(t)=0(i=0,1,2). If any solution x of (1.1) is either oscillatory, or satisfies the condition (1.7), or admits the asymptotic representation

x(i)=c(1+sin(t-α))(i)+εi(t),(i=0,1,2,3),

where c0 and α are constants, the continuous functions εi(i=0,1,2,3) vanish at infinity and ε0 satisfies the inequality cε0(t)>0 for large t, then we say that (1.1) has weak property A.

For n=3, the results in  deal with the equation

x′′′(t)+x(t)+r(t)f(x(t))=0, and read as follows.

Theorem 1.1 (see [<xref ref-type="bibr" rid="B12">16</xref>, Theorem  1.5]).

Let f be a nondecreasing function satisfying 1duf(u)<,--1duf(u)<. Then the condition 0r(t)dt= is necessary and sufficient in order that (1.9) has weak property A.

Theorem 1.2 (see [<xref ref-type="bibr" rid="B12">16</xref>, Corollary  1.5]).

Let for some K>0 and a>0r(t)|f(u)|Kt-1|u|foru,ta. Then (1.9) has property A.

In our previous paper  we have investigated (1.1) without deviating argument (i.e., φ(t)=t), especially when (1.3) is nonoscillatory. More precisely, the nonexistence of possible types of nonoscillatory solutions is examined, independently on the oscillation of (1.3).

Motivated by [1, 16], here we continue such a study, by giving necessary and sufficient conditions in order that all solutions of (1.1) are either oscillatory or satisfy lim inftx(t)=0. The property A for (1.1) is also considered and an extension to (1.1) of Theorem 1.1 is presented.

The role of the deviating argument φ and some phenomena for (1.1), which do not occur when (1.3) is nonoscillatory, are presented. Our results depend on a a priori classification of nonoscillatory solutions which is based on the concept of phase function  and on a suitable energy function. A fixed point method is also employed and sharp upper and lower estimates for bounded nonoscillatory solutions of (1.1) are established by means of a suitable “cut” function. This approach enables us to assume rL1[0,) instead of R(t)L1[0,), where

R(t)=tr(τ)dτ.

2. Classification of Nonoscillatory Solutions

A function g, defined in a neighborhood of infinity, is said to change its sign, if there exists a sequence {tk} such that g(tk)g(tk+1)<0.

The following theorem shows the possible types of nonoscillatory solutions for (1.1). It is worth noting that here q can change sign.

Theorem 2.1.

Any nonoscillatory solution x of (1.1) either satisfies x(t)x(t)0,x(t)0forlarget or x(t)0forlarget,x(t)changesitssign.

Proof.

Without loss of generality suppose that there exists a solution x of (1.1) and T0 such that x(t)>0, x(φ(t))>0, x(t)0 on [T,).

O. Boruvka  proved that if (1.3) is oscillatory, then there exists a continuously differentiable function α:[T,)[T,), called a phase function, such that α(t)>0 and

3(α′′(t))24(α(t))4-α′′′(t)2(α(t))3+q(t)α2(t)=1. Using this result, we can consider the change of variables s=α(t)  ,x(t)=1α(t)Ẋ(s),(·=dds)   for t[T,), s[T*,), T*=α(T). Thus, t=α-1(s) and x′′(t)=-12(α(t))-3/2α′′(t)Ẋ(s)+(α(t))1/2Ẍ(s),x′′′(t)=-12(α(t))-3/2α′′′(t)Ẋ(s)+34(α(t))-5/2(α′′(t))2Ẋ(s)+(α(t))3/2X  (s). Substituting into (1.1), we obtain X(s)+Ẋ(s)(34(α(t))-4(α′′(t))2-12(α(t))-3α′′′(t)+q(t)(α(t))-2)+(α(t))-3/2r(t)f(x(φ(t)))=0. From here and (2.3) we obtain X(s)+Ẋ(s)+(α(t))-3/2r(t)f(x(φ(t)))=0. Because limtφ(t)=, we have for large tX(s)+Ẋ(s)=-(α(t))-3/2r(t)f(x(φ(t)))<0. Since x(t)0, (2.4) yields Ẋ(s)0 and so X(s)<0, that is, Ẍ is decreasing. If there exists s1T such that Ẍ(s1)<0, Ẋ becomes eventually negative, which is a contradiction. Then Ẍ(s)0 and Ẋ(s) is nondecreasing. Let T1T* be such that Ẋ(s)>0 on [T1,). Thus, using (2.8) we obtain Ẍ(s)-Ẍ(T1)=T1sX(u)du-T1sẊ(u)du-Ẋ(T1)(s-T1). Hence, limsẌ(s)=-, which contradicts the positivity of Ẍ(s). Finally, the case Ẋ(s)0 on [T*,) cannot occur, because, if x(t)0 on [T,), then x0, which is a contradiction.

Remark 2.2.

Theorem 2.1 extends [1, Proposition  2] for (1.1) with φ(t)=t and improves [11, Theorem  3.2] for (1.5).

The following lemma is similar to [16, Lemma  2.2].

Lemma 2.3.

Any solution x of (1.1) satisfies lim inft|x(t)|=0,lim inft|x′′(t)|=0.

Proof.

In view of Theorem 2.1, it is sufficient to prove (2.10) and (2.11) for solutions x of (1.1) such that x(t)>0,x(t)0 for large t. If (2.10) does not hold, then, in view of Theorem 2.1, a positive constant C1 exists such that x(t)-C1 for large t and, hence, limtx(t)=-, a contradiction.

Now let us prove (2.11) and, without loss of generality, assume x(t)>0, x(φ(t))>0 for tt0. Suppose, for the sake of contradiction, that lim inft|x′′(t)|=2C>0. Then there exists t1t0 such that either

x′′(t)Corx′′(t)-Cfortt1. If x′′(t)C, then limtx(t)=, which is a contradiction with Theorem 2.1. Thus x′′(t)-C on [t1,). From this and the Taylor theorem we obtain x(t)=x(t1)+x(t1)(t-t1)+t1t(t-σ)x′′(σ)dσx(t1)+x(t1)(t-t1)-C2(t-t1)2, which gives limtx(t)=-, a contradiction to the positivity of x and so (2.11) holds.

Lemma 2.4.

Let q(t)0 for large t. If x is a solution of (1.1) such that x(t)>0 for large t and the function w(t)=x′′(t)+q(t)x(t) is nonincreasing for large t, then limtw(t)=w00.

Proof.

By contradiction, assume w0<0. Then for large tx′′(t)<-q(t)x(t)0. If there exists T0 such that x(T)<0, then x(t)<0 for large t, which is a contradiction. Thus x(t)0 for large t, which is again a contradiction to Theorem 2.1.

In view of Theorem 2.1, any nonoscillatory solution x of (1.1) is one of the following types.

x satisfies for large t

x(t)x(t)0,limtx(t)=0.

x satisfies for large t

|x(t)|mx>0.

x satisfies for large t

x(t)0,lim inft|x(t)|=0,xchangesitssign.

Remark 2.5.

Nonoscillatory solution x of (1.1), such that x changes its sign, is usually called weakly oscillatory solution. Note that weakly oscillatory solutions can be either of Type II or Type III. When φ(t)t, in [1, Theorem  6] conditions are given under which (1.1) does not have weakly oscillatory solutions, especially solutions of Type III. This result will be used later for proving that the only nonoscillatory solutions of (1.1) are of Type I.

3. Necessary Condition for Oscillation

Our main result here deals with the existence of solutions of Type II.

Theorem 3.1.

Assume q is continuously differentiable and is bounded away from zero, that is, q(t)q0>0. If 0|q(t)|dt<,0r(t)dt<, then for any c{0} there exists a solution x of (1.1) satisfying limtx(t)=c,limtx(i)(t)=0,i=1,2.

Proof.

We prove the existence of solutions of (1.1) satisfying (3.3) for c=1.

Let u and v be two linearly independent solutions of (1.3) with Wronskian d=1. By assumptions on q, all solutions of (1.3) and their derivatives are bounded; see, for example, [18, Theorem  2]. Put

β=max1/2u3/2f(u),M=suptt0{|u(t)|,|u(t)|,|v(t)|,|v(t)|} and denote h(s,t)=u(s)v(t)-u(t)v(s). Thus, there exists k1>0 such that (1+2|h(s,t)|)f(u)k1 for any u[1/2,3/2]. Hence2(|h(s,t)|)f(u)k1,(1+|h(s,t)|)f(u)k1. Let t0 be large so that 4k13q0t0r(t)dt12,1q0t0|q(s)|ds14. Let t̅t0 be such that φ(t)t0 for tt̅. Define φ̅(t)={φ(t),iftt̅,φ(t̅),ift0tt̅. Denote by C[t0,) the Fréchet space of all continuous functions on [t0,), endowed with the topology of uniform convergence on compact subintervals of [t0,). Consider the set ΩC[t0,) given by Ω={xC[t0,):12x(t)32}. Let T[t0,) be fixed and let I=[t0,T]. For any xΩ, consider the “cut” function Gx(t)=-tTτ(r(s)f(x(φ̅(s))))(u(s)v(τ)-u(τ)v(s))dsdτ,tI. Then Gx(t)=t  (r(s)f(x(φ̅(s))))(u(s)v(t)-u(t)v(s))ds,Gx′′(t)=t(r(s)f(x(φ̅(s))))(u(s)v(t)-u(t)v(s))ds,Gx′′′(t)=-r(t)f(x(φ̅(t)))-q(t)t(r(s)f(x(φ̅(s))))(u(s)v(t)-u(t)v(s))ds,=-r(t)f(x(φ̅(t)))-Gx(t)q(t). Consider the function g(t)=Gx′′(t)+q(t)Gx(t),tI. Then g(T)=Gx′′(T) and g(t)=Gx′′′(t)+q(t)Gx(t)+q(t)Gx(t)=-r(t)f(x(φ̅(t)))+q(t)Gx(t),tI. Integrating from t to T we have g(t)=g(T)-tTg(s)ds=g(T)+tT(r(s)f(x(φ̅(s))))ds-tTq(s)Gx(s)ds=Gx′′(x)(T)+tT(r(s)f(x(φ̅(s))))ds-tTq(s)Gx(s)ds. From here and (3.11) we get Gx(t)=1q(t)(Gx′′(T)-Gx′′(t)+tT(r(s)f(x(φ̅(s))))ds-tTq(s)Gx(s)ds)=1q(t)tT(r(s)f(x(φ̅(s))))(1-h(s,t))ds+1q(t)T(r(s)f(x(φ̅(s))))(h(s,T)-h(s,t))ds-1q(t)tTq(s)Gx(s)ds. Thus, in view of (3.5), we obtain |Gx(t)|k1q(t)tr(s)ds+maxtsT|Gx(s)|  1q(t)t0|q(s)|ds, or, in view of (3.6), |Gx(t)|k1q0tr(s)ds+14maxtsT|Gx(s)|, that is, maxtσT|Gx(σ)|k1q0tr(s)ds+14maxtσT|Gx(σ)|. Hence |Gx(t)|maxtσT|Gx(σ)|4k13q0tr(s)ds, that is, |tTτ(r(s)f(x(φ̅(s))))(u(s)v(τ)-u(τ)v(s))dsdτ|4k13q0tr(s)ds. In view of (3.19), using the Cauchy criterion, the limit limTtTτ(r(s)f(x(φ̅(s))))(u(s)v(τ)-u(τ)v(s))dsdτ exists finitely for any fixed t. This fact means that the operator H(x)(t)=1-tτ(r(s)f(x(φ̅(s))))(u(s)v(τ)-u(τ)v(s))dsdτ is well defined for any xΩ. Moreover, from (3.19) we have for tt0|tτ(r(s)f(x(φ̅(s))))(u(s)v(τ)-u(τ)v(s))dsdτ|4k13q0tr(s)ds and so, in view of (3.6), H maps Ω into itself.

Let us show that H(Ω) is relatively compact, that is, H(Ω) consists of functions which are equibounded and equicontinuous on every compact interval of [t0,). Because H(Ω)Ω, the equiboundedness follows. Moreover, for any xΩ we have

H(x)(t)=t(r(s)f(x(φ̅(s))))(u(s)v(t)-u(t)v(s))ds, and so |H(x)(t)|2βM2tr(s)ds, which proves the equicontinuity of the elements of H(Ω).

Now we prove the continuity of H on Ω. Let {xn},n, be a sequence in Ω which converges uniformly on every compact interval of [t0,) to xΩ. Because H(Ω) is relatively compact, the sequence {H(xn)} admits a subsequence, denoted again by {H(xn)} for sake of simplicity, which is convergent to y¯Ω. From (3.4) we obtain

|τ(r(s)f(xn(φ̅(s))))(u(s)v(τ)-u(τ)v(s))ds  |2βM2R(τ), where R is defined in (1.13). Hence, in virtue of the Lebesgue dominated convergence theorem, {Gxn(t)} converges pointwise to Gx(t) on [t0,T], that is, limnGxn(t)=Gx(t). Choosing a sufficiently large T and using (3.22), we obtain |H(xn)(t)-H(x)(t)|=|H(xn)(t)-Gxn(t)+Gxn(t)-Gx(t)+Gx(t)-H(x)(t)||H(xn)(T)-1|+|Gxn(t)-Gx(t)|+|H(x)(T)-1||Gxn(t)-Gx(t)|+8k13q0Tr(s)ds. Then, from (3.26), {H(xn)} converges point-wise to H(x) and, in view of the uniqueness of the limit, H(x)=y¯ is the only cluster point of the compact sequence {H(xn)}, that is, the continuity of H in Ω. Applying the Tychonov fixed point theorem, there exists a solution x of the integral equation x(t)=H(x)(t), which is a solution of (1.1) with the required properties.

Remark 3.2.

Theorem 3.1 partially extends [16, Theorem  1.4] for n=3.

4. Sufficient Condition for Oscillation

In this section we give a sufficient condition for oscillation of (1.1) in the sense that any solution is either oscillatory or satisfies lim inftx(t)=0.

Theorem 4.1.

Assume lim infu±|f(u)|>0, and q continuously differentiable for large t satisfying q(t)>0,q(t)0. If r(t)dt=, then any (nonoscillatory) solution x of (1.1) satisfies lim inft|x(t)|=0. Moreover, any nonoscillatory solution with x(t)x(t)0 for large t satisfies limtx(i)(t)=0 for i=0,1,2.

Proof.

To prove the first assertion, it is sufficient to show that (1.1) does not have solutions of Type II. By contradiction, let x be a solution of (1.1) such that x(t)>m>0,x(φ(t))>m for tt0.

Consider the function w given by (2.14). Then

w(t)=x′′′(t)+q(t)x(t)+q(t)x(t)=-r(t)f(x(φ(t)))+q(t)x(t)0. By Lemma 2.4, we have limtw(t)=w,0w<. Since f(x(φ(t)))infumf(u)=K>0fortt0, we get f(x(φ(t)))-q(t)r(t)x(t)K. Consequently, we have from (4.5) w(t0)w(t0)-w(t)=t0tr(s)(f(x(φ(s)))-q(s)r(s)x(s))dsKt0tr(s)ds, and, as t, we get a contradiction. Hence, any nonoscillatory solution satisfies (4.4).

Now let x be a solution of (1.1) such that x(t)>0,  x(φ(t))>0,x(t)0 for tt10. Hence x is of Type I and

limtx(t)=0. Because q is bounded and w is nonincreasing, (2.11) and (4.9) yield limtw(t)=0, and so limtx′′(t)=0. Consider the function F defined by F(t)=F(x)(t)=2x′′(t)x(t)-(x(t))2+q(t)x2(t). Then we have for tt1F(t)=-2r(t)f(x(φ(t)))x(t)+q(t)x2(t)<0. Since F is decreasing and q is bounded, in view of (2.10) we obtain limtF(t)=0 and so limtx(t)=0.

Remark 4.2.

Theorem 4.1generalizes Theorem  1.1. In [11, Theorem  3.8] the first part of Theorem 4.1 is proved, by a different method, for the particular (1.5) under assumptions (4.2) and (4.3).

Applying Theorems 3.1 and 4.1 we get the following result which extendsTheorem  1.1for (1.9).

Corollary 4.3.

Assume (4.1) and that q is continuously differentiable for large t satisfying (4.2) then, condition (4.3) is necessary and sufficient in order to every solution of (1.1) is either oscillatory or satisfies (4.4).

From Theorem 4.1 and its proof, we have the following results.

Corollary 4.4.

Assume that q is continuously differentiable for large t satisfying (4.2). If (4.1) and (4.3) are satisfied, then for any nonoscillatory solution of (1.1) one has F(t)>0,F(t)<0forlarget, where F is defined by (4.11).

In addition, if φ(t)t, then x is a nonoscillatory solution of (1.1) if and only if F(t)>0 for large t.

Proof.

By Theorem 4.1, any nonoscillatory solution of (1.1) is either of Type I or Type III. If x is of Type I, then limtx(i)(t)=0 for i=0,1,2 and so limtF(t)=0. Hence, by using the argument in the proof of Theorem 4.1, we obtain F(t)<0 for large t.

If x is of Type III, then there exists a sequence {tk} of zeros of x tending to such that x′′(tk)0 and so F(tk)>0. Similarly, reasoning as in the proof of Theorem 4.1, one has that F(t)<0 for this solution. Thus, in both cases, the monotonicity of F gives (4.13).

Finally, if φ(t)t and x is an oscillatory solution, then f(x(t))x(t)0 for any t and so F(t)0 for large t. Since F(τk)0 for some sequence {τk}, we get F(t)<0 for large t.

Theorem 4.5.

Let φ(t)t and q is continuously differentiable for t0 satisfying (4.2). If (4.1) and (4.3) are satisfied, for any nonoscillatory solution x of (1.1) defined for ttx, one has x(t)0forttx. In particular, any continuable solution with zero is oscillatory.

Proof.

Assume that x(t)>0 for t>T>tx and x(T)=0. Since φ(t)t, we have f(x(φ(t)))x(t)>0 for t>T. For the function F defined by (4.11), from (4.12) we obtain F(t)0 for t>T. Because F(T)0, we obtain F(t)0 for t>T. This is a contradiction with Corollary 4.4 and the assertion follows.

We conclude this section with the following result on the continuability of solutions of (1.1).

Proposition 4.6.

Assume q is continuously differentiable for t0 satisfying (4.2). Then any solution of (1.1) which is not continuable at infinity has an infinite number of zeros.

Proof.

Let x be a solution of (1.1) defined on [tx,τ),τ<. If x has a finite number of zeros on [tx,τ), then there exists T,tx<T<τ, such that x(t)0, x(φ(t))0 on J=[T,τ). Suppose x(t)>0 on J. Consider the function w given by (2.14). Then w(t)0.

Now consider these two cases. (a) Let x′′(t)>0 in the left neighborhood of τ or x′′ changes sign in this neighborhood. Then 0w(t)w(T) for tJ. Integrating this inequality on J we obtain

x(t)x(T)+w(T)(τ-T)-Ttq(s)x(s)dsx(T)+w(T)(τ-T)=c, Thus x(t)x(T)+c(t-T) and x is bounded on J. From here and the boundedness of w we have that x′′ is bounded on J. Since x(i) (i=0,1,2) are bounded, the solution x can be extended beyond τ, which is a contradiction.

(b) Let x′′(t)<0 in the left neighborhood of τ, say J1J. Then x is decreasing and x is bounded on J1. We claim that x(t) does not tend to - as tτ-. Indeed, if x(τ-)=-, then there exists a sequence {tk},tkτ-, such that limkx′′′(tk)=-, which is a contradiction with the boundedness of x. Thus x′′′ is bounded and so x′′ is bounded on J1 and x is continuable, which is again a contradiction.

Remark 4.7.

Theorem 4.5 and Proposition 4.6 improve [11, Corollary  3.4] and [11, Theorem  1.2] for (1.5), respectively.

5. Property A

In this section we study Property A for (1.1), that is, any solution either is oscillatory or tends to zero as t.

We start with the following result on the boundedness of nonoscillatory solutions, which extends [11, Theorem  3.12] for (1.5).

Theorem 5.1.

Assume that q is continuously differentiable for large t and there exists a constant q0 such that q(t)q0>0,q(t)0forlarget. Then every nonoscillatory solution x of (1.1) satisfies lim supt|x(i)(t)|<,i=0,1,2.

Proof.

Without loss of generality, assume that x is a solution of (1.1) such that x(t)>0, x(φ(t))>0 for tT0. Consider the function w defined by (2.14). Then w(t)=-r(t)f(x(φ(t)))+q(t)x(t)q(t)x(t)0, that is, w is nonincreasing. By Lemma 2.4, limtw(t)=w00. Moreover, from (5.3) we have T|q(t)|x(t)dt-Tw(t)dt=w(T)-w0, or limtt|q(t)|x(t)dt=0  ,limtt|w(t)|dt=0. Let u,v be two linearly independent solutions of (1.3) with Wronskian d. Because all solutions of (1.3) together with their derivatives are bounded, see, for example, [19, Ch.  XIV, Theorem  3.1], from (5.5) we get limtt(w(s)-q(s)x(s))(u(s)v(t)-u(t)v(s))ds=0. Moreover, from the equality x′′′(t)+q(t)x(t)=w(t)-q(t)x(t), using the variation of constants formula, we have x(t)=c1u(t)+c2v(t)-1dt(w(s)-q(s)x(s))(u(s)v(t)-u(t)v(s))ds, where c1 and c2 are real constants. Thus x is bounded. Moreover, x′′(t)=c1u(t)+c2v(t)-t(w(s)-q(s)x(s))(u(s)v(t)-u(t)v(s))ds. Since w is bounded, u,v,u,v are bounded, and (5.5) holds, x′′ is bounded, too. From here and (2.14) the boundedness of x follows.

The next result describes the asymptotic properties of nonoscillatory solutions and will be used later.

Theorem 5.2.

Assume (4.1), (4.3), and q is continuously differentiable satisfying (5.1). If lim infu0f(u)u>0,0r(t)|t-φ(t)|dt<, then any nonoscillatory solution x of (1.1) defined on [tx,) satisfies txr(t)x2(t)dt<. If, moreover, q(t)r(t)C<forlarget, then x satisfies tx(x(t))2dt<.

Proof.

Theorem 4.1 yields that x is not of Type II. Then there exists Ttx such that for tT either x(t)>0,x(φ(t))>0,x(t)0,limtx(t)=0, or x(t)>0,x(φ(t))>0,xchangesitssign.

By Theorem 5.1, x and x are bounded, that is, there exists m>0 such that |x(t)|m,|x(t)|m for ttx. By Corollary 4.4, the function F, given by (4.11), is positive for large t, say tT, and has a finite limit as t. Moreover

>F(T)-F()=TF(t)dt2Tr(t)x(t)f(x(φ(t)))dt2KTr(t)x(t)x(φ(t))dt. From the boundedness of x we get x(φ(t))=x(t)-x(ξ)(t-φ(t))x(t)-m|t-φ(t)|, and so Tr(t)x(t)x(φ(t))dtTr(t)x2(t)dt-m2T1r(t)|t-φ(t)|dt. Hence, in view of (5.11) and (5.17), we obtain (5.12).

In order to complete the proof, define for tT

z(t)=2x(t)x(t)-3Tt(x(s))2ds+Ttq(s)x2(s)ds. Then z(t)=F(t)>0 for tT and so z is increasing. If x satisfies either (5.15) or (5.16), there exists a sequence {τk} tending to such that x(τk)x(τk)0  ,k=1,2,. Moreover, Tq(s)x2(s)ds=Tq(s)r(s)r(s)x2(s)dsCTr(s)x2(s)ds=C1<. Now, in view of (5.20) and (5.21), we obtain 3Tτk(x(t))2dt=-z(τk)+2x(τk)x(τk)+Tτkq(s)x2(s)ds-z(τ1)+C1, and so (5.14) is satisfied.

Using the previous results, we obtain a sufficient condition for property A.

Theorem 5.3.

Assume (4.1), (5.10), (5.11), and q is continuously differentiable satisfying (5.1). If there exists M0>0 such that r(t)M0forlarget, then (1.1) has property A, that is, any nonoscillatory solution x of (1.1) satisfies limtx(i)(t)=0,i=0,1,2.

Proof.

Let x be a nonoscillatory solution of (1.1) defined for ttx0. Because (5.24) implies (4.3), by Theorem 4.1, x is of Type I or Type III. If x is of Type I, the assertion follows applying again Theorem 4.1. Now let x be of Type III and assume x(t)>0,  x(φ(t))>0 for tTtx. By applying Theorem 5.2, in view of (5.24), we obtain txx2(t)dt<,tx(x(t))2dt<. According to the inequality of Nagy, see, for example, [20, V, § 2, Theorem  1], we get limtx(t)=0. Now consider the function w given by (2.14). Then w(t)=-r(t)f(x(φ(t)))+q(t)x(t)<0 for tT. For a sequence {tk} of zeros of x′′ tending to , we have limkw(tk)=0. Hence, limtw(t)=0 and limtx′′(t)=0. By Corollary 4.4, the function F, given by (4.11), is positive decreasing for large t. From here, (2.10), (5.27), and (5.28) we obtain limtx(t)=0.

Corollary 5.4.

Suppose assumptions of Theorem 5.3, φ(t)=t, and for some K>0|f(u)|K|u|foru. then any nonoscillatory solution x of (1.1) satisfies x(t)x(t)<0 for large t.

Proof.

The assertion follows from Theorem 5.3 and [1, Theorem  6].

Acknowledgments

The first and third authors are supported by the Research Project 0021622409 of the Ministry of Education of the Czech Republic and Grant 201/08/0469 of the Czech Grant Agency. The fourth author is supported by the Research Project PRIN07-Area 01, no. 37 of the Italian Ministry of Education.

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