Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales

Abstract

We study the higher-order neutral dynamic equation {𝑎(𝑡)[(𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡)))^Δ𝑚]^𝛼}^Δ+𝑓(𝑡,𝑥(𝛿(𝑡)))=0 for 𝑡∈[𝑡₀,∞)_𝕋 and obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.

1. Introduction

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers. Thus, ℝ,ℤ,andℕ, that is, the real numbers, the integers, and the natural numbers, are examples of time scales. We assume throughout that the time scale 𝕋 has the topology that it inherits from the real numbers with the standard topology.

The theory of time scale, which has recently received a lot of attention, was introduced by Hilger's landmark paper [1], a rapidly expanding body of literature has sought to unify, extend, and generalize ideas from discrete calculus, quantum calculus, and continuous calculus to arbitrary time scale calculus, where a time scale is a nonempty closed subset of the real numbers, and the cases when this time scale is equal to the real numbers or to the integers represent the classical theories of differential and difference equations. Many other interesting time scales exist, and they give rise to many applications (see [2]). Not only the new theory of the so-called “dynamic equations” unifies the theories of differential equations and difference equations but also extends these classical cases to cases “in between”, for example, to the so-called 𝑞-difference equations when 𝕋={1,𝑞,𝑞²,…}, which has important applications in quantum theory (see [3]).

On a time scale 𝕋, the forward jump operator, the backward jump operator and the graininess function are defined as 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡},𝜇(𝑡)=𝜎(𝑡)−𝑡, (1.1) respectively. We refer the reader to [2, 4] for further results on time scale calculus.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to the papers of [5–20].

In [21] Zhu and Wang studied the existence of nonoscillatory solutions to neutral dynamic equation [𝑥(𝑡)+𝑝(𝑡)𝑥(𝑔(𝑡))]^Δ+𝑓(𝑡,𝑥(ℎ(𝑡)))=0. (1.2)

Karpuz [22] studied the asymptotic behavior of delay dynamic equations having the following form:[𝑥(𝑡)+𝐴(𝑡)𝑥(𝛼(𝑡))]^Δ+𝐵(𝑡)𝐹(𝑥(𝛽(𝑡)))−𝐶(𝑡)𝐺(𝑥(𝛾(𝑡)))=𝜑(𝑡). (1.3) Furthermore, Karpuz in [23] obtained necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of the following higher-order nonlinear forced neutral dynamic equation [𝑥(𝑡)+𝐴(𝑡)𝑥(𝛼(𝑡))]^Δ𝑛+𝑓(𝑡,𝑥(𝛽(𝑡)),𝑥(𝛾(𝑡)))=𝜑(𝑡) (1.4) and also studied in [24] oscillation of unbounded solutions of a similar type of equations.

Li et al. [25] considered the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form [𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏₀(𝑡))]^ΔΔ+𝑞₁(𝑡)𝑥(𝜏₁(𝑡))−𝑞₂(𝑡)𝑥(𝜏₂(𝑡))=𝑒(𝑡). (1.5)

In [26, 27], Zhang et al. obtained some sufficient conditions for the existence of nonoscillatory solutions for the following higher-order equations: [𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]^Δ𝑛+𝑓(𝑡,𝑥(𝑡−𝜏₁(𝑡)),…,𝑥(𝑡−𝜏_𝑘(𝑡)))=0,[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]^Δ𝑛+𝑓(𝑡,𝑥(𝜏₁(𝑡)),…,𝑥(𝜏_𝑘(𝑡)))=0, (1.6)

respectively.

Motivated by the above studies, in this paper, we investigate the existence of nonoscillatory solutions of the following higher order neutral dynamic equation:{𝑎(𝑡)[(𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡)))^Δ𝑚]^𝛼}^Δ+𝑓(𝑡,𝑥(𝛿(𝑡)))=0for𝑡∈[𝑡₀,∞)_𝕋, (1.7) where 𝑚∈ℕ, 𝛼 is the quotient of odd positive integers, 𝑡₀∈𝕋, the time scale interval [𝑡₀,∞)_𝕋={𝑡∈𝕋∶𝑡≥𝑡₀}, 𝑎∈𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,(0,∞)), 𝑝∈𝐶([𝑡₀,∞)_𝕋,ℝ), 𝜏,𝛿∈𝐶(𝕋,𝕋) with lim_𝑡→∞𝜏(𝑡)=lim_𝑡→∞𝛿(𝑡)=∞ and 𝑓∈𝐶([𝑡₀,∞)_𝕋×ℝ,ℝ) satisfying the following conditions:(i)𝑢𝑓(𝑡,𝑢)>0 for any 𝑡∈[𝑡₀,∞)_𝕋 and 𝑢≠0.(ii)𝑓(𝑡,𝑢) is nondecreasing in 𝑢 for any 𝑡∈[𝑡₀,∞)_𝕋.

Since we are interested in the oscillatory behavior of solutions near infinity, we assume that sup𝕋=∞. By a solution of (1.7) we mean a nontrivial real-valued function 𝑥∈𝐶_𝑟𝑑([𝑇_𝑥,∞)_𝕋,ℝ),𝑇_𝑥≥𝑡₀, such that 𝑎(𝑡)[(𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡)))^Δ𝑚]^𝛼∈𝐶¹_𝑟𝑑([𝑇_𝑥,∞)_𝕋,ℝ) and satisfies (1.7) on [𝑇_𝑥,∞). The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution 𝑥 of (1.7) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

2. Auxiliary Results

We state the following conditions, which are needed in the sequel:

(𝐻₁)∫^∞_𝑡0(𝑎(𝑡))^−1/𝛼Δ𝑡=∞;

(𝐻₂) there exist constants 𝑎₁,𝑏₁∈[0,1) with 𝑎₁+𝑏₁<1 such that −𝑎₁≤𝑝(𝑡)≤𝑏₁ for all 𝑡∈[𝑡₀,∞)_𝕋;

(𝐻₃) there exist constants 𝑎₂,𝑏₂∈(1,∞) such that −𝑎₂≤𝑝(𝑡)≤−𝑏₂ for all 𝑡∈[𝑡₀,∞)_𝕋;

(𝐻₄) there exist constants 𝑎₃,𝑏₃∈(1,∞) such that 𝑎₃≤𝑝(𝑡)≤𝑏₃ for all 𝑡∈[𝑡₀,∞)_𝕋.

Let 𝑘 be a nonnegative integer and 𝑠,𝑡∈𝕋; we define two sequences of functions ℎ_𝑘(𝑡,𝑠) and 𝑔_𝑘(𝑡,𝑠) as follows: ℎ_𝑘(𝑡,𝑠)=1if𝑘=0,_𝑡𝑠ℎ_𝑘−1𝑔(𝜏,𝑠)Δ𝜏if𝑘≥1,_𝑘(𝑡,𝑠)=1if𝑘=0,_𝑡𝑠𝑔_𝑘−1(𝜎(𝜏),𝑠)Δ𝜏if𝑘≥1. (2.1)

By Theorems 1.112 and 1.60 of [2], we have ℎ_𝑘(𝑡,𝑠)=(−1)^𝑘𝑔_𝑘ℎ(𝑠,𝑡)forall𝑡,𝑠∈𝕋,_Δ𝑡𝑘ℎ(𝑡,𝑠)=0if𝑘=0,_𝑘−1𝑔(𝑡,𝑠)if𝑘≥1,_Δ𝑡𝑘𝑔(𝑡,𝑠)=0if𝑘=0,_𝑘−1(𝜎(𝑡),𝑠)if𝑘≥1, (2.2)

where 𝑔^Δ𝑡_𝑘(𝑡,𝑠) and ℎ^Δ𝑡_𝑘(𝑡,𝑠) denote for each fixed 𝑠 the derivatisve of 𝑔_𝑘(𝑡,𝑠) and ℎ_𝑘(𝑡,𝑠) with respect to 𝑡, respectively.

Lemma 2.1 (see [23, 24]). Assume that 𝑠,𝑡∈𝕋 and 𝑔∈𝐶_𝑟𝑑(𝕋×𝕋,ℝ), then ^𝑡_𝑠[^𝑡_𝜂𝑔(𝜂,𝜁)Δ𝜁]Δ𝜂=^𝑡_𝑠[_𝑠^𝜎(𝜁)g(𝜂,𝜁)Δ𝜂]Δ𝜁. (2.3)

Lemma 2.2 (see [23, 24]). Let 𝑛 be a nonnegative integer, ℎ∈𝐶_𝑟𝑑(𝕋,[0,∞)), and 𝑠∈𝕋. Then each of the following is true:(i)∫^∞_𝑠𝑔_𝑛(𝜎(𝜃),𝑠)ℎ(𝜃)Δ𝜃<∞ implies that ∫^∞_𝑡𝑔_𝑛(𝜎(𝜃),𝑡)ℎ(𝜃)Δ𝜃<∞ for all 𝑡∈𝕋;(ii)∫^∞_𝑠𝑔_𝑛(𝜎(𝜃),𝑠)ℎ(𝜃)Δ𝜃=∞ implies that ∫^∞_𝑡𝑔_𝑛(𝜎(𝜃),𝑡)ℎ(𝜃)Δ𝜃=∞ for all 𝑡∈𝕋.

Lemma 2.3 (see [23]). Let 𝑛 be a nonnegative integer, ℎ∈𝐶_𝑟𝑑(𝕋,[0,∞)) and 𝑠∈𝕋. Then ^∞_𝑠𝑔_𝑛(𝜎(𝜃),𝑠)ℎ(𝜃)Δ𝜃<∞ (2.4)
implies that each of the following is true:
(i)∫^∞_𝑡𝑔_𝑗(𝜎(𝜃),𝑡)ℎ(𝜃)Δ𝜃 is decreasing for all 𝑡∈𝕋 and all 0≤𝑗≤𝑛.(ii)lim_𝑡→∞∫^∞_𝑡𝑔_𝑗(𝜎(𝜃),𝑡)ℎ(𝜃)Δ𝜃=0 for all 0≤𝑗≤𝑛.(iii)∫^∞_𝑡𝑔_𝑗(𝜎(𝜃),𝑡)ℎ(𝜃)Δ𝜃<∞ for all 𝑡∈𝕋 and all 0≤𝑗≤𝑛−1.

Lemma 2.4 (see [28]). Let 𝑚∈𝐍 and 𝑓^Δ𝑚∈𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ). Then(1)liminf_𝑡→∞𝑓^Δ𝑚(𝑡)>0 implies lim_𝑡→∞𝑓^Δ𝑖(𝑡)=∞ for all 0≤𝑖≤𝑚−1.(2)limsup_𝑡→∞𝑓^Δ𝑚(𝑡)<0 implies lim_𝑡→∞𝑓^Δ𝑖(𝑡)=−∞ for all 0≤𝑖≤𝑚−1.

Lemma 2.5 (see [29]). Let 𝑧(𝑡) be bounded for 𝑡∈[𝑡₀,∞)_𝕋 with 𝑧^Δ𝑛(𝑡)>0, where 𝑛∈𝐍. Then (−1)^𝑛−𝑖𝑧^Δ𝑖(𝑡)>0 for 1≤𝑖≤𝑛 and lim_𝑡→∞𝑧^Δ𝑖(𝑡)=0for1≤𝑖≤𝑛−1. (2.5)
In the sequel, write 𝑦(𝑡)=𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡)). (2.6)

Lemma 2.6. Assume that 𝑝(𝑡) is bounded and (𝐻₁) holds. If 𝑥(𝑡) is a bounded nonoscillatory solution of (1.7), then 𝑥(𝑡)𝑦^Δ𝑚(𝑡)>0 eventually.

Proof. Without loss of generality, assume that there is some 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0 and 𝑥(𝛿(𝑡))>0 for 𝑡≥𝑡₁. From (1.7) we have {𝑎(𝑡)[𝑦^Δ𝑚(𝑡)]^𝛼}^Δ=−𝑓(𝑡,𝑥(𝛿(𝑡)))<0for𝑡≥𝑡₁. (2.7) Thus, 𝑅(𝑡)=𝑎(𝑡)[𝑦^Δ𝑚(𝑡)]^𝛼 is strictly decreasing on [𝑡₁,∞)_𝕋. If there exists 𝑡₂≥𝑡₁ such that 𝑅(𝑡₂)<0, then 𝑦^Δ𝑚(𝑡)≤[𝑅(𝑡₂)]𝑎(𝑡)^1/𝛼for𝑡≥𝑡₂. (2.8) Therefore, we have 𝑦^Δ𝑚−1(𝑡)−𝑦^Δ𝑚−1(𝑡₂)≤^𝑡_𝑡2[𝑅(𝑡₂)]𝑎(𝑠)^1/𝛼Δ𝑠. (2.9) By condition (𝐻₁), we obtain lim_𝑡→∞𝑦^Δ𝑚−1(𝑡)=−∞. Then it follows from Lemma 2.4 that lim_𝑡→∞𝑦(𝑡)=−∞, which is a contradiction since 𝑥(𝑡) and 𝑝(𝑡) are bounded. Hence, 𝑦^Δ𝑚(𝑡)=[𝑅(𝑡)/𝑎(𝑡)]^1/𝛼>0 for all 𝑡≥𝑡₁. The proof is completed.
Let 𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ) be the Banach space of all bounded rd-continuous functions on [𝑡₀,∞)_𝕋 with sup norm ‖𝑥‖=sup_𝑡≥𝑡0|𝑥(𝑡)|. Let 𝑋⊂𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ), we say that 𝑋 is uniformly Cauchy if for any given 𝜀>0, there exists 𝑡₁>𝑡₀ such that for any 𝑥∈𝑋, |𝑥(𝑢)−𝑥(𝑣)|<𝜀 for all 𝑢,𝑣∈[𝑡₁,∞)_𝕋. 𝑋 is said to be equicontinuous on [𝑎,𝑏]_𝕋 if, for any given 𝜀>0, there exists 𝛿>0 such that, for any 𝑥∈𝑋 and 𝑢,𝑣∈[𝑎,𝑏]_𝕋 with |𝑢−𝑣|<𝛿, |𝑥(𝑢)−𝑥(𝑣)|<𝜀. 𝑆∶𝑋→𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ) is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.

Lemma 2.7 (see [21]). Suppose that 𝑋⊂𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ) is bounded and uniformly Cauchy. Further, suppose that 𝑋 is equi-continuous on [𝑡₀,𝑡₁]_𝕋 for any 𝑡₁∈[𝑡₀,∞)_𝕋. Then 𝑋 is relatively compact.

Lemma 2.8 (see [21]). Suppose that 𝑋 is a Banach space and Ω is a bounded, convex, and closed subset of 𝑋. Suppose further that there exist two operators 𝑈 and 𝑆∶Ω→𝑋 such that(i)𝑈𝑥+𝑆𝑦∈Ω for all 𝑥,𝑦∈Ω,(ii)𝑈 is a contraction mapping,(iii)𝑆 is completely continuous. Then 𝑈+𝑆 has a fixed point in Ω.

3. Main Results and Examples

Now, we state and prove our main results.

Theorem 3.1. Assume that (𝐻₁) and (𝐻₂) hold. Then (1.7) has a nonoscillatory bounded solution 𝑥(𝑡) with liminf_𝑡→∞|𝑥(𝑡)|>0 if and only if there exists some constant 𝐾≠0 such that ^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀1)[𝑎(𝑠)^∞_𝑠𝑓(𝜃,|𝐾|)Δ𝜃]^1/𝛼Δ𝑠<∞. (3.1)

Proof. Sufficiency. Assume that (1.7) has a nonoscillatory bounded solution 𝑥(𝑡) on [𝑡₀,∞)_𝕋 with liminf_𝑡→∞|𝑥(𝑡)|>0. Without loss of generality, we assume that there is a constant 𝐾>0 and some 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>𝐾 and 𝑥(𝛿(𝑡))>𝐾 for 𝑡≥𝑡₁. It follows from Lemma 2.6 that 𝑦^Δ𝑚(𝑡)>0 for 𝑡≥𝑡₁. By assumption that 𝑥(𝑡) is bounded and condition (𝐻₂), we see that 𝑦(𝑡) is bounded. Thus, by Lemma 2.5 we get that there exists 𝑡₂≥𝑡₁ such that (−1)^𝑚−𝑘𝑦^Δ𝑘(𝑡)>0for𝑡≥𝑡≥𝑡₂,1≤𝑘≤𝑚. (3.2) Integrating (1.7) from 𝑡(≥𝑡₂) to ∞, we have 𝑎(𝑡)(𝑦^Δ𝑚(𝑡))^𝛼≥^∞_𝑡𝑓(𝑠,𝑥(𝛿(𝑠)))Δ𝑠≥^∞_𝑡𝑓(𝑠,𝐾)Δ𝑠for𝑡≥𝑡₂. (3.3) Therefore, it follows from (3.2) and (3.3) that, for 𝑡≥𝑡₂, ^∞_𝑡𝑔_𝑚−11(𝜎(𝜃),𝑡)[𝑎(𝜃)^∞_𝜃𝑓(𝑠,𝐾)Δ𝑠]^1/𝛼≤Δ𝜃_∞𝑡𝑦^Δ𝑚(𝜃)𝑔_𝑚−1(𝜎(𝜃),𝑡)Δ𝜃=𝑦^Δ𝑚−1(𝜃)𝑔_𝑚−1(𝜃,𝑡)|^∞_𝑡−^∞_𝑡𝑦^Δ𝑚−1(𝜃)𝑔_𝑚−2(𝜎(𝜃),𝑡)Δ𝜃=−lim_𝜃→∞(−1)^{𝑚−(𝑚−1)}𝑦^Δ𝑚−1(𝜃)𝑔_𝑚−1(𝜃,𝑡)−^∞_𝑡𝑦^Δ𝑚−1(𝜃)𝑔_𝑚−2(𝜎(𝜃),𝑡)Δ𝜃≤−^∞_𝑡𝑦^Δ𝑚−1(𝜃)𝑔_𝑚−2(𝜎(𝜃),𝑡)Δ𝜃=−𝑦^Δ𝑚−2(𝜃)𝑔_𝑚−2(𝜃,𝑡)|^∞_𝑡+(−1)^2∞_𝑡𝑦^Δ𝑚−2(𝜃)𝑔_𝑚−3(𝜎(𝜃),𝑡)Δ𝜃≤(−1)^2∞_𝑡𝑦^Δ𝑚−2(𝜃)𝑔_𝑚−3(𝜎(𝜃),𝑡)Δ𝜃⋯⋯⋯⋯≤(−1)^𝑚−1∞_𝑡𝑦^Δ(𝜃)Δ𝜃=(−1)^𝑚−1𝑦(𝜃)|^∞_𝑡<∞. (3.4) By Lemma 2.2, we see that (3.1) holds.
Necessity. Suppose that there exists some constant 𝐾>0 such that ^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠<∞, (3.5) where 𝐴(𝑠)=[∫^∞_𝑠𝑓(𝜃,𝐾)Δ𝜃]^1/𝛼/𝑎(𝑠). Then by Lemma 2.3 there exists 𝑡₁≥𝑡₀ such that ^∞_𝑡𝑔_𝑚−1(𝜎(𝑠),𝑡)𝐴(𝑠)Δ𝑠<1−𝑏₁𝑛𝐾 (3.6) and min{𝛿(𝑡),𝜏(𝑡)}≥𝑡₀ for 𝑡≥𝑡₁, where 𝑛>2(1−𝑏₁)/(1−𝑏₁−𝑎₁) is a constant. Let Ω={𝑥∈𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ)∶𝑛(1−𝑏₁−𝑎₁)−2(1−𝑏₁)𝑛𝐾≤𝑥(𝑡)≤𝐾for𝑡≥𝑡₀}. (3.7) It is easy to verify that Ω is a bounded, convex, and closed subset of 𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ).
Now we define two operators 𝑆 and 𝑇∶Ω→𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ) as follows: (𝑆𝑥)(𝑡)=𝑝(𝑡^∗)𝑥(𝜏(𝑡^∗)),(𝑇𝑥)(𝑡)=(𝑛−1)(1−𝑏₁)𝑛𝐾+(−1)^𝑚∞_𝑡∗𝑔_𝑚−1(𝜎(𝑠),𝑡^∗)𝐴(𝑠,𝑥)Δ𝑠, (3.8) where 𝑢^∗=max{𝑢,𝑡₁} for any 𝑢∈[𝑡₀,∞)_𝕋 and 𝐴(𝑠,𝑧)=[∫^∞_𝑠𝑓(𝜃,z(𝛿(𝜃)))Δ𝜃]^1/𝛼/𝑎(𝑠) for any 𝑧∈Ω. Now we show that 𝑆 and 𝑇 satisfy the conditions in Lemma 2.8.
(1) We will show that 𝑆𝑥+𝑇𝑦∈Ω for any 𝑥,𝑦∈Ω. In fact, for any 𝑥,𝑦∈Ω and 𝑡≥𝑡₀, 𝑥(𝑡),𝑦(𝑡)∈[[𝑛(1−𝑏₁−𝑎₁)−2(1−𝑏₁)]𝐾/𝑛,𝐾], (𝑆𝑥)(𝑡)+(𝑇𝑦)(𝑡)=(𝑛−1)(1−𝑏₁)𝑛𝐾+𝑝(𝑡^∗)𝑥(𝜏(𝑡^∗))+(−1)^𝑚∞_𝑡∗𝑔_𝑚−1(𝜎(𝑠),𝑡^∗≤)𝐴(𝑠,𝑥)Δ𝑠(𝑛−1)(1−𝑏₁)𝑛𝐾+𝑏₁𝐾+1−𝑏₁𝑛𝐾=𝐾,(𝑆𝑥)(𝑡)+(𝑇𝑦)(𝑡)=(𝑛−1)(1−𝑏₁)𝑛𝐾+𝑝(𝑡^∗)𝑥(𝜏(𝑡^∗))+(−1)^𝑚∞_𝑡∗𝑔_𝑚−1(𝜎(𝑠),𝑡^∗≥)𝐴(𝑠,𝑥)Δ𝑠(𝑛−1)(1−𝑏₁)𝑛𝐾−𝑎₁𝐾−1−𝑏₁𝑛𝐾=𝑛(1−𝑏₁−𝑎₁)−2(1−𝑏₁)𝑛𝐾, (3.9) which implies that 𝑆𝑥+𝑇𝑦∈Ω for any 𝑥,𝑦∈Ω.
(2) We will show that 𝑆 is a contraction mapping. Indeed, for any 𝑥,𝑦∈Ω and 𝑡≥𝑡₀, we have |(𝑆𝑥)(𝑡)−(𝑆𝑦)(𝑡)|=|𝑝(𝑡^∗)𝑥(𝜏(𝑡^∗))−𝑝(𝑡^∗)𝑦(𝜏(𝑡^∗))|≤max{𝑎₁,𝑏₁}‖𝑥−𝑦‖. (3.10) Therefore, we have ‖𝑆𝑥−𝑆𝑦‖≤max{𝑎₁,𝑏₁}‖𝑥−𝑦‖, (3.11) which implies that 𝑆 is a contraction mapping.
(3) We will show that 𝑇 is a completely continuous mapping.
(i)By the proof of (1), we see that [𝑛(1−𝑏₁−𝑎₁)−2(1−𝑏₁)]𝐾/𝑛≤(𝑇𝑥)(𝑡)≤𝐾 for 𝑡∈[𝑡₀,∞)_𝕋. That is, 𝑇Ω⊂Ω. (ii)We consider the continuity of 𝑇. Let 𝑥_𝑛∈Ω and ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞, then 𝑥∈Ω and |𝑥_𝑛(𝑡)−𝑥(𝑡)|→0 as 𝑛→∞ for any 𝑡∈[𝑡₀,∞)_𝕋. Consequently, for any 𝑠∈[𝑡₁,∞)_𝕋, we have lim_𝑛→∞|𝑔_𝑚−1(𝜎(𝑠),𝑡₁)[𝐴(𝑠,𝑥_𝑛)−𝐴(𝑠,𝑥)]|=0. (3.12) Since |𝑔_𝑚−1(𝜎(𝑠),𝑡₁)[𝐴(𝑠,𝑥_𝑛)−𝐴(𝑠,𝑥)]|≤2𝑔_𝑚−1(𝜎(𝑠),𝑡₁)𝐴(𝑠) (3.13) and, for any 𝑡∈[𝑡₀,∞)_𝕋, |(𝑇𝑥_𝑛)(𝑡)−(𝑇𝑥)(𝑡)|≤^∞_𝑡1𝑔_𝑚−1(𝜎(𝑠),𝑡₁)|𝐴(𝑠,𝑥_𝑛)−𝐴(𝑠,𝑥)|Δ𝑠, (3.14) we have ‖𝑇𝑥_𝑛−𝑇𝑥‖≤^∞_𝑡1𝑔_𝑚−1(𝜎(𝑠),𝑡₁)|𝐴(𝑠,𝑥_𝑛)−𝐴(𝑠,𝑥)|Δ𝑠. (3.15) By Chapter 5 in [4], we see that the Lebesgue dominated convergence theorem holds for the integral on time scales. Then lim_𝑛→∞‖𝑇𝑥_𝑛−𝑇𝑥‖=0, (3.16) which implies that 𝑇 is continuous on Ω. (iii)We show that 𝑇Ω is uniformly Cauchy. In fact, for any 𝜀>0, take 𝑡₂>𝑡₁ so that ^∞_𝑡2𝑔_𝑚−1(𝜎(𝑠),𝑡₂)𝐴(𝑠)Δ𝑠<𝜀. (3.17) Then for any 𝑥∈Ω and 𝑢,𝑣∈[𝑡₂,∞)_𝕋, we have |(𝑇𝑥)(𝑢)−(𝑇𝑥)(𝑣)|<2𝜀, (3.18) which implies that 𝑇Ω is uniformly Cauchy. (iv)We show that 𝑇Ω is equicontinuous on [𝑡₀,𝑡₂]_𝕋 for any 𝑡₂∈[𝑡₀,∞)_𝕋. Without loss of generality, we assume 𝑡₂≥𝑡₁. For any 𝜀>0, choose 𝛿=𝜀/(1+∫^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠), then when 𝑢,𝑣∈[𝑡₀,𝑡₂] with |𝑢−𝑣|<𝛿, we have by Lemma 2.1 that for any 𝑥∈Ω, |(𝑇𝑥)(𝑢)−(𝑇𝑥)(𝑣)|=|^∞_𝑢∗𝑔_𝑚−1(𝜎(𝑠),𝑢^∗)𝐴(𝑠,𝑥)Δ𝑠−^∞_𝑣∗𝑔_𝑚−1(𝜎(𝑠),𝑣^∗)𝐴(𝑠,𝑥)Δ𝑠|=|^∞_𝑢∗ℎ_𝑚−1(𝑢^∗,𝜎(𝑠))𝐴(𝑠,𝑥)Δ𝑠−^∞_𝑣∗ℎ_𝑚−1(𝑣^∗,𝜎(𝑠))𝐴(𝑠,𝑥)Δ𝑠|=|^∞_𝑢∗[^𝑢∗_𝜎(𝑠)ℎ_𝑚−2−(𝜃,𝜎(𝑠))𝐴(𝑠,𝑥)Δ𝜃]Δ𝑠_∞𝑣∗[^𝑣∗_𝜎(𝑠)ℎ_𝑚−2(𝜃,𝜎(𝑠))𝐴(𝑠,𝑥)Δ𝜃]Δ𝑠|=|^∞_𝑢∗[^∞_𝜃ℎ_𝑚−2−(𝜃,𝜎(𝑠))𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃_∞𝑣∗[^∞_𝜃ℎ_𝑚−2(𝜃,𝜎(𝑠))𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃|=|^∞_𝑢∗[^∞_𝜃𝑔_𝑚−2−(𝜎(𝑠),𝜃)𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃_∞𝑣∗[^∞_𝜃𝑔_𝑚−2(𝜎(𝑠),𝜃)𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃|=|^𝑣∗_𝑢∗[^∞_𝜃𝑔_𝑚−2(𝜎(𝑠),𝜃)𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃|≤|^𝑣∗_𝑢∗[^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀)𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃|=|𝑢^∗−𝑣^∗|^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀)𝐴(𝑠,𝑥)Δ𝑠≤𝛿^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠<𝜀, (3.19)which implies that 𝑇Ω is equi-continuous on [𝑡₀,𝑡₂]_𝕋 for any 𝑡₂∈[𝑡₀,∞)_𝕋.
By Lemma 2.7, we see that 𝑇 is a completely continuous mapping. It follows from Lemma 2.8 that there exists 𝑥∈Ω such that (𝑈+𝑆)𝑥=𝑥, which is the desired bounded solution of (1.7) with liminf_𝑡→∞|𝑥(𝑡)|>0. The proof is completed.

Theorem 3.2. Assume that (𝐻₁) and (𝐻₃) hold, and that 𝜏 has the inverse 𝜏⁻¹∈𝐶(𝕋,𝕋). Then (1.7) has a nonoscillatory bounded solution 𝑥(𝑡) with liminf_𝑡→∞|𝑥(𝑡)|>0 if and only if there exists some constant 𝐾≠0 such that (3.1) holds.

Proof. The proof of sufficiency is similar to that of Theorem 3.1.
Necessity. Suppose that there exists some constant 𝐾>0 such that ^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠<∞, (3.20) where 𝐴(𝑠)=[∫^∞_𝑠𝑓(𝜃,𝐾)Δ𝜃]^1/𝛼/𝑎(𝑠). Then by Lemma 2.3 there exists 𝑡₁≥𝑡₀ such that ^∞_{𝜏−1(𝑡)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹𝑏(𝑡))𝐴(𝑠)Δ𝑠<²𝑛𝐾, (3.21) and min{𝛿(𝜏⁻¹(𝑡)),𝜏⁻¹(𝑡)}≥𝑡₀ for 𝑡≥𝑡₁, where 𝑛>2𝑏₂/(𝑏₂−1) is a constant. Let Ω={𝑥∈𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ)∶(𝑛−2)𝑏₂−𝑛𝑎₂𝑛𝐾≤𝑥(𝑡)≤𝐾for𝑡≥𝑡₀}. (3.22) It is easy to verify that Ω is a bounded, convex and closed subset of 𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ).
Now we define two operators 𝑆 and 𝑇∶Ω→𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ) as follows: (𝑆𝑥)(𝑡)=𝑥(𝜏⁻¹(𝑡^∗))𝑝(𝜏⁻¹(𝑡^∗+))(𝑛−1)𝑏₂𝐾−𝑛𝑝(𝜏⁻¹(𝑡^∗1))(𝑇x)(𝑡)=𝑝(𝜏⁻¹(𝑡^∗))(−1)^𝑚−1∞_{𝜏−1(𝑡∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑡^∗))𝐴(𝑠,𝑥)Δ𝑠, (3.23) where 𝑢^∗=max{𝑢,𝑡₁} for any 𝑢∈[𝑡₀,∞)_𝕋 and 𝐴(𝑠,𝑧)=[∫^∞_𝑠𝑓(𝜃,𝑧(𝛿(𝜃)))Δ𝜃]^1/𝛼/𝑎(𝑠) for any 𝑧∈Ω. Now we show that 𝑆 and 𝑇 satisfy the conditions in Lemma 2.8.
We will show that 𝑆𝑥+𝑇𝑦∈Ω for any 𝑥,𝑦∈Ω. In fact, for any 𝑥,𝑦∈Ω and 𝑡≥𝑡₀, 𝑥(𝑡),𝑦(𝑡)∈[[(𝑛−2)𝑏₂−𝑛]𝐾/𝑎₂𝑛,𝐾], (𝑆𝑥)(𝑡)+(𝑇𝑦)(𝑡)=𝑥(𝜏⁻¹(𝑡^∗))𝑝(𝜏⁻¹(𝑡^∗+1))−𝑝(𝜏⁻¹(𝑡^∗[))(𝑛−1)𝑏₂𝑛𝐾+(−1)^𝑚∞_{𝜏−1(𝑡∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑡^∗≤1))𝐴(𝑠,𝑥)Δ𝑠]𝑏₂[(𝑛−1)𝑏₂𝑛𝑏𝐾+²𝑛𝐾]=𝐾,(𝑆𝑥)(𝑡)+(𝑇𝑦)(𝑡)=𝑥(𝜏⁻¹(𝑡^∗))𝑝(𝜏⁻¹(𝑡^∗+1))−𝑝(𝜏⁻¹(𝑡^∗[))(𝑛−1)𝑏₂𝑛𝐾+(−1)^𝑚∞_{𝜏−1(𝑡∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑡^∗≥1))𝐴(𝑠,𝑥)Δ𝑠]𝑎₂[(𝑛−1)𝑏₂𝑛𝑏𝐾−𝐾−²𝑛=𝐾](𝑛−2)𝑏₂−𝑛𝑎₂𝑛𝐾, (3.24) and |(𝑇𝑥)(𝑡)|≤𝑏₂𝐾/𝑛, which implies that 𝑆𝑥+𝑇𝑦∈Ω for any 𝑥,𝑦∈Ω and 𝑇Ω is uniformly bounded.
Now we show that 𝑇Ω is equicontinuous on [𝑡₀,𝑡₂]_𝕋 for any 𝑡₂∈[𝑡₀,∞)_𝕋. Without loss of generality, we assume that 𝑡₂≥𝑡₁. Since 1/𝑝(𝜏⁻¹(𝑡)),𝜏⁻¹(𝑡) are continuous on [𝑡₀,𝑡₂]_𝕋, so they are uniformly continuous on [𝑡₀,𝑡₂]_𝕋. For any 𝜀>0, choose 𝛿>0 such that when 𝑢,𝑣∈[𝑡₀,𝑡₂]_𝕋 with |𝑢−𝑣|<𝛿, we have |1𝑝(𝜏⁻¹−1(𝑢))𝑝(𝜏⁻¹𝜀(𝑣))|<1+∫^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠|𝜏⁻¹(𝑢)−𝜏⁻¹𝜀(𝑣)|<1+∫^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀.)𝐴(𝑠)Δ𝑠 (3.25) Then, it follows from Lemma 2.1 that, for any 𝑥∈Ω, 1|(𝑇𝑥)(𝑢)−(𝑇𝑥)(𝑣)|=|𝑝(𝜏⁻¹(𝑢^∗))^∞_𝜏⁻¹_(𝑢^∗₎𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑢^∗−1))𝐴(𝑠,𝑥)Δ𝑠𝑝(𝜏⁻¹(𝑣^∗))^∞_𝜏⁻¹_(𝑣^∗₎𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑣^∗1))𝐴(𝑠,𝑥)Δ𝑠|≤|[𝑝(𝜏⁻¹(𝑢^∗−1))𝑝(𝜏⁻¹(𝑣^∗]))^∞_𝜏⁻¹_(𝑢^∗₎𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑢^∗1))𝐴(𝑠,𝑥)Δ𝑠|+|𝑝(𝜏⁻¹(𝑣^∗[))^∞_𝜏⁻¹_(𝑢^∗₎𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑢^∗−))𝐴(𝑠,𝑥)Δ𝑠_{∞𝜏−1(𝑣∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑣^∗1))𝐴(𝑠,𝑥)Δ𝑠]|=|[𝑝(𝜏⁻¹(𝑢^∗−1))𝑝(𝜏⁻¹(𝑣^∗]))^∞_𝜏⁻¹_(𝑢^∗₎𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑢^∗1))𝐴(𝑠,𝑥)Δ𝑠|+|𝑝(𝜏⁻¹(𝑣^∗))^{𝜏−1(𝑣∗)}_𝜏⁻¹_(𝑢^∗₎[^∞_𝜃𝑔_𝑚−21(𝜎(𝑠),𝜃)𝐴(𝑠,𝑥)Δ𝑠]Δ𝜃|≤|[𝑝(𝜏⁻¹(𝑢^∗−1))𝑝(𝜏⁻¹(𝑣^∗]))^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀1)𝐴(𝑠)Δ𝑠|+|𝑝(𝜏⁻¹(𝑣^∗))^{𝜏−1(𝑣∗}_𝜏⁾⁾⁻¹_(𝑢^∗₎[^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠]Δ𝜃|≤𝜀+|𝜏⁻¹(𝑢^∗)−𝜏⁻¹(𝑣^∗)|^∞_𝑡0𝑔_𝑚−2(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠<2𝜀, (3.26) which implies that 𝑇Ω is equi-continuous on [𝑡₀,𝑡₂]_𝕋 for any 𝑡₂∈[𝑡₀,∞)_𝕋.
The rest of the proof is similar to that of Theorem 3.1. The proof is completed.

Theorem 3.3. Assume that (𝐻₁) and (𝐻₄) hold and that 𝜏 has the inverse 𝜏⁻¹∈𝐶(𝕋,𝕋). Then (1.7) has a nonoscillatory bounded solution 𝑥(𝑡) with liminf_𝑡→∞|𝑥(𝑡)|>0 if and only if there exists some constant 𝐾≠0 such that (3.1) holds.

Proof. The proof of sufficiency is similar to that of Theorem 3.1.
Necessity. Suppose that there exists some constant 𝐾>0 such that ^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀)𝐴(𝑠)Δ𝑠<∞, (3.27) where 𝐴(𝑠)=[∫^∞_𝑠𝑓(𝜃,𝐾)Δ𝜃]^1/𝛼/𝑎(𝑠). Then by Lemma 2.3 there exists 𝑡₁≥𝑡₀ such that ^∞_{𝜏−1(𝑡)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹𝑎(𝑡))𝐴(𝑠)Δ𝑠<³−1𝑛𝐾 (3.28) and min{𝛿(𝜏⁻¹(𝑡)),𝜏⁻¹(𝑡)}≥𝑡₀ for 𝑡≥𝑡₁, where 𝑛>2 is a constant. Let Ω={𝑥∈𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ)∶(𝑛−2)(𝑎₃−1)𝑛(𝑏₃−1)𝐾≤𝑥(𝑡)≤𝐾for𝑡≥𝑡₀}. (3.29) It is easy to verify that Ω is a bounded, convex, and closed subset of 𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ).
Now we define two operators 𝑆 and 𝑇∶Ω→𝐵𝐶_𝑟𝑑([𝑡₀,∞)_𝕋,ℝ) as follows: (𝑆𝑥)(𝑡)=𝑥(𝜏⁻¹(𝑡^∗))𝑝(𝜏⁻¹(𝑡^∗+))(𝑛−1)(𝑎₃−1)𝐾𝑛𝑝(𝜏⁻¹(𝑡^∗,1))(𝑇𝑥)(𝑡)=𝑝(𝜏⁻¹(𝑡^∗))(−1)^𝑚−1∞_{𝜏−1(𝑡∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑡^∗))𝐴(𝑠,𝑥)Δ𝑠, (3.30) where 𝑢^∗=max{𝑢,𝑡₁} for any 𝑢∈[𝑡₀,∞)_𝕋 and 𝐴(𝑠,𝑧)=[∫^∞_𝑠𝑓(𝜃,𝑧(𝛿(𝜃)))Δ𝜃]^1/𝛼/𝑎(𝑠) for any 𝑧∈Ω. Now we show that 𝑆 and 𝑇 satisfy the conditions in Lemma 2.8.
We will show that 𝑆𝑥+𝑇𝑦∈Ω for any 𝑥,𝑦∈Ω. In fact, for any 𝑥,𝑦∈Ω and 𝑡≥𝑡₀, 𝑥(𝑡),𝑦(𝑡)∈[(𝑛−2)(𝑎₃−1)𝐾/𝑛(𝑏₃−1),𝐾], (𝑆𝑥)(𝑡)+(𝑇𝑦)(𝑡)=𝑥(𝜏⁻¹(𝑡^∗))𝑝(𝜏⁻¹(𝑡^∗+1))𝑝(𝜏⁻¹(𝑡^∗))×[(𝑛−1)(𝑎₃−1)𝑛𝐾+(−1)^𝑚−1∞_{𝜏−1(𝑡∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑡^∗≤1))𝐴(𝑠,𝑥)Δ𝑠]𝑎₃[(𝑛−1)(𝑎₃−1)𝑛𝑎𝐾+𝐾+³−1𝑛𝐾]=𝐾,(𝑆𝑥)(𝑡)+(𝑇𝑦)(𝑡)=𝑥(𝜏⁻¹(𝑡^∗))𝑝(𝜏⁻¹(𝑡^∗+1))𝑝(𝜏⁻¹(𝑡^∗))×[(𝑛−1)(𝑎₃−1)𝑛𝐾+(−1)^𝑚−1∞_{𝜏−1(𝑡∗)}𝑔_𝑚−1(𝜎(𝑠),𝜏⁻¹(𝑡^∗≥1))𝐴(𝑠,𝑥)Δ𝑠]𝑏₃[(𝑛−1)(𝑎₃−1)𝑛𝐾+(𝑛−2)(𝑎₃−1)𝐾𝑛(𝑏₃−𝑎−1)³−1𝑛=𝐾](𝑛−2)(𝑎₃−1)𝐾𝑛(𝑏₃−1) (3.31) and |(𝑇𝑥)(𝑡)|≤(𝑎₃−1)𝐾/𝑛, which implies that 𝑆𝑥+𝑇𝑦∈Ω for any 𝑥,𝑦∈Ω and 𝑇Ω is uniformly bounded. The rest of the proof is similar to that of Theorem 3.2. The proof is completed.
According to the proofs of Theorem 3.1, Theorem 3.2, and Theorem 3.3 in [23], we have

Lemma 3.4. Assume that (𝐻₁) holds. Suppose that one of the following holds:(i)𝑝(𝑡) satisfies condition (𝐻₂),(ii)𝑝(𝑡) satisfies condition (𝐻₃) and 𝜏 has the inverse 𝜏⁻¹∈𝐶(𝕋,𝕋),(iii)𝑝(𝑡) satisfies condition (𝐻₄) and 𝜏 has the inverse 𝜏⁻¹∈𝐶(𝕋,𝕋). If 𝑥(𝑡) is a bounded nonoscillatory solution of (1.7), then liminf_𝑡→∞|𝑥(𝑡)|=0 implies that lim_𝑡→∞𝑥(𝑡)=0.
By Lemma 3.4, Theorem 3.1,Theorem 3.2 and Theorem 3.3, we have.

Corollary 3.5. Assume that (𝐻₁) holds. Suppose that one of the following holds:(i)𝑝(𝑡) satisfies condition (𝐻₂),(ii)𝑝(𝑡) satisfies condition (𝐻₃) and 𝜏 has the inverse 𝜏⁻¹∈𝐶(𝕋,𝕋),(iii)𝑝(𝑡) satisfies condition (𝐻₄) and 𝜏 has the inverse 𝜏⁻¹∈𝐶(𝕋,𝕋). Then every bounded solution of (1.7) is oscillation or converges to zero at infinity if and only if there exists some constant 𝐾≠0 such that ^∞_𝑡0𝑔_𝑚−1(𝜎(𝑠),𝑡₀1)[𝑎(𝑠)^∞_𝑠𝑓(𝜃,|𝐾|)Δ𝜃]^1/𝛼Δ𝑠=∞. (3.32)

Example 3.6. Let 𝕋={𝑞^𝑛∶𝑛∈ℤ}∪{0} with 𝑞>1. Consider the following higher order dynamic equation: {𝑡^𝛼[(𝑥(𝑡)−𝑝_𝑘(𝑡)𝑥(𝑞𝑡))^Δ𝑚]^𝛼}^Δ+𝑞𝑔^𝛼_𝑚−1(𝑞²𝑡,0)−𝑔^𝛼_𝑚−1(𝑞𝑡,0)(𝑞−1)𝑞𝑡²𝑔^𝛼_𝑚−1(𝑞²𝑡,0)𝑔^𝛼_𝑚−1𝑥(𝑞𝑡,0)^2𝑟+1(𝑞^𝑟+3𝑡)=0for𝑡∈[1,∞)_𝕋, (3.33) where 𝑚,𝑟∈𝐍, 𝛼 is the quotient of odd positive integers,𝑝_𝑘(𝑡)=−2[(−1)^𝑘𝑘²+(−1)^log𝑞𝑡]/5,, where 𝑘=1,2,3, 𝑎(𝑡)=𝑡^𝛼, 𝜏(𝑡)=𝑞𝑡, 𝛿(𝑡)=𝑞^3+𝑟𝑡 and 𝑓(𝑡,𝑢)=(𝑞𝑔^𝛼_𝑚−1(𝑞²𝑡,0)−𝑔^𝛼_𝑚−1(𝑞𝑡,0)/(𝑞−1)𝑞𝑡²𝑔^𝛼_𝑚−1(𝑞²𝑡,0)𝑔^𝛼_𝑚−1(𝑞𝑡,0))𝑢^2𝑟+1.
It is easy to verify that 𝑝_𝑘(𝑡) satisfies condition (𝐻_𝑘+1) and 𝜏⁻¹∈𝐶(𝕋,𝕋). On the other hand, we have ^∞₁[1]𝑎(𝑡)^1/𝛼Δ𝑡=^∞₁[1𝑡^𝛼]^1/𝛼Δ𝑡=∞ (3.34) andfor any constant 𝐾>0, ^∞₁𝑔_𝑚−11(𝜎(𝑠),1)[𝑎(𝑠)^∞_𝑠𝑓(𝜃,𝐾)Δ𝜃]^1/𝛼Δ𝑠=𝐾^{(2𝑟+1)/𝛼∞}₁𝑔_𝑚−11(𝜎(𝑠),1){𝑠^𝛼∞_𝑠[−1𝜃𝑔^𝛼_𝑚−1](𝜎(𝜃),0)^Δ}^1/𝛼Δ𝑠=𝐾^{(2𝑟+1)/𝛼∞}₁𝑔_𝑚−11(𝜎(𝑠),1)[𝑠^𝛼1𝑠𝑔^𝛼_𝑚−1](𝜎(𝑠),0)^1/𝛼Δ𝑠=𝐾^{(2𝑟+1)/𝛼∞}₁1𝑠^1+1/𝛼Δ𝑠=𝐾2𝑟+1𝛼𝑞^1/𝛼(𝑞−1)𝑞^1/𝛼−1<∞. (3.35) That is, conditions (𝐻₁) and (3.1) hold. By Theorem 3.1, Theorem 3.2, and Theorem 3.3, we see that (3.33) has a nonoscillatory bounded solution 𝑥(𝑡) with liminf_𝑡→∞|𝑥(𝑡)|>0.

Remark 3.7. Results of this paper can be extended to the case with several delays easily.

Acknowledgment

Project Supported by NSF of China (10861002) and NSF of Guangxi (2010GXNSFA013106, 2011GXNSFA014781) and SF of Education Department of Guangxi (200911MS212) and Innovation Project of Guangxi Graduate Education 2010105930701M43.