Oscillation for Third-Order Nonlinear Differential Equations with Deviating Argument

Abstract

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 ℒ𝑥=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.(1.1)

The following assumptions will be made. 𝑞,𝑟,𝜑 are continuous functions for 𝑡≥0, 𝑟(𝑡)>0, 𝜑∶[0,∞)→[0,∞), 𝜑(0)=0, lim_𝑡→∞𝜑(𝑡)=∞, and 𝑓 is a continuous function, 𝑓∶ℝ→ℝ, such that

𝑓(𝑢)𝑢>0for𝑢≠0.(1.2) 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 ℒ𝑥=𝑥^′′′+𝑞(𝑡)𝑥^′ is oscillatory, that is, the second-order equation

ℎ^′′(𝑡)+𝑞(𝑡)ℎ(𝑡)=0(1.3) is oscillatory.

It is well known, see, for example, [1], 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,(1.4) where 𝑎_𝑖,𝑖=1,2, are continuous positive functions for 𝑡≥0.

Asymptotic properties of equations of type (1.4) have been widely investigated in the literature. We refer to [1–7] in case when ∫^∞₀1/𝑎_𝑖(𝑡)𝑑𝑡=∞ (𝑖=1,2), that is, the disconjugate differential operator ℒ₁𝑥=(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,(1.5) 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(1.6)

have been presented in [9] 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 lim_𝑡→∞𝑥(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 ℒ𝑥=𝑥^′′′+𝑞(𝑡)𝑥^′ is oscillatory, then very little is known. According to Kiguradze [16], we say that (1.1) has property A if each of its solutions either is oscillatory, or satisfies the condition

lim_𝑡→∞𝑥^(𝑖)(𝑡)=0(𝑖=0,1,2).(1.7) 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)+𝜀_𝑖(𝑡),(𝑖=0,1,2,3),(1.8)

where 𝑐≠0 and 𝛼 are constants, the continuous functions 𝜀_𝑖 (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 [16] deal with the equation

𝑥^′′′(𝑡)+𝑥^′(𝑡)+𝑟(𝑡)𝑓(𝑥(𝑡))=0,(1.9) and read as follows.

Theorem 1.1 (see [16, Theorem  1.5]). Let 𝑓 be a nondecreasing function satisfying ^∞₁𝑑𝑢𝑓(𝑢)<∞,⁻¹_−∞𝑑𝑢𝑓(𝑢)<∞.(1.10) Then the condition ^∞₀𝑟(𝑡)𝑑𝑡=∞(1.11) is necessary and sufficient in order that (1.9) has weak property A.

Theorem 1.2 (see [16, Corollary  1.5]). Let for some 𝐾>0 and 𝑎>0𝑟(𝑡)|𝑓(𝑢)|≥𝐾𝑡⁻¹|𝑢|for𝑢∈ℝ,𝑡≥𝑎.(1.12) Then (1.9) has property A.

In our previous paper [1] 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 liminf_𝑡→∞𝑥(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 [17] 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 𝑟∈𝐿¹[0,∞) instead of 𝑅(𝑡)∈𝐿¹[0,∞), where

𝑅(𝑡)=^∞_𝑡𝑟(𝜏)𝑑𝜏.(1.13)

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 {𝑡_𝑘}→∞ such that 𝑔(𝑡_𝑘)𝑔(𝑡_𝑘+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(2.1) or 𝑥(𝑡)≠0𝑓𝑜𝑟𝑙𝑎𝑟𝑔𝑒𝑡,𝑥^′(𝑡)𝑐ℎ𝑎𝑛𝑔𝑒𝑠𝑖𝑡𝑠𝑠𝑖𝑔𝑛.(2.2)

Proof. Without loss of generality suppose that there exists a solution 𝑥 of (1.1) and 𝑇≥0 such that 𝑥(𝑡)>0, 𝑥(𝜑(𝑡))>0, 𝑥^′(𝑡)≥0 on [𝑇,∞).
O. Boruvka [17] 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)𝛼^′2(𝑡)=1.(2.3) Using this result, we can consider the change of variables 𝑠=𝛼(𝑡),𝑥^′1(𝑡)=𝛼^′̇𝑑(𝑡)𝑋(𝑠),(⋅=)𝑑𝑠(2.4) for 𝑡∈[𝑇,∞), 𝑠∈[𝑇^∗,∞), 𝑇^∗=𝛼(𝑇). Thus, 𝑡=𝛼⁻¹(𝑠) and 𝑥^′′1(𝑡)=−2(𝛼^′(𝑡))^−3/2𝛼^′′(𝑡)𝑋̇(𝑠)+(𝛼^′(𝑡))^1/2̈𝑥𝑋(𝑠),_′′′1(𝑡)=−2(𝛼^′(𝑡))^−3/2𝛼^′′′̇3(𝑡)𝑋(𝑠)+4(𝛼^′(𝑡))^−5/2(𝛼^′′(𝑡))²𝑋̇(𝑠)+(𝛼^′(𝑡))^3/2𝑋⃛(𝑠).(2.5) Substituting into (1.1), we obtain 𝑋⃛(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.(2.6) From here and (2.3) we obtain 𝑋⃛(𝑠)+𝑋̇(𝑠)+(𝛼^′(𝑡))^−3/2𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))=0.(2.7) Because lim_𝑡→∞𝜑(𝑡)=∞, we have for large 𝑡𝑋⃛(𝑠)+𝑋̇(𝑠)=−(𝛼^′(𝑡))^−3/2𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))<0.(2.8) Since 𝑥^′(𝑡)≥0, (2.4) yields 𝑋̇(𝑠)≥0 and so 𝑋⃛(𝑠)<0, that is, 𝑋̈ is decreasing. If there exists 𝑠₁≥𝑇 such that 𝑋̈(𝑠₁)<0, 𝑋̇ becomes eventually negative, which is a contradiction. Then 𝑋̈(𝑠)≥0 and 𝑋̇(𝑠) is nondecreasing. Let 𝑇₁≥𝑇^∗ be such that 𝑋̇(𝑠)>0 on [𝑇₁,∞). Thus, using (2.8) we obtain 𝑋̈(𝑠)−𝑋̈(𝑇₁)=^𝑠_𝑇1𝑋⃛(𝑢)𝑑𝑢≤−^𝑠_𝑇1𝑋̇(𝑢)𝑑𝑢≤−𝑋̇(𝑇₁)(𝑠−𝑇₁).(2.9) Hence, lim_𝑠→∞𝑋̈(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 𝑥≡0, 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 liminf_𝑡→∞|𝑥^′(𝑡)|=0,(2.10)liminf_𝑡→∞|𝑥^′′(𝑡)|=0.(2.11)

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, lim_𝑡→∞𝑥(𝑡)=−∞, a contradiction.
Now let us prove (2.11) and, without loss of generality, assume 𝑥(𝑡)>0, 𝑥(𝜑(𝑡))>0 for 𝑡≥𝑡₀. Suppose, for the sake of contradiction, that liminf_𝑡→∞|𝑥^′′(𝑡)|=2𝐶>0. Then there exists 𝑡₁≥𝑡₀ such that either
𝑥^′′(𝑡)≥𝐶or𝑥^′′(𝑡)≤−𝐶for𝑡≥𝑡₁.(2.12) If 𝑥^′′(𝑡)≥𝐶, then lim_𝑡→∞𝑥^′(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)+^𝑡_𝑡1(𝑡−𝜎)𝑥^′′(𝜎)𝑑𝜎≤𝑥(𝑡₁)+𝑥^′(𝑡₁)(𝑡−𝑡₁𝐶)−2(𝑡−𝑡₁)²,(2.13) which gives lim_𝑡→∞𝑥(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)(2.14) is nonincreasing for large 𝑡, then lim_𝑡→∞𝑤(𝑡)=𝑤₀≥0.(2.15)

Proof. By contradiction, assume 𝑤₀<0. Then for large 𝑡𝑥^′′(𝑡)<−𝑞(𝑡)𝑥(𝑡)≤0.(2.16) If there exists 𝑇≥0 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.

Type I:𝑥 satisfies for large 𝑡

𝑥(𝑡)𝑥^′(𝑡)≤0,lim_𝑡→∞𝑥(𝑡)=0.(2.17)

Type II:𝑥 satisfies for large 𝑡

|𝑥(𝑡)|≥𝑚_𝑥>0.(2.18)

Type III:𝑥 satisfies for large 𝑡

𝑥(𝑡)≠0,liminf_𝑡→∞|𝑥(𝑡)|=0,𝑥^′changesitssign.(2.19)

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.(3.1) If ^∞₀|𝑞^′(𝑡)|𝑑𝑡<∞_∞0,𝑟(𝑡)𝑑𝑡<∞,(3.2) then for any 𝑐∈ℝ⧵{0} there exists a solution 𝑥 of (1.1) satisfying lim_𝑡→∞𝑥(𝑡)=𝑐,lim_𝑡→∞𝑥^(𝑖)(𝑡)=0,𝑖=1,2.(3.3)

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=max_{1/2≤𝑢≤3/2}𝑓(𝑢),𝑀=sup_𝑡≥𝑡0{|𝑢(𝑡)|,|𝑢^′(𝑡)|,|𝑣(𝑡)|,|𝑣^′(𝑡)|}(3.4) 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.(3.5) Let 𝑡₀ be large so that 4𝑘₁3𝑞₀^∞_𝑡01𝑟(𝑡)𝑑𝑡≤2,1𝑞₀^∞_𝑡0|𝑞^′1(𝑠)|𝑑𝑠≤4.(3.6) Let 𝑡≥𝑡₀ be such that 𝜑(𝑡)≥𝑡₀ for 𝑡≥𝑡. Define 𝜑(𝑡)=𝜑(𝑡),if𝑡≥𝑡,𝜑(𝑡),if𝑡₀≤𝑡≤𝑡.(3.7) 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 Ω⊂𝐶[𝑡₀,∞) given by Ω={𝑥∈𝐶[𝑡₀1,∞)∶23≤𝑥(𝑡)≤2}.(3.8) Let 𝑇∈[𝑡₀,∞) be fixed and let 𝐼=[𝑡₀,𝑇]. For any 𝑥∈Ω, consider the “cut” function 𝐺_𝑥(𝑡)=−^𝑇_𝑡^∞_𝜏(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝜏)−𝑢(𝜏)𝑣(𝑠))𝑑𝑠𝑑𝜏,𝑡∈𝐼.(3.9) Then 𝐺^′_𝑥(𝑡)=^∞_𝑡(𝑟(𝑠)𝑓(𝑥(𝐺𝜑(𝑠))))(𝑢(𝑠)𝑣(𝑡)−𝑢(𝑡)𝑣(𝑠))𝑑𝑠,_𝑥′′(𝑡)=^∞_𝑡(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣^′(𝑡)−𝑢^′𝐺(𝑡)𝑣(𝑠))𝑑𝑠,_{𝑥′′′}(𝑡)=−𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))−𝑞(𝑡)^∞_𝑡(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝑡)−𝑢(𝑡)𝑣(𝑠))𝑑𝑠,=−𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))−𝐺^′_𝑥(𝑡)𝑞(𝑡).(3.10) Consider the function 𝑔(𝑡)=𝐺_𝑥^′′(𝑡)+𝑞(𝑡)𝐺_𝑥(𝑡),𝑡∈𝐼.(3.11) Then 𝑔(𝑇)=𝐺_𝑥^′′(𝑇) and 𝑔^′(𝑡)=𝐺_𝑥^′′′(𝑡)+𝑞(𝑡)𝐺^′_𝑥(𝑡)+𝑞^′(𝑡)𝐺_𝑥(𝑡)=−𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))+𝑞^′(𝑡)𝐺_𝑥(𝑡),𝑡∈𝐼.(3.12) Integrating from 𝑡 to 𝑇 we have 𝑔(𝑡)=𝑔(𝑇)−^𝑇_𝑡𝑔^′(𝑠)𝑑𝑠=𝑔(𝑇)+^𝑇_𝑡(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))𝑑𝑠−^𝑇_𝑡𝑞^′(𝑠)𝐺_𝑥(𝑠)𝑑𝑠=𝐺_𝑥^′′(𝑥)(𝑇)+^𝑇_𝑡(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))𝑑𝑠−^𝑇_𝑡𝑞^′(𝑠)𝐺_𝑥(𝑠)𝑑𝑠.(3.13) From here and (3.11) we get 𝐺_𝑥1(𝑡)=𝑞(𝑡)(𝐺_𝑥^′′(𝑇)−𝐺_𝑥^′′(𝑡)+^𝑇_𝑡(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))𝑑𝑠−^𝑇_𝑡𝑞^′(𝑠)𝐺_𝑥=1(𝑠)𝑑𝑠)𝑞(𝑡)^𝑇_𝑡(𝑟(𝑠)𝑓(𝑥(+1𝜑(𝑠))))(1−ℎ(𝑠,𝑡))𝑑𝑠𝑞(𝑡)^∞_𝑇(𝑟(𝑠)𝑓(𝑥(1𝜑(𝑠))))(ℎ(𝑠,𝑇)−ℎ(𝑠,𝑡))𝑑𝑠−𝑞(𝑡)^𝑇_𝑡𝑞^′(𝑠)𝐺_𝑥(𝑠)𝑑𝑠.(3.14) Thus, in view of (3.5), we obtain |𝐺_𝑥𝑘(𝑡)|≤¹𝑞(𝑡)^∞_𝑡𝑟(𝑠)𝑑𝑠+max_{𝑡≤𝑠≤𝑇}|𝐺_𝑥1(𝑠)|𝑞(𝑡)^∞_𝑡0|𝑞^′(𝑠)|𝑑𝑠,(3.15) or, in view of (3.6), |𝐺_𝑥𝑘(𝑡)|≤¹𝑞₀^∞_𝑡1𝑟(𝑠)𝑑𝑠+4max_{𝑡≤𝑠≤𝑇}|𝐺_𝑥(𝑠)|,(3.16) that is, max_{𝑡≤𝜎≤𝑇}|𝐺_𝑥𝑘(𝜎)|≤¹𝑞₀^∞_𝑡1𝑟(𝑠)𝑑𝑠+4max_{𝑡≤𝜎≤𝑇}|𝐺_𝑥(𝜎)|.(3.17) Hence |𝐺_𝑥(𝑡)|≤max_{𝑡≤𝜎≤𝑇}|𝐺_𝑥(𝜎)|≤4𝑘₁3𝑞₀^∞_𝑡𝑟(𝑠)𝑑𝑠,(3.18) that is, |^𝑇_𝑡^∞_𝜏(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝜏)−𝑢(𝜏)𝑣(𝑠))𝑑𝑠𝑑𝜏|≤4𝑘₁3𝑞₀^∞_𝑡𝑟(𝑠)𝑑𝑠.(3.19) In view of (3.19), using the Cauchy criterion, the limit lim_𝑇→∞^𝑇_𝑡^∞_𝜏(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝜏)−𝑢(𝜏)𝑣(𝑠))𝑑𝑠𝑑𝜏(3.20) exists finitely for any fixed 𝑡. This fact means that the operator H(𝑥)(𝑡)=1−^∞_𝑡^∞_𝜏(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝜏)−𝑢(𝜏)𝑣(𝑠))𝑑𝑠𝑑𝜏(3.21) is well defined for any 𝑥∈Ω. Moreover, from (3.19) we have for 𝑡≥𝑡₀|^∞_𝑡^∞_𝜏(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝜏)−𝑢(𝜏)𝑣(𝑠))𝑑𝑠𝑑𝜏|≤4𝑘₁3𝑞₀^∞_𝑡𝑟(𝑠)𝑑𝑠(3.22) 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 𝑥∈Ω we have
H^′(𝑥)(𝑡)=^∞_𝑡(𝑟(𝑠)𝑓(𝑥(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝑡)−𝑢(𝑡)𝑣(𝑠))𝑑𝑠,(3.23) and so |H^′(𝑥)(𝑡)|≤2𝛽𝑀^2∞_𝑡𝑟(𝑠)𝑑𝑠,(3.24) which proves the equicontinuity of the elements of H(Ω).
Now we prove the continuity of H on Ω. Let {𝑥_𝑛},𝑛∈ℕ, be a sequence in Ω which converges uniformly on every compact interval of [𝑡₀,∞) to 𝑥∈Ω. Because H(Ω) is relatively compact, the sequence {H(𝑥_𝑛)} admits a subsequence, denoted again by {H(𝑥_𝑛)} for sake of simplicity, which is convergent to 𝑦∈Ω. From (3.4) we obtain
|^∞_𝜏(𝑟(𝑠)𝑓(𝑥_𝑛(𝜑(𝑠))))(𝑢(𝑠)𝑣(𝜏)−𝑢(𝜏)𝑣(𝑠))𝑑𝑠|≤2𝛽𝑀²𝑅(𝜏),(3.25) where 𝑅 is defined in (1.13). Hence, in virtue of the Lebesgue dominated convergence theorem, {𝐺_𝑥𝑛(𝑡)} converges pointwise to 𝐺_𝑥(𝑡) on [𝑡₀,𝑇], that is, lim_𝑛→∞𝐺_𝑥𝑛(𝑡)=𝐺_𝑥(𝑡).(3.26) Choosing a sufficiently large 𝑇 and using (3.22), we obtain |H(𝑥_𝑛)(𝑡)−H(𝑥)(𝑡)|=|H(𝑥_𝑛)(𝑡)−𝐺_𝑥𝑛(𝑡)+𝐺_𝑥𝑛(𝑡)−𝐺_𝑥(𝑡)+𝐺_𝑥(𝑡)−H(𝑥)(𝑡)|≤|H(𝑥_𝑛)(𝑇)−1|+|𝐺_𝑥𝑛(𝑡)−𝐺_𝑥(𝑡)|+|H(𝑥)(𝑇)−1|≤|𝐺_𝑥𝑛(𝑡)−𝐺_𝑥(𝑡)|+8𝑘₁3𝑞₀^∞_𝑇𝑟(𝑠)𝑑𝑠.(3.27) Then, from (3.26), {H(𝑥_𝑛)} 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(𝑥_𝑛)}, 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),(3.28) 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 liminf_𝑡→∞𝑥(𝑡)=0.

Theorem 4.1. Assume liminf_𝑢→±∞|𝑓(𝑢)|>0,(4.1) and 𝑞 continuously differentiable for large 𝑡 satisfying 𝑞(𝑡)>0,𝑞^′(𝑡)≤0.(4.2) If ^∞∞𝑟(𝑡)𝑑𝑡=∞,(4.3) then any (nonoscillatory) solution 𝑥 of (1.1) satisfies liminf_𝑡→∞|𝑥(𝑡)|=0.(4.4) Moreover, any nonoscillatory solution with 𝑥(𝑡)𝑥^′(𝑡)≤0 for large 𝑡 satisfies lim_𝑡→∞𝑥^(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 𝑡≥𝑡₀.
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.(4.5) By Lemma 2.4, we have lim_𝑡→∞𝑤(𝑡) =𝑤_∞,0≤𝑤_∞<∞. Since 𝑓(𝑥(𝜑(𝑡)))≥inf_𝑢≥𝑚𝑓(𝑢)=𝐾>0for𝑡≥𝑡₀,(4.6) we get 𝑞𝑓(𝑥(𝜑(𝑡)))−^′(𝑡)𝑟(𝑡)𝑥(𝑡)≥𝐾.(4.7) Consequently, we have from (4.5) 𝑤(𝑡₀)≥𝑤(𝑡₀)−𝑤(𝑡)=^𝑡_𝑡0𝑞𝑟(𝑠)(𝑓(𝑥(𝜑(𝑠)))−^′(𝑠)𝑟(𝑠)𝑥(𝑠))𝑑𝑠≥𝐾^𝑡_𝑡0𝑟(𝑠)𝑑𝑠,(4.8) 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 𝑡≥𝑡₁≥0. Hence 𝑥 is of Type I and
lim_𝑡→∞𝑥(𝑡)=0.(4.9) Because 𝑞 is bounded and 𝑤 is nonincreasing, (2.11) and (4.9) yield lim_𝑡→∞𝑤(𝑡)=0,(4.10) and so lim_𝑡→∞𝑥^′′(𝑡)=0. Consider the function 𝐹 defined by 𝐹(𝑡)=𝐹(𝑥)(𝑡)=2𝑥^′′(𝑡)𝑥(𝑡)−(𝑥^′(𝑡))²+𝑞(𝑡)𝑥²(𝑡).(4.11) Then we have for 𝑡≥𝑡₁𝐹^′(𝑡)=−2𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))𝑥(𝑡)+𝑞^′(𝑡)𝑥²(𝑡)<0.(4.12) Since 𝐹 is decreasing and 𝑞 is bounded, in view of (2.10) we obtain lim_𝑡→∞𝐹(𝑡)=0 and so lim_𝑡→∞𝑥^′(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,(4.13) 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 lim_𝑡→∞𝑥^(𝑖)(𝑡)=0 for 𝑖=0,1,2 and so lim_𝑡→∞𝐹(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 {𝑡_𝑘} of zeros of 𝑥^′ tending to ∞ such that 𝑥^′′(𝑡_𝑘)≥0 and so 𝐹(𝑡_𝑘)>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(𝜏_𝑘)≤0 for some sequence {𝜏_𝑘}, we get 𝐹(𝑡)<0 for large 𝑡.

Theorem 4.5. Let 𝜑(𝑡)≥𝑡 and 𝑞 is continuously differentiable for 𝑡≥0 satisfying (4.2). If (4.1) and (4.3) are satisfied, for any nonoscillatory solution 𝑥 of (1.1) defined for 𝑡≥𝑡_𝑥, one has 𝑥(𝑡)≠0for𝑡≥𝑡_𝑥.(4.14) In particular, any continuable solution with zero is oscillatory.

Proof. Assume that 𝑥(𝑡)>0 for 𝑡>𝑇>𝑡_𝑥 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 𝑡≥0 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 [𝑡_𝑥,𝜏),𝜏<∞. If 𝑥 has a finite number of zeros on [𝑡_𝑥,𝜏), then there exists 𝑇,𝑡_𝑥<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 0≤𝑤(𝑡)≤𝑤(𝑇) for 𝑡∈𝐽. Integrating this inequality on 𝐽 we obtain
𝑥^′(𝑡)≤𝑥^′(𝑇)+𝑤(𝑇)(𝜏−𝑇)−^𝑡_𝑇𝑞(𝑠)𝑥(𝑠)𝑑𝑠≤𝑥^′(𝑇)+𝑤(𝑇)(𝜏−𝑇)=𝑐,(4.15) 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 𝐽₁⊂𝐽. Then 𝑥^′ is decreasing and 𝑥 is bounded on 𝐽₁. We claim that 𝑥^′(𝑡) does not tend to −∞ as 𝑡→𝜏−. Indeed, if 𝑥^′(𝜏−)=−∞, then there exists a sequence {𝑡_𝑘},𝑡_𝑘→𝜏−, such that lim_𝑘𝑥^′′′(𝑡_𝑘)=−∞, 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.(5.1) Then every nonoscillatory solution 𝑥 of (1.1) satisfies limsup_𝑡→∞|𝑥^(𝑖)(𝑡)|<∞,𝑖=0,1,2.(5.2)

Proof. Without loss of generality, assume that 𝑥 is a solution of (1.1) such that 𝑥(𝑡)>0, 𝑥(𝜑(𝑡))>0 for 𝑡≥𝑇≥0. 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,(5.3) that is, 𝑤 is nonincreasing. By Lemma 2.4, lim_𝑡→∞𝑤(𝑡)=𝑤₀≥0. Moreover, from (5.3) we have ^∞_𝑇|𝑞^′(𝑡)|𝑥(𝑡)𝑑𝑡≤−^∞_𝑇𝑤^′(𝑡)𝑑𝑡=𝑤(𝑇)−𝑤₀,(5.4) or lim_𝑡→∞^∞_𝑡|𝑞^′(𝑡)|𝑥(𝑡)𝑑𝑡=0,lim_𝑡→∞^∞_𝑡|𝑤^′(𝑡)|𝑑𝑡=0.(5.5) 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 lim_𝑡→∞^∞_𝑡(𝑤^′(𝑠)−𝑞^′(𝑠)𝑥(𝑠))(𝑢(𝑠)𝑣(𝑡)−𝑢(𝑡)𝑣(𝑠))𝑑𝑠=0.(5.6) Moreover, from the equality 𝑥^′′′(𝑡)+𝑞(𝑡)𝑥^′(𝑡)=𝑤^′(𝑡)−𝑞^′(𝑡)𝑥(𝑡),(5.7) using the variation of constants formula, we have 𝑥^′(𝑡)=𝑐₁𝑢(𝑡)+𝑐₂1𝑣(𝑡)−𝑑^∞_𝑡(𝑤^′(𝑠)−𝑞^′(𝑠)𝑥(𝑠))(𝑢(𝑠)𝑣(𝑡)−𝑢(𝑡)𝑣(𝑠))𝑑𝑠,(5.8) 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.(5.9) 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 liminf_𝑢→0𝑓(𝑢)𝑢>0,(5.10)_∞0𝑟(𝑡)|𝑡−𝜑(𝑡)|𝑑𝑡<∞,(5.11) then any nonoscillatory solution 𝑥 of (1.1) defined on [𝑡_𝑥,∞) satisfies ^∞_𝑡𝑥𝑟(𝑡)𝑥²(𝑡)𝑑𝑡<∞.(5.12) If, moreover, 𝑞(𝑡)𝑟(𝑡)≤𝐶<∞forlarge𝑡,(5.13) then 𝑥 satisfies ^∞_𝑡𝑥(𝑥^′(𝑡))²𝑑𝑡<∞.(5.14)

Proof. Theorem 4.1 yields that 𝑥 is not of Type II. Then there exists 𝑇≥𝑡_𝑥 such that for 𝑡≥𝑇 either 𝑥(𝑡)>0,𝑥(𝜑(𝑡))>0,𝑥^′(𝑡)≤0,lim_𝑡→∞𝑥(𝑡)=0,(5.15) or 𝑥(𝑡)>0,𝑥(𝜑(𝑡))>0,𝑥^′changesitssign.(5.16)
By Theorem 5.1, 𝑥 and 𝑥^′ are bounded, that is, there exists 𝑚>0 such that |𝑥(𝑡)|≤𝑚, |𝑥^′(𝑡)|≤𝑚 for 𝑡≥𝑡_𝑥. By Corollary 4.4, the function 𝐹, given by (4.11), is positive for large 𝑡, say 𝑡≥𝑇, and has a finite limit as 𝑡→∞. Moreover
∞>𝐹(𝑇)−𝐹(∞)=^∞_𝑇𝐹^′(𝑡)𝑑𝑡≥2^∞_𝑇𝑟(𝑡)𝑥(𝑡)𝑓(𝑥(𝜑(𝑡)))𝑑𝑡≥2𝐾^∞_𝑇𝑟(𝑡)𝑥(𝑡)𝑥(𝜑(𝑡))𝑑𝑡.(5.17) From the boundedness of 𝑥^′ we get 𝑥(𝜑(𝑡))=𝑥(𝑡)−𝑥^′(𝜉)(𝑡−𝜑(𝑡))≥𝑥(𝑡)−𝑚|𝑡−𝜑(𝑡)|,(5.18) and so ^∞_𝑇𝑟(𝑡)𝑥(𝑡)𝑥(𝜑(𝑡))𝑑𝑡≥^∞_𝑇𝑟(𝑡)𝑥²(𝑡)𝑑𝑡−𝑚^2∞_𝑇1𝑟(𝑡)|𝑡−𝜑(𝑡)|𝑑𝑡.(5.19) Hence, in view of (5.11) and (5.17), we obtain (5.12).
In order to complete the proof, define for 𝑡≥𝑇
𝑧(𝑡)=2𝑥^′(𝑡)𝑥(𝑡)−3^𝑡_𝑇(𝑥^′(𝑠))²𝑑𝑠+^𝑡_𝑇𝑞(𝑠)𝑥²(𝑠)𝑑𝑠.(5.20) Then 𝑧^′(𝑡)=𝐹(𝑡)>0 for 𝑡≥𝑇 and so 𝑧 is increasing. If 𝑥 satisfies either (5.15) or (5.16), there exists a sequence {𝜏_𝑘} tending to ∞ such that 𝑥(𝜏_𝑘)𝑥^′(𝜏_𝑘)≤0,𝑘=1,2,….(5.21) Moreover, ^∞_𝑇𝑞(𝑠)𝑥²(𝑠)𝑑𝑠=^∞_𝑇𝑞(𝑠)𝑟(𝑠)𝑟(𝑠)𝑥²(𝑠)𝑑𝑠≤𝐶^∞_𝑇𝑟(𝑠)𝑥²(𝑠)𝑑𝑠=𝐶₁<∞.(5.22) Now, in view of (5.20) and (5.21), we obtain 3^𝜏𝑘_𝑇(𝑥^′(𝑡))²𝑑𝑡=−𝑧(𝜏_𝑘)+2𝑥^′(𝜏_𝑘)𝑥(𝜏_𝑘)+^𝜏𝑘_𝑇𝑞(𝑠)𝑥²(𝑠)𝑑𝑠≤−𝑧(𝜏₁)+𝐶₁,(5.23) 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,(5.24) then (1.1) has property A, that is, any nonoscillatory solution 𝑥 of (1.1) satisfies lim_𝑡→∞𝑥^(𝑖)(𝑡)=0,𝑖=0,1,2.(5.25)

Proof. Let 𝑥 be a nonoscillatory solution of (1.1) defined for 𝑡≥𝑡_𝑥≥0. 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 𝑡≥𝑇≥𝑡_𝑥. By applying Theorem 5.2, in view of (5.24), we obtain ^∞_𝑡𝑥𝑥²(𝑡)𝑑𝑡<∞,^∞_𝑡𝑥(𝑥^′(𝑡))²𝑑𝑡<∞.(5.26) According to the inequality of Nagy, see, for example, [20, V, § 2, Theorem  1], we get lim_𝑡→∞𝑥(𝑡)=0.(5.27) Now consider the function 𝑤 given by (2.14). Then 𝑤^′(𝑡)=−𝑟(𝑡)𝑓(𝑥(𝜑(𝑡)))+𝑞^′(𝑡)𝑥(𝑡)<0 for 𝑡≥𝑇. For a sequence {𝑡_𝑘} of zeros of 𝑥^′′ tending to ∞, we have lim_𝑘→∞𝑤(𝑡_𝑘)=0. Hence, lim_𝑡→∞𝑤(𝑡)=0 and lim_𝑡→∞𝑥^′′(𝑡)=0.(5.28) 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 lim_𝑡→∞𝑥^′(𝑡)=0.

Corollary 5.4. Suppose assumptions of Theorem 5.3, 𝜑(𝑡)=𝑡, and for some 𝐾>0|𝑓(𝑢)|≥𝐾|𝑢|for𝑢∈ℝ.(5.29) 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.