Nonoscillatory Solutions of Second-Order Superlinear Dynamic Equations with Integrable Coefficients

Abstract

The asymptotic behavior of nonoscillatory solutions of the superlinear dynamic equation on time scales (𝑟(𝑡)𝑥^Δ(𝑡))^Δ+𝑝(𝑡)|𝑥(𝜎(𝑡))|^𝛾sgn𝑥(𝜎(𝑡))=0, 𝛾>1, is discussed under the condition that 𝑃(𝑡)=lim_𝜏→∞∫^𝜏_𝑡𝑝(𝑠)Δ𝑠 exists and 𝑃(𝑡)≥0 for large 𝑡.

1. Introduction

Consider the second-order superlinear dynamic equation𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)|𝑥(𝜎(𝑡))|^𝛾sgn𝑥(𝜎(𝑡))=0,𝛾>1, (1.1) where𝑃(𝑡)=lim_𝜏→∞^𝜏_𝑡𝑝(𝑠)Δ𝑠 (1.2) exists and is finite. 𝑃(𝑡)≥0 for large 𝑡.

When 𝕋=ℝ, 𝑟(𝑡)=1, (1.1) is the second-order superlinear differential equation𝑥^′′(𝑡)+𝑝(𝑡)|𝑥(𝑡)|^𝛾sgn𝑥(𝑡)=0,𝛾>1. (1.3)

When 𝕋=ℕ, 𝑟(𝑛)=1, (1.1) is the second-order superlinear difference equationΔ²𝑥(𝑛)+𝑝(𝑛)|𝑥(𝑛+1)|^𝛾sgn𝑥(𝑛+1)=0,𝛾>1. (1.4)

The following condition is introduced in [1].

Condition (H). We say that 𝕋 satisfies Condition (H) provided one of the following holds.(1)There exists a strictly increasing sequence {𝑡_𝑛}^∞_𝑛=0⊂𝕋 with lim_𝑛→∞𝑡_𝑛=∞ and for each 𝑛≥0 either 𝜎(𝑡_𝑛)=𝑡_𝑛+1 or the real interval [𝑡_𝑛,𝑡_𝑛+1]⊂𝕋; (2)𝕋∩ℝ=[𝑇^′,∞) for some 𝑇^′∈𝕋.

We note that time scales which satisfy Condition (H) include most of the important time scales, such as ℝ, ℤ, and 𝑞^ℕ0, where 𝑞>1 and ℕ₀ is the nonnegative integers and harmonic numbers {∑^𝑛_𝑘=11/𝑘∶𝑛∈ℕ} [2, Example 1.45].

In [3], Naito proved the following result.

Theorem 1.1. If 𝑃(𝑡)=∫^∞_𝑡𝑝(𝑠)𝑑𝑠≥0 for large 𝑡, then a nonoscillatory solution 𝑥(𝑡) of (1.3) satisfies exactly one of the following three asymptotic properties: lim_𝑡→∞𝑥(𝑡)=𝑐≠0,lim_𝑡→∞𝑥(𝑡)𝑡=0,lim_𝑡→∞𝑥(𝑡)=±∞,lim_𝑡→∞𝑥(𝑡)𝑡=𝑐≠0. (1.5)

In this paper, we extend Theorem 1.1 to superlinear dynamic equation (1.1) on time scale. As an application, we get the asymptotic behavior of each nonoscillation solution of the difference equationΔ²𝑥(𝑛)+𝑎𝑛^1+𝑐+(−1)^𝑛𝑏𝑛^𝑐|𝑥(𝑛+1)|^𝛾sgn𝑥(𝑛+1)=0,𝛾>1, (1.6) where 𝑏>0, 𝑐>1, and 𝑎/𝑐>𝑏/2.

2. Main Theorems

Consider the second-order nonlinear dynamic equation𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)|𝑥(𝜎(𝑡))|^𝛾sgn𝑥(𝜎(𝑡))=0,𝛾>0, (2.1) where 𝑟(𝑡),𝑝(𝑡)∈𝐶(𝕋,𝑅), 𝑟(𝑡)>0, 𝑡₀∈𝕋, and ∫^∞_𝑡0[𝑟(𝑡)]⁻¹𝑑𝑡=∞. lim_𝑡→∞∫^𝑡_𝑡0𝑝(𝑠)Δ𝑠 exists and is finite.

Lemma 2.1. Suppose that 𝕋 satisfies Condition (𝐻). If 𝑥(𝑡) is a positive solution of (1.1) on [𝑇,∞), then the integral equation 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)=𝛼+𝑃(𝑡)+^∞_𝑡𝑟(𝑠)𝛾∫¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠 (2.2) is satisfied for 𝑡≥𝑇, where 𝛼 is a nonnegative constant.

Proof. Suppose that 𝑥(𝑡) is a positive solution of (1.1) on [𝑇,∞).
In the first place, we will prove ^∞_𝑇𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠<∞, (2.3) where 𝑥_ℎ(𝑡)=𝑥(𝑡)+ℎ𝜇(𝑡)𝑥^Δ(𝑡)=(1−ℎ)𝑥(𝑡)+ℎ𝑥(𝜎(𝑡))>0.
Multiplying both sides of (1.1) by 1/𝑥^𝛾(𝜎(𝑡)), we get that 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)^Δ(𝑡)−𝑟(𝑡)=−𝑝𝛾∫¹₀𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)²𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡)). (2.4) Integrating from 𝑇 to 𝑡, 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)−𝑟(𝑇)𝑥^Δ(𝑇)𝑥^𝛾(𝑇)=−^𝑡_𝑇𝑝(𝑠)Δ𝑠−^𝑡_𝑇𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠. (2.5) If (2.3) fails to hold, that is, ^∞_𝑇𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠=∞, (2.6) from (2.5), we have lim_𝑡→∞𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)=−∞. (2.7) Without loss of generality, we can assume that for 𝑡≥𝑇𝑟(𝑇)𝑥^Δ(𝑇)𝑥^𝛾(𝑇)−^𝑡_𝑇𝑝(𝑠)Δ𝑠<−1. (2.8) Otherwise, let 𝐿=max_𝑡≥𝑇|∫^𝑡_𝑇𝑝(𝑠)Δ𝑠|. By (2.7), we can take a large 𝑇₁>𝑇 such that 𝑟(𝑇₁)𝑥^Δ(𝑇₁)/𝑥^𝛾(𝑇₁)<−(2𝐿+1). So we have 𝑟𝑇₁𝑥^Δ𝑇₁𝑥^𝛾𝑇₁−^𝑡_𝑇1𝑝(𝑠)Δ𝑠<−(2𝐿+1)−^𝑡_𝑇𝑝(𝑠)Δ𝑠−^𝑇1_𝑇𝑝(𝑠)()−[]Δ𝑠≤−2𝐿+1−2𝐿=−1. (2.9) So we can replace 𝑇 by 𝑇₁>𝑇 such that (2.8) still holds.
From (2.5) and (2.8), we get for 𝑡≥𝑇𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)+^𝑡_𝑇𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠<−1. (2.10) In particular, we have 𝑥^Δ(𝑡)<0,for𝑡≥𝑇. (2.11) Therefore, 𝑥(𝑡) is strictly decreasing.
Assume that 𝑡=𝑡_𝑖−1<𝑡_𝑖=𝜎(𝑡). Then, 𝑥(𝜎(𝑡))<𝑥(𝑡), and so 𝛾¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ=𝛾¹₀[(1−ℎ)𝑥(𝑠)+ℎ𝑥(𝜎(𝑠))]^𝛾−1=)𝑥(𝑠)(𝜎(𝑠𝑑ℎ[(1−ℎ+ℎ𝑥))]^𝛾1₀𝑥(𝜎(𝑠(𝑠)=𝑥))−𝑥^𝛾(𝜎(𝑠))−𝑥^𝛾(𝑠)𝑥(𝜎(𝑠(𝑠).))−𝑥 (2.12) If the real interval [𝑡_𝑖−1,𝑡_𝑖]⊂𝕋, then, for 𝑠∈[𝑡_𝑖−1,𝑡_𝑖], we have 𝛾¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ=𝛾𝑥^𝛾−1(𝑠). (2.13) Let 𝑦(𝑡)∶=1+^𝑡_𝑇𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾−1(𝜎(𝑠))Δ𝑠. (2.14) Hence, from (2.10), we get that −𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)>𝑦(𝑡). (2.15) From (2.14) and (2.15), we get that 𝑦^Δ(𝑡)=𝑟(𝑡)𝛾∫¹₀𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)²𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡(𝑡)𝛾∫))>𝑦¹⁰𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ−𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡.)) (2.16) Assume that 𝑡=𝑡_𝑖−1<𝑡_𝑖=𝜎(𝑡). From (2.16) and (2.12), we get that 𝑦(𝜎(𝑡))−𝑦(𝑡)𝑦(𝑡𝜎(𝑡))>𝑥)(−𝑡^𝛾(𝜎(𝑡))−𝑥^𝛾(𝑡)𝑥(𝜎(𝑡(𝑡)𝑥(𝑡)(𝜎(𝑡))−𝑥−𝑥))𝑥^𝛾(𝜎(𝑡))[𝜎(𝑡)−𝑡]. (2.17) So, 𝑦(𝜎(𝑡))𝑦(𝑡)>𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡)), (2.18) that is, 𝑦𝑡_𝑖𝑦𝑡_𝑖−1>𝑥^𝛾𝑡_𝑖−1𝑥^𝛾𝑡_𝑖. (2.19) If the real interval [𝑡_𝑖−1,𝑡_𝑖]⊂𝕋, then, for 𝑡∈(𝑡_𝑖−1,𝑡_𝑖], it follows from (2.16) and (2.13) that 𝑦^′(𝑡)𝑦(𝑡)>𝛾𝑥^𝛾−1(𝑡)−𝑥^′(𝑡)𝑥^𝛾(𝑡), (2.20) that is, ln𝑦(𝑡)^′>−ln𝑥^𝛾(𝑡)^′. (2.21) Integrating from 𝑡_𝑖−1 to 𝑡, we get that 𝑦(𝑡)𝑦𝑡_𝑖−1>𝑥^𝛾𝑡_𝑖−1𝑥^𝛾(𝑡),𝑡∈𝑡_𝑖−1,𝑡_𝑖. (2.22) Let 𝑇=𝑡_𝑛0, and let 𝑡∈(𝑇,∞)_𝕋. Then, there is an 𝑛>𝑛₀ such that 𝑡∈(𝑡_𝑛−1,𝑡_𝑛]_𝕋. From (2.22) and (2.19), we get that 𝑦(𝑡)𝑦𝑡_𝑛−1>𝑥^𝛾𝑡_𝑛−1𝑥^𝛾(𝑡),𝑦𝑡_𝑛−1𝑦𝑡_𝑛−2>𝑥^𝛾𝑡_𝑛−2𝑥^𝛾𝑡_𝑛−1𝑦𝑡,…,^𝑛0+1𝑦𝑡_𝑛0>𝑥^𝛾𝑡_𝑛0𝑥^𝛾𝑡_𝑛0+1. (2.23) Multiplying, we get that 𝑦(𝑡)𝑦𝑡_𝑛0>𝑥^𝛾𝑡_𝑛0𝑥^𝛾(𝑡). (2.24) Using (2.15) again, we get −𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)(𝑡)>𝑦𝑡>𝑦^𝑛0𝑥^𝛾𝑡_𝑛0𝑥^𝛾(𝑡). (2.25) If we set 𝐿∶=𝑦(𝑡_𝑛0)𝑥^𝛾(𝑡_𝑛0), we get 𝑥^Δ(𝑡)𝐿<−𝑟(𝑡). (2.26) Integrating from 𝑇 to 𝑡, we get that 𝑥(𝑡)−𝑥(𝑇)<−^𝑡_𝑇𝐿𝑟(𝑠)Δ𝑠⟶−∞,as𝑡⟶∞, (2.27) which contradicts 𝑥(𝑡)>0.
In (2.5), letting 𝑡→∞, replacing 𝑇 by 𝜏, and denoting 𝛼=lim_𝑡→∞𝑟(𝑡)𝑥^Δ(𝑡)/𝑥^𝛾(𝑡), we get that 𝛼+^∞_𝜏𝑝(𝑠)Δ𝑠+^∞_𝜏𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠𝑟(𝜏)𝑥))Δ𝑠=^Δ(𝜏)𝑥^𝛾(𝜏). (2.28)
We need to show that 𝛼≥0.
Suppose that 𝛼<0. Then, there exists a large 𝑇₁ such that, for 𝑡>𝑇₁, we have 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)≤𝛼2. (2.29) So, 𝑥^Δ(𝑡)≤𝛼𝑥^𝛾(𝑡)2𝑟(𝑡). (2.30)
Thus, 𝑀𝑇₁=∶^∞_𝑇1𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠𝛼))Δ𝑠≥−2^∞_𝑇1𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ−𝑥^Δ(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠. (2.31) Assume that 𝑡=𝑡_𝑖−1<𝑡_𝑖=𝜎(𝑡). From (2.12) and 𝑥^Δ(𝑡)<0, we have _𝑡^𝜎(𝑡)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ−𝑥^Δ(𝑠)𝑥^𝛾(𝜎(𝑠=))Δ𝑠𝛾∫¹₀𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ−𝑥^Δ(𝑡)(𝜎(𝑡)−𝑡)𝑥^𝛾(𝜎(𝑡=𝑥))_𝛾(𝑡)−𝑥^𝛾(𝜎(𝑡))𝑥^𝛾(𝜎(𝑡≥))_{𝑥𝛾𝑥(𝑡)𝛾(𝜎(𝑡))}1𝑣𝑥𝑑𝑣=ln^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡𝑥))=ln^𝛾𝑡_𝑖−1𝑥^𝛾𝑡_𝑖. (2.32) If the real interval [𝑡_𝑖−1,𝑡_𝑖]⊂𝕋, from (2.13) we have, for 𝑡∈(𝑡_𝑖−1,𝑡_𝑖], ^𝑡_𝑡𝑖−1𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ−𝑥^Δ(𝑠)𝑥^𝛾(𝜎(𝑠=))Δ𝑠_{𝑡𝑡𝑖−1}𝛾𝑥^𝛾−1(𝑠)−𝑥^′(𝑠)𝑥^𝛾(𝑠)𝑥𝑑𝑠=ln^𝛾𝑡_𝑖−1𝑥^𝛾(𝑡). (2.33) From (2.31), (2.32), (2.33) and the additivity of the integral, it is easy to get 𝑀𝑇₁𝛼≥−2lim_𝑢→∞𝑥ln^𝛾𝑇₁𝑥^𝛾(𝑢). (2.34) So, for large 𝑢, we have 𝑥ln^𝛾𝑇₁𝑥^𝛾(𝑢)𝑇≤−2𝑀₁𝛼+1. (2.35) Thus, 𝑥^𝛾(𝑢)≥𝑥^𝛾𝑇₁𝑇exp2𝑀₁𝛼−1. (2.36) By (2.30) and noticing that 𝛼<0, we get that 𝑥^Δ(𝑢)≤𝛼𝑥^𝛾𝑇₁(𝑢)𝑇2𝑟exp2𝑀₁𝛼−1. (2.37) Integrating (2.37), we get that 𝑥(𝑢)→−∞, which is a contradiction.
This completes the proof of the lemma.

Consider the second-order superlinear dynamic equation𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)|𝑥(𝜎(𝑡))|^𝛾sgn𝑥(𝜎(𝑡))=0,𝛾>1, (2.38) where 𝑟(𝑡)>0, ∫^∞_𝑇(1/𝑟(𝑠))Δ𝑠=∞,𝑃(𝑡)=lim_𝜏→∞^𝜏_𝑡𝑝(𝑠)Δ𝑠 (2.39) exists and is finite, and 𝑃(𝑡)≥0 for 𝑡≥𝑇.

Theorem 2.2. Suppose that 𝕋 satisfies Condition (𝐻) and 𝑃(𝑡)≥0 for 𝑡≥𝑇. Then each nonoscillatory solution 𝑥(𝑡) of (2.38) satisfies exactly one of the following three asymptotic properties: lim_𝑡→∞𝑥(𝑡)=𝑐≠0, (2.40)lim_𝑡→∞𝑥(𝑡)∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)=0,lim_𝑡→∞𝑥(𝑡)=±∞, (2.41)lim_𝑡→∞𝑥(𝑡)∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)=𝑐≠0. (2.42)

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (2.38), say, 𝑥(𝑡)>0 for 𝑡≥𝑇>0. From Lemma 2.1, it is known that 𝑥(𝑡) satisfies the equality 𝑟(𝑡)𝑥^Δ(𝑡)=𝛼𝑥^𝛾(𝑡)+𝑃(𝑡)𝑥^𝛾(𝑡)+𝛾𝑥^𝛾(𝑡)^∞_𝑡𝑟(𝑠)∫¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠 (2.43) for 𝑡≥𝑇. Therefore, we have 𝑟(𝑡)𝑥^Δ(𝑡)≥𝑃(𝑡)𝑥^𝛾(𝑡), (2.44) for 𝑡≥𝑇. Since 𝑃(𝑡)≥0 for 𝑡≥𝑇, it follows that 𝑥^Δ(𝑡)≥0 for 𝑡≥𝑇. An integration by parts of (2.38) gives 𝑟(𝑢)𝑥^Δ(𝑢)−𝑃(𝑢)𝑥^𝛾(𝑢)+𝛾^𝑢_𝑡𝑃(𝑠)𝑥^Δ(𝑠)¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1(𝑡)𝑥𝑑ℎΔ𝑠=𝑟^Δ(𝑡)−𝑃(𝑡)𝑥^𝛾(𝑡), (2.45) where 𝑢≥𝑡≥𝑇. Let 𝑡 be fixed. Since 𝑃(𝑠)𝑥^Δ(𝑠)∫¹₀[𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)]^𝛾−1𝑑ℎ is nonnegative, the integral term in (2.45) has a finite limit or diverges to ∞ as 𝑡→∞. If the latter case occurs, then 𝑟(𝑢)𝑥^Δ(𝑢)−𝑃(𝑢)𝑥^𝛾(𝑢)→−∞ as 𝑢→∞, which is a contradiction to (2.44). Thus, the former case occurs, that is, ^∞_𝑡𝑃(𝑠)𝑥^Δ(𝑠)¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1𝑑ℎΔ𝑠<∞. (2.46) Define the function 𝑘₁(𝑡) as 𝑘₁(𝑡)=^∞_𝑡𝑃(𝑠)𝑥^Δ(𝑠)¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1𝑑ℎΔ𝑠<∞ (2.47) and the finite constant 𝛽=lim_𝑢→∞[𝑟(𝑢)𝑥^Δ(𝑢)−𝑃(𝑢)𝑥^𝛾(𝑢)]. Then, equality (2.45) yields 𝑟(𝑡)𝑥^Δ(𝑡)=𝛽+𝑃(𝑡)𝑥^𝛾(𝑡)+𝛾𝑘₁(𝑡), (2.48) for 𝑡≥𝑇. Observe by (2.44) that 𝛽≥0. From (2.44) and (2.46), it follows that ^∞_𝑡𝑃²(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1𝑑ℎΔ𝑠≤𝑘₁(𝑡)<∞. (2.49) Next, we define the function 𝑘₂(𝑡) by (noticing that 𝛾>1) 𝑘₂(𝑡)=^∞_𝑡𝑃²(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)¹₀𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠)^𝛾−1=𝑑ℎΔ𝑠_∞𝑡𝑃²(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)¹₀[(1−ℎ)𝑥(𝑠)+ℎ𝑥(𝜎(𝑠))]^𝛾−1≥𝑑ℎΔ𝑠_∞𝑡𝑃²(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)¹₀[(1−ℎ)𝑥(𝑠)+ℎ𝑥(𝑠)]^𝛾−1𝑑ℎΔ𝑠=^∞_𝑡𝑃²(𝑠)𝑥^2𝛾−1(𝑠)𝑟(𝑠)Δ𝑠 (2.50) for 𝑡≥𝑇. Dividing (2.48) by 𝑟(𝑡) and integrating from 𝑇 to 𝑡, we get 𝑥(𝑡)=𝑥(𝑇)+^𝑡_𝑇𝛽𝑟(𝑠)Δ𝑠+^𝑡_𝑇𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(s)Δ𝑠+𝛾^𝑡_𝑇𝑘₁(𝑠)𝑟(𝑠)Δ𝑠. (2.51) By Schwartz’s inequality and the fact that 𝑥^Δ(𝑡)≥0 for 𝑡≥𝑇, the second integral term in (2.51) can be estimated as follows: ^𝑡_𝑇𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠≤^𝑡_𝑇𝑃²(𝑠)𝑥^2𝛾−1(𝑠)𝑟(𝑠)Δs^1/2𝑡_𝑇𝑥(𝑠)𝑟(𝑠)Δ𝑠^1/2≤𝑘₂^1/2(𝑇)^𝑡_𝑇Δ𝑠𝑟(𝑠)^1/2𝑥^1/2(𝑡), (2.52) for 𝑡≥𝑇. From (2.51) and (2.52) and noticing that 𝑘₁(𝑡) is decreasing, we get that 𝑥(𝑡)≤𝑥(𝑇)+^𝑡_𝑇𝛽𝑟(𝑠)Δ𝑠+𝑘₂^1/2(𝑇)^𝑡_𝑇Δ𝑠𝑟(𝑠)^1/2𝑥^1/2(𝑡)+𝛾𝑘₁(𝑇)^𝑡_𝑇Δ𝑠𝑟(𝑠). (2.53) The above inequality may be regarded as a quadratic inequality in 𝑥^1/2(𝑡). Then, we have 𝑥^1/2(𝑡)≤12𝑘₂^1/2(𝑇)^𝑡_𝑇Δ𝑠𝑟(𝑠)^1/2+12𝐷^1/2(𝑡) (2.54) for 𝑡≥𝑇, where 𝐷(𝑡)=𝑘₂(𝑇)^𝑡_𝑇Δ𝑠𝑟(𝑠)+4𝑥(𝑇)+𝛽+𝛾𝑘₁(𝑇)^𝑡_𝑇Δ𝑠𝑟(𝑠). (2.55) It is obvious that 𝐷(𝑡)=𝑂(∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)) as 𝑡→∞, and, consequently, there exists a positive constant 𝑚 such that 𝑥(𝑡)≤𝑚^𝑡_𝑇Δ𝑠𝑟(𝑠) (2.56) for 𝑡≥𝑇. Let 𝑇₁(≥𝑇) be an arbitrary number. It is clear that 0≤^𝑡_𝑇𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠=^𝑇1_𝑇𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠+^𝑡_𝑇1𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠 (2.57) for 𝑡≥𝑇₁. Arguing as in (2.52), we find ^𝑡_𝑇1𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠≤𝑘₂(𝑇₁)𝑥(𝑡)^𝑡_𝑇1Δ𝑠𝑟(𝑠)^1/2 (2.58) for 𝑡≥𝑇₁, which when combined with (2.56) yields ^𝑡_𝑇1𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠≤𝑚𝑘₂(𝑇₁)^𝑡_𝑇Δ𝑠𝑟(𝑠)^𝑡_𝑇1Δ𝑠𝑟(𝑠)^1/2≤𝑚𝑘₂𝑇₁^1/2𝑡_𝑇Δ𝑠𝑟(𝑠), (2.59) for 𝑡≥𝑇₁≥𝑇. Using (2.57) and (2.59) and noticing that ∫^∞_𝑇Δ𝑠/𝑟(𝑠)=∞, we obtain 0≤limsup_𝑡→∞1∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)^𝑡𝑇𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠≤𝑚𝑘₂𝑇₁^1/2. (2.60) Since 𝑇₁ is arbitary and 𝑘₂(𝑇₁)→0 as 𝑇₁→∞, letting 𝑇₁→∞ in (2.60), we get 0≤lim_𝑡→∞1∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)^𝑡𝑇𝑃(𝑠)𝑥^𝛾(𝑠)𝑟(𝑠)Δ𝑠=0. (2.61) Using L’Hospital’s rule of time scale (see Theorem 1.119 of [2]), we have lim_𝑡→∞∫^𝑡_𝑇𝑘₁(𝑠)/𝑟(𝑠)Δ𝑠∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)=lim_𝑡→∞𝑘₁(𝑡)=0. (2.62) In view of (2.51), (2.61), and (2.62), we find lim_𝑡→∞𝑥(𝑡)/∫^𝑡_𝑇Δ𝑠/𝑟(𝑠)=𝛽.
Recall that 𝑥(𝑡) is nondecreasing for 𝑡≥𝑇. Now, there are three cases to consider:(i)𝛽=0 and 𝑥(𝑡) is bounded above,(ii)𝛽=0 and 𝑥(𝑡) is unbounded,(iii)𝛽>0 (and hence 𝑥(𝑡) is unbounded).
Case (i) implies (2.40) with 𝑐=lim_𝑡→∞𝑥(𝑡)>0, while case (iii) implies (2.42) with 𝑐=𝛽>0. It is also clear that case (ii) implies (2.41). This completes the proof.

The following lemma is from [1].

Lemma 2.3. Suppose that 𝕋 satisfies Condition (H). 𝑥(𝑡)>0 is a solution of (1.1). Then, one has ^𝑡_𝑇𝑥^Δ(𝑠)𝑥^𝛾(𝜎(𝑠≤𝑥))Δ𝑠^−𝛾+1(𝑇)−𝑥^−𝛾+1(𝑡)≤𝑥𝛾−1^−𝛾+1(𝑇).𝛾−1 (2.63)

Using Lemma 2.1, we can prove the following corollary.

Corollary 2.4. Under the assumptions of Lemma 2.1, if 𝑥(𝑡) is a positive solution of (1.1) on [𝑇,∞), then the integral equation 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡(𝑡)))=𝛼+𝑃+^∞_𝜎(𝑡)𝑟(𝑠)∫¹₀𝛾𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠 (2.64) is satisfied for 𝑡≥𝑇, where 𝑃(𝑡)=∫^∞_𝑡𝑝(𝑠)Δ𝑠, 𝑥_ℎ(𝑠)=𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠).

Proof. In the left side of (2.2), using 1𝑥^𝛾(𝑡)=1𝑥^𝛾(𝜎(𝑡−1))𝑥^𝛾(𝑡)^Δ𝜇(𝑡)=1𝑥^𝛾(𝜎(𝑡+∫))¹⁰𝛾𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡𝜇(𝑡))) (2.65) and using (2.2), (2.65), the additivity of the integral, and [2, Theorem 1.75], we have that 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝑡)=𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡+𝑟(𝑡)∫))¹⁰𝛾𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)²𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡𝜇(𝑡)(𝑡)+))=𝛼+𝑃^∞_𝜎(𝑡)𝑟(𝑠)∫¹₀𝛾𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠+))Δ𝑠_{𝑡𝜎(𝑡)}𝑟(𝑠)∫¹₀𝛾𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠(𝑡)+))Δ𝑠=𝛼+𝑃^∞_𝜎(𝑡)𝑟(𝑠)∫¹₀𝛾𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠+𝑟(𝑡∫))Δ𝑠)¹₀𝛾𝑥_ℎ(𝑡)^𝛾−1𝑑ℎ𝑥^Δ(𝑡)²𝑥^𝛾(𝑡)𝑥^𝛾(𝜎(𝑡𝜇(𝑡).)) (2.66) From (2.66), we get (2.64).

The following theorem can be regarded as a time scale version of [4, Theorem 1].

Theorem 2.5. Suppose that 𝕋 satisfies Condition (H), 𝑟(𝑡)>0 with ∫^∞_𝑇[𝑟(𝑡)]⁻¹Δ𝑡=∞, and suppose that lim_𝑡→∞∫^𝑡_𝑇𝑝(𝑠)Δ𝑠 exists and is finite. Let 𝑃(𝑡)=∫^∞_𝑡𝑝(𝑠)Δ𝑠. Then, the superlinear dynamic equation (1.1) is oscillatory if limsup_𝑡→∞^𝑡_𝑇𝑃(𝑠)𝑟(𝑠)Δ𝑠=∞. (2.67)

Proof. Suppose that 𝑥(𝑡) is a nonoscillatory solution of (1.1) on [𝑇,∞). Without loss of generality, assume that 𝑥(𝑡) is positive for 𝑡∈[𝑇,∞). From Corollary 2.4, 𝑥(𝑡) satisfies the integral equation (2.64). Dropping the last integral term in (2.64), we have the inequality 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡(𝑡)))≥𝑃. (2.68) Dividing (2.68) by 𝑟(𝑡), integrating from 𝑇 to 𝑡, and using Lemma 2.3, we find 𝑥^−𝛾+1(𝑇)≥𝛾−1^𝑡_𝑇𝑥^Δ(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠≥^𝑡_𝑇𝑃(𝑠)𝑟(𝑠)Δ𝑠. (2.69) This contradicts (2.67), and so (1.1) is oscillatory.

Consider the second-order superlinear dynamic equation with forced term𝑟(𝑡)𝑥^Δ(𝑡)^Δ+𝑝(𝑡)|𝑥(𝜎(𝑡))|^𝛾sgn𝑥(𝜎(𝑡))=ℎ(𝑡),𝛾>1, (2.70) where 𝑟(𝑡)>0, ∫^∞_𝑇(1/𝑟(𝑠))Δ𝑠=∞, and𝑃(𝑡)=lim_𝜏→∞^𝜏_𝑡𝑝(𝑠)Δ𝑠 (2.71) exists and is finite.

Lemma 2.6. Suppose that ^∞_𝑇|ℎ(𝑠)|Δ𝑠<+∞. (2.72) If 𝑥(𝑡) is a positive solution of (2.70) and liminf_𝑡→∞𝑥(𝑡)>0, then ^∞_𝑇𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠<∞, (2.73)𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡))=𝛼+^∞_𝑡𝑝(𝑠)−ℎ(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠+^∞_𝜎(𝑡)𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠 (2.74) are satisfied for sufficiently large 𝑡, where 𝑥_ℎ(𝑠)=𝑥(𝑠)+ℎ𝜇(𝑠)𝑥^Δ(𝑠), 𝛼 is a nonnegative constant.

Proof. The fact that liminf_𝑡→∞𝑥(𝑡)>0 implies the existence of 𝑡₁≥𝑇 and 𝑚>0 such that 𝑥(𝑡)≥𝑚 for 𝑡≥𝑡₁. Then, using (2.72), we find ^𝑡_𝑡1ℎ(𝑠)𝑥^𝛾(𝜎(𝑠≤))Δ𝑠^𝑡_𝑡1ℎ(𝑠)𝑥^𝛾(𝜎(𝑠1))Δ𝑠≤𝑚^𝛾𝑡_𝑡1|ℎ(𝑠)|Δ𝑠≤𝑀,𝑡≥𝑡₁, (2.75) where 𝑀 is some finite positive constant.
So, lim_𝜏→∞∫^𝜏_𝑡[𝑝(𝑠)−ℎ(𝑠)/𝑥^𝛾(𝜎(𝑠))]Δ𝑠 exists and is finite.
Similar to the proof of Lemma 2.1 and Corollary 2.4, it is easy to know that (2.73) and (2.74) hold.

For subsequent results, we defineΦ₀(𝑡)=^∞_𝑡𝑝(𝑠)−𝑘|ℎ(𝑠)|Δ𝑠,𝑡≥𝑇, (2.76) where 𝑘 is a positive constant. It is noted that, if (2.71) and (2.72) hold, then Φ₀(𝑡) is finite for any 𝑘. Assume that Φ₀(𝑡)>0 for sufficiently large 𝑡. Define, for a positive integer 𝑛 and a positive constant 𝜌, the following functions:Φ₁(𝑡)=^∞_𝜎(𝑡)Φ₀(𝑠)²𝑟(𝑠)ΦΔ𝑠,_𝑛+1(𝑡)=^∞_𝜎(𝑡)Φ₀(𝑠)+𝜌Φ_𝑛(𝑠)²𝑟(𝑠)Δ𝑠. (2.77) We introduce the following condition.

Condition (A). For every 𝜌>0, there exists a positive integer 𝑁 such that Φ_𝑛(𝑡) is finite for 𝑛=1,2,…,𝑁−1 and Φ_𝑁(𝑡) is infinite.

Theorem 2.7. Suppose that (2.71), (2.72), and Condition (A) hold. Then, every solution 𝑥(𝑡) of (2.70) is either oscillatory or satisfies liminf_𝑡→∞𝑥(𝑡)=0. (2.78)

Proof. Suppose on the contrary that 𝑥(𝑡) is a nonoscillatory solution of (2.70) and liminf_𝑡→∞|𝑥(𝑡)|>0. Without loss of generality, let 𝑥(𝑡) be eventually positive. By Lemma 2.6, 𝑥(𝑡) satisfies (2.73) and (2.74). Further, there exist 𝑡₁≥𝑇 and 𝑚>0 such that 𝑥(𝑡)≥𝑚 for 𝑡≥𝑡₁. Let Φ₀(𝑡)=^∞_𝑡𝑝(𝑠)−|ℎ(𝑠)|𝑚^𝛾Δ𝑠. (2.79) Then, from (2.74) we find 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡))≥Φ₀(𝑡)+^∞_𝜎(𝑡)𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠≥Φ₀(𝑡)>0, (2.80) for 𝑡≥𝑡₁. From (2.80), we get 𝑥^Δ(𝑡)≥Φ₀(𝑡)𝑥^𝛾(𝜎(𝑡))𝑟(𝑡)>0,𝑡≥𝑡₁. (2.81) Applying (2.81) and noticing that 𝛾>1, we find for 𝑡≥𝑡₁^∞_𝜎(𝑡)𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠≥))Δ𝑠_{∞𝜎(𝑡)}𝛾[𝑥(𝑠)]^𝛾−1Φ₀(𝑠)𝑥^𝛾(𝜎(𝑠))²𝑟(𝑠)𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠≥𝛾𝑚^𝛾−1∞_𝜎(𝑡)Φ₀(𝑠)²𝑟(𝑠)Δ𝑠=𝛾𝑚^𝛾−1Φ₁(𝑡). (2.82) If 𝑁=1 in Condition (A), then the right side of (2.82) is infinite. This is a contradiction to (2.73).
Next, it follows from (2.80) and (2.82) that 𝑟(𝑡)𝑥^Δ(𝑡)𝑥^𝛾(𝜎(𝑡))≥Φ₀(𝑡)+𝛾𝑚^𝛾−1Φ₁(𝑡). (2.83) Using a similar technique and relations (2.83), we get ^∞_𝜎(𝑡)𝑟(𝑠)𝛾∫¹₀𝑥_ℎ(𝑠)^𝛾−1𝑑ℎ𝑥^Δ(𝑠)²𝑥^𝛾(𝑠)𝑥^𝛾(𝜎(𝑠))Δ𝑠≥𝛾𝑚^𝛾−1∞_𝜎(𝑡)Φ₀(𝑠)+𝛾𝑚^𝛾−1Φ₁(𝑠)²𝑟(𝑠)Δ𝑠=𝛾𝑚^𝛾−1Φ₂(𝑡),𝑡≥𝑡₁. (2.84) If 𝑁=2 in Condition (A), then the right side of (2.84) is infinite. This again contradicts (2.73).
A similar argument yields a contradiction for any integer 𝑁>2. This completes the proof of the theorem.

Example 2.8. We have Δ²𝑥(𝑛)+𝑎𝑛^1+𝑐+𝑏(−1)^𝑛𝑛^𝑐|𝑥(𝑛+1)|^𝛾sgn𝑥(𝑛+1)=0,𝛾>1, (2.85) where 𝑏>0, 𝑐>1, and 𝑎/𝑐>𝑏/2. It is easy to see that ^∞_𝑘=𝑛1𝑘^1+𝑐≥^∞_𝑛1𝑡^1+𝑐1𝑑𝑡=𝑐𝑛^𝑐. (2.86) By [5], we have ∑^∞_𝑘=𝑛(−1)^𝑛/𝑛^𝑐∼(−1)^𝑛/2𝑛^𝑐. So, ^∞_𝑘=𝑛(−1)^𝑘𝑘^𝑐=[(1()1+𝑜)]−1^𝑛2𝑛^𝑐. (2.87) Using (2.86) and (2.87), we get that, for large 𝑛, 𝑃(𝑛)=^∞_𝑘=𝑛𝑎𝑘^1+𝑐+𝑏(−1)^𝑘𝑘^𝑐≥(𝑎/𝑐)+(𝑏/2)[1+𝑜(1)](−1)^𝑛𝑛^𝑐>0. (2.88) By Theorem 2.2, each nonoscillatory solution 𝑥(𝑡) of (2.85) satisfies exactly one of the following three asymptotic properties: lim_𝑛→∞𝑥(𝑛)=𝑐≠0,lim_𝑛→∞𝑥(𝑛)𝑛=0,lim_𝑛→∞𝑥(𝑛)=±∞,lim_𝑛→∞𝑥(𝑛)𝑛=𝑐≠0. (2.89)

Acknowledgment

The paper is supported by the National Natural Science Foundation of China (no. 10971232).