Existence of Oscillatory Solutions of Singular Nonlinear Differential Equations

Abstract

Asymptotic properties of solutions of the singular differential equation (𝑝(𝑡)𝑢^(𝑡))^=𝑝(𝑡)𝑓(𝑢(𝑡)) are described. Here, f is Lipschitz continuous on ℝ and has at least two zeros 0 and 𝐿>0. The function p is continuous on [0, ∞) and has a positive continuous derivative on (0, ∞) and 𝑝(0)=0. Further conditions for f and p under which the equation has oscillatory solutions converging to 0 are given.

1. Introduction

For 𝑘∈ℕ, 𝑘>1, and 𝐿∈(0,∞), consider the equation𝑢^+𝑘−1𝑡𝑢^=𝑓(𝑢),𝑡∈(0,∞),(1.1) where𝑓∈Lip_loc(ℝ),𝑓(0)=𝑓(𝐿)=0,𝑓(𝑥)<0,𝑥∈(0,𝐿),(1.2)∃𝐵∈(−∞,0)∶𝑓(𝑥)>0,𝑥∈𝐵,0.(1.3) Let us put𝐹(𝑥)=−^𝑥0𝑓(𝑧)d𝑧for𝑥∈ℝ.(1.4) Moreover, we assume that 𝑓 fulfils𝐹𝐵=𝐹(𝐿),(1.5) and denote 𝐿₀=inf𝑥<𝐵∶𝑓(𝑥)>0≥−∞.(1.6) Due to (1.2)–(1.4), we see that 𝐹∈𝐶¹(ℝ) is decreasing and positive on (𝐿₀,0) and increasing and positive on (0,𝐿].

Equation (1.1) arises in many areas. For example, in the study of phase transitions of Van der Waals fluids [1–3], in population genetics, where it serves as a model for the spatial distribution of the genetic composition of a population [4, 5], in the homogenous nucleation theory [6], and in relativistic cosmology for description of particles which can be treated as domains in the universe [7], in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [8]. Numerical simulations of solutions of (1.1), where 𝑓 is a polynomial with three zeros, have been presented in [9–11]. Close problems about the existence of positive solutions can be found in [12–14].

In this paper, we investigate a generalization of (1.1) of the form𝑝(𝑡)𝑢^^=𝑝(𝑡)𝑓(𝑢),𝑡∈(0,∞),(1.7) where 𝑓 satisfies (1.2)–(1.5) and 𝑝 fulfils𝑝∈𝐶[0,∞)∩𝐶¹(0,∞),𝑝(0)=0,(1.8)𝑝^(𝑡)>0,𝑡∈(0,∞),lim_𝑡→∞𝑝^(𝑡)𝑝(𝑡)=0.(1.9) Equation (1.7) is singular in the sense that 𝑝(0)=0. If 𝑝(𝑡)=𝑡^𝑘−1, with 𝑘>1, then 𝑝 satisfies (1.8), (1.9), and (1.7) is equal to (1.1).

Definition 1.1. A function 𝑢∈𝐶¹[0,∞)∩𝐶²(0,∞) which satisfies (1.7) for all 𝑡∈(0,∞) is called a solution of (1.7).

Consider a solution 𝑢 of (1.7). Since 𝑢∈𝐶¹[0,∞), we have 𝑢(0),𝑢′(0)∈ℝ and the assumption, 𝑝(0)=0 yields 𝑝(0)𝑢′(0)=0. We can find 𝑀>0 and 𝛿>0 such that |𝑓(𝑢(𝑡))|≤𝑀 for 𝑡∈(0,𝛿). Integrating (1.7), we get ||𝑢^(𝑡)||=||||1𝑝(𝑡)^𝑡0𝑝(𝑠)𝑓(𝑢(𝑠))d𝑠||||≤𝑀𝑝(𝑡)^𝑡0𝑝(𝑠)d𝑠≤𝑀𝑡,𝑡∈(0,𝛿).(1.10) Consequently, the condition𝑢^(0)=0(1.11) is necessary for each solution of (1.7). Denote 𝑢_sup=sup{𝑢(𝑡)∶𝑡∈[0,∞)}.(1.12)

Definition 1.2. Let 𝑢 be a solution of (1.7). If 𝑢_sup<𝐿, then 𝑢 is called a damped solution.

If a solution 𝑢 of (1.7) satisfies 𝑢_sup=𝐿 or 𝑢_sup>𝐿, then we call 𝑢 a bounding homoclinic solution or an escape solution. These three types of solutions have been investigated in [15–18]. Here, we continue the investigation of the existence and asymptotic properties of damped solutions. Due to (1.11) and Definition 1.2, it is reasonable to study solutions of (1.7) satisfying the initial conditions𝑢(0)=𝑢₀∈𝐿₀,𝐿,𝑢^(0)=0.(1.13) Note that if 𝑢₀>𝐿, then a solution 𝑢 of the problem (1.7), (1.13) satisfies 𝑢_sup>𝐿, and consequently 𝑢 is not a damped solution. Assume that 𝐿₀>−∞, then 𝑓(𝐿₀)=0, and if we put 𝑢₀=𝐿₀, a solution 𝑢 of (1.7), (1.13) is a constant function equal to 𝐿₀ on [0,∞). Since we impose no sign assumption on 𝑓(𝑥) for 𝑥<𝐿₀, we do not consider the case 𝑢₀<𝐿₀. In fact, the choice of 𝑢₀ between two zeros 𝐿₀ and 0 of 𝑓 has been motivated by some hydrodynamical model in [11].

A lot of papers are devoted to oscillatory solutions of nonlinear differential equations. Wong [19] published an account on a nonlinear oscillation problem originated from earlier works of Atkinson and Nehari. Wong's paper is concerned with the study of oscillatory behaviour of second-order Emden-Fowler equations𝑦^(𝑥)+𝑎(𝑥)||𝑦(𝑥)||^𝛾−1𝑦(𝑥)=0,𝛾>0,(1.14) where 𝑎 is nonnegative and absolutely continuous on (0,∞). Both superlinear case (𝛾>1) and sublinear case (𝛾∈(0,1)) are discussed, and conditions for the function 𝑎 giving oscillatory or nonoscillatory solutions of (1.14) are presented; see also [20]. Further extensions of these results have been proved for more general differential equations. For example, Wong and Agarwal [21] or Li [22] worked with the equation𝑎(𝑡)𝑦^(𝑡)^𝜎^+𝑞(𝑡)𝑓(𝑦(𝑡))=0,(1.15) where 𝜎>0 is a positive quotient of odd integers, 𝑎∈𝐶¹(ℝ) is positive, 𝑞∈𝐶(ℝ), 𝑓∈𝐶¹(ℝ), 𝑥𝑓(𝑥)>0, 𝑓^(𝑥)≥0 for all 𝑥≠0. Kulenović and Ljubović [23] investigated an equation𝑟(𝑡)𝑔𝑦^(𝑡)′+𝑝(𝑡)𝑓(𝑦(𝑡))=0,(1.16) where 𝑔(𝑢)/𝑢≤𝑚, 𝑓(𝑢)/𝑢≥𝑘>0, or 𝑓′(𝑢)≥𝑘 for all 𝑢≠0. The investigation of oscillatory and nonoscillatory solutions has been also realized in the class of quasilinear equations. We refer to the paper [24] by Ho, dealing with the equation𝑡^𝑛−1Φ_𝑝𝑢^′+𝑡^𝑁𝑛−1_𝑖=1𝛼_𝑖𝑡^𝛽𝑖Φ_𝑞𝑖(𝑢)=0,(1.17) where 1<𝑝<𝑛, 𝛼_𝑖>0, 𝛽_𝑖≥−𝑝, 𝑞_𝑖>𝑝−1, 𝑖=1,…,𝑁, Φ_𝑝(𝑦)=|𝑦|^𝑝−2𝑦.

Oscillation results for the equation𝑎(𝑡)Φ_𝑝𝑥^′+𝑏(𝑡)Φ_𝑞(𝑥)=0,(1.18) where 𝑎,𝑏∈𝐶([0,∞)) are positive, can be found in [25]. We can see that the nonlinearity 𝑓(𝑦)=|𝑦|^𝛾−1𝑦 in (1.14) is an increasing function on ℝ having a unique zero at 𝑦=0.

Nonlinearities in all the other (1.15)–(1.18) have similar globally monotonous behaviour. We want to emphasize that, in contrast to the above papers, the nonlinearity 𝑓 in our (1.7) needs not be globally monotonous. Moreover, we deal with solutions of (1.7) starting at a singular point 𝑡=0, and we provide an interval for starting values 𝑢₀ giving oscillatory solutions (see Theorems 2.3, 2.10, and 2.16). We specify a behaviour of oscillatory solutions in more details (decreasing amplitudes—see Theorems 2.10 and 2.16), and we show conditions which guarantee that oscillatory solutions converge to 0 (Theorem 3.1).

The paper is organized in this manner: Section 2 contains results about existence, uniqueness, and other basic properties of solutions of the problem (1.7), (1.13). These results which mainly concern damped solutions are taken from [18] and extended or modified a little. We also provide here new conditions for the existence of oscillatory solutions in Theorem 2.16. Section 3 is devoted to asymptotic properties of oscillatory solutions, and the main result is contained in Theorem 3.1.

2. Solutions of the Initial Problem (1.7), (1.13)

Let us give an account of this section in more details. The main objective of this paper is to characterize asymptotic properties of oscillatory solutions of the problem (1.7), (1.13). In order to present more complete results about the solutions, we start this section with the unique solvability of the problem (1.7), (1.13) on [0,∞) (Theorem 2.1). Having such global solutions, we have proved (see papers [15–18]) that oscillatory solutions of the problem (1.7), (1.13) can be found just in the class of damped solutions of this problem. Therefore, we give here one result about the existence of damped solutions (Theorem 2.3). Example 2.5 shows that there are damped solutions which are not oscillatory. Consequently, we bring results about the existence of oscillatory solutions in the class of damped solutions. This can be found in Theorem 2.10, which is an extension of Theorem 3.4 of [18] and in Theorem 2.16, which are new. Theorems 2.10 and 2.16 cover different classes of equations which is illustrated by examples.

Theorem 2.1 (existence and uniqueness). Assume that (1.2)–(1.5), (1.8), (1.9) hold and that there exists 𝐶_𝐿∈(0,∞) such that 0≤𝑓(𝑥)≤𝐶_𝐿for𝑥≥𝐿(2.1) then the initial problem (1.7), (1.13) has a unique solution 𝑢. The solution 𝑢 satisfies 𝑢𝑢((𝑡𝑡))≥>𝑢₀if𝑢₀<0,𝐵if𝑢₀≥0,for𝑡∈[0,∞).(2.2)

Proof. Let 𝑢₀<0, then the assertion is contained in Theorem  2.1 of [18]. Now, assume that 𝑢₀∈[0,𝐿], then the proof of Theorem  2.1 in [18] can be slightly modified.

For close existence results, see also Chapters 13 and 14 of [26], where this kind of equations is studied.

Remark 2.2. Clearly, for 𝑢₀=0 and 𝑢₀=𝐿, the problem (1.7), (1.13) has a unique solution 𝑢≡0 and 𝑢≡𝐿, respectively. Since 𝑓∈Lip_loc(ℝ), no solution of the problem (1.7), (1.13) with 𝑢₀<0 or 𝑢₀∈(0,𝐿) can touch the constant solutions 𝑢≡0 and 𝑢≡𝐿.
In particular, assume that 𝐶∈{0,𝐿}, 𝑎>0, 𝑢 is a solution of the problem (1.7), (1.13) with 𝑢₀<𝐿, 𝑢₀≠0, and (1.2), (1.8), and (1.9) hold. If 𝑢(𝑎)=𝐶, then 𝑢^(𝑎)≠0, and if 𝑢^(𝑎)=0, then 𝑢(𝑎)≠𝐶.

The next theorem provides an extension of Theorem 2.4 in [18].

Theorem 2.3 (existence of damped solutions). Assume that (1.2)–(1.5), (1.8), and (1.9) hold, then for each 𝑢₀∈[𝐵,𝐿), the problem (1.7), (1.13) has a unique solution. This solution is damped.

Proof. First, assume that there exists 𝐶_𝐿>0 such that 𝑓 satisfies (2.1), then, by Theorem 2.1, the problem (1.7), (1.13) has a unique solution 𝑢 satisfying (2.2). Assume that 𝑢 is not damped, that is, sup{𝑢(𝑡)∶𝑡∈[0,∞)}≥𝐿.(2.3) By (1.3)–(1.5), the inequality 𝐹(𝑢₀)≤𝐹(𝐿) holds. Since 𝑢 fulfils (1.7), we have 𝑢^(𝑡)+𝑝^(𝑡)𝑝(𝑡)𝑢^(𝑡)=𝑓(𝑢(𝑡))for𝑡∈(0,∞).(2.4) Multiplying (2.4) by 𝑢′ and integrating between 0 and 𝑡>0, we get 0<𝑢^2(𝑡)2+^𝑡0𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠=𝐹𝑢₀−𝐹(𝑢(𝑡)),𝑡∈(0,∞),(2.5) and consequently 0<^𝑡0𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠≤𝐹𝑢₀−𝐹(𝑢(𝑡)),𝑡∈(0,∞).(2.6) By (2.3), we can find that 𝑏∈(0,∞] such that 𝑢(𝑏)≥𝐿, (𝑢(∞)=limsup_𝑡→∞𝑢(𝑡)), and hence, according to (1.5), 0<^𝑏0𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠≤𝐹𝑢₀−𝐹(𝑢(𝑏))≤𝐹(𝐵)−𝐹(𝐿)≤0,(2.7) which is a contradiction. We have proved that sup{𝑢(𝑡)∶𝑡∈[0,∞)}<𝐿, that is, 𝑢 is damped. Consequently, assumption (2.1) can be omitted.

Example 2.4. Consider the equation 𝑢^+2𝑡𝑢^=𝑢(𝑢−1)(𝑢+2),(2.8) which is relevant to applications in [9–11]. Here, 𝑝(𝑡)=𝑡², 𝑓(𝑥)=𝑥(𝑥−1)(𝑥+2), 𝐿₀=−2, and 𝐿=1. Hence 𝑓(x)<0 for 𝑥∈(0,1), 𝑓(𝑥)>0 for 𝑥∈(−2,0), and 𝐹(𝑥)=−^𝑥0𝑓(𝑧)d𝑧=−𝑥⁴4−𝑥3³+𝑥².(2.9) Consequently, 𝐹 is decreasing and positive on [−2,0) and increasing and positive on (0,1]. Since 𝐹(1)=5/12 and 𝐹(−1)=13/12, there exists a unique 𝐵∈(−1,0) such that 𝐹(𝐵)=5/12=𝐹(1). We can see that all assumptions of Theorem 2.3 are fulfilled and so, for each 𝑢₀∈[𝐵,1), the problem (2.8), (1.13) has a unique solution which is damped. We will show later (see Example 2.11), that each damped solution of the problem (2.8), (1.13) is oscillatory.

In the next example, we will show that damped solutions can be nonzero and monotonous on [0,∞) with a limit equal to zero at ∞. Clearly, such solutions are not oscillatory.

Example 2.5. Consider the equation 𝑢^+3𝑡𝑢^=𝑓(𝑢),(2.10) where 𝑓(𝑥)=⎧⎪⎨⎪⎩−𝑥³𝑥1−2fffooorrr𝑥𝑥𝑥∈≥≤13(1,.,3),(2.11) We see that 𝑝(𝑡)=𝑡³ in (2.10) and the functions 𝑓 and 𝑝 satisfy conditions (1.2)–(1.5), (1.8), and (1.9) with 𝐿=2. Clearly, 𝐿₀=−∞. Further, 𝐹(𝑥)=−^𝑥0𝑓(𝑧)d𝑧=⎧⎪⎪⎨⎪⎪⎩𝑥⁴−4𝑥for𝑥≤1,₂−𝑥2++123𝑥−54for𝑥∈(1,3),4for𝑥≥3.(2.12) Since 𝐹(𝐿)=𝐹(2)=3/4, assumption (1.5) yields 𝐹(𝐵)=𝐵⁴/4=3/4 and 𝐵=−3^1/4. By Theorem 2.3, for each 𝑢₀∈[−3^1/4,2), the problem (2.10), (1.13) has a unique solution 𝑢 which is damped. On the other hand, we can check by a direct computation that for each 𝑢₀≤1 the function 𝑢(𝑡)=8𝑢⁰8+𝑢²⁰𝑡²,𝑡∈[0,∞)(2.13) is a solution of equation (2.10) and satifies conditions (1.13). If 𝑢₀<0, then 𝑢<0,𝑢^>0 on (0,∞), and if 𝑢₀∈(0,1], then 𝑢>0, 𝑢^<0 on (0,∞). In both cases, lim_𝑡→∞𝑢(𝑡)=0.

In Example 2.5, we also demonstrate that there are equations fulfilling Theorem 2.3 for which all solutions with 𝑢₀<𝐿, not only those with 𝑢₀∈[𝐵,𝐿), are damped. Some additional conditions giving, moreover, bounding homoclinic solutions and escape solutions are presented in [15–17].

In our further investigation of asymptotic properties of damped solutions the following lemmas are useful.

Lemma 2.6. Assume (1.2), (1.8), and (1.9). Let 𝑢 be a damped solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,𝐿) which is eventually positive or eventually negative, then lim_𝑡→∞𝑢(𝑡)=0,lim_𝑡→∞𝑢^(𝑡)=0.(2.14)

Proof. Let 𝑢 be eventually positive, that is, there exists 𝑡₀≥0 such that 𝑢(𝑡)>0for𝑡∈𝑡₀,∞.(2.15) Denote 𝜃=inf{𝑡₀≥0∶𝑢(𝑡)>0,𝑡∈[𝑡₀,∞)}.
Let 𝜃>0, then 𝑢(𝜃)=0 and, by Remark 2.2, 𝑢′(𝜃)>0. Assume that 𝑢′>0 on (𝜃,∞), then 𝑢 is increasing on (𝜃,∞), and there exists lim_𝑡→∞𝑢(𝑡)=ℓ∈(0,𝐿). Multiplying (2.4) by 𝑢′, integrating between 𝜃 and 𝑡, and using notation (1.4), we obtain 𝑢^2(𝑡)2+^𝑡𝜃𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠=𝐹𝑢₀−𝐹(𝑢(𝑡)),𝑡∈(𝜃,∞).(2.16) Letting 𝑡→∞, we get lim_𝑡→∞𝑢^2(𝑡)2=−lim^𝑡→∞^𝑡𝜃𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠+𝐹𝑢₀−𝐹(ℓ).(2.17) Since the function ∫^𝑡𝜃(𝑝′(𝑠)/𝑝(𝑠))𝑢^2(𝑠)d𝑠 is positive and increasing, it follows that it has a limit at ∞, and hence there exists also lim_𝑡→∞𝑢^(𝑡)≥0. If lim_𝑡→∞𝑢′(𝑡)>0, then 𝐿>𝑙=lim_𝑡→∞𝑢(𝑡)=∞, which is a contradiction. Consequently lim_𝑡→∞𝑢^(𝑡)=0.(2.18) Letting 𝑡→∞ in (2.4) and using (1.2), (1.9) and ℓ∈(0,𝐿), we get lim_𝑡→∞𝑢^(𝑡)=𝑓(ℓ)<0, and so lim_𝑡→∞𝑢′(𝑡)=−∞, which is contrary to (2.18). This contradiction implies that the inequality 𝑢^>0 on (𝜃,∞) cannot be satisfied and that there exists 𝑎>𝜃 such that 𝑢^(𝑎)=0. Since 𝑢>0 on (𝑎,∞), we get by (1.2), (1.7), and (1.13) that (𝑝𝑢^)^<0 on (𝑎,∞). Due to 𝑝(𝑎)𝑢^(𝑎)=0, we see that 𝑢^<0 on (𝑎,∞). Therefore, 𝑢 is decreasing on (𝑎,∞) and lim_𝑡→∞𝑢(𝑡)=ℓ₀∈[0,𝐿). Using (2.16) with 𝑎 in place of 𝜃, we deduce as above that (2.18) holds and that lim_𝑡→∞𝑢^(𝑡)=𝑓(ℓ₀)=0. Consequently, ℓ₀=0. We have proved that (2.14) holds provided 𝜃>0.
If 𝜃=0, then we take 𝑎=0 and use the above arguments. If 𝑢 is eventually negative, we argue similarly.

Lemma 2.7. Assume (1.2)–(1.5), (1.8), (1.9), and 𝑝∈𝐶²(0,∞),limsup_𝑡→∞||||𝑝^(𝑡)lim_𝑥→0+𝑝^(𝑡)||||<∞,(2.19)𝑓(𝑥)𝑥<0.(2.20) Let 𝑢 be a solution of the problem (1.7), (1.13) with 𝑢₀∈(0,𝐿), then there exists 𝛿₁>0 such that 𝑢𝛿₁=0,𝑢^(𝑡)<0for𝑡∈0,𝛿₁.(2.21)

Proof. Assume that such 𝛿₁ does not exist, then 𝑢 is positive on [0,∞) and, by Lemma 2.6, 𝑢 satisfies (2.14). We define a function 𝑣(𝑡)=√𝑝(𝑡)𝑢(𝑡),𝑡∈[0,∞).(2.22) By (2.19), we have 𝑣∈𝐶²(0,∞) and 𝑣^(𝑡)=𝑝^(𝑡)𝑢(𝑡)2√𝑝(𝑡)+√𝑣𝑝(𝑡)𝑢^(𝑡),(2.23)_(𝑡)=𝑣(𝑡)12𝑝^(𝑡)𝑝(𝑡)−14𝑝^(𝑡)𝑝(𝑡)²+𝑓(𝑢(𝑡))𝑢(𝑡),𝑡∈(0,∞).(2.24) By (1.9) and (2.19), we get lim_𝑡→∞12𝑝^(𝑡)𝑝(𝑡)−14𝑝𝑝′((𝑡𝑡))²=12lim^𝑡→∞𝑝^(𝑡)𝑝^(𝑡)⋅𝑝^(𝑡)𝑝(𝑡)=0.(2.25) Since 𝑢 is positive on (0,∞), conditions (2.14) and (2.20) yield lim_𝑡→∞𝑓(𝑢(𝑡))𝑢(𝑡)=lim^𝑥→0+𝑓(𝑥)𝑥<0.(2.26) Consequently, there exist 𝜔>0 and 𝑅>0 such that 12𝑝^(𝑡)𝑝(𝑡)−14𝑝^(𝑡)𝑝(𝑡)²+𝑓(𝑢(𝑡))𝑢(𝑡)<−𝜔for𝑡≥𝑅.(2.27) By (2.22), 𝑣 is positive on (0,∞) and, due to (2.24) and (2.27), we get 𝑣^(𝑡)<−𝜔𝑣(𝑡)<0for𝑡≥𝑅.(2.28) Thus, 𝑣^ is decreasing on [𝑅,∞) and lim_𝑡→∞𝑣^(𝑡)=𝑉. If 𝑉<0, then lim_𝑡→∞𝑣(𝑡)=−∞, contrary to the positivity of 𝑣. If 𝑉≥0, then 𝑣′>0 on [𝑅,∞) and 𝑣(𝑡)≥𝑣(𝑅)>0 for 𝑡∈[𝑅,∞). Then (2.28) yields 0>−𝜔𝑣(𝑅)≥−𝜔𝑣(𝑡)>𝑣^(𝑡) for 𝑡∈[𝑅,∞). We get lim_𝑡→∞𝑣′(𝑡)=−∞ which contradicts 𝑉≥0. The obtained contradictions imply that 𝑢 has at least one zero in (0,∞). Let 𝛿₁>0 be the first zero of 𝑢. Then 𝑢>0 on [0,𝛿₁) and, by (1.2) and (1.7), 𝑢^<0 on (0,𝛿₁). Due to Remark 2.2, we have also 𝑢^(𝛿₁)<0.

For negative starting value, we can prove a dual lemma by similar arguments.

Lemma 2.8. Assume (1.2)–(1.5), (1.8), (1.9), (2.19) and lim_𝑥→0−𝑓(𝑥)𝑥<0.(2.29) Let 𝑢 be a solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0), then there exists 𝜃₁>0 such that 𝑢𝜃₁=0,𝑢^(𝑡)>0for𝑡∈0,𝜃₁.(2.30)

The arguments of the proof of Lemma 2.8 can be also found in the proof of Lemma 3.1 in [18], where both (2.20) and (2.29) were assumed. If one argues as in the proofs of Lemmas 2.7 and 2.8 working with 𝑎₁, 𝐴₁ and 𝑏₁, 𝐵₁ in place of 0, and 𝑢₀, one gets the next corollary.

Corollary 2.9. Assume (1.2)–(1.5), (1.8), (1.9), (2.19), (2.20), and (2.29). Let 𝑢 be a solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0)∪(0,𝐿).
(I) Assume that there exist 𝑏₁>0 and 𝐵₁∈(𝐿₀,0) such that 𝑢𝑏₁=𝐵₁,𝑢′𝑏₁=0,(2.31) then there exists 𝜃>𝑏₁ such that 𝑢(𝜃)=0,𝑢^(𝑡)>0for𝑡∈𝑏₁,𝜃.(2.32)
(II) Assume that there exist 𝑎₁>0 and 𝐴₁∈(0,𝐿) such that 𝑢𝑎₁=𝐴₁,𝑢^𝑎₁=0,(2.33) then there exists 𝛿>𝑎₁ such that 𝑢(𝛿)=0,𝑢^(𝑡)<0for𝑡∈𝑎₁,𝛿.(2.34)

Note that if all conditions of Lemmas 2.7 and 2.8 are satisfied, then each solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0)∪(0,𝐿) has at least one simple zero in (0,∞). Corollary 2.9 makes possible to construct an unbounded sequence of all zeros of any damped solution 𝑢. In addition, these zeros are simple (see the proof of Theorem 2.10). In such a case, 𝑢 has either a positive maximum or a negative minimum between each two neighbouring zeros. If we denote sequences of these maxima and minima by {𝐴_𝑛}^∞𝑛=1 and {𝐵_𝑛}^∞𝑛=1, respectively, then we call the numbers |𝐴_𝑛−𝐵_𝑛|,𝑛∈ℕ amplitudes of 𝑢.

In [18], we give conditions implying that each damped solution of the problem (1.7), (1.13) with 𝑢₀<0 has an unbounded set of zeros and decreasing sequence of amplitudes. Here, there is an extension of this result for 𝑢₀∈(0,𝐿).

Theorem 2.10 (existence of oscillatory solutions I). Assume that (1.2)–(1.5), (1.8), (1.9), (2.19), (2.20), and (2.29) hold, Then each damped solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0)∪(0,𝐿) is oscillatory and its amplitudes are decreasing.

Proof. For 𝑢₀<0, the assertion is contained in Theorem  3.4 of [18]. Let 𝑢 be a damped solution of the problem (1.7), (1.13) with 𝑢₀∈(0,𝐿). By (2.2) and Definition 1.2, we can find 𝐿₁∈(0,𝐿) such that 𝐵<𝑢(𝑡)≤𝐿₁for𝑡∈[0,∞).(2.35)Step 1. Lemma 2.7 yields 𝛿₁>0 satisfying (2.21). Hence, there exists a maximal interval (𝛿₁,𝑏₁) such that 𝑢^<0 on (𝛿₁,𝑏₁). If 𝑏₁=∞, then 𝑢 is eventually negative and decreasing. On the other hand, by Lemma 2.6, 𝑢 satisfies (2.14). But this is not possible. Therefore, 𝑏₁<∞ and there exists 𝐵₁∈(𝐵,0) such that (2.31) holds. Corollary 2.9 yields 𝜃₁>𝑏₁ satisfying (2.32) with 𝜃=𝜃₁. Therefore, 𝑢 has just one negative local minimum 𝐵₁=𝑢(𝑏₁) between its first zero 𝛿₁ and second zero 𝜃₁.Step 2. By (2.32) there exists a maximal interval (𝜃₁,𝑎₁), where 𝑢^>0. If 𝑎₁=∞, then 𝑢 is eventually positive and increasing. On the other hand, by Lemma 2.6, 𝑢 satisfies (2.14). We get a contradiction. Therefore 𝑎₁<∞ and there exists 𝐴₁∈(0,𝐿) such that (2.33) holds. Corollary 2.9 yields 𝛿₂>𝑎₁ satisfying (2.34) with 𝛿=𝛿₂. Therefore 𝑢 has just one positive maximum 𝐴₁=𝑢(𝑎₁) between its second zero 𝜃₁ and third zero 𝛿₂.Step 3. We can continue as in Steps 1 and 2 and get the sequences {𝐴_𝑛}^∞𝑛=1⊂(0,𝐿) and {𝐵_𝑛}^∞𝑛=1⊂[𝑢₀,0) of positive local maxima and negative local minima of 𝑢, respectively. Therefore 𝑢 is oscillatory. Using arguments of the proof of Theorem  3.4 of [18], we get that the sequence {𝐴_𝑛}^∞𝑛=1 is decreasing and the sequence {𝐵_𝑛}^∞𝑛=1 is increasing. In particular, we use (2.5) and define a Lyapunov function 𝑉_𝑢 by 𝑉_𝑢(𝑡)=𝑢^2(𝑡)2+𝐹(𝑢(𝑡))=𝐹𝑢⁰−^𝑡0𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠,𝑡∈(0,∞),(2.36) then 𝑉_𝑢(𝑡)>0,𝑉^𝑢(𝑡)=−𝑝^(𝑡)𝑝(𝑡)𝑢^2𝑉(𝑡)≤0for𝑡∈(0,∞),(2.37)_𝑢(𝑡)<0for𝑡∈(0,∞),𝑡≠𝑎_𝑛,𝑏_𝑛,𝑛∈ℕ.(2.38) Consequently, 𝑐_𝑢∶=lim_𝑡→∞𝑉_𝑢(𝑡)≥0.(2.39) So, sequences {𝑉_𝑢(𝑎_𝑛)}^∞𝑛=1={𝐹(𝐴_𝑛)}^∞𝑛=1 and {𝑉_𝑢(𝑏_𝑛)}^∞𝑛=1={𝐹(𝐵_𝑛)}^∞𝑛=1 are decreasing and lim_𝑛→∞𝐹𝐴_𝑛=lim_𝑛→∞𝐹𝐵_𝑛=𝑐_𝑢.(2.40) Finally, due to (1.4), the sequence {𝐴_𝑛}^∞𝑛=1 is decreasing and the sequence {𝐵_𝑛}^∞𝑛=1 is increasing. Hence, the sequence of amplitudes {𝐴_𝑛−𝐵_𝑛}^∞𝑛=1 is decreasing, as well.

Example 2.11. Consider the problem (1.7), (1.13), where 𝑝(𝑡)=𝑡² and 𝑓(𝑥)=𝑥(𝑥−1)(𝑥+2). In Example 2.4, we have shown that (1.2)–(1.5), (1.8), and (1.9) with 𝐿₀=−2, 𝐿=1 are valid. Since lim_𝑡→∞𝑝^(𝑡)𝑝^(𝑡)=lim^𝑡→∞1lim_𝑥→0𝑡=0,𝑓(𝑥)𝑥=lim^𝑥→0(𝑥−1)(𝑥+2)=−2<0,(2.41) we see that (2.19), (2.20), and (2.29) are satisfied. Therefore, by Theorem 2.10, each damped solution of (2.8), (1.13) with 𝑢₀∈(−2,0)∪(0,1) is oscillatory and its amplitudes are decreasing.

Example 2.12. Consider the problem (1.7), (1.13), where 𝑝(𝑡)=𝑡^𝑘1+𝑡^ℓ𝑓(𝑥)=𝑥𝑥((𝑥𝑥−−11))((,𝑥𝑥++𝑘34>)),,ℓ≥ffoo0rr,𝑥𝑥≤>00,,(2.42) then 𝐿₀=−3, 𝐿=1, lim_𝑡→∞𝑝^(𝑡)𝑝^(𝑡)=0,lim^𝑥→0−𝑓(𝑥)𝑥=−3,lim^𝑥→0+𝑓(𝑥)𝑥=−4.(2.43) We can check that also all remaining assumptions of Theorem 2.10 are satisfied, and this theorem is applicable here.

Assume that 𝑓 does not fulfil (2.20) and (2.29). It occurs, for example, if 𝑓(𝑥)=−|𝑥|^𝛼sign𝑥 with 𝛼>1 for 𝑥 in some neighbourhood of 0, then Theorem 2.10 cannot be applied. Now, we will give another sufficient conditions for the existence of oscillatory solutions. For this purpose, we introduce the following lemmas.

Lemma 2.13. Assume (1.2)–(1.5), (1.8), (1.9), and ^∞11∃𝜖>0∶𝑓∈𝐶₁𝑝(𝑠)d𝑠=∞,(2.44)(0,𝜖),𝑓^≤0on(0,𝜖).(2.45) Let 𝑢 be a solution of the problem (1.7), (1.13) with 𝑢₀∈(0,𝐿), then there exists 𝛿₁>0 such that 𝑢𝛿₁=0,𝑢^(𝑡)<0for𝑡∈0,𝛿₁.(2.46)

Proof. Assume that such 𝛿₁ does not exist, then 𝑢 is positive on [0,∞) and, by Lemma 2.6, 𝑢 satisfies (2.14). In view of (1.7) and (1.2), we have 𝑢^<0 on (0,∞). From (2.45), it follows that there exists 𝑡₀>0 such that 0<𝑢(𝑡)<𝜖,for𝑡∈𝑡₀,∞.(2.47) Motivated by arguments of [27], we divide (1.7) by 𝑓(𝑢) and integrate it over interval [𝑡₀,𝑡]. We get ^𝑡𝑡₀𝑝(𝑠)𝑢^(𝑠)′𝑓(𝑢(𝑠))d𝑠=^𝑡𝑡₀𝑝(𝑠)d𝑠for𝑡∈𝑡₀,∞.(2.48) Using the per partes integration, we obtain 𝑝(𝑡)𝑢^(𝑡)𝑓(𝑢(𝑡))+^𝑡𝑡₀𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠=𝑝𝑡⁰𝑢^𝑡₀𝑓𝑢𝑡₀+^𝑡𝑡₀𝑝(𝑠)d𝑠,𝑡∈𝑡₀,∞.(2.49) From (1.8) and (1.9), it follows that there exists 𝑡₁∈(𝑡₀,∞) such that 𝑝𝑡₀𝑢^𝑡₀𝑓𝑢𝑡₀+^𝑡𝑡₀𝑝(𝑠)d𝑠≥1,𝑡∈𝑡₁,∞,(2.50) and therefore 𝑝(𝑡)𝑢^(𝑡)𝑓(𝑢(𝑡))+^𝑡𝑡₀𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠≥1,𝑡∈𝑡¹,∞.(2.51) From the fact that 𝑓^(𝑢(𝑠))≤0 for 𝑠>𝑡₀ (see (2.45)), we have 𝑝(𝑡)𝑢^(𝑡)𝑓(𝑢(𝑡))+^𝑡𝑡₁𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠≥1,𝑡∈𝑡¹,∞,(2.52) then 𝑝(𝑡)𝑢′(𝑡)𝑓(𝑢(𝑡))≥1−^𝑡𝑡₁𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠>0,𝑡∈𝑡¹𝑝(𝑡)𝑢′(𝑡),∞,(2.53)𝑓(𝑢(𝑡))1−∫^𝑡𝑡₁𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓⁻²(𝑢(𝑠))d𝑠≥1,𝑡∈𝑡¹,∞.(2.54) Multiplying this inequality by −𝑓^(𝑢(𝑡))𝑢^(𝑡)/𝑓(𝑢(𝑡))≥0, we get ln1−^𝑡𝑡₁𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠^≥−ln||𝑓(𝑢(𝑡))||^,𝑡∈𝑡₁,∞,(2.55) and integrating it over [𝑡₁,𝑡], we obtain ln1−^𝑡𝑡₁𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠≥ln𝑓𝑢𝑡¹𝑓(𝑢(𝑡)),(2.56) and therefore, 1−^𝑡𝑡₁𝑝(𝑠)𝑓^(𝑢(𝑠))𝑢^2(𝑠)𝑓²(𝑢(𝑠))d𝑠≥𝑓𝑢𝑡¹𝑓(𝑢(𝑡)),𝑡∈𝑡¹,∞.(2.57) According to (2.53), we have 𝑝(𝑡)𝑢^(𝑡)𝑓(𝑢(𝑡))≥𝑓𝑢𝑡¹𝑓(𝑢(𝑡)),𝑡∈𝑡¹,∞,(2.58) and consequently, 𝑢^(𝑡)≤𝑓𝑢𝑡₁1𝑝(𝑡),𝑡∈𝑡¹,∞.(2.59) Integrating it over [𝑡₁,𝑡], we get 𝑢(𝑡)≤𝑢𝑡₁+𝑓𝑢𝑡₁^𝑡𝑡₁1𝑝(𝑠)d𝑠,𝑡∈𝑡¹,∞.(2.60) From (2.44), it follows that lim_𝑡→∞𝑢(𝑡)=−∞,(2.61) which is a contradiction.

By similar arguments, we can prove a dual lemma.

Lemma 2.14. Assume (1.2)–(1.5), (1.8), (1.9), (2.44), and ∃𝜖>0∶𝑓∈𝐶¹(−𝜖,0),𝑓^≤0on(−𝜖,0).(2.62) Let 𝑢 be a solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0), then, there exists 𝜃₁>0 such that 𝑢𝜃₁=0,𝑢^(𝑡)>0for𝑡∈0,𝜃₁.(2.63)

Following ideas before Corollary 2.9, we get the next corollary.

Corollary 2.15. Assume (1.2)–(1.5), (1.8), (1.9), (2.44), (2.45), and (2.62). Let 𝑢 be a solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0)∪(0,𝐿), then the assertions I and II of Corollary 2.9 are valid.

Now, we are able to formulate another existence result for oscillatory solutions. Its proof is almost the same as the proof of Theorem 2.10 for 𝑢₀∈(𝐿₀,0) and the proof of Theorem 3.4 in [18] for 𝑢₀∈(0,𝐿). The only difference is that we use Lemmas 2.13, 2.14, and Corollary 2.15, in place of Lemmas 2.7, 2.8, and Corollary 2.9, respectively.

Theorem 2.16 (existence of oscillatory solutions II). Assume that (1.2)–(1.5), (1.8), (1.9), (2.44), (2.45), and (2.62) hold, then each damped solution of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0)∪(0,𝐿) is oscillatory and its amplitudes are decreasing.

Example 2.17. Let us consider (1.7) with 𝑓(𝑥)=𝑝(⎧⎪⎨⎪⎩𝑡)−=|𝑥𝑡|^𝛼_𝜆,𝑡∈[0,∞),𝑥1,−2s,gn𝑥,𝑥𝑥𝑥∈≥≤31(1,,,3),(2.64) where 𝜆 and 𝛼 are real parameters.Case 1. Let 𝜆∈(1,∞) and 𝛼∈(0,1], then all assumptions of Theorem 2.16 are satisfied. Note that 𝑓 satisfies neither (2.20) nor (2.29) and hence Theorem 2.10 cannot be applied. Case 2. Let 𝜆=1 and 𝛼∈(0,∞), then all assumptions of Theorem 2.10 are satisfied. If 𝛼∈(0,1], then also all assumptions of Theorem 2.16 are fulfilled, but for 𝛼∈(1,∞), the function 𝑝 does not satisfy (2.44), and hence Theorem 2.16 cannot be applied.

3. Asymptotic Properties of Oscillatory Solutions

In Lemma 2.6 we show that if 𝑢 is a damped solution of the problem (1.7), (1.13) which is not oscillatory then 𝑢 converges to 0 for 𝑡→∞. In this section, we give conditions under which also oscillatory solutions converge to 0.

Theorem 3.1. Assume that (1.2)–(1.5), (1.8), and (1.9) hold and that there exists 𝑘₀>0 such that liminf_𝑡→∞𝑝(𝑡)𝑡^𝑘0>0,(3.1) then each damped oscillatory solution 𝑢 of the problem (1.7), (1.13) with 𝑢₀∈(𝐿₀,0)∪(0,𝐿) satisfies lim_𝑡→∞𝑢(𝑡)=0,lim_𝑡→∞𝑢^(𝑡)=0.(3.2)

Proof. Consider an oscillatory solution 𝑢 of the problem (1.7), (1.13) with 𝑢₀∈(0,𝐿).Step 1. Using the notation and some arguments of the proof of Theorem 2.10, we have the unbounded sequences {𝑎_𝑛}^∞𝑛=1, {𝑏_𝑛}^∞𝑛=1, {𝜃_𝑛}^∞𝑛=1, and {𝛿_𝑛}^∞𝑛=1, such that 0<𝛿₁<𝑏₁<𝜃₁<𝑎₁<𝛿₂<⋯<𝛿_𝑛<𝑏_𝑛<𝜃_𝑛<𝑎_𝑛<𝛿_𝑛+1<⋯,(3.3) where 𝑢(𝜃_𝑛)=𝑢(𝛿_𝑛)=0, 𝑢(𝑎_𝑛)=𝐴_𝑛>0 is a unique local maximum of 𝑢 in (𝜃_𝑛,𝛿_𝑛+1), 𝑢(𝑏_𝑛)=𝐵_𝑛<0 is a unique local minimum of 𝑢 in (𝛿_𝑛,𝜃_𝑛), 𝑛∈ℕ. Let 𝑉_𝑢 be given by (2.36) and then (2.39) and (2.40) hold and, by (1.2)–(1.4), we see that lim_𝑡→∞𝑢(𝑡)=0⟺𝑐_𝑢=0.(3.4) Assume that (3.2) does not hold. Then 𝑐_𝑢>0. Motivated by arguments of [28], we derive a contradiction in the following steps.Step 2 (estimates of 𝑢). By (2.36) and (2.39), we have lim_𝑛→∞𝑢^2𝛿_𝑛2=lim^𝑛→∞𝑢^2𝜃_𝑛2=𝑐^𝑢>0,(3.5) and the sequences {𝑢^2(𝛿_𝑛)}^∞𝑛=1 and {𝑢^2(𝜃_𝑛)}^∞𝑛=1 are decreasing. Consider 𝑛∈ℕ. Then 𝑢^2(𝛿_𝑛)/2>𝑐_𝑢 and there are 𝛼_𝑛,𝛽_𝑛 satisfying 𝑎_𝑛<𝛼_𝑛<𝛿_𝑛<𝛽_𝑛<𝑏_𝑛 and such that 𝑢^2𝛼_𝑛=𝑢^2𝛽_𝑛=𝑐_𝑢,𝑢^2(𝑡)>𝑐_𝑢,𝑡∈𝛼_𝑛,𝛽_𝑛.(3.6) Since 𝑉_𝑢(𝑡)>𝑐_𝑢 for 𝑡>0 (see (2.39)), we get by (2.36) and (3.6) the inequalities 𝑐_𝑢/2+𝐹(𝑢(𝛼_𝑛))>𝑐_𝑢 and 𝑐_𝑢/2+𝐹(𝑢(𝛽_𝑛))>𝑐_𝑢, and consequently 𝐹(𝑢(𝛼_𝑛))>𝑐_𝑢/2 and 𝐹(𝑢(𝛽_𝑛))>𝑐_𝑢/2. Therefore, due to (1.4), there exists 𝑐̃>0 such that 𝑢𝛼_𝑛>𝑐̃,𝑢𝛽_𝑛<−𝑐̃,𝑛∈ℕ.(3.7) Similarly, we deduce that there are 𝛼_𝑛, 𝛽̃_𝑛 satisfying 𝑏_𝑛<𝛼_𝑛<𝜃_𝑛<𝛽̃_𝑛<𝑎_𝑛+1 and such that 𝑢𝛼_𝑛<−𝑐̃,𝑢𝛽̃_𝑛>𝑐̃,𝑛∈ℕ.(3.8) The behaviour of 𝑢 and inequalities (3.7) and (3.8) yield ||𝑢(𝑡)||>𝑐̃,𝑡∈𝛽_𝑛,𝛼_𝑛∪𝛽̃_𝑛,𝛼_𝑛+1,𝑛∈ℕ.(3.9)Step 3 (estimates of 𝛽_𝑛−𝛼_𝑛). We prove that there exist 𝑐₀,𝑐₁∈(0,∞) such that 𝑐₀<𝛽_𝑛−𝛼_𝑛<𝑐₁,𝑛∈ℕ.(3.10) Assume on the contrary that there exists a subsequence satisfying lim_ℓ→∞(𝛽_ℓ−𝛼_ℓ)=0. By the mean value theorem and (3.7), there is 𝜉_ℓ∈(𝛼_ℓ,𝛽_ℓ) such that 0<2𝑐̃<𝑢(𝛼_ℓ)−𝑢(𝛽_ℓ)=|𝑢′(𝜉_ℓ)|(𝛽_ℓ−𝛼_ℓ). Since 𝐹(𝑢(𝑡))≥0 for 𝑡∈[0,∞), we get by (2.16) the inequality ||𝑢^(𝑡)||<2𝐹𝑢₀,𝑡∈[0,∞),(3.11) and consequently 0<2𝑐̃≤2𝐹𝑢₀lim_ℓ→∞𝛽_ℓ−𝛼_ℓ=0,(3.12) which is a contradiction. So, 𝑐₀ satisfying (3.10) exists. Using the mean value theorem again, we can find 𝜏_𝑛∈(𝛼_𝑛,𝛿_𝑛) such that 𝑢(𝛿_𝑛)−𝑢(𝛼_𝑛)=𝑢^(𝜏_𝑛)(𝛿_𝑛−𝛼_𝑛) and, by (3.6), 𝛿_𝑛−𝛼_𝑛=−𝑢𝛼^𝑛𝑢^𝜏_𝑛=𝑢𝛼^𝑛||𝑢^𝜏_𝑛||<√𝐴𝑐¹_𝑢.(3.13) Similarly, we can find 𝜂_𝑛∈(𝛿_𝑛,𝛽_𝑛) such that 𝛽_𝑛−𝛿_𝑛=𝑢𝛽^𝑛𝑢^𝜂_𝑛=||𝑢𝛽^𝑛||||𝑢^𝜂_𝑛||<||𝐵¹||√𝑐_𝑢.(3.14) If we put 𝑐₁=(𝐴₁+|𝐵₁|)/√𝑐_𝑢, then (3.10) is fulfilled. Similarly, we can prove 𝑐₀<𝛽̃_𝑛−𝛼_𝑛<𝑐₁,𝑛∈ℕ.(3.15)Step 4 (estimates of 𝛼_𝑛+1−𝛼_𝑛). We prove that there exist 𝑐₂∈(0,∞) such that 𝛼_𝑛+1−𝛼_𝑛<𝑐₂,𝑛∈ℕ.(3.16) Put 𝑚₁=min{𝑓(𝑥)∶𝐵₁≤𝑥≤−𝑐̃}>0. By (3.9), 𝐵₁≤𝑢(𝑡)<−𝑐̃ for 𝑡∈[𝛽_𝑛,𝛼_𝑛], 𝑛∈ℕ. Therefore, 𝑓(𝑢(𝑡))≥𝑚₁,𝑡∈𝛽_𝑛,𝛼_𝑛,𝑛∈ℕ.(3.17) Due to (1.9), we can find 𝑡₁>0 such that 𝑝^(𝑡)𝑝(𝑡)2𝐹𝑢₀<𝑚¹2,𝑡∈𝑡¹,∞.(3.18) Let 𝑛₁∈ℕ fulfil 𝛼_𝑛1≥𝑡₁, then, according to (2.4), (3.11), (3.17), and (3.18), we have 𝑢^(𝑡)>−𝑚¹2+𝑚¹=𝑚¹2,𝑡∈𝛽^𝑛,𝛼_𝑛,𝑛≥𝑛₁.(3.19) Integrating (3.19) from 𝑏_𝑛 to 𝛽_𝑛 and using (3.6), we get 2√𝑐_𝑢>𝑚₁(𝑏_𝑛−𝛽_𝑛) for 𝑛≥𝑛₁. Similarly we get 2√𝑐_𝑢>𝑚₁(𝛼_𝑛−𝑏_𝑛) for 𝑛≥𝑛₁. Therefore 4𝑚₁√𝑐_𝑢>𝛼_𝑛−𝛽_𝑛,𝑛≥𝑛₁.(3.20) By analogy, we put 𝑚₂=min{−𝑓(𝑥)∶𝑐̃≤𝑥≤𝐴₁}>0 and prove that there exists 𝑛₂∈ℕ such that 4𝑚₂√𝑐_𝑢>𝛼_𝑛+1−𝛽̃_𝑛,𝑛≥𝑛₂.(3.21) Inequalities (3.10), (3.15), (3.20), and (3.21) imply the existence of 𝑐₂ fulfilling (3.16).Step 5 (construction of a contradiction). Choose 𝑡₀>𝑐₁ and integrate the equality in (2.37) from 𝑡₀ to 𝑡>𝑡₀. We have 𝑉_𝑢(𝑡)=𝑉_𝑢𝑡₀−^𝑡𝑡₀𝑝^(𝜏)𝑝(𝜏)𝑢^2(𝜏)d𝜏,𝑡≥𝑡₀.(3.22) Choose 𝑛₀∈ℕ such that 𝛼_𝑛0>𝑡₀. Further, choose 𝑛∈ℕ, 𝑛>𝑛₀ and assume that 𝑡>𝛽_𝑛, then, by (3.6), ^𝑡𝑡₀𝑝′(𝜏)𝑝(𝜏)𝑢^2(𝜏)d𝜏>^𝑛_𝑗_=𝑛0^𝛽𝑗_𝛼𝑗𝑝^(𝜏)𝑝(𝜏)𝑢^2>𝑐_𝑢𝑛(𝜏)d𝜏_𝑗_=𝑛0^𝛽𝑗_𝛼𝑗𝑝^(𝜏)𝑝(𝜏)d𝜏=𝑐^𝑢𝑛_𝑗=𝑛0[ln𝑝(𝜏)]^𝛽𝑗_𝛼𝑗.(3.23) By virtue of (3.1) there exists 𝑐₃>0 such that 𝑝(𝑡)/𝑡^𝑘0>𝑐₃ for 𝑡∈[𝑡₀,∞). Thus, ln𝑝(𝑡)>ln𝑐₃+𝑘₀ln𝑡 and ^𝑡𝑡₀𝑝^(𝜏)𝑝(𝜏)𝑢^2(𝜏)d𝜏>𝑐_𝑢𝑛𝑗_=𝑛0[ln𝑐₃+𝑘₀ln𝑡)]^𝛽𝑗_𝛼𝑗=𝑐_𝑢𝑘_0𝑛𝑗_=𝑛0ln𝛽^𝑗𝛼_𝑗.(3.24) Due to (3.10) and 𝑐₁<𝛼_𝑛0, we have 1<𝛽^𝑗𝛼_𝑗<1+𝑐¹𝛼_𝑗<2,𝑗=𝑛₀,…,𝑛,(3.25) and the mean value theorem yields 𝜉_𝑗∈(1,2) such that ln𝛽^𝑗𝛼_𝑗=𝛽^𝑗𝛼_𝑗−11𝜉_𝑗>𝛽^𝑗−𝛼_𝑗2𝛼_𝑗,𝑗=𝑛₀,…,𝑛.(3.26) By (3.10) and (3.16), we deduce 𝛽_𝑗−𝛼_𝑗𝛼_𝑗>𝑐⁰𝛼_𝑗,𝛼_𝑗<𝑗𝑐₂+𝛼₁,𝑗=𝑛₀,…,𝑛.(3.27) Thus, 𝛽_𝑗−𝛼_𝑗𝛼_𝑗>𝑐⁰𝑗𝑐₂+𝛼₁,𝑗=𝑛₀,…,𝑛.(3.28) Using (3.24)–(3.28) and letting 𝑡 to ∞, we obtain ^∞𝑡₀𝑝′(𝜏)𝑝(𝜏)𝑢′²(𝜏)d𝜏≥𝑐_𝑢𝑘_0∞𝑛_=𝑛0ln𝛽^𝑛𝛼_𝑛≥12𝑐^𝑢𝑘_0∞𝑛_=𝑛0𝛽_𝑛−𝛼_𝑛𝛼_𝑛≥12𝑐^𝑢𝑘_0∞𝑛_=𝑛0𝑐₀𝑛𝑐₂+𝛼₁=∞.(3.29) Using it in (3.22), we get lim_𝑡→∞𝑉_𝑢(𝑡)=−∞, which is a contradiction. So, we have proved that 𝑐_𝑢=0.
Using (2.4) and (3.4), we have lim_𝑡→∞𝑢^2(𝑡)2+^𝑡0𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠=𝐹𝑢₀−𝐹(0)=𝐹𝑢₀.(3.30) Since the function ∫^𝑡0(𝑝^(𝑠)/𝑝(𝑠))𝑢^2(𝑠)d𝑠 is increasing, there exists lim_𝑡→∞^𝑡0𝑝^(𝑠)𝑝(𝑠)𝑢^2(𝑠)d𝑠≤𝐹𝑢₀.(3.31) Therefore, there exists lim_𝑡→∞𝑢^2(𝑡)=ℓ².(3.32) If ℓ>0, then lim_𝑡→∞|𝑢^(𝑡)|=ℓ, which contradicts (3.4). Therefore, ℓ=0 and (3.2) is proved.
If 𝑢₀∈(𝐿₀,0), we argue analogously.

Acknowledgments

The authors thank the referees for valuable comments and suggestions. This work was supported by the Council of Czech Government MSM 6198959214.