Asymptotic Properties of Solutions to Third-Order Nonlinear Neutral Differential Equations

Abstract

The aim of this work is to discuss asymptotic properties of a class of third-order nonlinear neutral functional differential equations. The results obtained extend and improve some related known results. Two examples are given to illustrate the main results.

1. Introduction

In this work, we study the asymptotic behavior of solutions of the third-order neutral differential equation 𝑎(𝑡)𝑏(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡))^′′′+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))𝑔𝑥^′(𝑡)=0. (1) We always assume that the following conditions hold:(H₁)𝑎(𝑡),𝑏(𝑡),𝑝(𝑡),𝑞(𝑡)∈𝐶([𝑡₀,∞),[0,∞)),0≤𝑝(𝑡)≤𝑝₀<1;(H₂)𝜎(𝑡),𝜏(𝑡)∈𝐶([𝑡₀,∞),[0,∞)), 𝜎(𝑡)≤𝑡, 𝜏(𝑡)≤𝑡, lim_𝑡→∞𝜎(𝑡)=lim_𝑡→∞𝜏(𝑡)=∞;(H₃)𝑓∈𝐶(𝑅,𝑅), 𝑓(𝑥)/𝑥≥𝐾>0, for all 𝑥≠0;(H₄)𝑔∈𝐶(𝑅,[𝐿,∞)), 𝐿>0;(H₅)∫^∞_𝑡0(1/𝑎(𝑡))d𝑡=∫^∞_𝑡0(1/𝑏(𝑡))d𝑡=∞.

Set 𝑧(𝑡)=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜎(𝑡)). By a solution of (1), we mean a nontrivial function 𝑥(𝑡)∈𝐶([𝑇_𝑥,∞),𝑅), 𝑇_𝑥≥𝑡₀, which has the properties 𝑧(𝑡)∈𝐶¹([𝑇_𝑥,∞),𝑅), 𝑏(𝑡)𝑧^′(𝑡)∈𝐶¹([𝑇_𝑥,∞)), and 𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′∈𝐶¹([𝑇_𝑥,∞)) and satisfies (1) on [𝑇_𝑥,∞). We consider only those solutions 𝑥(𝑡) of (1) which satisfies sup{|𝑥(𝑡)|∶𝑡≥𝑇}>0 for all 𝑇≥𝑇_𝑥. We assume that (1) possesses such a solution. A solution of (1) is called oscillatory if it has arbitrarily large zeros on [𝑇_𝑥,∞); otherwise, it is called nonoscillatory.

Recently, great attention has been devoted to the oscillation of various classes of differential equations. See, for example, [1–19]. Hartman and Wintner [1] and Erbe et al. [3] studied the third-order differential equation 𝑥^′′′(𝑡)+𝑞(𝑡)𝑥(𝑡)=0. (2)

Paper [5] studied the oscillation of third-order trinomial delay differential equation 𝑥^′′′(𝑡)+𝑝(𝑡)𝑥^′(𝑡)+𝑔(𝑡)𝑥(𝜏(𝑡))=0. (3)

Li et al. [7] discussed (1) with 𝑓(𝑥(𝜏(𝑡)))=𝑥(𝜏(𝑡)) and 𝑔(𝑥^′(𝑡))=1. Han [8] examined the oscillation of (1) with 𝑏(𝑡)=1.

In this work, we establish some oscillation criteria for (1) which extend and improve the results in [7, 8].

2. Main Results

In the following, all functional inequalities considered are assumed to hold eventually for all 𝑡 large enough. Without loss of generality, we deal only with the positive solutions of (1).

Theorem 1. Suppose that ^∞_𝑡01𝑏(𝑣)^∞_𝑣1𝑎(𝑢)^∞_𝑢𝑞(𝑠)d𝑠d𝑢d𝑣=∞. (4) If for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₂>𝑡₁>𝑡₀, one has limsup_𝑡→∞^𝑡_𝑡2𝜌(𝑠)𝑄(𝑠)−𝜌^′(𝑠)²𝑎(𝑠)4𝜌(𝑠)d𝑠=∞, (5) where 𝑄(𝑡)=𝐾𝐿1−𝑝₀∫_𝑡^𝜏(𝑡)₂∫^𝑣_𝑡1(1/(𝑎(𝑢)/𝑏(𝑣)))d𝑢d𝑣∫^𝑡_𝑡1((𝑢,1/𝑏))d𝑢 (6) then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof. Assume that 𝑥 is a positive solution of (1). Based on condition (𝐻₅), there are two possible cases:(1)𝑧(𝑡)>0, 𝑧^′(𝑡)>0, (𝑏(𝑡)𝑧^′(𝑡))^′>0, [𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0;(2)𝑧(𝑡)>0, 𝑧^′(𝑡)<0, (𝑏(𝑡)𝑧^′(𝑡))^′>0, [𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′<0.
First, consider that 𝑧(𝑡) satisfies (1). We have 𝑥(𝑡)=𝑧(𝑡)−𝑝(𝑡)𝑥(𝜎(𝑡))≥1−𝑝₀𝑧(𝑡). (7) From (1), (𝐻₃), and (𝐻₄), we get [𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′]^′≤−𝐾𝐿1−𝑝₀𝑞(𝑡)𝑧(𝜏(𝑡))≤0. (8) Define a function 𝜔 by 𝜔(𝑡)(𝑡)𝑎(𝑡)=𝜌(𝑏(𝑡)𝑧^′(𝑡))^′𝑏(𝑡)𝑧^′(𝑡),𝑡≥𝑡₁; (9) we obtain 𝜔(𝑡)>0. Then 𝜔^′(𝑡)=𝜌^′(𝑡)𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′𝑏(𝑡)𝑧^′(𝑡)+𝜌(𝑡)(𝑎(𝑡)(𝑏(𝑡)𝑧^′(𝑡))^′)^′𝑏(𝑡)𝑧^′(𝑡)(𝑡)𝑎(𝑡𝑏(𝑡)𝑧−𝜌)^′(𝑡)^′2𝑏(𝑡)𝑧^′(𝑡)²(𝑡)𝑎(𝑡)𝑏(𝑡)𝑧=𝜌^′(𝑡)^′′𝑏(𝑡)𝑧^′(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)−𝜔²(𝑡)𝜌(𝑡)𝑎(𝑡)≤−𝐾𝐿1−𝑝₀𝜌(𝑡)𝑞(𝑡)𝑧(𝜏(𝑡))𝑏(𝑡)𝑧^′(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)−𝜔²(𝑡)𝜌(𝑡)𝑎(𝑡)≤−𝐾𝐿1−𝑝₀𝜌(𝑡)𝑞(𝑡)𝑧(𝜏(𝑡))𝑏(𝑡)𝑧^′(𝑡)−𝜔(𝑡)−1𝜌(𝑡)𝑎(𝑡)2𝑎(𝑡)𝜌𝜌(𝑡)^′(𝑡)²+𝜌^′(𝑡)²𝑎(𝑡)(𝑡).4𝜌 (10) By the proof of [7, Theorem 2.1], we have 𝜔^′(𝑡)𝜌(𝑡)𝑄(𝑡)−𝜌≤−^′(𝑡)²𝑎(𝑡)(𝑡),4𝜌 (11) where 𝑄(𝑡) is defined as in (6). We obtain ^𝑡_𝑡1𝜌(𝑠)𝑄(𝑠)−𝜌^′(𝑠)²𝑎(𝑠)4𝜌(𝑠)d𝑠≤−^𝑡_𝑡1𝜔^′(𝑠)d𝑠. (12) That is, ^𝑡_𝑡1𝜌(𝑠)𝑄(𝑠)−𝜌^′(𝑠)²𝑎(𝑠)(𝑠)𝑡4𝜌d𝑠≤𝜔₁−𝜔(𝑡)<𝜔𝑡₁<∞, (13) which contradicts (5). Assume that case (2) holds. Using the similar proof of [8, Lemma 4], we can get lim_𝑡→∞𝑥(𝑡)=0. This completes the proof.

Based on Theorem 1, we present a Kamenev-type criterion for (1).

Theorem 2. Assume that (4) holds. If for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₁>𝑡₀, one has limsup_𝑡→∞1𝑡^𝑛𝑡_𝑡1(𝑡−𝑠)^𝑛𝜌(𝑠)𝑄(𝑠)−𝜌^′(𝑠)²𝑎(𝑠)4𝜌(𝑠)d𝑠=∞, (14) then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof. Assume that 𝑥(𝑡) is a positive solution of (1). Then by the proof of Theorem 1, we have cases (1) and (2). Let case (1) hold. Proceeding as in the proof of Theorem 1, we have (11). Then we have ^𝑡_𝑡1(𝑡−𝑠)^𝑛𝜌(𝑠)𝑄(𝑠)−𝜌^′(𝑠)²𝑎(𝑠)4𝜌(𝑠)d𝑠≤−^𝑡_𝑡1(𝑡−𝑠)^𝑛𝜔^′(𝑠)d𝑠. (15) That is, 1𝑡^𝑛𝑡_𝑡1(𝑡−𝑠)^𝑛𝜌(𝑠)𝑄(𝑠)−𝜌^′(𝑠)²𝑎(𝑠)(𝑠)𝑛4𝜌d𝑠≤−𝑡^𝑛𝑡_𝑡1(𝑡−𝑠)^𝑛−1𝜔(𝑠)𝑡d𝑠+1−¹𝑡^𝑛𝜔𝑡₁<𝑡1−¹𝑡^𝑛𝜔𝑡₁<∞, (16) which contradicts (14). Assume that case (2) holds. We can get lim_𝑡→∞𝑥(𝑡)=0. The proof is completed.

Next, we present a Philos-type criterion for (1). Let 𝐷=(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀,𝐷₀=(𝑡,𝑠)∶𝑡>𝑠≥𝑡₀. (17) We say that a function 𝐻∈𝐶(𝐷,𝑅) belongs to a function class 𝑃, if it satisfies(i)𝐻(𝑡,𝑡)=0, 𝑡≥𝑡₀; 𝐻(𝑡,𝑠)>0, (𝑡,𝑠)∈𝐷₀;(ii)𝐻 has a continuous and nonpositive partial derivative on 𝐷₀ with respect to the second variable, and such that −𝜕𝐻(𝑡,𝑠)()𝜕𝑠=ℎ𝑡,𝑠𝐻(𝑡,𝑠),(𝑡,𝑠)∈𝐷₀. (18)

Theorem 3. Assume that (4) holds. If for some function 𝜌∈𝐶¹([𝑡₀,∞),(0,∞)), for all sufficiently large 𝑡₁>𝑡₀, one has limsup_𝑡→∞1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()𝜌(𝑠)𝑄(𝑠)−ℎ𝑡,𝑠²¹(𝑡,𝑠)4𝐵(𝑠)d𝑠=∞, (19) where 𝑄(𝑡) is defined as in (6), 𝐵(𝑡)=1/𝜌(𝑡)𝑎(𝑡), and ℎ₁()()−𝜌𝑡,𝑠=ℎ𝑡,𝑠^′(𝑠)𝜌(𝑠)𝐻(𝑡,𝑠), (20) then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof. Assume that 𝑥(𝑡) is a positive solution of (1), and 𝑧(𝑡) has the case of (1); 𝜔(𝑡) is defined as in (9). Then 𝜔^′(𝑡)(𝑡)𝑄(𝑡)+𝜌≤−𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)−1𝑎(𝑡)𝜌(𝑡)𝜔²(𝑡). (21) Let 𝐵(𝑡)=1/𝜌(𝑡)𝑎(𝑡), we have 𝜌(𝑡)𝑄(𝑡)≤−𝜔^′(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)−𝐵(𝑡)𝜔²(𝑡). (22) We obtain ^𝑡_𝑡1𝐻()𝜌(𝑠)𝑄(𝑠)≤𝑡,𝑠d𝑠_𝑡𝑡1𝐻(𝑡,𝑠)−𝜔^′(𝑠)+𝜌^′(𝑠)𝜌(𝑠)𝜔(𝑠)−𝐵(𝑠)𝜔²(𝑠)d𝑠=𝐻𝑡,𝑡₁𝜔𝑡₁−^𝑡_𝑡1ℎ(𝑡,𝑠)𝐻()()𝜌𝑡,𝑠−𝐻𝑡,𝑠^′(𝑠)𝜌(𝑠)𝜔(𝑠)+𝐻(𝑡,𝑠)𝐵(𝑠)𝜔²(𝑠)d𝑠=𝐻𝑡,𝑡₁𝜔𝑡₁−^𝑡_𝑡1𝐻(𝑡,𝑠)ℎ₁(𝑡,𝑠)𝜔(𝑠)+𝐻(𝑡,𝑠)𝐵(𝑠)𝜔²(𝑠)d𝑠=𝐻𝑡,𝑡₁𝜔𝑡₁−^𝑡_𝑡1𝐻()𝐵(𝑠)𝜔(𝑠)+ℎ𝑡,𝑠¹(𝑡,𝑠)2𝐵(𝑠)²+d𝑠_𝑡𝑡1ℎ²₁(𝑡,𝑠)4𝐵(𝑠)d𝑠<𝐻𝑡,𝑡₁𝜔𝑡₁+^𝑡_𝑡1ℎ²₁(𝑡,𝑠)4𝐵(𝑠)d𝑠. (23) That is, 1𝐻𝑡,𝑡₁^𝑡_𝑡1𝐻()𝜌(𝑠)𝑄(𝑠)−ℎ𝑡,𝑠²¹(𝑡,𝑠)(𝑠)𝑡4𝐵d𝑠≤𝜔₁, (24) which contradicts (19). Assume that (2) holds. We can get lim_𝑡→∞𝑥(𝑡)=0. The proof is completed.

3. Examples

In this section, we will present two examples to illustrate the main results.

Example 1. Consider the third-order nonlinear neutral differential equation: 𝑡𝑥(𝑡)+𝑝₁𝑥𝑡2^′′′+𝜆𝑡²𝑥(𝑡)1+𝑥^′(𝑡)²=0,𝜆>0,𝑡≥1, (25) where 𝑝₁∈[0,1), and 𝐾=𝐿=1. Let 𝜌(𝑡)=𝑡. It follows from Theorem 1 that every solution 𝑥(𝑡) of (25) is oscillatory or convergent to zero asymptotically.

Example 2. Consider the third-order nonlinear neutral differential equation: 1𝑡𝑡^1/21(𝑥(𝑡)+2𝑥1𝑡−2^′′′+𝑡^𝜆𝜆2−cos𝑡𝑡()+2+sin𝑡×𝑥𝑡−11+𝑥²(𝑡−1)1+𝑥^′(𝑡)²=0, (26) where 𝜆>0, 𝑡≥1. We have ^𝑡_𝑡0𝑞(𝑠)d𝑠=^𝑡_𝑡1𝑠^𝜆𝜆2−cos𝑠𝑠≥+2+sin𝑠d𝑠_𝑡𝑡1𝑠^𝜆𝜆2−cos𝑠𝑠=+sin𝑠d𝑠_𝑡𝑡1𝑑𝑠^𝜆(2−cos𝑠)=𝑡^𝜆(2−cos𝑡)−𝑡^𝜆₀2−cos𝑡₀≥𝑡^𝜆−𝐾₀⟶∞(𝑡⟶∞); (27) we see that (4) and (H₁)–(H₅) hold. Let 𝐻(𝑡,𝑠)=(𝑡−𝑠)², 𝜌(𝑡)=1. Then ℎ₁(𝑡,𝑠)=2. It follows, from Theorem 3, that the solutions of (26) are oscillatory or convergent to zero asymptotically.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was partially supported by the NNSF of China (11171178).