Asymptotic Dichotomy in a Class of Fourth-Order Nonlinear Delay Differential Equations with Damping

Abstract

All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.

1. Introduction

It has been observed that the solutions of quite a few higher order functional differential equations oscillate or converge to zero (see, e.g., the recent paper [1]). Such a dichotomy may sometimes yield useful information in real problems (see the remark at the end of this note). In this paper, we report that the same phenomena occurs for fourth-order nonlinear delay differential equations of the form

𝑦⁽⁴⁾(𝑡)+𝑝(𝑡)𝑦^'(𝑡)+𝑞(𝑡)𝑓(𝑦(𝑔(𝑡)))=0,𝑡∈𝐼∶=[𝑎,∞),(1.1) where

(i)𝑎 is a nonnegative constant, 𝑝 is a nonnegative function in 𝐶(𝐼,𝑅) such that it does not vanish identically on any [𝑇,∞)⊆𝐼;(ii)𝑞 is a positive function in 𝐶(𝐼,𝑅);(iii)𝑔∈𝐶¹(𝐼,𝑅) and 0<𝑔(𝑡)<t,𝑔^'(𝑡)≥0 for 𝑡∈𝐼 and lim_𝑡→+∞𝑔(𝑡)=+∞;(iv)𝑓∈𝐶(𝑅,𝑅) and 𝑓(𝑢)/𝑢≥𝜅>0 for 𝑢≠0.

As is well known, fourth-order differential equations can be used to model the deflection of a beam (in particular, the function 𝑦(𝑡) in (1.1) may be interpreted as the deflection from the equilibrium position of a horizontal beam at the spatial coordinate 𝑡). Therefore, a short note reporting our observation may be of interest in the qualitative theory of fourth-order differential equations. Indeed, in this note, we will distinct ourselves by emphasizing the role played by the function 𝑝 in (1.1), since the term 𝑝(𝑡)𝑦^'(𝑡) acts as a control of the slope of the beam under consideration at the coordinate 𝑡.

We will restrict our attention to those solutions of (1.1) which exist on 𝐼 and satisfy the condition sup{|𝑦(𝑡)|∶𝑇≤𝑡<∞}>0for any 𝑇∈𝐼. Such a solution is called oscillatory if it has arbitrarily large zeros, and nonoscillatory otherwise.

Lemma 1.1. Assume that 𝑥∈𝐶^𝑛(𝐼,𝑅) such that 𝑥(𝑡)>0 for 𝑡≥𝑎 and 𝑥^(𝑛)(𝑡) is nonpositive for 𝑡≥𝑎 and does not vanish identically on any [𝑇,∞)⊆𝐼. If 𝑛 is even (or odd), then there exists 𝑙∈{1,3,…,𝑛−1} (resp., 𝑙∈{0,2,…,𝑛−1}) such that for all sufficiently large 𝑡,𝑥(𝑡)𝑥^(𝑗)(𝑡)>0 for 𝑗=0,1,…,𝑙, and (−1)^{𝑛+𝑗−1}𝑥(𝑡)𝑥^(𝑗)(𝑡)>0 for 𝑗=𝑙+1,𝑙+2,…,𝑛−1. Furthermore, |𝑥^'𝑔(𝑔(𝑡))|≥^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^{𝑛−𝑙−1}2^𝑙−1(𝑙−1)!(𝑛−1−𝑙)!|𝑥^(𝑛−1)(𝑡)|(1.2) for all sufficiently large 𝑡, where 𝑔∈𝐶¹(𝐼,𝑅) satisfies 0<𝑔(𝑡)<𝑡,𝑔^'(𝑡)≥0 and lim_𝑡→∞𝑔(𝑡)=+∞.

Proof. The existence of 𝑙 in the above result is due to Kiguradze and is well known (see, e.g., [2]). Next, we will prove the inequality (1.2) for even 𝑛 (the odd case being similar) as follows (cf. [3]). First we may assume that there is a 𝑇>0 such that 𝑥(𝑡)>0,𝑥^'(𝑡)>0,…,𝑥^(𝑙)(𝑡)>0,(−1)^𝑛+𝑙𝑥^(𝑙+1)(𝑡)>0,…,(−1)^2𝑛−2𝑥^(𝑛−1)(𝑡)>0 for 𝑡≥𝑇. By Taylor's formula, we have 𝑥^'(𝑠)=𝑥^'(𝑇)+𝑥^''𝑥(𝑇)(𝑠−𝑇)+⋯+^(𝑙)(𝑠^∗)(𝑙−1)!(𝑠−𝑇)^𝑙−1≥𝑥^(𝑙)(𝑠^∗)(𝑙−1)!(s−𝑇)^𝑙−1,(1.3) where 𝑇≤𝑠^∗≤𝑠. Furthermore, 𝑥^'𝑥(𝑠)≥^(𝑙)(𝑠)(𝑙−1)!(𝑠−𝑇)^𝑙−1,𝑠≥𝑇,(1.4) since 𝑥^(𝑙+1)(𝑡)<0 for 𝑡∈[𝑇,+∞). Next, we pick 𝑇₁≥𝑇 such that 1−𝑇/𝑔(𝑡)>1/2 for 𝑡≥𝑇₁. Then 𝑔(𝑡)>2𝑇 and 𝑥^'𝑥(𝑔(𝑡))≥^(𝑙)(𝑔(𝑡))𝑔(𝑙−1)!^𝑙−1𝑇(𝑡)(1−)𝑔(𝑡)^𝑙−1≥𝑔^𝑙−1(𝑡)2^𝑙−1𝑥(𝑙−1)!^(𝑙)(𝑔(𝑡)),𝑡≥𝑇₁.(1.5) By Taylor's formula again, we get 𝑥^(𝑙)(𝑔(𝑡))=𝑥^(𝑙)(𝑡)+(−1)𝑥^(𝑙+1)(𝑡)(𝑡−𝑔(𝑡))+⋯+(−1)^{𝑛−𝑙−1}𝑥^(𝑛−1)(𝑡^∗)(𝑛−𝑙−1)!(𝑡−𝑔(𝑡))^{𝑛−𝑙−1},(1.6) here 𝑔(𝑡)≤𝑡^∗≤𝑡. Hence 𝑥^(𝑙)𝑥(𝑔(𝑡))≥^(𝑛−1)(𝑡^∗)(𝑛−𝑙−1)!(𝑡−𝑔(𝑡))^{𝑛−𝑙−1}≥𝑥^(𝑛−1)(𝑡)(𝑛−𝑙−1)!(𝑡−𝑔(𝑡))^{𝑛−𝑙−1}.(1.7) Combining (1.7) and (1.5), we see that 𝑥^'𝑔(𝑔(𝑡))≥^𝑙−1(𝑡)2^𝑙−1𝑥(𝑙−1)!(𝑛−𝑙−1)!^(𝑛−1)(𝑡)(𝑡−𝑔(𝑡))^{𝑛−𝑙−1}(1.8) for all large 𝑡.

Lemma 1.2. Suppose the linear third-order differential equation 𝑧^'''+𝑝(𝑡)𝑧=0,𝑡∈𝐼(1.9) has an eventually positive increasing solution on 𝐼 and 𝑦 is a nonoscillatory solution of (1.1) on 𝐼. Then there exists 𝑡₀∈[𝑎,∞) such that 𝑦(𝑡)𝑦^'(𝑡)>0 or 𝑦(𝑡)𝑦^'(𝑡)<0 for 𝑡≥𝑡₀.

Proof. Without loss of generality, we may assume that 𝑦(𝑡)>0 and 𝑦(𝑔(𝑡))>0 for 𝑡≥𝑎. Then 𝑥(𝑡)=−𝑦^'(𝑡) is a solution of the third-order delay differential equation 𝑥^'''+𝑝(𝑡)𝑥=𝑞(𝑡)𝑓(𝑦(𝑔(𝑡))),𝑡≥𝑎.(1.10) We claim that, all solutions of (1.10) are nonoscillatory (so that 𝑦^'(𝑡) is eventually positive or eventually negative). To see this, let 𝑧 be the solution of (1.9) such that 𝑧(𝑡)>0 and 𝑧^'(𝑡)>0 for 𝑡≥𝑎. Then from Lemma 1.1, we have 𝑧^''(𝑡)>0 for 𝑡 greater than or equal to a positive number 𝑡₁.
Suppose to the contrary that 𝑥 is an oscillatory solution of (1.10). We assert that 𝑠=𝑥^'𝑧−𝑥𝑧^' oscillates. Indeed, since 𝑥 is oscillatory, for any 𝑇>𝑎, there are 𝑏 and 𝑐 such that 𝑇≤𝑏<𝑐, 𝑥(𝑏)=𝑥(𝑐)=0,𝑥^'(𝑏)≥0 and 𝑥^'(𝑐)≤0. Hence 𝑠(𝑏)=𝑥^'(𝑏)𝑧(𝑏)≥0 and 𝑠(𝑐)=𝑥^'(𝑐)𝑧(𝑐)≤0. If 𝑠(𝑏)=0 or 𝑠(𝑐)=0, then either 𝑏 or 𝑐 is a zero of 𝑠 in [𝑇,∞). Otherwise, 𝑠(𝑏)>0 and 𝑠(𝑐)<0 so that there is 𝑑∈(𝑏,𝑐) such that 𝑠(𝑑)=0. These show that 𝑠 has arbitrarily large zeros.
Now that 𝑠 is oscillatory, either 𝑠(𝑡)≤0 for all 𝑡∈[𝑡₁,∞) or else there are strictly increasing and divergent sequences {𝑏_𝑛} and {𝑐_𝑛} such that 𝑠(𝑡)>0 for 𝑡∈(𝑏_𝑛,𝑐_𝑛)⊂(𝑡₁,∞),𝑠(𝑏_𝑛)=𝑠(𝑐_𝑛)=0,𝑠^'(𝑏_𝑛)≥0 and 𝑠^'(𝑐_𝑛)≤0.
Suppose the former case holds. Then there are (𝑏,𝑐)⊂(𝑡₁,∞) such that 𝑠(𝑡)≤0 for 𝑡∈(b,𝑐) and 𝑠^'(𝑏)≥0 and 𝑠^'(𝑐)≤0. From (1.9) and (1.10), we have 𝑧𝑞(𝑡)𝑓(𝑦(𝑔(𝑡)))=𝑥^'''𝑧−𝑧^'''𝑥=𝑠^''−𝑧^'𝑧𝑠^'+𝑧^''𝑧𝑠,(1.11) or, (𝑠^'𝑧)^'𝑧=𝑞(𝑡)𝑓(𝑦(𝑔(𝑡)))−^''𝑠𝑧².(1.12) Hence we have 0≥^𝑐_𝑏(𝑠^'(𝑡))𝑧(𝑡)^'𝑑𝑡=^𝑐_𝑏𝑧{𝑞(𝑡)𝑓(𝑦(𝑔(𝑡)))−^''(𝑡)𝑠(𝑡)𝑧²(𝑡)}𝑑𝑡>0,(1.13) which is a contradiction.
Suppose the latter case holds. From (1.9) and (1.10), we have 𝑥^'''𝑧^'−𝑧^'''𝑥^'+𝑝(𝑡)(𝑥𝑧^'−𝑥^'𝑧)=𝑧^'𝑞(𝑡)𝑓(𝑦(𝑔(𝑡))),(1.14) from which by 𝑥^'''𝑧^'−𝑧^'''𝑥^'𝑧=(^'𝑧𝑠^'−𝑧^''𝑧𝑠)^',(1.15) we obtain (𝑧^'𝑧𝑠^'−𝑧^''𝑧𝑠)^'−𝑝(𝑡)𝑠=𝑧^'𝑞(𝑡)𝑓(𝑦(𝑔(𝑡))).(1.16) Since 𝑧^'(𝑡)>0,𝑠^'(𝑐_𝑛)≤0,𝑠^'(𝑏_𝑛)≥0,𝑠(𝑐_𝑛)=𝑠(𝑏_𝑛)=0,𝑠(𝑡)>0,𝑡∈(𝑏_𝑛,𝑐_𝑛), we see that 0<^𝑐𝑛_𝑏𝑛𝑧^'=(𝑡)𝑞(𝑡)𝑓(𝑦(𝑔(𝑡)))𝑑𝑡_{𝑐𝑛𝑏𝑛}𝑧{(^'(𝑡)𝑠𝑧(𝑡)^'𝑧(𝑡)−^''(𝑡)𝑧(𝑡)𝑠(𝑡))^'=𝑧−𝑝(𝑡)𝑠(𝑡)}𝑑𝑡_'(𝑐_𝑛)𝑧(𝑐_𝑛)𝑠^'(𝑐_𝑛𝑧)−^'(𝑏_𝑛)𝑧(𝑏_𝑛)𝑠^'(𝑏_𝑛)−^𝑐𝑛_𝑏𝑛𝑝(𝑡)𝑠(𝑡)𝑑𝑡≤0,(1.17) which is a contradiction.

2. Asymptotic Dichotomy

We are now ready to state and prove our main result.

Theorem 2.1. Suppose that(i)equation (1.9) has an eventually positive increasing solution 𝑧 on [𝑎,∞),(ii)there is a differentiable function 𝜌∶[𝑎,∞)→(0,∞) such that limsup_𝑡→∞^𝑡_𝑇{𝜅𝜌(𝑠)𝑞(𝑠)−(𝑙−1)!(3−𝑙)!(𝜌^'(𝑠))²2^3−𝑙𝑔^𝑙−1(𝑠)(𝑠−𝑔(𝑠))^3−𝑙𝑔^'(𝑠)𝜌(𝑠)}𝑑𝑠=+∞,𝑙=1and3,(2.1) for every 𝑇≥𝑎,(iii)limsup_𝑡→∞^𝑡_𝑇^𝑠_𝑇{𝜅𝑞(𝑢)−𝑝(𝑢)}𝑑𝑢𝑑𝑠=+∞.(2.2)Then every solution 𝑦 of (1.1) either oscillates or converges to 0.

Proof. Let 𝑦 be a solution of (1.1) which does not converge to 0. Suppose to the contrary that 𝑦 is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that y(𝑡)>0 and 𝑦(𝑔(𝑡))>0 for 𝑡≥𝑎. Furthermore, by Lemma 1.2, there is 𝑡₀≥𝑎 such that 𝑦^'(𝑡)>0 or 𝑦^'(𝑡)<0 for 𝑡≥𝑡₀.
Suppose 𝑦^'(𝑡)>0 for 𝑡≥𝑡₀. Then by (1.1), 𝑦⁽⁴⁾(𝑡)=−𝑝(𝑡)𝑦^'(𝑡)−𝑞(𝑡)𝑓(𝑦(𝑔(𝑡)))<0,𝑡≥𝑡₀.(2.3) By Lemma 1.1, we may suppose further that 𝑦^'''(𝑡)>0 and (1.2) holds for 𝑡≥𝑡₀ as well. Let 𝜔(𝑡)∶=𝜌(𝑡)𝑦^'''(𝑡)𝑦(𝑔(𝑡))>0,𝑡≥𝑡₀.(2.4) Then by (1.1) and (1.2), we see that 𝜔^'𝜌(𝑡)≤−𝜅𝜌(𝑡)𝑞(𝑡)+^'(𝑡)𝜌(𝑡)𝜔(𝑡)−𝜌(𝑡)𝑦^'''(𝑡)𝑦^'(𝑔(𝑡))𝑔^'(𝑡)𝑦²𝜌(𝑔(𝑡))≤−𝜅𝜌(𝑡)𝑞(𝑡)+^'(𝑡)𝑔𝜌(𝑡)𝜔(𝑡)−^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^3−𝑙𝑔^'(𝑡)2^𝑙−1𝜔(𝑙−1)!(3−𝑙)!𝜌(𝑡)²𝑔(𝑡)=−𝜅𝜌(𝑡)𝑞(𝑡)−^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^3−𝑙𝑔^'(𝑡)2^𝑙−12(𝑙−1)!(3−𝑙)!𝜌(𝑡)(𝜔−^𝑙−1(𝑙−1)!(3−𝑙)!𝜌^'(𝑡)2𝑔^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^3−𝑙𝑔^')(𝑡)²+(𝑙−1)!(3−𝑙)!(𝜌^'(𝑡))²2^3−𝑙𝑔^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^3−𝑙𝑔^'(𝑡)𝜌(𝑡)≤−{𝜅𝜌(𝑡)𝑞(𝑡)−(𝑙−1)!(3−𝑙)!(𝜌^'(𝑡))²2^3−𝑙𝑔^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^3−𝑙𝑔^'(𝑡)𝜌(𝑡)},𝑡≥𝛼.(2.5) Hence ^𝑡_𝑡0{𝜅𝜌(𝑡)𝑞(𝑡)−(𝑙−1)!(3−𝑙)!(𝜌^'(𝑡))²2^3−𝑙𝑔^𝑙−1(𝑡)(𝑡−𝑔(𝑡))^3−𝑙𝑔^'(𝑡)𝜌(𝑡)}𝑑𝑠≤−^𝑡_𝑡0𝜔^'(𝑠)𝑑𝑠=𝜔(𝑡₀)−𝜔(𝑡)≤𝜔(𝑡₀)(2.6) for all 𝑡≥𝑡₀, which is contrary to our assumption (2.1).
Suppose 𝑦^'(𝑡)<0for 𝑡≥𝑡₀. Since 𝑦 is eventually positive, eventually decreasing, and does not converge to 0, we easily see that (i) 𝑦^'' cannot be eventually nonpositive, (ii) lim_𝑡→∞𝑦(𝑡)=𝜆>0, (iii) lim_𝑡→∞𝑦^'(𝑡)=0, and (iv) there is 𝛼≥𝑎 such that −𝑦^'(𝑡)<𝜆<𝑦(𝑡) for 𝑡≥𝛼.
In view of (i) above, either 𝑦^''is oscillatory or 𝑦^'' is eventually nonnegative. If 𝑦^''is oscillatory, then𝑦^''' is also oscillatory. Hence there exists some number 𝛽≥𝛼 such that 𝑦^'''(𝛽)=0 and 𝑦(𝑔(𝑡))≥𝜆 for 𝑡≥𝛽. From (1.1) and (2.2), we have 𝑦^'''(𝑡)=−^𝑡_𝛽𝑦(𝑔(𝑠))𝑞(𝑠)𝑓(𝑦(𝑔(𝑠)))𝑦(𝑔(𝑠))𝑑𝑠−^𝑡_𝛽𝑝(𝑠)𝑦^'(𝑠)𝑑𝑠≤−𝜆^𝑡_𝛽[𝜅𝑞(𝑠)−𝑝(𝑠)]𝑑𝑠.(2.7) It follows that 𝑦^''(𝑡)≤𝑦^''(𝛽)−𝜆^𝑡_𝛽^𝑠_𝛽[𝜅𝑞(𝑢)−𝑝(𝑢)]𝑑𝑢𝑑𝑠.(2.8) Since the right-hand side tends to −∞ as 𝑡 tends to +∞,𝑦^'' is eventually negative which is contrary to our assumption on 𝑦^''.
As a consequence, we must have (𝑦(𝑡)>0,𝑦^'(𝑡)<0 and) 𝑦^''(𝑡)≥0, say for 𝑡≥𝛼≥𝑎. There are three cases to consider:
Case 1. If 𝑦^'''(𝑡)≥0 and does not vanish identically for 𝑡≥𝛼, then 𝑦^'(𝑡)>0 for all large 𝑡, a contradiction. If 𝑦^'''(𝑡)≡0 for 𝑡≥𝛼, then 𝑦^' is a linear function. Since lim_𝑡→∞𝑦^'(𝑡)=0, we get 𝑦^'(𝑡)=0 for 𝑡≥𝛼, which is contrary to our assumption that 𝑦^'(𝑡)<0 for 𝑡≥𝛼.Case 2. If 𝑦^'''(𝑡)≤0 for 𝑡≥𝛼, then from (1.1), we get 𝑦^'''(𝛼)−𝑦^'''(𝑡)=^𝑡_𝛼𝑞(𝑠)𝑓(𝑦(𝑔(𝑠)))𝑑𝑠+^𝑡_𝛼𝑝(𝑠)𝑦^'(𝑠)𝑑𝑠≥𝜆^𝑡_𝛼(𝜅𝑞(𝑠)−𝑝(𝑠))𝑑𝑠,0≥−𝑦^''(𝑡)≥−𝑦^''(𝛼)+𝜆^𝑡_𝛼^𝑠_𝛼(𝜅𝑞(𝜏)−𝑝(𝜏))𝑑𝜏𝑑𝑠,(2.9) which is contrary to (2.2).Case 3. If 𝑦^''' oscillates, then (2.8) holds so that 𝑦^'' is eventually negative, which is contrary to our assumption on 𝑦^''. This completes the proof.

3. Remarks and Example

We remark that the condition (2.2) in the above result can, in case 𝑝 is differentiable, be replaced by the alternate condition

limsup_𝑡→∞^𝑡_𝑇{𝜅𝑞(𝑢)−𝑝^'(𝑢)}𝑑𝑢=+∞(3.1) for every 𝑇≥0. Indeed, we only need to note that

^𝑡_𝛾𝑞(𝑠)𝑓(𝑦(𝑔(𝑠)))𝑑𝑠+^𝑡_𝛾𝑝(𝑠)𝑦^'=(𝑠)𝑑𝑠_𝑡𝛾𝑦(𝑔(𝑠))[𝑞(𝑠)𝑓(𝑦(𝑔(𝑠)))𝑦(𝑔(𝑠))]𝑑𝑠+𝑝(𝑠)𝑦(𝑠)∣^𝑡_𝛾−^𝑡_𝛾𝑦(𝑠)𝑝^'(𝑠)𝑑𝑠≥𝜆^𝑡_𝛾[𝜅𝑞(𝑠)−𝑝^'(𝑠)]𝑑𝑠−𝑝(𝛾)𝑦(𝛾),(3.2) where 𝛾=𝛼 or 𝛽, and then we may follow the rest of the above arguments.

Example 3.1. Consider the fourth-order delay differential equation 𝑦⁽⁴⁾+1𝑡³𝑦(ln𝑡−1)^'+1𝑡²1(1−𝑡(ln𝑡−1))𝑦(𝑡−3𝜋2)=0,𝑡>3𝜋2.(3.3) Here, 𝑝(𝑡)=1/𝑡³(ln𝑡−1),𝑞(𝑡)=(1/𝑡²)(1−1/𝑡(ln𝑡−1)), and 𝑓(𝑢)=𝑢with 𝜅=1. It is clear that (2.2) is satisfied. The equation 𝑧^'''+(1/𝑡³(ln𝑡−1))𝑧=0has an eventually positive and eventually increasing solution 𝑧=𝑡(ln𝑡−1). By choosing 𝜌(𝑡)=𝑡 and any large 𝑇,we have limsup_𝑡→∞^𝑡_𝑇1[(𝑠²−1𝑠³(ln𝑠−1))𝑠−(𝑙−1)!(3−𝑙)!2^3−𝑙𝑠(𝑠−3𝜋/2)^𝑙−1(3𝜋/2)^3−𝑙]𝑑𝑠=+∞,𝑙=1and3.(3.4) Consequently, condition (2.1) is satisfied. Hence by Theorem 2.1, any solution of (3.3) is oscillatory or satisfies lim_𝑡→∞𝑦(𝑡)=0.

As our final remark, suppose we have a (sufficiently) long elastic horizontal beam modeled by (1.1). Our Theorem says that (under appropriate conditions) if the right end of the beam is fixed at some nonzero vertical position, then there must be places where there are positive as well as negative displacements of the beam from its equilibrium position. This shows that the dichotomy in our Theorem has implications in the elastic beam deflection problem.