Asymptotic Dichotomy in a Class of Odd-Order Nonlinear Differential Equations with Impulses

Abstract

We investigate the oscillatory and asymptotic behavior of a class of odd-order nonlinear differential equations with impulses. We obtain criteria that ensure every solution is either oscillatory or (nonoscillatory and) zero convergent. We provide several examples to show that impulses play an important role in the asymptotic behaviors of these equations.

1. Introduction

Impulsive effect, likewise, exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments in time, involving such fields as medicine and biology, economics, mechanics, electronics, telecommunications, and so forth. It has been observed that the solutions of quite a few first-or second-order impulsive differential equations are either oscillatory or (nonoscillatory and) zero convergent (see, e.g., [1–10]). For example, Bainov et al. studied the oscillation properties of first-order impulsive differential equations with deviating arguments [3]. Especially in [4], Chen investigated oscillations of second-order nonlinear differential with impulses, and he promposed that the impulses may change the oscillatory behavior of an equation. Based on [4], the authors were devoted to oscillations of impulsive differential equations (see, e.g., [5–10]). Such a dichotomy may yield useful information in real problems. The implications of this dichotomy are applied to the deflection of an elastic beam [11]. Thus, it is of interest to see whether similar dichotomies occur in different types of impulsive differential equations.

One such type consists of impulsive differential equations which are important in the simulation of processes with jump conditions. But papers devoted to the study of asymptotic behaviors of higher order equations with impulses are quite rare. For this reason, Wen et al. studied in [12] the dichotomous properties of the following third-order nonlinear differential equation with impulses: 𝑟(𝑡)𝑥^′′(𝑡)^′+𝑓(𝑡,𝑥)=0,𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥^(𝑖)𝑡⁺_𝑘=𝑔_𝑘^[𝑖]𝑥^(𝑖)𝑡_𝑘𝑥,𝑖=0,1,2;𝑘=1,2,…,_(𝑖)𝑡⁺₀=𝑥₀^(𝑖),𝑖=0,1,2, (1) where 𝑥⁽⁰⁾(𝑡)=𝑥(𝑡), 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯ such that lim_𝑘→∞𝑡_𝑘=∞. On the other hand, in [13], Chen and Wen investigated the oscillatory and asymptotic behaviors for odd-order nonlinear differential equations with impulses of the form 𝑥^(2𝑛+1)(𝑡)+𝑝(𝑡)𝑥(𝑡)=0,𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥^(𝑖)𝑡⁺_𝑘=𝑎_𝑘^[𝑖]𝑥^(𝑖)𝑡_𝑘𝑥,𝑖=0,1,…,2𝑛;𝑘=1,2,…,_(𝑖)𝑡⁺₀=𝑥₀^(𝑖),𝑖=0,1,…,2𝑛, (2) where 𝑛 is a positive integer and 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯ such that lim_𝑘→+∞𝑡_𝑘=+∞. They obtained some interesting results for assuring that every bounded solution of (2) is either oscillatory or nonoscillatory and zero convergent.

In this paper, we will study a class of odd-order nonlinear differential equations with impulses of the form 𝑟(𝑡)𝑥^(2𝑛)(𝑡)^′+𝑓(𝑡,𝑥)=0,𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥^(𝑖)𝑡⁺_𝑘=𝑔_𝑘^[𝑖]𝑥^(𝑖)𝑡_𝑘𝑥,𝑖=0,1,…,2𝑛;𝑘=1,2,…,_(𝑖)𝑡⁺₀=𝑥₀^[𝑖],𝑖=0,1,…,2𝑛, (3) where 𝑥⁽⁰⁾(𝑡)=𝑥(𝑡),𝑛 is a positive integer, and 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯ such that lim_𝑘→+∞𝑡_𝑘=+∞, 𝑥^(𝑖)𝑡_𝑘=lim_ℎ→0−𝑥^(𝑖−1)𝑡_𝑘+ℎ−𝑥^(𝑖−1)𝑡_𝑘ℎ,𝑥^(𝑖)𝑡⁺_𝑘=lim_ℎ→0+𝑥^(𝑖−1)𝑡_𝑘+ℎ−𝑥^(𝑖−1)𝑡⁺_𝑘ℎ. (4) By a solution of (3), we mean a real function 𝑥=𝑥(𝑡) defined on [𝑡₀,+∞) such that(I)𝑥^(𝑖)(𝑡⁺₀)=𝑥₀^[𝑖] for 𝑖=0,1,…,2𝑛;(II)𝑥^(𝑖)(𝑡),𝑖=0,1,…,2𝑛, and (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′ are continuous on [𝑡₀,+∞)⧵{𝑡_𝑘}; for 𝑖=0,1,…,2𝑛,𝑥^(𝑖)(𝑡⁺_𝑘) and 𝑥^(𝑖)(𝑡⁻_𝑘) exist, 𝑥^(𝑖)(𝑡⁻_𝑘)=𝑥^(𝑖)(𝑡_𝑘) and 𝑥^(𝑖)(𝑡⁺_𝑘)=𝑔_𝑘^[𝑖](𝑥^(𝑖)(𝑡_𝑘)) for any 𝑡_𝑘;(III)𝑥(𝑡) satisfies (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′+𝑓(𝑡,𝑥(𝑡))=0 at each point 𝑡∈[𝑡₀,+∞)⧵{𝑡_𝑘}.

A solution of (3) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it is said to be oscillatory.

We will establish oscillatory and asymptotic results of (3) based on combinations of the following conditions.(i)𝑟(𝑡)>0 and 𝑟(𝑡) is continuous on [𝑡₀,+∞);𝑓(𝑡,𝑥) is continuous on [𝑡₀,+∞)×(−∞,+∞),𝑥𝑓(𝑡,𝑥)>0 for 𝑥≠0, and 𝑓(𝑡,𝑥)/𝜑(𝑥)⩾𝑝(𝑡), where 𝑝(𝑡) is positive and continuous on [𝑡₀,+∞) and 𝜑 is differentiable in (−∞,+∞) such that 𝜑^′(𝑥)≥0 for 𝑥∈(−∞,+∞).(ii)For 𝑘=1,2,…, 𝑔_𝑘^[𝑖](𝑥) are continuous in (−∞,+∞) and there exist positive numbers 𝑎_𝑘^[𝑖],𝑏_𝑘^[𝑖] such that 𝑎_𝑘^[𝑖]≤𝑔_𝑘^[𝑖](𝑥)/𝑥≤𝑏_𝑘^[𝑖],𝑖=0,1,2,…,2𝑛. (5)(iii)_𝑡^+∞_{0𝑡0<𝑡𝑘<𝑠}𝑎_𝑘^[𝑖]𝑏_𝑘^[𝑖−1]𝑑𝑠=+∞;𝑖=1,2,…,2𝑛−1,_𝑡+∞01𝑟(𝑠)_{𝑡0<𝑡𝑘<𝑠}𝑎_𝑘^[2𝑛]𝑏_𝑘^[2𝑛−1]𝑑𝑠=+∞. (6)

Our plan is the following. We first obtain three theorems (Theorems 1–3) to ensure every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent. We will also illustrate our results with several examples. As applications of our results, we state three corollaries (Corollaries 5–7). These corollaries are new even for the special case (2).

2. Main Results

The main results of the paper are as follows.

Theorem 1. Assume that the conditions (i)–(iii) hold. Suppose further that there exists a positive integer 𝑘₀ such that for 𝑘≥𝑘₀, 𝑎_𝑘^[0]≥1, ^+∞_𝑘=1𝑏_𝑘^[0]−1<+∞,_{𝑡+∞0𝑡0<𝑡𝑘<𝑠}1𝑏_𝑘^[2𝑛]𝑝(𝑠)𝑑𝑠=+∞. (7) Then every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent.

Theorem 2. Assume that the conditions (i)–(iii) hold and that 𝜑(𝑎𝑏)≥𝜑(𝑎)𝜑(𝑏) for any 𝑎𝑏>0. Suppose further that there exists a positive integer 𝑘₀ such that for 𝑘≥𝑘₀, 𝑏_𝑘^[0]≤1,𝑏_𝑘^[2𝑛]≤1,𝑏_𝑘^[2𝑛]≤𝜑𝑎_𝑘^[0],_𝑡+∞0𝑝(𝑠)𝑑𝑠=+∞. (8) Then every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent.

Theorem 3. Assume that the conditions (i)–(iii) hold and that 𝜑(𝑎𝑏)≥𝜑(𝑎)𝜑(𝑏) for any 𝑎𝑏>0. Suppose further that 𝑏_𝑘^[2𝑛]≤𝑎_𝑘^[0],{∏^𝑛_𝑘=1𝑏_𝑘^[0]} is bounded, max𝑎^+∞_𝑘=1𝑘^[0]−1,𝑏_𝑘^[0]−1<+∞,_+∞𝑘=1𝑏_𝑘^[2𝑛]−1<+∞,_𝑡+∞0𝑝(𝑠)𝑑𝑠=+∞. (9) Then every solution of (3) is either oscillatory or (nonoscillatory and) zero convergent.

Remark 4. When 𝑛=1, (3) reduces to (1). Our Theorems 1–3 are Theorem 2.1, Theorem 2.3, Theorem 2.4 in [12], respectively. So our results generalize and contain results in [12].
Next, for (2), we will also be able to obtain some new results. It is easy to see that (2) has the form of (3) by setting 𝑟(𝑡)=1,𝑓(𝑡,𝑥(𝑡))=𝑝(𝑡)𝑥(𝑡) and 𝑔_𝑘^[𝑖](𝑥^(𝑖)(𝑡_𝑘))=𝑎_𝑘^[𝑖]𝑥^(𝑖)(𝑡_𝑘) for 𝑖=0,1,…,2𝑛 and 𝑘=1,2,…. Let 𝑎_𝑘^[𝑖]=𝑏_𝑘^[𝑖],𝑖=0,1,…,2𝑛;𝑘=1,2,…,𝜑(𝑥)=𝑥. (10) Our Theorems 1–3 can directly lead us to the following corollaries for (2).

Corollary 5. Assume that the conditions (i)–(iii) hold. Suppose further that there exists a positive integer 𝑘₀ such that for 𝑘≥𝑘₀,𝑎_𝑘^[0]≥1, ^+∞_𝑘=1𝑎_𝑘^[0]−1<+∞,_{𝑡+∞0𝑡0<𝑡𝑘<𝑠}1𝑎_𝑘^[2𝑛]𝑝(𝑠)𝑑𝑠=+∞. (11) Then every solution of (2) is either oscillatory or (nonoscillatory and) zero convergent.

Corollary 6. Assume that the conditions (i)–(iii) hold. Suppose further that there exists a positive integer 𝑘₀ such that for 𝑘≥𝑘₀, 𝑎_𝑘^[0]≤1,𝑎_𝑘^[2𝑛]≤1,𝑎_𝑘^[2𝑛]≤𝑎_𝑘^[0],_𝑡+∞0𝑝(𝑠)𝑑𝑠=+∞. (12) Then every solution of (2) is either oscillatory or (nonoscillatory and) zero convergent.

Corollary 7. Assume that the conditions (i)–(iii) hold. Suppose further that 𝑎_𝑘^[2𝑛]≤𝑎_𝑘^[0], {∏^𝑛_𝑘=1𝑎_𝑘^[0]} is bounded, ^+∞_𝑘=1𝑎_𝑘^[0]−1<+∞,_+∞𝑘=1𝑎_𝑘^[2𝑛]−1<+∞,_𝑡+∞0𝑝(𝑠)𝑑𝑠=+∞. (13)
Then every solution of (2) is either oscillatory or (nonoscillatory and) zero convergent.

Remark 8. We note that the above corollaries for ensuring every solution of (2) to be either oscillatory or tend to zero with fixed sign eventually is also new for (2).

Example 9. Consider the equation 𝑥^′′′(𝑡)+38𝑡𝑥³(𝑡)1=0,𝑡≥2𝑥,𝑡≠𝑘,_(𝑖)𝑘⁺=11+𝑘²𝑥^(𝑖)(𝑘)𝑥1,𝑖=0,1,2;𝑘=1,2,…,2=𝑥₀,𝑥^′12=𝑥₀^[1],𝑥^′′12=𝑥₀^[2], (14) where 𝑎_𝑘^[𝑖]=𝑏_𝑘^[𝑖]=(1+(1/𝑘²))≥1 for 𝑖=0,1,2;𝑟(𝑡)=1,𝑝(𝑡)=(3/8)𝑡,𝑡_𝑘=𝑘, and 𝜑(𝑥)=𝑥³. It is not difficult to see that conditions (i)–(iii) are satisfied. Furthermore, max𝑎^+∞_𝑘=1𝑘^[0]−1,𝑏_𝑘^[0]−1=^+∞_𝑘=11𝑘²<+∞,_{𝑡+∞0𝑡0<𝑡𝑘<𝑠}1𝑏_𝑘^[2]𝑝(𝑠)𝑑𝑠=^+∞_1/238_{1/2<𝑡𝑘<𝑠}𝑘²𝑘²+1𝑠𝑑𝑠=+∞. (15) Thus by Theorem 1, every solution of (14) is either oscillatory or (nonoscillatory and) zero convergent.

Example 10. Consider the equation (2−sin𝑡)𝑔(𝑡)𝑥⁽⁴⁾(𝑡)^′+𝑡𝑥³(𝑡)1=0,𝑡≥2𝑥,𝑡≠𝑘,_(𝑖)𝑘⁺=11+𝑘²𝑥^(𝑖)(𝑘),𝑥⁽⁴⁾𝑘⁺=11−𝑘²𝑥⁽⁴⁾(𝑘)𝑥,𝑖=0,1,2,3;𝑘=1,2,…,_(𝑗)12=𝑥₀^[𝑗],𝑗=0,1,2,3,4, (16) where 𝑎_𝑘^[𝑖]=𝑏_𝑘^[𝑖]=(1+(1/𝑘²)) for 𝑖=0,1,2,3,𝑎_𝑘^[4]=𝑏_𝑘^[4]=(1−(1/𝑘²));𝑝(𝑡)=𝑡,𝑡_𝑘=𝑘,𝜑(𝑥)=𝑥³, and 𝑟(𝑡)=(2−sin𝑡)𝑔(𝑡), (17) where 𝑔(𝑡)=1𝑡−𝑘−2+1,𝑡∈[𝑘,𝑘+1),𝑘=1,2,…. (18) Here, we do not assume that 𝑟(𝑡) is bounded, monotonic, or differentiable. It is not difficult to see that conditions (i)–(iii) are satisfied. Furthermore, 𝑏_𝑘^[4]=11−𝑘²≤11+𝑘²=𝑎_𝑘^[0],_+∞𝑘=1max𝑎_𝑘^[0]−1,𝑏_𝑘^[0]=−1_+∞𝑘=11𝑘²<+∞,_+∞𝑘=1𝑏_𝑘^[4]−1=^+∞_𝑘=11𝑘²<+∞_𝑡+∞0𝑝(𝑠)𝑑𝑠=^+∞_1/21𝑠𝑑𝑠=+∞. (19) Thus by Theorem 3, every solution of (16) is either oscillatory or (nonoscillatory and) zero convergent.

Example 11. Consider the equation 𝑡⁻¹𝑥⁽⁴⁾(𝑡)^′+𝑡⁻¹−𝑡⁻²𝑥(𝑡)1=0,𝑡≥2𝑥𝑘,𝑡≠𝑘,₊=𝑥(𝑘),𝑥^′𝑘⁺=𝑥^′(𝑘),𝑥^(𝑖)𝑘⁺=𝑘𝑥𝑘+1^(𝑖)(𝑘)𝑥,𝑖=2,3,4;𝑘=1,2,…,_(𝑗)12=𝑥₀^[𝑗],𝑗=0,1,2,3,4, (20) where 𝑎_𝑘^[0]=𝑏_𝑘^[0]=𝑎_𝑘^[1]=𝑏_𝑘^[1]=1,𝑎_𝑘^[𝑖]=𝑏_𝑘^[𝑖]=𝑘/(𝑘+1) for 𝑖=2,3,4;𝑝(𝑡)=𝑡⁻¹−𝑡⁻²,𝑟(𝑡)=𝑡⁻¹,𝑡_𝑘=𝑘;𝜑(𝑥)=𝑥. It is not difficult to see that conditions (i)–(iii) are satisfied. Furthermore, 𝑏_𝑘^[2𝑛]=𝑘𝑎𝑘+1≤1=𝜑_𝑘^[0],_𝑡+∞0𝑝(𝑠)𝑑𝑠=^+∞_1/21𝑠−1𝑠²𝑑𝑠=+∞. (21) Thus, by Theorem 2, every solution of (20) is either oscillatory or (nonoscillatory and) zero convergent. But the ordinary differential equation 𝑡⁻¹𝑥⁽⁴⁾(𝑡)^′+𝑡⁻¹−𝑡⁻²𝑥(𝑡)=0, (22) has a solution 𝑥(𝑡)=−𝑒^𝑡 which tends to −∞ as 𝑡→+∞. This example shows that impulses play an important role in oscillatory and asymptotic behaviors of equations.

3. Proofs

To prove our Theorems, we need the following Lemmas.

Lemma 12 (Lakshmikantham et al. [1]). Assume that(H₀)𝑚∈𝑃𝐶^′(𝑅⁺,𝑅) and 𝑚(𝑡) is left-continuous at 𝑡_𝑘 for 𝑘=1,2,…(H₁)for 𝑡_𝑘,𝑘=1,2,… and 𝑡≥𝑡₀, 𝑚^′(𝑡)≤𝑝(𝑡)𝑚(𝑡)+𝑞(𝑡),𝑡≠𝑡_𝑘,𝑚𝑡⁺_𝑘≤𝑑_𝑘𝑚𝑡_𝑘+𝑏_𝑘, (23)where 𝑝,𝑞∈𝑃𝐶(𝑅⁺,𝑅),𝑑_𝑘≥0, and 𝑏_𝑘 are real constants. Then for 𝑡≥𝑡₀, 𝑚(𝑡)≤𝑚𝑡_{0𝑡0<𝑡𝑘<𝑡}𝑑_𝑘exp^𝑡_𝑡0𝑝(𝑠)𝑑𝑠+_{𝑡0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝑑_𝑗exp^𝑡_𝑡𝑘𝑝(𝑠)𝑑𝑠𝑏_𝑘+^𝑡_{𝑡0𝑠<𝑡𝑘<𝑡}𝑑_𝑘exp^𝑡_𝑠𝑝(𝜎)𝑑𝜎𝑞(𝑠)𝑑𝑠. (24)

Remark 13. If the inequalities in (23) are reversed, then the inequality in (24) should be reversed as well.

Lemma 14. Suppose that conditions (i)–(iii) hold and 𝑥(𝑡) is a solution of (3). One has the following statements:(a)if there exists some 𝑇≥𝑡₀ such that 𝑥^(2𝑛)(𝑡)>0 and (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′≥0 for 𝑡≥𝑇, then there exists some 𝑇₁≥𝑇 such that 𝑥^(2𝑛−1)(𝑡)>0 for 𝑡≥𝑇₁;(b)if there exists 𝑖∈{1,2,…,2𝑛−1} and some 𝑇≥𝑡₀ such that 𝑥^(𝑖)(𝑡)>0 and 𝑥^(𝑖+1)(𝑡)≥0 for 𝑡≥𝑇, then there exists some 𝑇₁≥𝑇 such that 𝑥^(𝑖−1)(𝑡)>0 for 𝑡≥𝑇₁.

Proof. First of all, we will prove the result of (a) to be true. Without loss of generality, we may assume that 𝑥^(2𝑛)(𝑡)>0 and (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′≥0 for 𝑡≥𝑡₀. We assert that there exists some 𝑗 such that 𝑥^(2𝑛−1)(𝑡_𝑗)>0 for 𝑡_𝑗≥𝑡₀. If this is not true, then for any 𝑡_𝑘>𝑡₀, we have 𝑥^(2𝑛−1)(𝑡_𝑘)≤0. Since 𝑥^(2𝑛−1)(𝑡) is increasing on intervals of the form (𝑡_𝑘,𝑡_𝑘+1], we see that 𝑥^(2𝑛−1)(𝑡)≤0 for 𝑡≥𝑡₀. Since 𝑟(𝑡)𝑥^(2𝑛)(𝑡) is increasing on intervals of the form (𝑡_𝑘,𝑡_𝑘+1], we see that for 𝑡∈(𝑡₁,𝑡₂], 𝑟(𝑡)𝑥^(2𝑛)(𝑡)≥𝑟𝑡₁𝑥^(2𝑛)𝑡⁺₁; (25) that is, 𝑥^(2𝑛)(𝑡)≥𝑟𝑡₁𝑟(𝑡)𝑥^(2𝑛)𝑡⁺₁. (26) In particular, 𝑥^(2𝑛)𝑡₂≥𝑟𝑡₁𝑟𝑡₂𝑥^(2𝑛)𝑡⁺₁. (27) Similarly, for 𝑡∈(𝑡₂,𝑡₃], we have 𝑥^(2𝑛)(𝑡)≥𝑟𝑡₂𝑟(𝑡)𝑥^(2𝑛)𝑡⁺₂≥𝑟𝑡₂𝑟(𝑡)𝑎₂^(2𝑛)𝑥^(2𝑛)𝑡₂≥𝑟𝑡₁𝑟(𝑡)𝑎₂^(2𝑛)𝑥^(2𝑛)𝑡⁺₁. (28) By induction, we know that 𝑥^(2𝑛)(𝑡)≥𝑟𝑡₁𝑟(𝑡)_{𝑡1<𝑡𝑘<𝑡}𝑎_𝑘^(2𝑛)𝑥^(2𝑛)𝑡⁺₁,𝑡≠𝑡_𝑘. (29) From condition (ii), we have 𝑥^(2𝑛−1)𝑡⁺_𝑘≥𝑏_𝑘^[2𝑛−1]𝑥^(2𝑛−1)𝑡_𝑘,𝑡>𝑡₁,𝑘=2,3,…. (30) It follows from (29), (30), and Lemma 12, that for 𝑡≥𝑡₁, 𝑥^(2𝑛−1)(𝑡)≥_{𝑡1<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛−1]𝑥^(2𝑛−1)𝑡⁺₁+𝑥^(2𝑛)𝑡⁺₁𝑟𝑡₁×^𝑡_𝑡11𝑟(𝑠)_{𝑡1<𝑡𝑘<𝑠}𝑎_𝑘^[2𝑛]𝑏_𝑘^[2𝑛−1]𝑑𝑠. (31) Note that 𝑎_𝑘^[𝑖]>0, 𝑏_𝑘^[𝑖]>0, and the second equality of condition (iii) hold. Thus, we get 𝑥^(2𝑛−1)(𝑡)>0 for all sufficiently large 𝑡, which is contrary to 𝑥^(2𝑛−1)(𝑡)≤0 for 𝑡≥𝑡₀. So there exists some 𝑗 such that 𝑡_𝑗≥𝑡₀ and 𝑥^(2𝑛−1)(𝑡_𝑗)>0. Since 𝑥^(2𝑛−1)(𝑡) is increasing on intervals of the form (𝑡_𝑗+𝜆,𝑡_𝑗+𝜆+1] for each 𝜆=0,1,2,…, then for 𝑡∈(𝑡_𝑗+𝜆,𝑡_𝑗+𝜆+1], we have 𝑥^(2𝑛−1)(𝑡)≥_{𝑡𝑗≤𝑡𝑘<𝑡}𝑎_𝑘^[2𝑛−1]𝑥^(2𝑛−1)𝑡_𝑗>0. (32) The assertion (a) is thus proved.
Next, we will prove the result of (b) to be true. Without loss of generality, we may assume that 𝑥^(𝑖)(𝑡)>0 and 𝑥^(𝑖+1)(𝑡)≥0 for 𝑡≥𝑡₀. We assert there exists some 𝑗 such that 𝑥^(𝑖−1)(𝑡_𝑗)>0 for 𝑡_𝑗≥𝑡₀. If this is not true, then for any 𝑡_𝑘>𝑡₀, we have 𝑥^(𝑖−1)(𝑡_𝑘)≤0. Since 𝑥^(𝑖−1)(𝑡) is increasing on intervals of the form (𝑡_𝑘,𝑡_𝑘+1], we see that 𝑥^(𝑖−1)(𝑡)≤0 for 𝑡≥𝑡₀. By 𝑥^(𝑖)(𝑡)>0,𝑥^(𝑖+1)(𝑡)≥0, and 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we have that 𝑥^(𝑖)(𝑡) is nondecreasing on (𝑡_𝑘,𝑡_𝑘+1]. For 𝑡∈(𝑡₁,𝑡₂], we have 𝑥^(𝑖)(𝑡)≥𝑥^(𝑖)𝑡⁺₁. (33) In particular, 𝑥^(𝑖)𝑡₂≥𝑥^(𝑖)𝑡⁺₁. (34) Similarly, for 𝑡∈(𝑡₂,𝑡₃], we have 𝑥^(𝑖)(𝑡)≥𝑥^(𝑖)𝑡⁺₂≥𝑎₂^(𝑖)𝑥^(𝑖)𝑡₂≥𝑎₂^(𝑖)𝑥^(𝑖)𝑡⁺₁. (35) By induction, we know that 𝑥^(𝑖)(𝑡)≥_{𝑡1<𝑡𝑘<𝑡}𝑎_𝑘^(𝑖)𝑥^(𝑖)𝑡⁺₁,𝑡>𝑡₁,𝑡≠𝑡_𝑘. (36) From condition (ii), we have 𝑥^(𝑖−1)𝑡⁺_𝑘≥𝑏_𝑘^(𝑖−1)𝑥^(𝑖−1)𝑡_𝑘,𝑘=2,3,…. (37) Set 𝑢(𝑡)=−𝑥^(𝑖−1)(𝑡). Then from (36) and (37), we see that 𝑢^′(𝑡)≤−_{𝑡1<𝑡𝑘<𝑡}𝑎_𝑘^(𝑖)𝑥^(𝑖)𝑡⁺₁,𝑡>𝑡₁,𝑡≠𝑡_𝑘,𝑢𝑡⁺_𝑘≤𝑏_𝑘^(𝑖−1)𝑢𝑡_𝑘,𝑘=2,3,…. (38) It follows from Lemma 12 that 𝑢(𝑡)≤𝑢𝑡⁺_{1𝑡1<𝑡𝑘<𝑡}𝑏_𝑘^(𝑖−1)−𝑥^(𝑖)𝑡⁺₁^𝑡_{𝑡1𝑠<𝑡𝑘<𝑡}𝑏_𝑘^(𝑖−1)_{𝑡1<𝑡𝑘<𝑡}𝑎_𝑘^(𝑖)𝑑𝑠=_{𝑡1<𝑡𝑘<𝑡}𝑏_𝑘^(𝑖−1)𝑢𝑡⁺₁−𝑥^(𝑖)𝑡⁺₁^𝑡_{𝑡1𝑡1<𝑡𝑘<𝑠}𝑎_𝑘^(𝑖)𝑏_𝑘^(𝑖−1)𝑑𝑠; (39) that is, 𝑥^(𝑖−1)(𝑡)≥_{𝑡1<𝑡𝑘<𝑡}𝑏_𝑘^(𝑖−1),×𝑥^(𝑖−1)𝑡⁺₁+𝑥^(𝑖)𝑡⁺₁^𝑡_{𝑡1𝑡1<𝑡𝑘<𝑠}𝑎_𝑘^(𝑖)𝑏_𝑘^(𝑖−1)𝑑𝑠. (40) Note that 𝑎_𝑘^(𝑖)>0,𝑏_𝑘^(𝑖)>0, and the first equality of condition (iii) holds. Thus, we get 𝑥^(𝑖−1)(𝑡)>0 for all sufficiently large 𝑡. The relation 𝑥^(𝑖−1)(𝑡)≤0 leads to a contradiction. So there exists some 𝑗 such that 𝑡_𝑗>𝑡₀ and 𝑥^(𝑖−1)(𝑡_𝑗)>0. Then 𝑥^(𝑖−1)𝑡⁺_𝑗≥𝑎_𝑗^(𝑖−1)𝑥^(𝑖−1)𝑡_𝑗>0. (41) Since 𝑥^(𝑖)(𝑡)>0, then for every positive integer 𝑚, we have that 𝑥^(𝑖−1)(𝑡) is increasing on (𝑡_{𝑗+𝑚−1},𝑡_𝑗+𝑚]. For (𝑡_𝑗,𝑡_𝑗+1], we have 𝑥^(𝑖−1)(𝑡)≥𝑥^(𝑖−1)𝑡⁺_𝑗>0. (42) In particular, 𝑥^(𝑖−1)𝑡_𝑗+1≥𝑥^(𝑖−1)𝑡⁺_𝑗>0. (43) Similarly, for 𝑡∈(𝑡_𝑗+1,𝑡_𝑗+2], we have 𝑥^(𝑖−1)(𝑡)≥𝑥^(𝑖−1)𝑡⁺_𝑗+1≥𝑎^(𝑖−1)_𝑗+1𝑥^(𝑖−1)𝑡_𝑗+1>0. (44) By induction, for (𝑡_{𝑗+𝑚−1},𝑡_𝑗+𝑚], we have 𝑥^(𝑖−1)(𝑡)>0. (45) Summing up the above discussion, we know that there exists some 𝑇₁≥𝑇 such that 𝑥^(𝑖−1)(𝑡)>0,𝑡≥𝑇₁. (46) The proof of Lemma 14 is complete.

Remark 15. We may prove in similar manners the following statements.(a^′)If we replace the condition (a) in Lemma 14 “𝑥^(2𝑛)(𝑡)>0 and (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′≥0 for 𝑡≥𝑇” with “𝑥^(2𝑛)(𝑡)<0 and (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′≤0 for 𝑡≥𝑇,” then under conditions (i)–(iii), there exists some 𝑇₁≥𝑇 such that 𝑥^(2𝑛−1)(𝑡)<0 for 𝑡≥𝑇₁.(b^′)If we replace the condition (b) in Lemma 14 “𝑥^(𝑖)(𝑡)>0 and 𝑥^(𝑖+1)(𝑡)≥0 for 𝑡≥𝑇” with “𝑥^(𝑖)(𝑡)<0 and 𝑥^(𝑖+1)(𝑡)≤0 for 𝑡≥𝑇,” then under conditions (i)–(iii), there exists some 𝑇₁≥𝑇 such that 𝑥^(𝑖−1)(𝑡)<0 for 𝑡≥𝑇₁.

Lemma 16. Suppose conditions (i)–(iii) hold. Let 𝑥=𝑥(𝑡) be a solution of (3).(a)If there exists some 𝑇≥𝑡₀ such that 𝑥(𝑡)>0 and (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′<0 for 𝑡≥𝑇, then 𝑥^(2𝑛)(𝑡)>0 for all sufficiently large 𝑡.(b)If there exists 𝑖∈{1,2,…,2𝑛} and some 𝑇≥𝑡₀ such that 𝑥(𝑡)>0 and 𝑥^(𝑖)(𝑡)<0 for 𝑡≥𝑇, then 𝑥^(𝑖−1)(𝑡)>0 for all sufficiently large 𝑡.

Proof. First of all, we will prove the result of (a) to be true. We assert that 𝑥^(2𝑛)(𝑡)>0 for any 𝑡_𝑘≥𝑇. If this is not true, then there exists some 𝑡_𝑗≥𝑇 such that 𝑥^(2𝑛)(𝑡_𝑗)≤0. Since 𝑟(𝑡)>0 and 𝑟(𝑡)𝑥^(2𝑛)(𝑡) are strictly decreasing on (𝑡_𝑗+𝑚,𝑡_𝑗+𝑚+1] for 𝑚=0,1,2,… and for 𝑡∈(𝑡_𝑗,𝑡_𝑗+1], we have 𝑟(𝑡)𝑥^(2𝑛)(𝑡)<𝑟𝑡_𝑗𝑥^(2𝑛)𝑡⁺_𝑗≤𝑎_𝑗^(2𝑛)𝑟𝑡_𝑗𝑥^(2𝑛)𝑡_𝑗≤0. (47) Similarly, for 𝑡∈(𝑡_𝑗+1,𝑡_𝑗+2], we have 𝑟(𝑡)𝑥^(2𝑛)(𝑡)<𝑟𝑡_𝑗+1𝑥^(2𝑛)𝑡⁺_𝑗+1≤𝑎_𝑗^(2𝑛)𝑎^(2𝑛)_𝑗+1𝑟𝑡_𝑗𝑥^(2𝑛)𝑡_𝑗≤0. (48) We can easily prove that, for any positive integer 𝑛≥2 and 𝑡∈(𝑡_𝑗+𝑛,𝑡_𝑗+𝑛+1], we have 𝑟(𝑡)𝑥^(2𝑛)(𝑡)<𝑎_𝑗^(2𝑛)𝑎^(2𝑛)_𝑗+1…𝑎^(2𝑛)_𝑗+𝑛𝑟𝑡_𝑗𝑥^(2𝑛)𝑡_𝑗≤0. (49) Hence, 𝑥^(2𝑛)(𝑡)<0 for 𝑡≥𝑡_𝑗+1. By the result (a) of Remark 13, for sufficiently large 𝑡, we have 𝑥^(2𝑛−1)(𝑡)<0. Using the result (b) of Remark 13 repeatedly, for all sufficiently large 𝑡, we get 𝑥(𝑡)<0, which is contrary to 𝑥(𝑡)>0 for 𝑡≥𝑇. Hence we have 𝑥^(2𝑛)(𝑡_𝑘)>0 for any 𝑡_𝑘≥𝑇. So we get 𝑥^(2𝑛)(𝑡)>0 for all sufficiently large 𝑡.
Next, we will prove the result of (b) to be true. We assert that 𝑥^(𝑖−1)(𝑡_𝑘)>0 for any 𝑡_𝑘≥𝑇. If this is not true, then there exists some 𝑡_𝑗≥𝑇 such that 𝑥^(𝑖−1)(𝑡_𝑗)<0. Since 𝑥^(𝑖−1)(𝑡) is strictly monotony decreasing on (𝑡_𝑗+𝑛,𝑡_𝑗+𝑛+1] for 𝑛=0,1,2,… and for 𝑡∈(𝑡_𝑗,𝑡_𝑗+1], we have 𝑥^(𝑖−1)(𝑡)<𝑥^(𝑖−1)𝑡⁺_𝑗≤𝑎_𝑗^(𝑖−1)𝑥^(𝑖−1)𝑡_𝑗≤0. (50) Similarly, for 𝑡∈(𝑡_𝑗+1,𝑡_𝑗+2], we have 𝑥^(𝑖−1)(𝑡)<𝑥^(𝑖−1)𝑡⁺_𝑗+1≤𝑎_𝑗^(𝑖−1)𝑎^(𝑖−1)_𝑗+1𝑥^(𝑖−1)𝑡_𝑗≤0. (51) We can easily prove that for any positive integer 𝑛≥2 and 𝑡∈(𝑡_𝑗+𝑛,𝑡_𝑗+𝑛+1], we have 𝑥^(𝑖−1)(𝑡)<𝑎_𝑗^(𝑖−1)…𝑎^(𝑖−1)_𝑗+1^(𝑖−1)_𝑗+𝑛𝑥^(𝑖−1)𝑡_𝑗≤0. (52) Hence, 𝑥^(𝑖−1)(𝑡)<0 for 𝑡≥𝑡_𝑗+1. By the result (b^′) of Remark 8, for sufficiently large 𝑡, we have 𝑥^(𝑖−2)(𝑡)<0. Similarly, by using the result (b^′) of Remark 8 again, we can conclude that for all sufficiently large 𝑡, 𝑥(𝑡)<0. That is contrary with 𝑥(𝑡)>0 for 𝑡≥𝑇. Hence, we have 𝑥^(𝑖−1)(𝑡_𝑘)>0 for any 𝑡_𝑘≥𝑇. So we get 𝑥^(𝑖−1)(𝑡)>0 for all sufficiently large 𝑡.
The proof of Lemma 16 is complete.

Lemma 17. Suppose conditions (i)–(iii) hold. Let 𝑥=𝑥(𝑡) be a solution of (3). Suppose that 𝑇≥𝑡₀ and 𝑥(𝑡)>0 for 𝑡≥𝑇. Then there exists some 𝑇^′≥𝑇 and 𝑙∈{0,2,4,…,2𝑛} such that for 𝑡≥𝑇^′, 𝑥^(𝑖)(𝑡)()>0,𝑖=0,1,…,𝑙;−1^(𝑖−1)𝑥^(𝑖)(𝑡)𝑟(𝑡)𝑥>0,𝑖=𝑙+1,…,2𝑛;_(2𝑛)(𝑡)^′<0. (53)

Proof. Let 𝑥(𝑡)>0 for 𝑡≥𝑇≥𝑡₀. By (3) and condition (i), we have 𝑟(𝑡)𝑥^(2𝑛)(𝑡)^′=−𝑓(𝑡,𝑥)≤−𝑝(𝑡)𝜑(𝑥)<0. (54) By the result (a) of Lemma 16, there exists some 𝑇₁≥𝑇 such that 𝑥^(2𝑛)(𝑡)>0 for 𝑡≥𝑇₁. Without loss of generality, we may let 𝑥^(2𝑛)(𝑡)>0 for 𝑡≥𝑡₀. Then 𝑥^(2𝑛−1)(𝑡) is strictly increasing on (𝑡_𝑘,𝑡_𝑘+1]. If for any 𝑡_𝑘,𝑥^(2𝑛−1)(𝑡_𝑘)<0, then 𝑥^(2𝑛−1)(𝑡)<0 for 𝑡≥𝑇₁. If there exists some 𝑡_𝑗 such that 𝑥^(2𝑛−1)(𝑡_𝑗)≥0, since 𝑥^(2𝑛−1)(𝑡) is strictly monotony increasing and 𝑎_𝑘^[2𝑛−1]>0, then 𝑥^(2𝑛−1)(𝑡)>0 for 𝑡>𝑡_𝑗. Thus, there exists some 𝑇₂≥𝑇₁ such that 𝑥^(2𝑛−1)(𝑡)>0 for 𝑡≥𝑇₂. So one of the following statements holds: 𝐴₁𝑥^(2𝑛)(𝑡)>0,𝑥^(2𝑛−1)(𝑡)>0,𝑡≥𝑇₂;𝐵₁𝑥^(2𝑛)(𝑡)>0,𝑥^(2𝑛−1)(𝑡)<0,𝑡≥𝑇₂. (55)(56)
If (55) holds, then by the result (b) of Lemma 14, 𝑥^(2𝑛−2)(𝑡)>0 for all sufficiently large 𝑡. Using the result (b) of Lemma 14 repeatedly, for all sufficiently large 𝑡, we can conclude that 𝑥^(2𝑛)(𝑡)>0,𝑥^(2𝑛−1)(𝑡)>0,…,𝑥^′(𝑡)>0,𝑥(𝑡)>0. (57)
If (56) holds, by Lemma 16, we have 𝑥^(2𝑛−3)(𝑡)<0 for all sufficiently large 𝑡. Similarly, there exists some 𝑇₃≥𝑇₂ such that one of the following statements holds: 𝐴₂𝑥^(2𝑛−2)(𝑡)>0,𝑥^(2𝑛−3)(𝑡)>0,𝑡≥𝑇₃;𝐵₂𝑥^(2𝑛−2)(𝑡)>0,𝑥^(2𝑛−3)(𝑡)<0,𝑡≥𝑇₃. (58) Repeating the discussion above, we can eventually get that there exist some 𝑇^′≥𝑇 and 𝑙∈{0,2,4,…,2𝑛} such that for 𝑡≥𝑇^′, 𝑥^(𝑖)(𝑡)()>0,𝑖=0,1,…,𝑙;−1^(𝑖−1)𝑥^(𝑖)(𝑡)𝑟(𝑡)𝑥>0,𝑖=𝑙+1,…,2𝑛;_(2𝑛)(𝑡)^′<0. (59) The proof of Lemma 17 is complete.

Lemma 18 (see [13]). Suppose for 𝑡>0,𝑥=𝑥(𝑡) is continuous at 𝑡≠𝑡_𝑘, left-continuous at 𝑡=𝑡_𝑘, and lim_𝑡→𝑡⁺_𝑘𝑥(𝑡) exists for 𝑘=1,2,…. Further, assume that(H₂)there exists 𝑡∈𝑅⁺, such that 𝑥(𝑡)>0(<0) for 𝑡≥𝑡;(H₃)𝑥(𝑡) is nonincreasing (resp., nondecreasing) on (𝑡_𝑘,𝑡_𝑘+1] for 𝑘=1,2,…;(H₄)∑^+∞_𝑘=1[𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘)] is convergent.Then lim_𝑡→∞𝑥(𝑡)=𝑟 exists and 𝑟≥0 (resp. ≤0).
We now turn to the proof of Theorem 1. Without loss of generality, we may assume that 𝑘₀=1. If (3) has a nonoscillatory solution 𝑥=𝑥(𝑡), we may assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀. By Lemma 17, there exists a 𝑇≥𝑡₀ and 𝑙∈{0,2,4,…,2𝑛} such that for 𝑡≥𝑇, (53) holds. Next, we rewrite (53) as 𝑥^′(𝑡)<0,…,𝑥^(2𝑛)(𝑡)𝑟(𝑡)𝑥>0,_(2𝑛)(𝑡)^′𝑥<0𝑓𝑜𝑟𝑙=0;_′(𝑡)>0,…,𝑥^(𝑙)(𝑡)>0,(−1)^(𝑖)𝑥^(𝑖)(𝑡)𝑟(𝑡)𝑥>0,_(2𝑛)(𝑡)^′<0𝑓𝑜𝑟𝑙≥2. (60)(61)
If (60) holds, then we see that the conditions (H₂) and (H₃) of Lemma 18 are satisfied. Furthermore, note that ∑^+∞_𝑘=1(𝑏_𝑘^[0]−1)<+∞ and 𝑏_𝑘^[0]≥𝑎_𝑘^[0]≥1, then one has ^+∞_𝑘=1𝑏_𝑘^[0]<+∞. (62)
Since 𝑥^′(𝑡)<0 and  𝑡≥𝑇, one obtains for any 𝑡_𝑘>𝑇, 𝑥𝑡_𝑘≤_{𝑇<𝑡𝑗<𝑡𝑘}𝑏_𝑗^[0]𝑥𝑇⁺. (63) By (62) and (63), we know {𝑥(𝑡_𝑘)} is bounded. Thus, there exists 𝑀>0 such that |𝑥(𝑡_𝑘)|≤𝑀. It follows from condition (ii) that 𝑥𝑡⁺_𝑘−𝑥𝑡_𝑘≤𝑏_𝑘^[0]−1𝑥𝑡_𝑘≤𝑀𝑏_𝑘^[0]−1. (64) From (64) and note that ∑^+∞_𝑘=1(𝑏_𝑘^[0]−1) is convergent, we know that ∑^+∞_𝑘=1[𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘)] is convergent. Therefore, the condition (H₄) of Lemma 18 is also satisfied. By Lemma 18, we have lim_𝑡→+∞𝑥(𝑡)=𝑟≥0. We assert that 𝑟=0. If 𝑟>0, then there exists 𝑇₁≥𝑇, such that for any 𝑡≥𝑇₁,𝑥(𝑡)>𝑟/2>0. Note that 𝜑^′(𝑥)≥0; hence we may see that 𝜑(𝑥(𝑡))≥𝜑(𝑟/2) for 𝑡≥𝑇₁. Let Φ(𝑡)=𝑟(𝑡)𝑥^(2𝑛)(𝑡). Then Φ(𝑡)>0 for 𝑡≥𝑇₁. By conditions (i), (ii), and (3), one has for 𝑡≥𝑇₁, Φ^′(𝑡)()(𝑡)𝜑(𝑥(𝑡𝑟=−𝑓𝑡,𝑥≤−𝑝))≤−𝜑2𝑝(𝑡),𝑡≠𝑡_𝑘,Φ𝑡⁺_𝑘=𝑟𝑡⁺_𝑘𝑥^(2𝑛)𝑡⁺_𝑘≤𝑏_𝑘^[2𝑛]𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘≤𝑏_𝑘^[2𝑛]Φ𝑡_𝑘,𝑡_𝑘≥𝑇₁. (65) From (65) and Lemma 12, one gets for 𝑡≥𝑇₁, Φ(𝑡)≤Φ𝑇⁺_{1𝑇1<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]𝑟−𝜑2^𝑡_{𝑇1𝑠<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]𝑝(𝑠)𝑑𝑠=_{𝑇1<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]Φ𝑇⁺₁𝑟−𝜑2^𝑡_{𝑇1𝑇1<𝑡𝑘<𝑠}1𝑏_𝑘^[2𝑛]𝑝(𝑠)𝑑𝑠. (66) It is easy to see from (7) and (66) that Φ(𝑡)<0 for sufficiently large 𝑡. This is contrary to Φ(𝑡)>0 for 𝑡≥𝑇₁. Thus, 𝑟=0; that is, lim_𝑡→+∞𝑥(𝑡)=0.
If (61) holds, then let Ψ(𝑡)=𝑟(𝑡)𝑥^(2𝑛)(𝑡)/𝜑(𝑥(𝑡)). Then Ψ(𝑡)>0 for 𝑡≥𝑇. By (3) and condition (i), one has for 𝑡≥𝑇, Ψ^′(𝑡)=−𝑓(𝑡,𝑥(𝑡))𝜑(𝑥(𝑡−𝑟(𝑡)𝑥))^(2𝑛)(𝑡)𝜑^′(𝑥(𝑡))𝑥^′(𝑡)𝜑²(𝑥(𝑡≤((𝑡))−𝑓𝑡,𝑥))𝜑(𝑥(𝑡(𝑡)))≤−𝑝,𝑡≠𝑡_𝑘. (67) From the conditions (i) and (ii) and note that 𝑎_𝑘^[0]≥1, we know that Ψ𝑡⁺_𝑘=𝑟𝑡⁺_𝑘𝑥^(2𝑛)𝑡⁺_𝑘𝜑𝑥𝑡⁺_𝑘≤𝑟𝑡_𝑘𝑏_𝑘^[2𝑛]𝑥^(2𝑛)𝑡_𝑘𝜑𝑎_𝑘^[0]𝑥𝑡_𝑘≤𝑏_𝑘^[2𝑛]𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘𝜑𝑥𝑡_𝑘=𝑏_𝑘^[2𝑛]Ψ𝑡_𝑘. (68) From (67), (68), and Lemma 12, one gets for 𝑡≥𝑇, Ψ(𝑡)≤Ψ𝑇⁺_{𝑇<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]−^𝑡_{𝑇𝑢<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]𝑝(𝑢)𝑑𝑢=_{𝑇<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]Ψ𝑇⁺−^𝑡_{𝑇𝑇<𝑡𝑘<𝑢}1𝑏_𝑘^[2𝑛]𝑝(𝑢)𝑑𝑢. (69) It is easy to see from (7) and (69) that Ψ(𝑡)<0 for sufficiently large 𝑡. This is contrary to Ψ(𝑡)>0 for 𝑡≥𝑇. Thus, every solution of (3) is oscillatory. The proof of Theorem 1 is complete.
We now give the proof of Theorem 2. Without loss of generality, we may assume that 𝑘₀=1. If (3) has a nonoscillatory solution 𝑥=𝑥(𝑡), we may assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀. By Lemma 17, there exists a 𝑇≥𝑡₀ and 𝑙∈{0,2,4,…,2𝑛} such that for 𝑡≥𝑇, (53) holds. Next, we may rewrite (53) as (60) and (61).
If (60) holds, then note that 𝑏_𝑘^[0]≤1. Since 𝑥(𝑡) is decreasing on (𝑡_𝑗+𝑙,𝑡_𝑗+𝑙+1] for 𝑡_𝑗≥𝑇 and each 𝑙=0,1,2,…, one has for 𝑡∈(𝑡_𝑗+𝑙,𝑡_𝑗+𝑙+1], 𝑥(𝑡)<𝑥𝑡⁺_𝑗+𝑙≤⋯≤𝑥𝑡_𝑗+1<𝑥𝑡⁺_𝑗≤𝑏_𝑗^[0]𝑥𝑡_𝑗≤𝑥𝑡_𝑗. (70) So, since 𝑥(𝑡) is decreasing and bounded on (𝑡_𝑗,+∞), we know that 𝑥(𝑡) is convergent as 𝑡→+∞. Let lim_𝑡→+∞𝑥(𝑡)=𝑟. Then 𝑟≥0. We assert that 𝑟=0. If 𝑟>0, then there exists 𝑇₁≥𝑇 such that for 𝑡≥𝑇₁,𝑥(𝑡)>𝑟/2>0. Since 𝜑^′(𝑥)≥0, one has 𝜑(𝑥(𝑡))≥𝜑(𝑟/2). By (3) and condition (i), one has for 𝑡≥𝑇, 𝑟(𝑡)𝑥^(2𝑛)(𝑡)^′()(𝑡)𝜑(𝑥(𝑡𝑟=−𝑓𝑡,𝑥≤−𝑝))≤−𝜑2𝑝(𝑡)<0,𝑡≠𝑡_𝑘. (71) From condition (ii) and noting that 𝑏_𝑘^[2𝑛]≤1, one has 𝑟𝑡⁺_𝑘𝑥^(2𝑛)𝑡⁺_𝑘≤𝑏_𝑘^[2𝑛]𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘≤𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘,𝑡_𝑘≥𝑇₁. (72) By (71), (72), and Lemma 12, one has for 𝑡≥𝑇₁, 𝑟(𝑡)𝑥^(2𝑛)(𝑡)≤𝑟𝑇₁𝑥^(2𝑛)𝑇⁺₁𝑟−𝜑2^𝑡_𝑇1𝑝(𝑠)𝑑𝑠. (73) It is easy to see from (8) and (73) that 𝑥^(2𝑛)(𝑡)<0 for sufficiently large 𝑡. This is contrary to 𝑥^(2𝑛)(𝑡)>0 for 𝑡≥𝑇. Thus 𝑟=0, that is, lim_𝑡→+∞𝑥(𝑡)=0;
If (61) holds, then let Ψ(𝑡)=𝑟(𝑡)𝑥^(2𝑛)(𝑡)/𝜑(𝑥(𝑡)). We see that Ψ(𝑡)>0 for 𝑡≥𝑇. By (3) and the condition (i), one gets for 𝑡≥𝑇Ψ^′(𝑡)𝑓((𝑡=−𝑡,𝑥))𝜑(𝑥(𝑡−𝑟(𝑡)𝑥))^(2𝑛)(𝑡)𝜑^′(𝑥(𝑡))𝑥^′(𝑡)𝜑²(𝑥(𝑡𝑓((𝑡))≤−𝑡,𝑥))𝜑(𝑥(𝑡(𝑡)))≤−𝑝,𝑡≠𝑡_𝑘. (74) From the conditions (i) and (ii), we know that Ψ𝑡⁺_𝑘=𝑟𝑡_𝑘𝑥^(2𝑛)𝑡⁺_𝑘𝜑𝑥𝑡⁺_𝑘≤𝑟𝑡_𝑘𝑏_𝑘^[2𝑛]𝑥^(2𝑛)𝑡_𝑘𝜑𝑎_𝑘^[0]𝑥𝑡_𝑘≤𝑏_𝑘^[2𝑛]𝜑𝑎_𝑘^[0]𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘𝜑𝑥𝑡_𝑘≤Ψ𝑡_𝑘. (75) From (74), (75), and Lemma 12, one gets for 𝑡≥𝑇, Ψ(𝑡)≤Ψ𝑇⁺−^𝑡_𝑇𝑝(𝑠)𝑑𝑠. (76) It is easy to see from (8) and (76) that Ψ(𝑡)<0 for sufficiently large 𝑡. This is contrary to Ψ(𝑡)>0 for 𝑡≥𝑇. Thus, every solution of (3) is oscillatory. The proof of Theorem 2 is complete.
Finally, we give the proof of Theorem 3. Without loss of generality, we may assume that 𝑘₀=1. If (3) has a nonoscillatory solution 𝑥=𝑥(𝑡), we may assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀. By Lemma 17, there exists a 𝑇≥𝑡₀ and 𝑙∈{0,2,4,…,2𝑛} such that for 𝑡≥𝑇, (53) hold. Next, we rewrite (53) as (60) and (61).
If (60) holds, then we see that the conditions (H₂) and (H₃) of Lemma 18 are satisfied. Furthermore, since 𝑥^′(𝑡)<0 for 𝑡≥𝑇, we see that there exists some 𝑡_𝑖≥𝑇 such that for any 𝑡_𝑘>𝑡_𝑖, 𝑥𝑡_𝑘≤_{𝑡𝑖<𝑡𝑗<𝑡𝑘}𝑏_𝑗^[0]𝑥𝑡⁺_𝑖. (77) Since {∏^𝑛_𝑘=1𝑏_𝑘^[0]} is bounded, in view of (77), we know that {𝑥(𝑡_𝑘)} is bounded. Thus, there exists 𝑀₁>0, such that |𝑥(𝑡_𝑘)|≤𝑀₁. It follows from the condition (ii) that 𝑥𝑡⁺_𝑘−𝑥𝑡_𝑘≤max𝑎_𝑘^[0]−1,𝑏_𝑘^[0]−1𝑥𝑡_𝑘≤𝑀₁max𝑎_𝑘^[0]−1,𝑏_𝑘^[0]−1. (78) By (78), we know that ∑^+∞_𝑘=1[𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘)] is convergent. Therefore, the condition (H₄) of Lemma 18 is also satisfied. By Lemma 18, we know that lim_𝑡→+∞𝑥(𝑡)=𝑟≥0. We assert that 𝑟=0. If 𝑟>0, then there exists 𝑇₁≥𝑇, such that for 𝑡≥𝑇₁,𝑥(𝑡)>𝑟/2>0. Since 𝜑^′(𝑥)≥0, we see that 𝜑(𝑥(𝑡))≥𝜑(𝑟/2) for 𝑡≥𝑇₁. By 𝑏_𝑘^[2𝑛]≤𝑎_𝑘^[0], the boundedness of {∏^𝑛_𝑘=1𝑏_𝑘^[0]} and the condition (ii), we know that {∏^𝑛_𝑘=1𝑏_𝑘^[2𝑛]} is bounded. Note that (𝑟(𝑡)𝑥^(2𝑛)(𝑡))^′<0 for 𝑡≥𝑇₁. So we may also see that there exists 𝑀₂>0 such that |𝑟(𝑡_𝑘)𝑥^(2𝑛)(𝑡_𝑘)|≤𝑀₂. Therefore, one has 𝑏_𝑘^[2𝑛]−1𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘≤𝑀₂𝑏_𝑘^[2𝑛]−1. (79) By (3) and the condition (i), one has 𝑟(𝑡)𝑥^(2𝑛)(𝑡)^′=−𝑓(𝑡,𝑥)≤−𝑝(𝑡)𝜑(𝑥(𝑡))<0,𝑡≠𝑡_𝑘. (80) Integrating (80) from 𝑇₁ to 𝑡, it follows from (79) and the condition (ii) that 𝑟(𝑡)𝑥^(2𝑛)(𝑡)−𝑟𝑇₁𝑥^(2𝑛)𝑇⁺₁≤_{𝑇1<𝑡𝑘<𝑡}𝑟𝑡_𝑘𝑥^(2𝑛)𝑡⁺_𝑘−𝑥^(2𝑛)𝑡_𝑘𝑟−𝜑2^𝑡_𝑇1𝑝(𝑠)𝑑𝑠≤_{𝑇1<𝑡𝑘<𝑡}𝑏_𝑘^[2𝑛]−1𝑟𝑡_𝑘𝑥^(2𝑛)𝑡_𝑘𝑟−𝜑2^𝑡_𝑇1𝑝(𝑠)𝑑𝑠≤_{𝑇1<𝑡𝑘<𝑡}𝑀₂𝑏_𝑘^[2𝑛]𝑟−1−𝜑2^𝑡_𝑇1𝑝(𝑠)𝑑𝑠. (81)
Since ∑^+∞_𝑘=1|𝑏_𝑘^[2𝑛]−1| is convergent, it is easy to see from (9) and (81) that 𝑥^(2𝑛)(𝑡)<0 for sufficiently large 𝑡. This is contrary to 𝑥^(2𝑛)(𝑡)>0 for 𝑡≥𝑇. Thus 𝑟=0; that is, lim_𝑡→+∞𝑥(𝑡)=0.
If (61) holds, then let Ψ(𝑡)=𝑟(𝑡)𝑥^(2𝑛)(𝑡)/𝜑(𝑥(𝑡)). We see that Ψ(𝑡)>0 for 𝑡≥𝑇. Similar to the proof of (76), we also obtain Ψ(𝑡)≤Ψ𝑇⁺−^𝑡_𝑇𝑝(𝑠)𝑑𝑠. (82) It is easy to see from (9) and (82) that Ψ(𝑡)<0 for sufficiently large 𝑡. This is contrary to Ψ(𝑡)>0 for 𝑡≥𝑇. Thus, every solution of (3) is oscillatory. The proof of Theorem 3 is complete.