Oscillation and Asymptotic Behavior of Higher-Order Nonlinear Differential Equations

Abstract

The aim of this paper is to offer a generalization of the Philos and Staikos lemma. As a possible application of the lemma in the oscillation theory, we study the asymptotic properties and oscillation of the 𝑛th order delay differential equations (𝐸) (𝑟(𝑡)[𝑥^(𝑛−1)(𝑡)]^𝛾)^+𝑞(𝑡)𝑥^𝛾(𝜏(𝑡))=0. The results obtained utilize also the comparison theorems.

1. Introduction

In this paper, we will study the asymptotic and oscillation behavior of the solutions of the higher-order advanced differential equations: 𝑟(𝑡)𝑥^(𝑛−1)(𝑡)^𝛾^+𝑞(𝑡)𝑥^𝛾(𝜏(𝑡))=0.(𝐸)

Throughout the paper, we will assume 𝑞,𝜏,𝑟∈𝐶([𝑡₀,∞)), and (𝐻₁)𝑛≥3, 𝛾 is the ratio of two positive odd integers, (𝐻₂)𝑟(𝑡)>0, 𝑞(𝑡)>0, 𝜏(𝑡)≤𝑡, lim_𝑡→∞𝜏(𝑡)=∞.

Whenever, it is assumed 𝑅(𝑡)=^𝑡𝑡₀𝑟^−1/𝛾(𝑠)d𝑠⟶∞as𝑡⟶∞.(1.1)

By a solution of (𝐸) we mean a function 𝑥(𝑡)∈𝐶^𝑛−1([𝑇_𝑥,∞)),  𝑇_𝑥≥𝑡₀, which has the property 𝑟(𝑡)(𝑥^(𝑛−1)(𝑡))^𝛾∈𝐶¹([𝑇_𝑥,∞)) and satisfies  (𝐸) on [𝑇_𝑥,∞). We consider only those solutions 𝑥(𝑡) of (𝐸) which satisfy sup{|𝑥(𝑡)|∶𝑡≥𝑇}>0 for all 𝑇≥𝑇_𝑥. We assume that (𝐸) possesses such a solution. A solution of (𝐸) is called oscillatory if it has arbitrarily large zeros on [𝑇_𝑥,∞), and otherwise it is called to be nonoscillatory.

The problem of the oscillation of higher-order differential equations has been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for studied equations (see, e.g., [1–19]).

Philos and Staikos lemma (see [16, 17]) essentially simplifies the examination of 𝑛th-order differential equations of the form 𝑦^(𝑛)(𝑡)+𝑞(𝑡)𝑦^𝛾(𝜏(𝑡))=0,(1.2) since it provides needed relationship between 𝑦(𝑡) and 𝑦^(𝑛−1)(𝑡), and this fact permit us to establish just one condition for asymptotic behavior of (1.2). If we try to apply the Philos and Staikos lemma to (𝐸), the strong condition 𝑟′(𝑡)≥0 appears (see, e.g., [2, 18, 20]). In this paper we offer such generalization of the Philos and Staikos lemma, where this restriction is relaxed. Moreover, the obtained lemma yields many applications in the oscillation theory. As an example of it, we offer its disposal in the comparison theory and we establish new oscillation criteria for (𝐸).

2. Main Results

The following result is a well-known lemma of Kiguradze, see, for example, [6] or [13].

Lemma 2.1. Let 𝑧(𝑡)∈𝐶^𝑘−1([𝑡₀,∞)) and 𝑟(𝑡)(𝑧^(𝑘−1)(𝑡))^𝛾∈𝐶¹([𝑡₀,∞)) with 𝑧(𝑡)>0, (𝑟(𝑡)(𝑧^(𝑘−1)(𝑡))^𝛾)^≤0, and not identically zero on a subray of [𝑡₀,∞). Then there exist a 𝑡₁≥𝑡₀ and an integer ℓ, 0≤ℓ≤𝑘−1, with 𝑘+ℓ odd so that 𝑧(−1)^ℓ+𝑗𝑧^(𝑗)(𝑡)>0,𝑗=ℓ,…,𝑘−1,_(𝑖)(𝑡)>0,𝑖=1,…,ℓ−1,𝑤ℎ𝑒𝑛ℓ>1,(2.1) on [𝑡₁,∞).

Now we are prepared to provide a generalization of the Philos and Staikos lemma.

Lemma 2.2. Let 𝑧(𝑡) be as in Lemma 2.1 and numbers 𝑡₁ and ℓ assigned to 𝑧(𝑡) by Lemma 2.1. Then for 0<ℓ<𝑘−1, 𝑧(𝑡)≥𝑟^1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)(𝑘−2)!^𝑡𝑡₁𝑟^−1/𝛾(𝑠)𝑠−𝑡₁^𝑘−2d𝑠,(2.2) for ℓ=𝑘−1; 𝑧(𝑡)≥𝑟^1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)(𝑘−2)!^𝑡𝑡₁𝑟^−1/𝛾(𝑠)(𝑡−𝑠)^𝑘−2d𝑠,(2.3) for 𝑡≥𝑡₁.

Proof. Let ℓ be the integer assigned to function 𝑧(𝑡) as in Lemma 2.1. Assume that ℓ<𝑘−1, then, for any 𝑠,𝑡 with 𝑡≥𝑠≥𝑡₁, −𝑧^(𝑘−2)(𝑠)≥≥𝑟^𝑡𝑠_1/𝛾𝑧^(𝑘−1)(𝑢)d𝑢=^𝑡𝑠𝑟^1/𝛾(𝑢)𝑧^(𝑘−1)(𝑢)𝑟^−1/𝛾(𝑢)d𝑢(𝑡)𝑧^(𝑘−1)(𝑡)^𝑡𝑠𝑟^−1/𝛾(𝑢)d𝑢.(2.4) Repeated integration in 𝑠 from 𝑠 to 𝑡 yields 𝑧^(ℓ)(𝑠)≥𝑟^1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)^𝑡𝑠𝑟^−1/𝛾(𝑢)(𝑢−𝑠)^{𝑘−2−ℓ}(𝑘−2−ℓ)!d𝑢.(2.5) On the other hand, if ℓ≥1, then, for every 𝑡≥𝑡₁, 𝑧^(ℓ−1)(𝑡)≥^𝑡𝑡₁𝑧^(ℓ)(𝑠)d𝑠.(2.6) Repeated integration from 𝑡₁ to 𝑡 leads to 𝑧(𝑡)≥1(ℓ−1)!^𝑡𝑡₁𝑧^(ℓ)(𝑠)(𝑡−𝑠)^ℓ−1d𝑠.(2.7) Setting (2.5) into (2.7), one gets 𝑧(𝑡)≥𝑟^1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)(ℓ−1)!(𝑘−ℓ−2)!^𝑡𝑡₁(𝑡−𝑠)^ℓ−1^𝑡𝑠≥𝑟𝑟^−1/𝛾(𝑢)(𝑢−𝑠)^{𝑘−2−ℓ}d𝑢d𝑠_1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)(𝑘−3)!^𝑡𝑡₁^𝑡𝑠=𝑟𝑟^−1/𝛾(𝑢)(𝑢−𝑠)^𝑘−3d𝑢d𝑠_1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)(𝑘−2)!^𝑡𝑡₁𝑟^−1/𝛾(𝑢)𝑢−𝑡₁^𝑘−2d𝑠.(2.8) We have verified the first part of the lemma. Now assume that ℓ=𝑘−1. It follows from (2.7) that 𝑧(𝑡)≥1(𝑘−2)!^𝑡𝑡₁≥1𝑧^(𝑘−1)(𝑠)(𝑡−𝑠)^𝑘−2d𝑢(𝑘−2)!^𝑡𝑡₁=𝑟𝑟^1/𝛾(𝑠)𝑧^(𝑘−1)(𝑠)𝑟^−1/𝛾(𝑠)(𝑡−𝑠)^𝑘−2d𝑠_1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)(𝑘−2)!^𝑡𝑡₁𝑟^−1/𝛾(𝑠)(𝑡−𝑠)^𝑘−2d𝑠.(2.9) The proof is complete now.

Employing additional conditions, we are able to joint (2.5) and (2.7) to just one estimate.

Lemma 2.3. Let 𝑧(𝑡) be as in Lemma 2.1 and lim_𝑡→∞𝑧(𝑡)≠0. Let 𝑟^(𝑡)≥0,  Then for any 𝜆∈(0,1) there exists some 𝑡_𝜆≥𝑡₁ such that 𝑧(𝑡)≥𝜆(𝑘−1)!𝑡^𝑘−1𝑧^(𝑘−1)(𝑡),(2.10) for 𝑡≥𝑡_𝜆.

Proof. Note that 𝑟′(𝑡)≥0 implies that 𝑟^−1/𝛾(𝑡) is nonincreasing. Assume that ℓ is the integer associated with 𝑧(𝑡) in Lemma 2.1. If 0<ℓ<𝑘−1, then using (2.2), we have 𝑧(𝑡)≥𝑧^(𝑘−1)(𝑡)(𝑘−2)!^𝑡𝑡₁𝑠−𝑡₁^𝑘−2d𝑠=𝑧^(𝑘−1)(𝑡)(𝑘−1)!𝑡−𝑡¹^𝑘−1.(2.11) It is easy to see that for any 𝜆∈(0,1) there exists a 𝑡_𝜆≥𝑡₁ such that 𝑡−𝑡₁≥𝜆^1/(𝑘−1)𝑡 for 𝑡≥𝑡_𝜆, which in view of (2.11) yields (2.10).
If ℓ=𝑘−1, then proceeding similarly as above it can be shown that (2.3) implies (2.10).
If ℓ=0, then it follows from (2.5) that 𝑧(𝑠)≥𝑟^1/𝛾(𝑡)𝑧^(𝑘−1)(𝑡)^𝑡𝑠𝑟^−1/𝛾(𝑢)(𝑢−𝑠)^𝑘−2(𝑘−2)!d𝑢=𝑧^(𝑘−1)(𝑡)(𝑘−1)!(𝑡−𝑠)^𝑘−1.(2.12) Setting 𝑠=(1−𝜆^{1/2(𝑘−1)})𝑡, we have 𝑥1−𝜆^{1/2(𝑘−1)}𝑡≥𝑧^(𝑘−1)(𝑡)(𝑘−1)!𝜆^1/2𝑡^𝑘−1.(2.13) Moreover, lim_𝑡→∞𝑥(𝑡)𝑥1−𝜆^{1/2(𝑘−1)}𝑡=1>𝜆^1/2.(2.14) Therefore, for all large 𝑡, 𝑥(𝑡)≥𝜆^1/2𝑥1−𝜆^{1/2(𝑘−1)}𝑡≥𝜆(𝑘−1)!𝑡^𝑘−1𝑧^(𝑘−1)(𝑡).(2.15) The proof is complete now.

In the following result, we compare (2.2) and (2.3) to verify that both estimates are independent; that is, one does not result from the other, although for 𝑟(𝑡)≡1 they are equivalent.

Remark 2.4. Let 𝑧(𝑡) be as in Lemma 2.1, such that 𝑧(𝑡) is increasing. At first we consider 𝑟(𝑡)=𝑡³, 𝛾=3, 𝑡₁=1, and 𝑘=3 then (2.2), (2.3), and (2.10) yield 𝑧(𝑡)≥𝑧^(𝑡)𝑡²−𝑡ln𝑡−𝑡,𝑧(𝑡)≥𝑧^(𝑡)𝑡²ln𝑡−𝑡²+𝑡,𝑧(𝑡)≥𝑧^(𝑡)𝜆𝑡²2,(2.16) respectively.
Now we modify 𝑟(𝑡)=𝑡⁻³. Then (2.2) and (2.3) reduce to 𝑧(𝑡)≥𝑧^(𝑡)𝑡²3−2𝑡+61𝑡,𝑧(𝑡)≥𝑧^(𝑡)𝑡²6−21+31𝑡,(2.17) respectively, and (2.10) is not applicable.

3. Applications

Lemma 2.2 can be applied in various techniques for investigations of the higher-order differential equations. We offer one such application in comparison theory.

Theorem 3.1. Assume that both first-order delay differential equations 𝑦′(𝑡)+𝑞(𝑡)((𝑛−2)!)^𝛾^{𝑡𝜏(𝑡)}₁𝑦′(𝑡)+𝑞(𝑡)𝑟^−1/𝛾(𝑠)𝑠−𝑡₁^𝑛−2d𝑠^𝛾𝑦(𝜏(𝑡))=0,𝐸₁((𝑛−2)!)^𝛾^{𝑡𝜏(𝑡)}₁𝑟^−1/𝛾(𝑠)(𝜏(𝑡)−𝑠)^𝑛−2d𝑠^𝛾𝑦(𝜏(𝑡))=0𝐸₂ are oscillatory. Moreover, for 𝑛-odd assume that ^∞𝑡₀𝑢^𝑛−2𝑟^−1/𝛾(𝑢)^∞𝑢𝑞(𝑠)d𝑠^1/𝛾d𝑢=∞.𝑃₀ Then (i)for 𝑛 even, (𝐸) is oscillatory; (ii)for 𝑛 odd, every nonoscillatory solution 𝑥(𝑡) of (𝐸) satisfies lim_𝑡→∞𝑥(𝑡)=0.

Proof. Assume that 𝑥(𝑡) is a nonoscillatory solution of (𝐸); let us say positive. Then (𝑟(𝑡)[𝑥^(𝑛−1)(𝑡)]^𝛾)^<0, and there exist a 𝑡₁≥𝑡₀ and an integer ℓ with 𝑛+ℓ odd such that (2.1) holds.
If 0<ℓ<𝑛−1, then by Lemma 2.2𝑥(𝑡)≥𝑟^1/𝛾(𝑡)𝑥^(𝑛−1)(𝑡)(𝑛−2)!^𝑡𝑡₁𝑟^−1/𝛾(𝑠)𝑠−𝑡₁^𝑛−2d𝑠.(3.1)Setting to (𝐸), we get 𝑟(𝑡)𝑥^(𝑛−1)(𝑡)^𝛾^+𝑞(𝑡)𝑟(𝜏(𝑡))𝑥^(𝑛−1)(𝜏(𝑡))^𝛾((𝑛−2)!)^𝛾×^{𝑡𝜏(𝑡)}₁𝑟^−1/𝛾(𝑠)𝜏(𝑠)−𝑡₁^𝑛−2d𝑠^𝛾≤0.(3.2) Then 𝑦(𝑡)=𝑟(𝑡)[𝑥^(𝑛−1)(𝑡)]^𝛾 is positive and satisfies the differential inequality: 𝑦^(𝑡)+𝑞(𝑡)((𝑛−2)!)^𝛾^{𝑡𝜏(𝑡)}₁𝑟^−1/𝛾(𝑠)𝑠−𝑡₁^𝑛−2d𝑠^𝛾𝑦(𝜏(𝑡))≤0.(3.3) By Theorem 1 in [15], the corresponding equation 𝐸₁ has also a positive solution. A contradiction.
If ℓ=𝑛−1, then by Lemma 2.2𝑥(𝑡)≥𝑟^1/𝛾(𝑡)𝑥^(𝑛−1)(𝑡)(𝑛−2)!^𝑡𝑡₁𝑟^−1/𝛾(𝑠)(𝑡−𝑠)^𝑘−2d𝑠,(3.4) and proceeding as above, we find out that 𝐸₂ has a positive solution. A contradiction and the proof are finished for 𝑛 even.
Assume that ℓ=0; note that it is possible only for 𝑛 is odd. Since 𝑥′(𝑡)<0, then there exists a finite lim_𝑡→∞𝑥(𝑡)=𝑐≥0. We claim that 𝑐=0. If not, then 𝑥(𝜏(𝑡))≥𝑐>0, eventually, let us say, for 𝑡≥𝑡₂. An integration of (𝐸) from 𝑡 to ∞ yields 𝑟(𝑡)𝑥^(𝑛−1)(𝑡)^𝛾≥^∞𝑡𝑞(𝑠)𝑥^𝛾(𝜏(𝑠))d𝑠(3.5) or equivalently 𝑥^(𝑛−1)(𝑡)≥𝑟^−1/𝛾(𝑡)^∞𝑡𝑞(𝑠)𝑥^𝛾(𝜏(𝑠))d𝑠^1/𝛾.(3.6) Integrating 𝑛−1 times from 𝑡 to ∞, we get 𝑥(𝑡)≥^∞𝑡(𝑢−𝑡)^𝑛−2(𝑛−2)!𝑟^−1/𝛾(𝑢)^∞𝑢𝑞(𝑠)𝑥^𝛾(𝜏(𝑠))d𝑠^1/𝛾d𝑢,(3.7) then 𝑥𝑡₂≥𝑐(𝑛−2)!^∞𝑡₂𝑢−𝑡₂^𝑛−2𝑟^−1/𝛾(𝑢)^∞𝑢𝑞(𝑠)d𝑠^1/𝛾d𝑢,(3.8) which contradicts 𝑃₀. The proof is complete.

Employing any result (e.g., Theorem 2.1.1 in [13]) for the oscillation of 𝐸₁ and 𝐸₂, we immediately obtain criteria for studied properties of (𝐸).

Corollary 3.2. Let liminf_𝑡→∞^{𝑡𝜏(𝑡)}𝑞(𝑢)^{𝑡𝜏(𝑢)}₁𝑟^−1/𝛾(𝑠)𝑠−𝑡₁^𝑛−2d𝑠^𝛾d𝑢>((𝑛−2)!)^𝛾liminf_𝑡→∞e,^{𝑡𝜏(𝑡)}𝑞(𝑢)^{𝑡𝜏(𝑢)}₁𝑟^−1/𝛾(𝑠)(𝜏(𝑠)−𝑡)^𝑛−2d𝑠^𝛾d𝑢>((𝑛−2)!)^𝛾e.(3.9) Moreover, for 𝑛-odd assume that 𝑃₀ holds. Then (i)for 𝑛 even, (𝐸) is oscillatory; (ii)for 𝑛 odd, every nonoscillatory solution 𝑥(𝑡) of (𝐸) satisfies lim_𝑡→∞𝑥(𝑡)=0.

If we replace 𝑃₀ by stronger condition, we can establish oscillation of (𝐸) even if 𝑛 is odd.

Theorem 3.3. Let 𝜏^(𝑡)≥0. Assume that both first-order delay differential equations 𝐸₁ and 𝐸₁ are oscillatory. Moreover, for 𝑛-odd assume that limsup_𝑡→∞^{𝑡𝜏(𝑡)}(𝑢−𝜏(𝑡))^𝑛−2𝑟^−1/𝛾(𝑢)^∞𝑢𝑞(𝑠)d𝑠^1/𝛾d𝑢>(𝑛−2)!.𝑃₁ Then (𝐸) is oscillatory.

Proof. Assume that 𝑥(𝑡) is a positive solution of (𝐸). Then there exists an integer ℓ assigned with 𝑥(𝑡) by Lemma 2.1. If 1≤ℓ≤𝑛−1, then proceeding as in the proof of Theorem 3.1 we eliminate 𝑥(𝑡). If ℓ=0, then it follows from (3.7) that 𝑥(𝜏(𝑡))≥^∞𝜏(_t)(𝑢−𝜏(𝑡))^𝑛−2(𝑛−2)!𝑟^−1/𝛾≥𝑥(𝜏(𝑡))_{𝑡𝜏(𝑡)}(𝑢)^∞𝑢𝑞(𝑠)𝑥^𝛾(𝜏(𝑠))d𝑠^1/𝛾d𝑢(𝑢−𝜏(𝑡))^𝑛−2(𝑛−2)!𝑟^−1/𝛾(𝑢)^𝑡𝑢𝑞(𝑠)d𝑠^1/𝛾d𝑢,(3.10) which contradicts 𝑃₁. The proof is complete.

Corollary 3.4. Let (3.9) hold. Moreover, for 𝑛-odd assume that 𝑃₁ holds. Then (𝐸) is oscillatory.

Example 3.5. We consider the forth-order delay differential equation: 𝑡𝑥^(𝑡)³^+𝑡𝑎₆𝑥³(𝜆𝑡)=0,𝑎>0,0<𝜆<1,𝑡≥1.𝐸₃ It is easy to verify that 𝑃₀ holds for 𝐸₃; moreover, conditions (3.9) reduce to 𝑎𝜆⁵ln1𝜆>1e53³𝑎27𝜆₂,(3.11)1𝜆2⁶5ln𝜆1−3𝜆³(1−𝜆)50+31−𝜆²40−(12−4𝜆𝜆³)³>1e,(3.12) respectively. Thus, by Corollary 3.4, if both (3.11) and (3.12) hold, then all nonoscillatory solutions of 𝐸₃ tend to zero. For 𝑎=𝛼³(𝛼+1)³(3𝛼+5)𝜆^3𝛼, with 𝛼>0, one such solution is 𝑥(𝑡)=𝑡^−𝛼. On the other hand, condition 𝑃₁ takes the form 𝑎ln1𝜆+1−𝜆³>5.(3.13) Therefore, by Corollary 3.4, 𝐸₃ is oscillatory, provided that all (3.11), (3.12), and (3.13) hold.

Acknowledgments

This work is the result of the project implementation: Development of the Center of Information and Communication Technologies for Knowledge Systems (ITMS project code: 26220120030) supported by the Research & Development Operational Program funded by the ERDF.