Bounded Oscillation of a Forced Nonlinear Neutral Differential Equation

Abstract

This paper is concerned with the 𝑛th-order forced nonlinear neutral differential equation [𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡))]^(𝑛)+∑^𝑚𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖(𝑥(𝜎_𝑖1(𝑡)),𝑥(𝜎_𝑖2(𝑡)),…,𝑥(𝜎_𝑖𝑘𝑖(𝑡)))=𝑔(𝑡),𝑡≥𝑡₀. Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.

1. Introduction

Consider the following 𝑛th-order forced nonlinear neutral differential equation:[𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡))]^(𝑛)+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝜎_𝑖1(𝑡),𝑥𝜎_𝑖2(𝑡),…,𝑥𝜎_𝑖𝑘𝑖(𝑡)=𝑔(𝑡),𝑡≥𝑡₀,(1.1) where 𝑡₀∈ℝ and 𝑛,𝑚,𝑘_𝑖∈ℕ are constants for 1≤𝑖≤𝑚. In what follows, we assume that(𝐴1)  𝑝,𝑔,𝜏,𝜎_𝑖𝑗∈𝐶([𝑡₀,+∞),ℝ) and 𝑞_𝑖∈𝐶([𝑡₀,+∞),ℝ⁺) satisfy that lim_𝑡→+∞𝜏(𝑡)=lim_𝑡→+∞𝜎_𝑖𝑗(𝑡)=+∞,1≤𝑗≤𝑘_𝑖,1≤𝑖≤𝑚,(1.2) and there exists 1≤𝑖₀≤𝑚 such that 𝑞_𝑖0 is positive eventually:(𝐴2)  𝜏 is strictly increasing and 𝜏(𝑡)<𝑡 in [𝑡₀,+∞);(𝐴3)  𝑓_𝑖∈𝐶(ℝ^𝑘𝑖,ℝ) satisfies that 𝑓𝑓_𝑖𝑖𝑢₁,𝑢₂,…,𝑢_𝑘𝑖>0,∀𝑢₁,𝑢₂,…,𝑢_𝑘𝑖∈ℝ⁺⧵{0}^𝑘𝑖,𝑢₁,𝑢₂,…,𝑢_𝑘𝑖<0,∀𝑢₁,𝑢₂,…,𝑢_𝑘𝑖∈ℝ₋⧵{0}^𝑘𝑖(1.3) for 1≤𝑖≤𝑚.

During the last decades, the oscillation criteria and the existence results of nonoscillatory solutions for various linear and nonlinear differential equations have been studied extensively, for example, see [1–28] and the references cited therein. In particular, Zhang and Yan [25] obtained some sufficient conditions for the oscillation of the first-order linear neutral delay differential equation with positive and negative coefficients:[𝑥(𝑡)−𝑝(𝑡)𝑥(𝑡−𝜏)]^+𝑞(𝑡)𝑥(𝑡−𝜎)−𝑟(𝑡)𝑥(𝑡−𝛿)=0,𝑡≥𝑡₀,(1.4) where 𝑝,𝑞,𝑟∈𝐶([𝑡₀,+∞),ℝ⁺),𝜏>0, and 𝜎≥𝛿≥0. Das and Misra [7] studied the nonhomogeneous neutral delay differential equation:[𝑥(𝑡)−𝑐𝑥(𝑡−𝜏)]^+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=𝑔(𝑡),𝑡≥𝑡₀,(1.5) where 𝑞,𝑔∈𝐶([𝑇,+∞),ℝ⁺⧵{0}),𝜎>0,𝜏>0,𝑐∈[0,1),𝑓∶ℝ→ℝ, 𝑡𝑓(𝑡)>0 for 𝑡≠0, 𝑓 is nondecreasing, Lipschitzian, and satisfies ∫^𝑘0(1/𝑓(𝑡))𝑑𝑡<+∞ for every 𝑘>0, and they obtained a necessary and sufficient condition for the solutions of (1.5) to be oscillatory or tend to zero asymptotically. Parhi and Rath [18] extended Das and Misra’s result to the following forced first-order neutral differential equation with variable coefficients:[𝑥(𝑡)−𝑝(𝑡)𝑥(𝑡−𝜏)]^+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=𝑔(𝑡),𝑡≥0,(1.6) where 𝑝∈𝐶(ℝ⁺,ℝ), and they got necessary and sufficient conditions which ensures every solution of (1.6) is oscillatory or tends to zero or to ±∞ as 𝑡→+∞. By using Banach’s fixed point theorem, Zhang et al. [24] proved the existence of a nonoscillatory solution for the first-order linear neutral delay differential equation:[𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏)]^+^𝑛_𝑖=1𝑓_𝑖(𝑡)𝑥𝑡−𝜎_𝑖=0,𝑡≥𝑡₀,(1.7) where 𝑝∈𝐶([𝑡₀,+∞),ℝ),𝜏>0,𝜎_𝑖∈ℝ⁺, and 𝑓_𝑖∈𝐶([𝑡₀,+∞),ℝ) for 1≤𝑖≤𝑚. Çakmak and Tiryaki [6] showed several sufficient conditions for the oscillation of the forced second-order nonlinear differential equations with delayed argument in the form:𝑥^(𝑡)+𝑝(𝑡)𝑓(𝑥(𝛼(𝑡)))=𝑔(𝑡),𝑡≥𝑡₀≥0,(1.8) where 𝑝,𝛼,𝑔∈𝐶([𝑡₀,+∞),ℝ), 𝛼(𝑡)≤𝑡,lim_𝑡→+∞𝛼(𝑡)=+∞, and 𝑓∈𝐶(ℝ,ℝ). Travis [20] investigated the oscillatory behavior of the second-order differential equation with functional argument:𝑥^(𝑡)+𝑝(𝑡)𝑓(𝑥(𝑡),𝑥(𝛼(𝑡)))=0,𝑡≥𝑡₀,(1.9) where 𝑝,𝛼∈𝐶([𝑡₀,+∞),ℝ) and 𝑓∈𝐶(ℝ²,ℝ) satisfies that 𝑓(𝑠,𝑡) has the same sign of 𝑠 and 𝑡 when they have the same sign. Lin [12] got some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral differential equation:[𝑥(𝑡)−𝑝(𝑡)𝑥(𝑡−𝜏)]^+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=0,𝑡≥0,(1.10) where 𝑝,𝑞∈𝐶(ℝ⁺,ℝ),𝑝∈[0,1) with 0≤𝑝(𝑡)≤𝑝 eventually, 𝑓∈𝐶(ℝ,ℝ), 𝑓 is nondecreasing and 𝑡𝑓(𝑡)>0 for 𝑡≠0. Kulenović and Hadžiomerspahić [9] deduced the existence of a nonoscillatory solution for the neutral delay differential equation of second order with positive and negative coefficients:[𝑥(𝑡)+𝑐𝑥(𝑡−𝜏)]^+𝑞₁(𝑡)𝑥𝑡−𝜎₁−𝑞₂(𝑡)𝑥𝑡−𝜎₂=0,𝑡≥𝑡₀,(1.11) where 𝑐≠±1, 𝜏>0,𝜎_𝑖∈ℝ⁺,𝑞_𝑖∈𝐶([𝑡₀,+∞),ℝ⁺), and ∫^𝑡+∞₀𝑞_𝑖(𝑡)𝑑𝑡<+∞ for 𝑖∈{1,2}. Utilizing the fixed point theorems due to Banach, Schauder and Krasnoselskii, and Zhou and Zhang [27], and Zhou et al. [28] established some sufficient conditions for the existence of a nonoscillatory solution of the following higher-order neutral functional differential equations:[𝑥(𝑡)[+𝑥(𝑐𝑡𝑥)+(𝑡−𝑝(𝜏𝑡))]𝑥(^(𝑛)𝑡+−(𝜏−)1])^𝑛+1_(𝑛)[𝑃(𝑡)𝑥(𝑡−𝜎)−𝑄(𝑡)𝑥(𝑡−𝛿)]=0,𝑡≥𝑡₀,+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝑡−𝜎_𝑖=𝑔(𝑡),𝑡≥𝑡₀,(1.12) where 𝑐∈ℝ⧵{±1},𝜏,𝜎,𝛿,𝜎_𝑖∈ℝ⁺,𝑃,𝑄∈𝐶([𝑡₀,+∞),ℝ⁺), and 𝑝,𝑔,𝑓_𝑖∈𝐶([𝑡₀,+∞),ℝ) for 1≤𝑖≤𝑚. Li et al. [11] investigated the existence of an unbounded positive solution, bounded oscillation, and nonoscillation criteria for the following even-order neutral delay differential equation with unstable type:[𝑥(𝑡)−𝑝(𝑡)𝑥(𝑡−𝜏)]^(𝑛)−𝑞(𝑡)||𝑥(𝑡−𝜎)||^𝛼−1𝑥(𝑡−𝜎)=0,𝑡≥𝑡₀,(1.13) where 𝜏>0,𝜎>0,𝛼≥1, and 𝑝,𝑞∈𝐶([𝑡₀,+∞),ℝ⁺). Zhang and Yan [22] obtained some sufficient conditions for oscillation of all solutions of the even-order neutral differential equation with variable coefficients and delays:[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]^(𝑛)+𝑞(𝑡)𝑥(𝜎(𝑡))=0,𝑡≥𝑡₀,(1.14) where 𝑛 is even, 𝑝,𝑞,𝜏,𝜎∈𝐶([𝑡₀,+∞),ℝ⁺),𝑝(𝑡)<1,𝜏(𝑡)≤𝑡 and 𝜎(𝑡)≤𝑡 for 𝑡∈[𝑡₀,+∞), and lim_𝑡→+∞𝜏(𝑡)=lim_𝑡→+∞𝜎(𝑡)=+∞. Yilmaz and Zafer [21] discussed sufficient conditions for the existence of positive solutions and the oscillation of bounded solutions of the 𝑛th-order neutral type differential equations:[𝑥([𝑡)𝑥+(𝑡)𝑝+(𝑡)𝑐𝑥𝑥((𝜏𝜏((𝑡𝑡))))]]^(𝑛)_(𝑛)+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0,𝑡≥𝑡₀,+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=𝑔(𝑡),𝑡≥𝑡₀,(1.15) where 𝑐∈ℝ⧵{±1}, 𝜏,𝜎∈𝐶([𝑡₀,+∞),ℝ⁺),𝑝,𝑞,𝑔∈𝐶([𝑡₀,+∞),ℝ), and 𝑓∈𝐶(ℝ,ℝ). Bolat and Akin [4, 5] got sufficient criteria for oscillatory behaviour of solutions for the higher-order neutral type nonlinear forced differential equations with oscillating coefficients:[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]^(𝑛)+^𝑚_𝑖=1[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]_(𝑛)𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝜎_𝑖(𝑡)=0,𝑡≥𝑡₀,+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝜎_𝑖(𝑡)=𝑔(𝑡),𝑡≥𝑡₀,(1.16) where 𝑛∈ℕ⧵{1},𝑚∈ℕ,𝑝,𝑓_𝑖,𝑔,𝜏,𝜎_𝑖∈𝐶([𝑡₀,+∞),ℝ), 𝑓_𝑖 is nondecreasing and 𝑢𝑓_𝑖(𝑢)>0 for 𝑢≠0,𝜎_𝑖∈𝐶¹([𝑡₀,+∞),ℝ),𝜎^𝑖(𝑡)>0,𝜎_𝑖(𝑡)≤𝑡 for 𝑡∈[𝑡₀,+∞), lim_𝑡→+∞𝜏(𝑡)=lim_𝑡→+∞𝜎_𝑖(𝑡)=+∞ for 1≤𝑖≤𝑚, and 𝑝 and 𝑔 are oscillating functions. Zhou and Yu [26] attempted to extend the result of Bolat and Akin [4] and established a necessary and sufficient condition for the oscillation of bounded solutions of the higher-order nonlinear neutral forced differential equation of the form:[𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡))]^(𝑛)+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝜎_𝑖(𝑡)=𝑔(𝑡),𝑡≥𝑡₀,(1.17) where 𝑛∈ℕ⧵{1},𝑚∈ℕ, and(𝐶₁)  𝑝,𝑞_𝑖,𝜏,𝑔∈𝐶([𝑡₀,+∞),ℝ) for 𝑖=1,2,…,𝑚 and lim_𝑡→+∞𝜏(𝑡)=+∞;(𝐶₂)  𝑝 and 𝑔 are oscillating functions;(𝐶₃)  𝜎_𝑖∈𝐶([𝑡₀,+∞),ℝ),𝜎^𝑖(𝑡)>0,𝜎_𝑖(𝑡)≤𝑡 and lim_𝑡→+∞𝜎_𝑖(𝑡)=+∞ for 𝑖=1,2,…,𝑚;(𝐶₄)  𝑓_𝑖∈𝐶(ℝ,ℝ) is nondecreasing function, 𝑢𝑓_𝑖(𝑢)>0 for 𝑢≠0 and 𝑖=1,2,…,𝑚.

That is, they claimed the following result.

Theorem 1.1 (see [26, Theorem 2.1]). Assume that(𝐶₅) there is an oscillating function 𝑟∈𝐶([𝑡₀,+∞),ℝ) such that 𝑟^(𝑛)(𝑡)=𝑔(𝑡) and lim_𝑡→+∞𝑟(𝑡)=0;(𝐶₆)  𝑝 is an oscillating function and |𝑝(𝑡)|≤𝑝₀<1/2;(𝐶₇)  𝑞_𝑖(𝑡)≥0, 𝑖=1,2,…,𝑚.Then, every bounded solution of (1.17) either oscillates or tends to zero if and only if ^𝑡+∞₀𝑠^𝑛−1𝑞_𝑖(𝑠)𝑑𝑠=+∞,𝑖=1,2,…,𝑚.(1.18)

We, unfortunately, point out that the necessary part in Theorem 1.1 is false, see Remark 4.2 and Example 4.7 below. It is clear that (1.1) includes (1.4)–(1.17) as special cases. To the best of our knowledge, there is no literature referred to the oscillation and existence of uncountably many bounded nonoscillatory solutions of (1.1). The aim of this paper is to establish the bounded oscillation and the existence of uncountably many bounded positive and negative solutions for (1.1) without the monotonicity of the nonlinear term 𝑓_𝑖. Our results extend and improve substantially some known results in [4, 5, 9, 10, 20, 24, 26–28] and correct Theorem  2.1 in [26].

The paper is organized as follows. In Section 2, a few notation and lemmas are introduced and proved, respectively. In Section 3, by employing Krasnoselskii’s fixed point theorem and some techniques, the existence of uncountably many bounded positive and negative solutions for (1.1) are given, and some necessary and sufficient conditions for all bounded solutions of (1.1) to be oscillatory or tend to zero as 𝑡→+∞ are provided. In Section 4, a number of examples which clarify advantages of our results are constructed.

2. Preliminaries

It is assumed throughout this paper that ℝ=(−∞,+∞),ℝ⁺=[0,+∞),ℝ₋=(−∞,0] and𝛽=min𝑡₀,inf𝜏(𝑡),𝜎_𝑖𝑗(𝑡)∶𝑡∈𝑡₀,+∞,1≤𝑗≤𝑖_𝑘,1≤𝑖≤𝑚.(2.1) By a solution of (1.1), we mean a function 𝑥∈𝐶([𝛽,+∞),ℝ) for some 𝑇≥𝑡₀+𝛽, such that 𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡)) is 𝑛 times continuously differentiable in [𝑇,+∞) and such that (1.1) is satisfied for 𝑡≥𝑇. As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is nonoscillatory, that is, if it is eventually positive or eventually negative. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Let 𝐵𝐶([𝛽,+∞),ℝ) stand for the Banach space of all bounded continuous functions in [𝛽,+∞) with the norm ‖𝑥‖=sup_𝑡≥𝛽|𝑥(𝑡)| for each 𝑥∈𝐵𝐶([𝛽,+∞),ℝ) and𝐴(𝑁,𝑀)={𝑥∈𝐵𝐶([𝛽,+∞),ℝ)∶𝑁≤𝑥(𝑡)≤𝑀,𝑡≥𝛽}for𝑀,𝑁∈ℝwith𝑀>𝑁.(2.2) It is easy to see that 𝐴(𝑁,𝑀) is a bounded closed and convex subset of the Banach space 𝐵𝐶([𝛽,+∞),ℝ).

Lemma 2.1. Let 𝑛∈ℕ and 𝑥∈𝐶^𝑛([𝑡₀,+∞),ℝ) be bounded. If 𝑥^(𝑛)(𝑡)≤0 eventually, then(a)lim_𝑡→+∞𝑥(𝑡) exists and lim_𝑡→+∞𝑥^(𝑖)(𝑡)=0 for 1≤𝑖≤𝑛−1; furthermore, there exists 𝜃=0 for 𝑛 odd and 𝜃=1 for 𝑛 even such that(b)(−1)^𝜃+𝑖𝑥^(𝑖)(𝑡)≥0 eventually for 1≤𝑖≤𝑛;(c)(−1)^𝜃+𝑖𝑥^(𝑖) is nonincreasing eventually for 0≤𝑖≤𝑛−1.

Proof. Now, we consider two possible cases below.
Case 1. Assume that 𝑛=1. Let 𝜃=0. Note that 𝑥′(𝑡)≤0 eventually. It follows that there exists a constant 𝑡₁>𝑡₀ satisfying 𝑥^(𝑡)≤0, for all 𝑡≥𝑡₁, which yields that 𝑥 is nonincreasing in [𝑡₁,+∞). Since 𝑥 is bounded in [𝑡₀,+∞), it follows that lim_𝑡→+∞𝑥(𝑡) exists.
Case 2. Assume that 𝑛≥2. Notice that 𝜃+𝑛 is odd. It follows that (−1)^𝜃+𝑛𝑥^(𝑛)(𝑡)≥0 eventually, which implies that there exists a constant 𝑡₁>𝑡₀ satisfying (−1)^𝜃+𝑛𝑥^(𝑛)(𝑡)≥0,∀𝑡≥𝑡₁,(2.3) which means that (−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑡)isnonincreasingin𝑡₁,+∞.(2.4)
Suppose that there exists a constant 𝑡₂≥𝑡₁ satisfying (−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑡₂)<0, which together with (2.4) gives that (−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑡)≤(−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)𝑡₂<0,∀𝑡≥𝑡₂,(2.5) which guarantees that (−1)^{𝜃+𝑛−2}𝑥^(𝑛−2)(𝑡) is increasing in [𝑡₂,+∞) and (−1)=^{𝜃+𝑛−1}_𝑡𝑡2𝑥^(𝑛−2)(𝑡)−(−1)^{𝜃+𝑛−1}𝑥^(𝑛−2)𝑡₂(−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑠)𝑑𝑠≤(−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)𝑡₂𝑡−𝑡₂⟶−∞as𝑡⟶+∞,(2.6) that is, lim_𝑡→+∞𝑥^(𝑛−2)(𝑡)=−∞,(2.7) which means that lim_𝑡→+∞𝑥^(𝑛−3)(𝑡)=lim_𝑡→+∞𝑥^(𝑛−4)(𝑡)=⋅⋅⋅=lim_𝑡→+∞𝑥^(𝑡)=lim_𝑡→+∞𝑥(𝑡)=−∞,(2.8) which contradicts the boundedness of 𝑥. Consequently, we have (−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑡)≥0,∀𝑡≥𝑡₁.(2.9) Combining (2.4) and (2.9), we conclude easily that there exists a constant 𝐿≥0 with lim_𝑡→+∞(−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑡)=𝐿.(2.10)
Next, we claim that 𝐿=0. Otherwise, there exists a constant 𝑏>𝑡₁ satisfying (−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑡)≥𝐿2>0,∀𝑡≥𝑏,(2.11) which yields that (−1)=^{𝜃+𝑛−1}𝑥^(𝑛−2)(𝑡)−(−1)^{𝜃+𝑛−1}𝑥^(𝑛−2)(𝑏)_𝑡𝑏(−1)^{𝜃+𝑛−1}𝑥^(𝑛−1)(𝑠)𝑑𝑠≥𝐿(𝑡−𝑏)2⟶+∞as𝑡⟶+∞,(2.12) which gives that lim_𝑡→+∞𝑥^(𝑛−2)(𝑡)=+∞,(2.13) which means that lim_𝑡→+∞𝑥^(𝑛−3)(𝑡)=lim_𝑡→+∞𝑥^(𝑛−4)(𝑡)=⋅⋅⋅=lim_𝑡→+∞𝑥^(𝑡)=lim_𝑡→+∞𝑥(𝑡)=+∞,(2.14) which contradicts the boundedness of 𝑥 in [𝑡₀,+∞). Hence, 𝐿=0, that is, lim_𝑡→+∞𝑥^(𝑛−1)(𝑡)=0.(2.15)
Repeating the proof of (2.3)–(2.15), we deduce similarly that (−1)^𝜃+𝑗𝑥^(𝑗)islnimon_𝑡→+∞increasingandnonnegativein𝑡₁,+∞,𝑥^(𝑗)(𝑡)=0,1≤𝑗≤𝑛−1,(2.16) which together with the boundedness of 𝑥 implies that (−1)^𝜃𝑥 is nonincreasing in [𝑡₁,+∞) and lim_𝑡→+∞𝑥(𝑡) exists.
Thus, (2.3) and (2.16) yield (a)–(c). This completes the proof.

Lemma 2.2. Let 𝑥,𝑝,𝜏,𝑟,𝑦∈𝐶([𝑡₀,+∞),ℝ) satisfy (𝐴2) and 𝑦(𝑡)𝑥=is𝑥b(o𝑡)u−nd𝑝e(d𝑡)a𝑥nd(𝜏(li𝑡m))_𝑡→+∞−𝑟(𝑡),∀𝑡≥𝑡₀;(2.17)lim_𝑡→+∞𝜏(𝑡)=+∞;(2.18)𝑦(𝑡)=lim_𝑡→+∞𝑟(𝑡)=0,||𝑝(𝑡)||≥𝑝₀>1eventually,(2.19) where 𝑝₀ is a fixed constant. Then, lim_𝑡→+∞𝑥(𝑡)=0.

Proof. Since 𝜏 is a strictly increasing continuous function, 𝜏(𝑡)<𝑡 in [𝑡₀,+∞) and lim_𝑡→+∞𝜏(𝑡)=+∞, it follows that the inverse function 𝜏⁻¹ of 𝜏 is also strictly increasing continuous, 𝜏⁻¹(𝑡)>𝑡 in [𝜏(𝑡₀),+∞) and lim_𝑗→∞𝜏^−𝑗(𝑡)=+∞, where 𝜏^−𝑗=𝜏^{−(𝑗−1)}(𝜏⁻¹) for all 𝑗∈ℕ. Equation (2.18) implies that there exists a constant 𝐵>0 with ||𝑥(𝑡)||≤𝐵,∀𝑡≥𝑡₀.(2.20) Using (2.18) and (2.19), we deduce that, for any 𝜀>0, there exist sufficiently large numbers 𝑇>1+|𝑡₀| and 𝐾∈ℕ satisfying 𝐵𝑝^𝐾0<𝜀4,max||𝑦(𝑡)||,||𝑟(𝑡)||<𝜀𝑝⁰−14,||𝑝(𝑡)||≥𝑝⁰,∀𝑡≥𝑇.(2.21) In view of (2.17), (2.20), and (2.21), we infer that for all 𝑡≥𝑇||𝑥(𝑡)||=||𝑥𝜏⁻¹(𝑡)−𝑦𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)||≤||𝑥𝜏||𝑝𝜏⁻¹(𝑡)₋₁||(𝑡)||+||𝑦𝜏⁻¹(𝑡)||+||𝑟𝜏⁻¹(𝑡)||<1||𝑝𝜏⁻¹(𝑡)||𝑝₀||𝑥𝜏⁻¹(𝑡)||+𝜀𝑝⁰−12𝑝₀≤1𝑝₀𝑝1₀||𝑥𝜏⁻²(𝑡)||+𝜀𝑝⁰−12𝑝₀+𝜀𝑝⁰−12𝑝₀=1𝑝²⁰||𝑥𝜏⁻²(𝑡)||+𝜀𝑝⁰−1≤≤⋅1⋅⋅2𝑝₀1+𝑝1₀𝑝^𝐾0||𝑥𝜏^−𝐾(𝑡)||+𝜀𝑝⁰−12𝑝₀1+𝑝1₀+⋅⋅⋅+1≤𝐵𝑝^0𝐾−1𝑝^𝐾0+𝜀𝑝⁰−12𝑝₀⋅11−1/𝑝₀<𝜀,(2.22) which gives that lim_𝑡→+∞𝑥(𝑡)=0. This completes the proof.

Lemma 2.3. Let 𝑥,𝑝,𝜏,𝑟, and 𝑦 be in 𝐶([𝑡₀,+∞),ℝ) satisfying (𝐴2), (2.17), (2.18), and 𝑝lim_𝑡→+∞||𝑦(𝑡)||=𝑑>0,lim_𝑡→+∞𝑟(𝑡)=0;(2.23)₁≥||𝑝(𝑡)||≥𝑝₀>1eventually,𝑝²⁰>𝑝₀+𝑝₁,(2.24) where 𝑑,𝑝₀, and 𝑝₁ are constants. Then, there exists 𝐿>0 such that |𝑥(𝑡)|≥𝐿 eventually.

Proof. Obviously, (2.20) holds. It follows from (2.18), (2.23), and (2.24) that for 𝜀=𝑑[𝑝₀(𝑝₀−1)−𝑝₁]/(𝑝₀(𝑝₀−1)+𝑝₁)>0, there exist 𝐾∈ℕ and 𝑇>1+|𝑡₀| satisfying 𝐵𝑝^𝐾0<𝜀4𝑝₁,𝑑−𝜀4<||𝑦(𝑡)||<𝑑+4𝜀,||𝑟(𝑡)||<4𝜀𝑝₀,𝑝₁≥||𝑝(𝑡)||≥𝑝₀,∀𝑡≥𝑇.(2.25) Put 𝐿=𝑑[𝑝₀(𝑝₀−1)−𝑝₁]/2𝑝₁𝑝₀(𝑝₀−1). In light of (2.17), we conclude that for each 𝑡≥𝑇𝑥(𝑡)=𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑦𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)=1𝑝𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)𝑥𝜏⁻²(𝑡)𝑝𝜏⁻²(𝑡)−𝑦𝜏⁻²(𝑡)𝑝𝜏⁻²(𝑡)−𝑟𝜏⁻²(𝑡)𝑝𝜏⁻²(𝑡)−𝑦𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)=𝑥𝜏𝑝𝜏⁻¹(𝑡)₋₂(𝑡)Π^2𝑖=1𝑝(𝜏^−𝑖(𝑡))−²_𝑗=1𝑦𝜏^−𝑗(𝑡)Π^𝑗𝑖=1𝑝(𝜏^−𝑖(𝑡))−²_𝑗=1𝑟𝜏^−𝑗(𝑡)==⋅⋅𝑥⋅𝜏_−𝐾Π^𝑗𝑖=1𝑝(𝜏^−𝑖(𝑡))(𝑡)Π^𝐾𝑖=1𝑝(𝜏^−𝑖(𝑡))−^𝐾_𝑗=1𝑦𝜏^−𝑗(𝑡)Π^𝑗𝑖=1𝑝(𝜏^−𝑖(𝑡))−^𝐾_𝑗=1𝑟𝜏^−𝑗(𝑡)Π^𝑗𝑖=1𝑝(𝜏^−𝑖(𝑡)),(2.26) which together with (2.20) and (2.25) yields that for any 𝑡≥𝑇||𝑥(𝑡)||≥||𝑦𝜏⁻¹(𝑡)||||𝑝𝜏⁻¹(𝑡)||−||𝑥𝜏^−𝐾(𝑡)||Π^𝐾𝑖=1||𝑝(𝜏^−𝑖(𝑡))||−^𝐾_𝑗=2||𝑦𝜏^−𝑗(𝑡)||Π^𝑗𝑖=1||𝑝(𝜏^−𝑖(𝑡))||−^𝐾_𝑗=1||𝑟𝜏^−𝑗(𝑡)||≥𝑑−𝜀/4Π^𝑗𝑖=1||𝑝(𝜏^−𝑖(𝑡))||𝑝₁−𝐵𝑝^𝐾0−𝑑+𝜀4^𝐾_𝑗=21𝑝^𝑗0−𝜀4𝑝_0𝐾𝑗=11𝑝^𝑗0≥𝑑−𝜀/4𝑝₁−𝜀4𝑝₁−𝑑+𝜀4⋅1−1/𝑝1²⁰/𝑝₀−𝜀4𝑝₀⋅1/𝑝⁰1−1/𝑝₀=𝑑−𝜀/2𝑝₁−𝑑+𝜀/2𝑝₀𝑝₀−1=𝑑𝑝⁰𝑝₀−1−𝑝₁−(𝜀/2)𝑝₀𝑝₀−1+𝑝₁=𝐿.𝑝₁𝑝₀𝑝₀−1(2.27) This completes the proof.

Similar to the proof of Lemma  3.2 in [26], we have the following two lemmas.

Lemma 2.4. Let 𝑥,𝑝,𝜏,𝑟, and 𝑦 be in 𝐶([𝑡₀,+∞),ℝ) satisfying (𝐴2), (2.17), (2.18), and ||𝑝lim(𝑡)_𝑡→+∞||𝑦≤(𝑡𝑝)₀=lim_𝑡→+∞𝑟(𝑡)=0;(2.28)<12eventually,(2.29) where 𝑝₀ is a constant. Then, lim_𝑡→+∞𝑥(𝑡)=0.

Lemma 2.5. Let 𝑥, 𝑝, 𝜏, 𝑟, and 𝑦 be in 𝐶([𝑡₀,+∞),ℝ) satisfying (𝐴2), (2.17), (2.18), (2.23), and (2.29). Then, there exists 𝐿>0 such that |𝑥(𝑡)|≥𝐿 eventually.

Lemma 2.6 (Krasnoselskii’s fixed point theorem). Let 𝑋 be a Banach space, let 𝑌 be a nonempty bounded closed convex subset of 𝑋, and let 𝑓, 𝑔 be mappings of 𝑌 into 𝑋 such that 𝑓𝑥+𝑔𝑦∈𝑌 for every pair 𝑥,𝑦∈𝑌. If 𝑓 is a contraction mapping and 𝑔 is completely continuous, then the mapping 𝑓+𝑔 has a fixed point in 𝑌.

3. Main Results

First, we use the Krasnoselskii’s fixed point theorem to show the existence and multiplicity of bounded positive and negative solutions of (1.1).

Theorem 3.1. Let (𝐴1),(𝐴2), and (𝐴3) hold. Assume that there exist 𝑝₀,𝑝₁∈ℝ⁺⧵{0},𝑟₀,𝑟₁∈ℝ⁺, and 𝑟∈𝐶^𝑛([𝑡₀,+∞),ℝ) satisfying 𝑝𝑟₁≥𝑝(𝑡)≥𝑝₀>1eventually,𝑝²⁰>𝑝₀+𝑝₁;(3.1)_(𝑛)(𝑡)=𝑔(𝑡),−𝑟₀≤𝑟(𝑡)≤𝑟₁eventually;(3.2)_𝑡+∞0𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<+∞.(3.3) Then, the following hold:(a)for arbitrarily positive constants 𝑀 and 𝑁 with 𝑝₀−1𝑀>𝑝₁−1𝑁+𝑝¹𝑟₁𝑝₀+𝑟₀,(3.4) equation (1.1) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀;(3.5)(b)for arbitrarily positive constants 𝑀 and 𝑁 with 𝑝₀−1𝑁>𝑝₁−1𝑀+𝑝¹𝑟₀𝑝₀+𝑟₁,(3.6) equation (1.1) has uncountably many bounded negative solutions 𝑥∈𝐴(−𝑁,−𝑀) with −𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤−𝑀.(3.7)

Proof. It follows from (3.1) and (3.2) that there exists an enough large constant 𝑇₀ with 𝜏⁻¹(𝑇₀)>1+|𝑡₀|+|𝛽| satisfying 𝑝₀≤𝑝(𝑡)≤𝑝₁,𝑟^(𝑛)(𝑡)=𝑔(𝑡),−𝑟₀≤𝑟(𝑡)≤𝑟₁,∀𝑡≥𝑇₀.(3.8)
(a) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.4). Let 𝐷∈((𝑝₁−1)𝑁+(𝑝₁𝑟₁/𝑝₀),(𝑝₀−1)𝑀−𝑟₀). First of all, we prove that there exist two mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(𝑁,𝑀)→𝐵𝐶([𝛽,+∞),ℝ) and a constant 𝑇_𝐷>𝜏⁻¹(𝑇₀) such that 𝐹_𝐷+𝐺_𝐷 has a fixed point 𝑥∈𝐴(𝑁,𝑀), which is also a bounded positive solution of (1.1) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀. Put 𝐵=max||𝑓_𝑖𝑢₁,𝑢₂,…,𝑢_𝑘𝑖||∶𝑢_𝑗∈[𝑁,𝑀],1≤𝑗≤𝑘_𝑖,1≤𝑖≤𝑚.(3.9) In light of (3.3), (3.9), and (𝐴2), we infer that there exists a sufficiently large number 𝑇_𝐷>𝜏⁻¹(𝑇₀) satisfying 𝐵𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<min𝑀−𝐷+𝑀+𝑟⁰𝑝₀,𝐷+𝑁𝑝₁−𝑟¹𝑝₀−𝑁.(3.10) Define two mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(𝑁,𝑀)→𝐶([𝛽,+∞),ℝ) by 𝐹_𝐷𝑥(𝑡)=⎧⎪⎨⎪⎩𝑝𝜏𝐷⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡),𝑡≥𝑇^𝐷𝐺𝐹_𝐷𝑥𝑇_𝐷,𝛽≤𝑡<𝑇_𝐷,(3.11)_𝐷𝑥(𝑡)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩(−1)^𝑛𝑝×𝜏⁻¹(𝑡)(𝑛−1)!_{𝜏+∞−1(𝑡)}𝑠−𝜏⁻¹(𝑡)^𝑛−1×^𝑚_𝑖=1𝐺_𝐷𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠,𝑡≥𝑇_𝐷,𝑥𝑇_𝐷,𝛽≤𝑡<𝑇_𝐷,(3.12) for each 𝑥∈𝐴(𝑁,𝑀). In view of (3.1), (3.8), and (3.10)–(3.12), we conclude that for any 𝑥,𝑢∈𝐴(𝑁,𝑀) and 𝑡≥𝑇_𝐷||𝐹_𝐷𝑥(𝑡)−𝐹_𝐷𝑢(𝑡)||=||||𝑥𝜏⁻¹(𝑡)−𝑢𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)||||≤𝑝1₀𝐹‖𝑥−𝑢‖,_𝐷𝑥=(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≤𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₀+𝑀𝑝₀+𝑟⁰𝑝₀+𝐵𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1<𝐷+𝑀+𝑟₀𝑞_𝑖(𝑠)𝑑𝑠𝑝₀+min𝑀−𝐷+𝑀+𝑟⁰𝑝₀,𝐷+𝑁𝑝₁−𝑟¹𝑝₀𝐹≤𝑀,_𝐷−𝑁𝑥=(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≥𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₁+𝑁𝑝₁−𝑟¹𝑝₀−𝐵𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1>𝐷+𝑁𝑞_𝑖(𝑠)𝑑𝑠𝑝₁−𝑟¹𝑝₀−min𝑀−𝐷+𝑀+𝑟⁰𝑝₀,𝐷+𝑁𝑝₁−𝑟¹𝑝₀||𝐺≥𝑁,_𝐷−𝑁=𝑢||||(𝑡)||(−1)_𝑛𝑝×𝜏⁻¹(𝑡)(𝑛−1)!_{𝜏+∞−1(𝑡)}≤𝐵𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠|||||𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1<min𝑀−𝐷+𝑀+𝑟₀𝑞_𝑖(𝑠)𝑑𝑠𝑝₀,𝐷+𝑁𝑝₁−𝑟¹𝑝₀<𝑀,−𝑁(3.13) which ensures that ‖‖𝐹_𝐷𝑥−𝐹_𝐷𝑢‖‖=sup_{𝑡≥𝑇𝐷}||𝐹_𝐷𝑥(𝑡)−𝐹_𝐷𝑢(𝑡)||≤1𝑝₀𝐹‖𝑥−𝑢‖,∀𝑥,𝑢∈𝐴(𝑁,𝑀),(3.14)_𝐷𝑥+𝐺‖‖𝐺_𝐷𝑢∈𝐴(𝑁,𝑀),∀𝑥,𝑢∈𝐴(𝑁,𝑀),(3.15)_𝐷𝑢‖‖≤𝑀,∀𝑢∈𝐴(𝑁,𝑀).(3.16) It follows from (3.11), (3.12), (3.15), and (3.16) that 𝐹_𝐷 and 𝐺_𝐷 map 𝐴(𝑁,𝑀) into 𝐵𝐶([𝛽,+∞),ℝ), respectively.
Now, we show that 𝐺_𝐷 is continuous in 𝐴(𝑁,𝑀). Let {𝑥_𝑙}_𝑙∈ℕ⊂𝐴(𝑁,𝑀) and 𝑥∈𝐴(𝑁,𝑀) with lim_𝑙→∞𝑥_𝑙=𝑥, given 𝜀>0. It follows from the uniform continuity of 𝑓_𝑖 in [𝑁,𝑀]^𝑘𝑖 for 1≤𝑖≤𝑚 and lim_𝑙→∞𝑥_𝑙=𝑥 that there exist 𝛿>0 and 𝐾∈ℕ satisfying ||𝑓_𝑖𝑢<_𝑖1,𝑢_𝑖2,…,𝑢_𝑖𝑘𝑖−𝑓_𝑖𝑣_𝑖1𝜀,𝑣_𝑖2,…,𝑣_𝑖𝑘𝑖||1+1/𝑝₀(𝑛−1)!∫^𝜏+∞_{−1𝑇𝐷}𝑠^𝑛−1∑^𝑚𝑖=1𝑞_𝑖(𝑠)𝑑𝑠,∀𝑢^𝑖𝑗||𝑢,𝑣_𝑖𝑗∈[𝑁,𝑀_𝑖𝑗],‖‖𝑥−𝑣_𝑖𝑗||<𝛿,1≤𝑗≤𝑘_𝑖,1≤𝑖≤𝑚,_𝑙−𝑥‖‖<𝛿,∀𝑙≥𝐾.(3.17) In view of (3.8), (3.12), (3.17), we arrive at ‖𝐺_𝐷𝑥=_𝑙s−up𝐺_{𝐷𝑡≥𝑇𝐷}𝑥‖||||(−1)^𝑛×𝑝𝜏⁻¹(𝑡)(𝑛−1)!_{𝜏+∞−1(𝑡)}𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(−𝑠)𝑓_𝑖𝑓_𝑖𝑥_𝑙𝜎_𝑖1(𝑠),𝑥_𝑙𝜎_𝑖2(𝑠),…,𝑥_𝑙𝜎_𝑖𝑘𝑖(𝑠)≤sup_{𝑡≥𝑇𝐷}𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠|||||1×𝑝₀(𝑛−1)!_{𝜏+∞−1(𝑡)}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)−||𝑓𝑓_𝑖𝑖𝑥_𝑙𝜎_𝑖1(𝑠),𝑥_𝑙𝜎_𝑖2(𝑠),…,𝑥_𝑙𝜎_𝑖𝑘𝑖(𝑠)≤1𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)||𝑑𝑠𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠⋅𝜀<𝜀,∀𝑙≥𝐾,1+1/𝑝₀(𝑛−1)!∫^𝜏+∞_{−1𝑇𝐷}𝑠^𝑛−1∑^𝑚𝑖=1𝑞_𝑖(𝑠)𝑑𝑠(3.18) which means that 𝐺_𝐷 is continuous in 𝐴(𝑁,𝑀).
Next, we show that 𝐺_𝐷(𝐴(𝑁,𝑀)) is equicontinuous in [𝛽,+∞). Let 𝜀>0. Taking into account (3.3) and (𝐴2), we know that there exists 𝑇^∗>𝑇_𝐷 satisfying 1𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑇^∗₎𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<𝜀4.(3.19) Put 𝐵₁=max𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)∶𝜏⁻¹𝑇_𝐷≤𝑠≤𝜏⁻¹𝑇^∗.(3.20) It follows from the uniform continuity of 𝑝𝜏⁻¹ and 𝜏⁻¹ in [𝑇_𝐷,𝑇^∗] that there exists 𝛿>0 satisfying ||𝑝𝜏⁻¹𝑡₁−𝑝𝜏⁻¹𝑡₂||<𝜀𝑝²⁰(𝑛−1)!∀𝑡₁41+𝐵∫^𝜏+∞_{−1𝑇𝐷}𝑠^𝑛−1∑^𝑚𝑖=1𝑞_𝑖(𝑠)𝑑𝑠,||𝜏,𝑡₂∈𝑇_𝐷,𝑇^∗with||𝑡₁−𝑡₂||<𝛿;₋₁𝑡₁−𝜏⁻¹𝑡₂||<𝜀𝑝⁰(𝑛−1)!4𝐵1+∀𝐵𝑡₁₁+(𝑛−1)∫^𝜏+∞_{−1𝑇𝐷}𝑢^𝑛−1∑^𝑚𝑖=1𝑞_𝑖(𝑠)𝑑𝑠,,𝑡₂∈𝑇_𝐷,𝑇^∗with||𝑡₁−𝑡₂||<𝛿.(3.21) Let 𝑥∈𝐴(𝑁,𝑀) and 𝑡₁,𝑡₂∈[𝛽,+∞) with |𝑡₁−𝑡₂|<𝛿. We consider three possible cases.
Case 1. Let 𝑡₁,𝑡₂∈[𝑇^∗,+∞). In view of (3.8), (3.9), (3.12), and (3.19), we conclude that ||𝐺_𝐷=𝑥𝑡₁1−𝐺_𝐷𝑥𝑡₂||(𝑛−1)×!||||𝑝𝜏⁻¹1𝑡₁_{𝜏+∞−1𝑡1}−1𝑠−𝜏⁻¹𝑡₁^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠×𝑝𝜏⁻¹𝑡₂_{𝜏+∞−1𝑡2}≤𝐵𝑠−𝜏⁻¹𝑡₂^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠|||||𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑡1}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠+^𝜏+∞_{−1𝑡2}𝑠^𝑚𝑛−1_𝑖=1<𝜀𝑞_𝑖(𝑠)𝑑𝑠2.(3.22)
Case 2. Let 𝑡₁,𝑡₂∈[𝑇_𝐷,𝑇^∗]. In terms of (3.8), (3.9), (3.12), (3.21), we arrive at ||𝐺_𝐷=𝑥𝑡₁1−𝐺_𝐷𝑥𝑡₂||(𝑛−1)×!||||𝑝𝜏⁻¹1𝑡₁_{𝜏+∞−1𝑡1}−1𝑠−𝜏⁻¹𝑡₁^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠×𝑝𝜏⁻¹𝑡₂_{𝜏+∞−1𝑡2}≤1𝑠−𝜏⁻¹𝑡₂^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠|||||(𝑛−1)×!||||𝑝𝜏⁻¹1𝑡₁−𝑝𝜏⁻¹1𝑡₂_{𝜏+∞−1𝑡1}||||+1𝑠−𝜏⁻¹𝑡₁^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠×𝑝⎡⎢⎢⎣|||||𝜏⁻¹𝑡₂_{𝜏−1(𝑡2)𝜏−1𝑡1}+𝑠−𝜏⁻¹𝑡₁^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠_{𝜏+∞−1(𝑡2)}||||||||𝑠−𝜏₋₁𝑡₁^𝑛−1−𝑠−𝜏⁻¹𝑡₂^𝑛−1×_𝑚|||_𝑖=1≤𝐵𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠(𝑛−1)!||𝑝𝜏⁻¹𝑡₁−𝑝𝜏⁻¹𝑡₂||𝑝𝜏⁻¹𝑡₁𝑝𝜏⁻¹𝑡₂^𝜏+∞−1_{𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠+1𝑝₀×⎡⎢⎢⎣|||||^{𝜏−1(𝑡2)𝜏}_{−1𝑡1}𝑠^𝑚𝑛−1_𝑖=1+𝑞_𝑖(𝑠)𝑑𝑠_{𝜏+∞−1𝑡2}|||||(𝑛−1)𝑠^{max{𝑛−2,0}}||𝜏⁻¹𝑡₁−𝜏⁻¹𝑡₂||^𝑚_𝑖=1≤𝐵𝑞_𝑖(𝑠)𝑑𝑠𝑝²⁰(𝑛−1)!||𝑝𝜏⁻¹+𝐵𝑡₁−𝑝𝜏⁻¹𝑡₂||^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠𝑝₀(𝑛−1)!𝐵¹+(𝑛−1)^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1<𝜀𝑞_𝑖(𝑠)𝑑𝑠||𝜏⁻¹𝑡₁−𝜏⁻¹𝑡₂||2.(3.23)
Case 3. Let 𝑡₁,𝑡₂∈[𝛽,𝑇_𝐷]. By (3.12), we have ||𝐺_𝐷𝑥𝑡₁−𝐺_𝐷𝑥𝑡₂||=||𝐺_𝐷𝑥𝑇_𝐷−𝐺_𝐷𝑥𝑇_𝐷||=0<𝜀.(3.24) Thus, 𝐺_𝐷(𝐴(𝑁,𝑀)) is equicontinuous in [𝛽,+∞). Consequently, 𝐺_𝐷(𝐴(𝑁,𝑀)) is relatively compact by (3.16) and the continuity of 𝐺_𝐷. By means of (3.14), (3.15), and Lemma 2.6, we infer that 𝐹_𝐷+𝐺_𝐷 possesses a fixed point 𝑥∈𝐴(𝑁,𝑀), that is, 𝑥(𝑡)=𝐷𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠,∀𝑡≥𝑇_𝐷,(3.25) which gives that 𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡))=−𝐷+𝑟(𝑡)+(−1)^𝑛−1×(𝑛−1)!_𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎∀_𝑖2𝑡(𝑠≥)𝜏,…₋₁,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠,[𝑥(𝑡)−𝑝(𝑡)𝑥(𝜏(𝑡))]_(𝑛)𝑇_𝐷,=𝑔(𝑡)−^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝜎_𝑖1(𝑡),𝑥𝜎_𝑖2(𝑡),…,𝑥𝜎_𝑖𝑘𝑖(𝑡),∀𝑡≥𝜏⁻¹𝑇_𝐷,(3.26) which mean that 𝑥∈𝐴(𝑁,𝑀) is a bounded positive solution of (1.1) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀.(3.27)
Let 𝐷₁ and 𝐷₂ be two arbitrarily different numbers in ((𝑝₁−1)𝑁+(𝑝₁𝑟₁/𝑝₀),(𝑝₀−1)𝑀−𝑟₀). Similarly, we conclude that for each 𝑙∈{1,2} there exist two mappings 𝐹_𝐷𝑗,𝐺_𝐷𝑗∶𝐴(𝑁,𝑀)→𝐵𝐶([𝛽,+∞),ℝ) and a sufficiently large number 𝑇_𝐷𝑙>𝜏⁻¹(𝑇₀) satisfying (3.8)–(3.12), where 𝐷,𝑇_𝐷,𝐹_𝐷, and 𝐺_𝐷 are replaced by 𝐷_𝑙,𝑇_𝐷𝑙,𝐹_𝐷𝑙, and 𝐺_𝐷𝑙, respectively, and 𝐹_𝐷𝑙+𝐺_𝐷𝑙 has a fixed point 𝑥_𝑙∈𝐴(𝑁,𝑀), which is also a bounded positive solution with 𝑁≤liminf_𝑡→+∞𝑥_𝑙(𝑡)≤limsup_𝑡→+∞𝑥_𝑙(𝑡)≤𝑀, that is, 𝑥_𝑙(𝑡)=𝐷^𝑙𝑝𝜏⁻¹(𝑡)+𝑥^𝑙𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥_𝑙𝜎_𝑖1(𝑠),𝑥_𝑙𝜎_𝑖2(𝑠),…,𝑥_𝑙𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠,∀𝑡≥𝑇_𝐷𝑙.(3.28) It follows from (3.3) that there exists 𝑇₃>max{𝑇_𝐷1,𝑇_𝐷2} satisfying 𝐵𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇3}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<||𝐷¹−𝐷₂||4𝑝₁.(3.29) Combining (3.8), (3.28), and (3.29), we conclude easily that ||𝑥₁(𝑡=)−||||𝐷𝑥₂(𝑡)₁||−𝐷₂𝑝𝜏⁻¹(𝑡)+𝑥¹𝜏⁻¹(𝑡)−𝑥₂𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}𝑠−𝜏⁻¹(𝑡)^𝑛−1×^𝑚_𝑖=1𝑞_𝑖(𝑠)𝑓−_𝑖𝑓𝑥_1𝑖𝜎_𝑖1(𝑠),𝑥₁𝜎_𝑖2(𝑠),…,𝑥₁𝜎_𝑖𝑘𝑖(𝑠)≥||𝐷𝑥₂𝜎_𝑖1(𝑠),𝑥₂𝜎_𝑖2(𝑠),…,𝑥₂𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠₁||||−𝐷₂||𝑝𝜏⁻¹(𝑡)−||𝑥¹𝜏⁻¹(𝑡)−𝑥₂𝜏⁻¹(𝑡)||𝑝𝜏⁻¹(𝑡)−𝑝𝜏⁻¹(𝑡2)𝐵(𝑛−1)!^𝜏+∞−1_(𝑡)𝑠^𝑚𝑛−1_𝑖=1≥||𝐷𝑞_𝑖(𝑠)𝑑𝑠₁−𝐷₂||𝑝₁−‖‖𝑥¹−𝑥₂‖‖𝑝₀−2𝐵𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇3}𝑠^𝑚𝑛−1_𝑖=1>||𝐷𝑞_𝑖(𝑠)𝑑𝑠₁−𝐷₂||𝑝₁−‖‖𝑥¹−𝑥₂‖‖𝑝₀−||𝐷¹−𝐷₂||2𝑝₁=||𝐷¹−𝐷₂||2𝑝₁−‖‖𝑥¹−𝑥₂‖‖𝑝₀,∀𝑡≥𝑇₃,(3.30) which guarantees that ‖‖𝑥₁−𝑥₂‖‖≥𝑝⁰||𝐷₁−𝐷₂||2𝑝₁1+𝑝₀>0,(3.31) that is, 𝑥₁≠𝑥₂. Hence, (1.1) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀.
(b) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.6) and put 𝐵₂=max||𝑓_𝑖𝑢₁,𝑢₂,…,𝑢_𝑘𝑖||∶𝑢_𝑗∈[−𝑁,−𝑀],1≤𝑗≤𝑘_𝑖,1≤𝑖≤𝑚.(3.32) Let 𝐷∈((1−𝑝₀)𝑁+𝑟₁,(1−𝑝₁)𝑀−(𝑝₁𝑟₀/𝑝₀)). It follows from (3.3), (3.8), (3.32), and (𝐴2) that there exists 𝑇_𝐷>𝜏⁻¹(𝑇₀) satisfying 𝐵₂𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<min−𝑀+𝑀−𝐷𝑝₁−𝑟⁰𝑝₀,𝑁+𝐷−𝑁−𝑟¹𝑝₀.(3.33) Let the mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(−𝑁,−𝑀)→𝐶([𝛽,+∞),ℝ) be defined by (3.11) and (3.12), respectively.
Using (3.1), (3.8), (3.11), (3.12), and (3.33), we deduce that for any 𝑥,𝑢∈𝐴(−𝑁,−𝑀) and 𝑡≥𝑇_𝐷𝐹_𝐷𝑥=(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≤𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₁−𝑀𝑝₁+𝑟⁰𝑝₀+𝐵²𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1<𝐷−𝑀𝑞_𝑖(𝑠)𝑑𝑠𝑝₁+𝑟⁰𝑝₀+min−𝑀+𝑀−𝐷𝑝₁−𝑟⁰𝑝₀,𝑁+𝐷−𝑁−𝑟¹𝐹≤−𝑀,_𝐷𝑝₀𝑥=(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≥𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₀−𝑁𝑝₀−𝑟¹𝑝₀−𝐵²𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1>𝐷−𝑁−𝑟₁𝑞_𝑖(𝑠)𝑑𝑠𝑝₀−min−𝑀+𝑀−𝐷𝑝₁−𝑟⁰𝑝₀,𝑁+𝐷−𝑁−𝑟¹≥−𝑁,𝑝₀(3.34) which give that 𝐹_𝐷𝑥+𝐺_𝐷𝑢∈𝐴(−𝑁,−𝑀),∀𝑥,𝑢∈𝐴(−𝑁,−𝑀).(3.35) The rest of the proof is similar to the proof of (a) and is omitted. This completes the proof.

Theorem 3.2. Let (𝐴1),(𝐴2), and (𝐴3), hold. Assume that there exist 𝑝₀,𝑝₁∈ℝ⁺⧵{0},𝑟₀,𝑟₁∈ℝ⁺, and 𝑟∈𝐶^𝑛([𝑡₀,+∞),ℝ) satisfying (3.2), (3.3), and 𝑝₁≥−𝑝(𝑡)≥𝑝₀>1eventually.(3.36) Then, the following hold:(a)for arbitrarily positive constants 𝑀 and 𝑁 with 𝑁<𝑀,𝑝²⁰−𝑝₁𝑀>𝑝₁−𝑝⁰𝑝₁𝑝⁰𝑁+𝑝₀𝑟₁+𝑝₁𝑟₀,(3.37) equation (1.1) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀;(3.38)(b)for arbitrarily positive constants 𝑀 and 𝑁 with 𝑀<𝑁,𝑝²⁰−𝑝₁𝑁>𝑝₁−𝑝⁰𝑝₁𝑝⁰𝑀+𝑝₁𝑟₁+𝑝₀𝑟₀,(3.39) equation (1.1) has uncountably many bounded negative solutions 𝑥∈𝐴(−𝑁,−𝑀) with −𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤−𝑀.(3.40)

Proof. It follows from (3.2) and (3.36) that there exists a constant 𝑇₀ with 𝜏(𝑇₀)>1+|𝑡₀|+|𝛽| satisfying 𝑝₀≤−𝑝(𝑡)≤𝑝₁,𝑟^(𝑛)(𝑡)=𝑔(𝑡),−𝑟₀≤𝑟(𝑡)≤𝑟₁,∀𝑡≥𝑇₀.(3.41)
(a) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.37). Let 𝐷∈(𝑝₁((𝑀+𝑟₀)/𝑝₀+𝑁),𝑝₀(𝑁/𝑝₁+𝑀)−𝑟₁) and 𝐵 be defined by (3.9). In light of (3.3), (3.9), and (𝐴2), there exists a sufficiently large number 𝑇_𝐷>𝜏⁻¹(𝑇₀) satisfying 𝐵𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<min𝑀−𝐷+𝑟¹𝑝₀+𝑁𝑝₁,𝐷𝑝₁−𝑀+𝑟⁰𝑝₀−𝑁.(3.42) Define two mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(𝑁,𝑀)→𝐶([𝛽,+∞),ℝ) by (3.12) and 𝐹_𝐷𝑥(𝑡)=⎧⎪⎨⎪⎩−𝑝𝜏𝐷⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡),𝑡≥𝑇^𝐷𝐹_𝐷𝑥𝑇_𝐷,𝛽≤𝑡<𝑇_𝐷(3.43) for each 𝑥∈𝐴(𝑁,𝑀). In view of (3.12), (3.36), and (3.41)–(3.43), we conclude that for any 𝑥,𝑢∈𝐴(𝑁,𝑀) and 𝑡≥𝑇_𝐷𝐹_𝐷𝑥=−(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≤𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₀−𝑁𝑝₁+𝑟¹𝑝₀+𝐵𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1<𝐷𝑞_𝑖(𝑠)𝑑𝑠𝑝₀−𝑁𝑝₁+𝑟¹𝑝₀+min𝑀−𝐷+𝑟¹𝑝₀+𝑁𝑝₁,𝐷𝑝₁−𝑀+𝑟⁰𝑝₀𝐹≤𝑀,_𝐷−𝑁𝑥=−(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≥𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₁−𝑀𝑝₀−𝑟⁰𝑝₀−𝐵𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1>𝐷𝑞_𝑖(𝑠)𝑑𝑠𝑝₁−𝑀+𝑟⁰𝑝₀−min𝑀−𝐷+𝑟¹𝑝₀+𝑁𝑝₁,𝐷𝑝₁−𝑀+𝑟⁰𝑝₀≥𝑁,−𝑁(3.44) which imply (3.15). The rest of the proof is similar to that of Theorem 3.1 and is omitted.
(b) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.39). Let 𝐷∈(−𝑝₀(𝑁+(𝑀/𝑝₁))𝑀+𝑟₀,−𝑀𝑝₁−(𝑝₁/𝑝₀)(𝑁+𝑟₁)) and 𝐵₂ be defined by (3.32). Note that (3.3), (3.32), and (𝐴2) yield that there exists a sufficiently large number 𝑇_𝐷>𝜏⁻¹(𝑇₀) satisfying 𝐵₂𝑝₀(𝑛−1)!^𝜏+∞_{−1𝑇𝐷}𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<min−𝑀−𝐷𝑝₁−𝑁+𝑟¹𝑝₀,𝑁+𝐷−𝑟⁰𝑝₀+𝑀𝑝₁.(3.45) Let the mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(−𝑁,−𝑀)→𝐶([𝛽,+∞),ℝ) be defined by (3.12) and (3.43), respectively.
Using (3.12), (3.36), (3.41), and (3.45), we infer that for any 𝑥,𝑢∈𝐴(𝑁,𝑀) and 𝑡≥𝑇_𝐷𝐹_𝐷𝑥=−(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≤𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₁+𝑁𝑝₀+𝑟¹𝑝₀+𝐵²𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1<𝐷𝑞_𝑖(𝑠)𝑑𝑠𝑝₁+𝑁𝑝₀+𝑟¹𝑝₀+min−𝑀−𝐷𝑝₁−𝑁+𝑟¹𝑝₀,𝑁+𝐷−𝑟⁰𝑝₀+𝑀𝐹≤−𝑀,_𝐷𝑝₁𝑥=−(𝑡)+𝐷𝐺_𝐷𝑢(𝑡)𝑝𝜏⁻¹(𝑡)+𝑥𝜏⁻¹(𝑡)𝑝𝜏⁻¹(𝑡)−𝑟𝜏⁻¹(𝑡)×𝑝𝜏⁻¹(𝑡)+𝑝𝜏⁻¹((−𝑡)1)(^𝑛𝑛−1)!_{𝜏+∞−1(𝑡)}≥𝐷𝑠−𝜏⁻¹(𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑝₀+𝑀𝑝₁−𝑟⁰𝑝₀−𝐵²𝑝₀(𝑛−1)!^𝜏+∞_−1(𝑡)𝑠^𝑚𝑛−1_𝑖=1>𝐷𝑞_𝑖(𝑠)𝑑𝑠𝑝₀+𝑀𝑝₁−𝑟⁰𝑝₀−min−𝑀−𝐷𝑝₁−𝑁+𝑟¹𝑝₀,𝑁+𝐷−𝑟⁰𝑝₀+𝑀≥−𝑁,𝑝₁(3.46) which give (3.15). The rest of the proof is similar to the proof of Theorem 3.1 and is omitted. This completes the proof.

Theorem 3.3. Let (𝐴1) and (𝐴3) hold. Assume that there exist 𝑝₀,𝑝₁∈ℝ⁺⧵{0}, 𝑟₀,𝑟₁∈ℝ⁺, and 𝑟∈𝐶^𝑛([𝑡₀,+∞),ℝ) satisfying (3.2), (3.3), and −𝑝₀≤𝑝(𝑡)≤𝑝₁eventually,𝑝₀+𝑝₁<1.(3.47) Then, the following hold:
(a) for arbitrarily positive constants 𝑀 and 𝑁 with 𝑟₀+𝑟₁+𝑁<1−𝑝₀−𝑝₁𝑀,(3.48) equation (1.1) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀;(3.49) for arbitrarily positive constants 𝑀 and 𝑁 with 𝑟₀+𝑟₁+𝑀<1−𝑝₀−𝑝₁𝑁,(3.50) equation (1.1) has uncountably many bounded negative solutions 𝑥∈𝐴(−𝑁,−𝑀) with −𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤−𝑀.(3.51)

Proof. It follows from (3.2) and (3.47) that there exists a constant 𝑇₀>1+|𝑡₀|+|𝛽| satisfying −𝑝₀≤𝑝(𝑡)≤𝑝₁,𝑟^(𝑛)(𝑡)=𝑔(𝑡),−𝑟₀≤𝑟(𝑡)≤𝑟₁,∀𝑡≥𝑇₀.(3.52)
(a) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.48). Let 𝐷∈(𝑝₀𝑀+𝑟₀+𝑁,(1−𝑝₁)𝑀₁−𝑟₁) and 𝐵 be defined by (3.9). In light of (3.3), (3.9), and (𝐴2), we infer that there exists a sufficiently large number 𝑇_𝐷>max{𝑇₀,𝜏(𝑇₀)} satisfying 𝐵𝑝₀(𝑛−1)!^𝑇+∞_𝐷𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<min𝑀−𝐷−𝑝₁𝑀−𝑟₁,𝐷−𝑝₀𝑀−𝑟₀−𝑁.(3.53) Define two mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(𝑁,𝑀)→𝐶([𝛽,+∞),ℝ) by 𝐹_𝐷𝑥(𝑡)=𝐷+𝑝(𝑡)𝑥(𝜏(𝑡))+𝑟(𝑡),𝑡≥𝑇^𝐷𝐹_𝐷,𝐺𝑥𝑇_𝐷,𝛽≤𝑡<𝑇_𝐷,(3.54)_𝐷𝑥(𝑡)⎧⎪⎪⎨⎪⎪⎩(−1)^𝑛−1(𝑛×−1)!_𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠,𝑡≥𝑇_𝐷𝐺_𝐷𝑥𝑇_𝐷,𝛽≤𝑡<𝑇_𝐷,(3.55) for each 𝑥∈𝐴(𝑁,𝑀). In view of (3.47) and (3.52)–(3.55), we conclude that for any 𝑥,𝑢∈𝐴(𝑁,𝑀) and 𝑡≥𝑇_𝐷||𝐹𝐹_𝐷𝑥(𝑡)−𝐹_𝐷𝑢(𝑡)||≤||𝑝(𝑡)(𝑥(𝜏(𝑡))−𝑢(𝜏(𝑡)))||≤𝑝₀+𝑝₁‖𝑥−𝑢‖,_𝐷𝑥(𝑡)+𝐺_𝐷𝑢(𝑡)=𝐷+𝑝(𝑡)𝑥(𝜏(𝑡))+𝑟(𝑡)+(−1)^𝑛−1×(𝑛−1)!_𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1≤𝐷+𝑝₁𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑀+𝑟₁+𝐵(𝑛−1)!^𝑡+∞𝑠^𝑚𝑛−1_𝑖=1<𝐷+𝑝₁𝑞_𝑖(𝑠)𝑑𝑠𝐹𝑀+𝑟₁+min𝑀−𝐷−𝑝₁𝑀−𝑟₁,𝐷−𝑝₀𝑀−𝑟₀−𝑁≤𝑀,_𝐷𝑥(𝑡)+𝐺_𝐷𝑢(𝑡)=𝐷+𝑝(𝑡)𝑥(𝜏(𝑡))+𝑟(𝑡)+(−1)^𝑛−1×(𝑛−1)!_𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1≥𝐷−𝑝₀𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑀−𝑟₀−𝐵(𝑛−1)!^𝑡+∞𝑠^𝑚𝑛−1_𝑖=1>𝐷−𝑝₀𝑞_𝑖(𝑠)𝑑𝑠≥𝑁,𝑀−𝑟₀−min𝑀−𝐷−𝑝₁𝑀−𝑟₁,𝐷−𝑝₀𝑀−𝑟₀−𝑁(3.56) which yield (3.15). The rest of the proof is similar to that of Theorem 3.1 and is omitted.
(b) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.50). Let 𝐷∈(𝑟₀−(1−𝑝₁)𝑁−𝑀−𝑁,𝑝₀−𝑟₁) and 𝐵₂ be defined by (3.32). In light of (3.3), (3.32), and (𝐴2), we infer that there exists a sufficiently large number 𝑇_𝐷>max{𝑇₀,𝜏(𝑇₀)} satisfying 𝐵₂𝑝₀(𝑛−1)!^𝑇+∞_𝐷𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<min−𝑀−𝐷−𝑝₀𝑁−𝑟₁,𝐷+𝑁1−𝑝₁−𝑟₀.(3.57) Define two mappings 𝐹_𝐷,𝐺_𝐷∶𝐴(−𝑁,−𝑀)→𝐶([𝛽,+∞),ℝ) by (3.54) and (3.55). In view of (3.47), (3.52), (3.54), (3.55), and (3.57), we conclude that (3.56) holds and 𝐹_𝐷𝑥(𝑡)+𝐺_𝐷𝑢(𝑡)=×𝐷+𝑝(𝑡)𝑥(𝜏(𝑡))+𝑟(𝑡)+((𝑛−−1)1)^𝑛−1!_𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1≤𝐷+𝑝₀𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑁+𝑟₁+𝐵(𝑛−1)!^𝑡+∞𝑠^𝑚𝑛−1_𝑖=1<𝐷+𝑝₀𝑞_𝑖(𝑠)𝑑𝑠≤−𝑀,𝑁∀+𝑥,𝑟𝑢₁∈+𝐴m(in𝑁,𝑀−𝑀),−𝑡𝐷≥𝑇−_𝐷𝑝₀𝑁−𝑟₁,𝐷+𝑁1−𝑝₁−𝑟₀𝐹_𝐷,𝑥(𝑡)+𝐺_𝐷𝑢(𝑡)=𝐷+𝑝(𝑡)𝑥(𝜏(𝑡))+𝑟(𝑡)+(−1)^𝑛−1×(𝑛−1)!_𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1≥𝐷−𝑝₁𝑞_𝑖(𝑠)𝑓_𝑖𝑢𝜎_𝑖1(𝑠),𝑢𝜎_𝑖2(𝑠),…,𝑢𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠𝑁−𝑟₀−𝐵(𝑛−1)!^𝑡+∞𝑠^𝑚𝑛−1_𝑖=1>𝐷−𝑝₁𝑞_𝑖(𝑠)𝑑𝑠≥−𝑁,𝑁∀𝑥−,𝑢𝑟₀∈−𝐴m(i𝑁n,𝑀−𝑀),−𝑡≥𝐷𝑇−_𝐷𝑝₀𝑁−𝑟₁,𝐷+𝑁1−𝑝₁−𝑟₀.(3.58) Thus, (3.15) follows from (3.58). The rest of the proof is similar to that of Theorem 3.1 and is omitted. This completes the proof.

Second, we provide necessary and sufficient conditions for the oscillation of bounded solutions of (1.1).

Theorem 3.4. Let (𝐴1),(𝐴2), and (𝐴3) hold. Assume that there exist 𝑝₀,𝑝₁∈ℝ⁺⧵{0} and 𝑟∈𝐶^𝑛([𝑡₀,+∞),ℝ) satisfying (2.24) and lim_𝑡→+∞𝑟(𝑡)=0,𝑟^(𝑛)(𝑡)=𝑔(𝑡)eventually.(3.59) Then, each bounded solution of (1.1) either oscillates or tends to 0 as 𝑡→+∞ if and only if ^𝑡+∞₀𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠=+∞.(3.60)

Proof. Sufficiency. Suppose, without loss of generality, that (1.1) possesses a bounded eventually positive solution 𝑥 with limsup_𝑡→+∞𝑥(𝑡)>0, which together with (𝐴1),(𝐴3), (2.17), (2.24), and (3.60), yields that there exist constants 𝑀>0 and 𝑇>1+|𝑡₀|+|𝛽| satisfying 0<𝑥(𝑡)≤𝑀,∀𝑡≥𝑇;(3.61)𝑦^(𝑛)(𝑡)=−^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑥𝜎_𝑖1(𝑡),𝑥𝜎_𝑖2(𝑡),…,𝑥𝜎_𝑖𝑘𝑖(𝑡)<0,∀𝑡≥𝑇.(3.62) Obviously (2.17), (2.24), (3.59), and the boundedness of 𝑥 imply that 𝑦 is bounded. It follows from (2.17), (3.62), Lemmas 2.1 and 2.2 that there exists a constant 𝐿 satisfying lim_𝑡→+∞𝑦(𝑡)=𝐿≠0,lim_𝑡→+∞𝑦^(𝑖)(𝑡)=0,1≤𝑖≤𝑛−1.(3.63) Thus, (𝐴1), (3.61), (3.63), and Lemma 2.3 imply that there exist constants 𝑁 and 𝑇₁≥𝑇₀≥𝑇 satisfying inf0𝜎<_𝑖𝑗𝑁(𝑡)≤∶𝑥𝑡(≥𝑡)𝑇,₁,1||𝑦≤(𝑡)𝑗−≤𝐿𝑘||_𝑖<,11≤,𝑖∀≤𝑡𝑚≥𝑇≥₁𝑇₀,.(3.64) Put 𝐵₃=min𝑓_𝑖𝑢₁,𝑢₂,…,𝑢_𝑘𝑖∶𝑢_𝑗∈[𝑁,𝑀],1≤𝑗≤𝑘_𝑖,1≤𝑖≤𝑚.(3.65) Clearly, (𝐴3) guarantees that 𝐵₃>0. Integrating (3.62) from 𝑡 to +∞, by (3.63) and (3.64), we have 𝑦^(𝑛−1)(𝑡)=(−1)²^𝑡𝑚+∞_𝑖=1𝑞_𝑖𝑢₁𝑓_𝑖𝑥𝜎_𝑖1𝑢₁,𝑥𝜎_𝑖2𝑢₁,…,𝑥𝜎_𝑖𝑘𝑖𝑢₁𝑑𝑢₁,∀𝑡≥𝑇₁,(3.66) repeating this procedure, we obtain that 𝑦^(𝑛−2)(𝑡)=(−1)³^𝑡+∞𝑑𝑢₂^𝑢+∞₂^𝑚_𝑖=1𝑦′(𝑡)=(⋯−1)_𝑛𝑞_𝑖𝑢₁𝑓_𝑖𝑥𝜎_𝑖1𝑢₁,𝑥𝜎_𝑖2𝑢₁,…,𝑥𝜎_𝑖𝑘𝑖𝑢₁𝑑𝑢₁,∀𝑡≥𝑇₁,^𝑡+∞𝑑𝑢_𝑛−1^𝑢+∞_𝑛−1×_{𝑢+∞2𝑚}𝑑𝑢_𝑛−2⋯_𝑖=1𝐿−𝑦(𝑡)=lim_𝑢→+∞𝑞_𝑖𝑢₁𝑓_𝑖𝑥𝜎_𝑖1𝑢₁,𝑥𝜎_𝑖2𝑢₁,…,𝑥𝜎_𝑖𝑘𝑖𝑢₁𝑑𝑢₁,∀𝑡≥𝑇₁,=(−1)_𝑛𝑦(𝑢)−𝑦(𝑡)^𝑡+∞𝑑𝑢_𝑛^𝑢+∞_𝑛×_{𝑢+∞2𝑚}𝑑𝑢_𝑛−1⋯_𝑖=1𝑞_𝑖𝑢₁𝑓_𝑖𝑥𝜎_𝑖1𝑢₁,𝑥𝜎_𝑖2𝑢₁,…,𝑥𝜎_𝑖𝑘𝑖𝑢₁𝑑𝑢₁=(−1)^𝑛(𝑛−1)!^𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠,∀𝑡≥𝑇₁,(3.67) which together with (3.64) and (𝐴3) means that 1>||𝐿−𝑦(𝑡)||=|||||(−1)^𝑛(𝑛−1)!^𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1≥𝐵𝑞_𝑖(𝑠)𝑓_𝑖𝑥𝜎_𝑖1(𝑠),𝑥𝜎_𝑖2(𝑠),…,𝑥𝜎_𝑖𝑘𝑖(𝑠)𝑑𝑠₃|||||(𝑛−1)!^𝑡+∞(𝑠−𝑡)^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠,∀𝑡≥𝑇₁,(3.68) which gives that ^𝑇+∞₁𝑠^𝑚𝑛−1_𝑖=1𝑞_𝑖(𝑠)𝑑𝑠<+∞,(3.69) which contradicts (3.60).
Necessity. Suppose that (3.60) does not hold. Observe that lim_𝑡→+∞𝑟(𝑡)=0 implies that there exist two positive constants 𝑟₀ and 𝑟₁ satisfying −𝑟₀≤𝑟(𝑡)≤𝑟₁eventually.(3.70) It follows from Theorem 3.1 or Theorem 3.2 that, for any positive constants 𝑀 and 𝑁 satisfying (3.4) or (3.37), (1.1) possesses uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑀≥limsup_𝑡→+∞𝑥(𝑡)≥liminf_𝑡→+∞𝑥(𝑡)≥𝑁. This is a contradiction. This completes the proof.

As in the proof of Theorem 3.4, by means of Lemmas 2.1, 2.4, and 2.5, we have

Theorem 3.5. Let (𝐴1) and (𝐴3) hold. Assume that there exist 𝑝₀∈ℝ⁺⧵{0} and 𝑟∈𝐶^𝑛([𝑡₀,+∞),ℝ) satisfying (2.29) and (3.59). Then, each bounded solution of (1.1) either oscillates or tends to 0 as 𝑡→+∞ if and only if (3.60) holds.

4. Remarks and Examples

Now, we compare the results in Section 3 with some known results in the literature. In order to illustrate the advantage and applications of our results, five nontrivial examples are constructed.

Remark 4.1. Theorems 3.1–3.3 extend and improve the Theorem in [9], Theorem  8.4.2 in [10], Theorem  1 in [21], Theorems  1–3 in [24], Theorem  2.2 in [26], and Theorems  1–4 in [27, 28].

Remark 4.2. The sufficient part of Theorem 3.5 is a generalization of Theorem  3.1 in [4, 5]. Theorem 3.5 corrects and perfects Theorem  2.1 in [26].

The examples below show that our results extend indeed the corresponding results in [4, 5, 9, 10, 21, 24, 26–28]. Notice that none of the known results can be applied to these examples.

Example 4.3. Consider the 𝑛th-order forced nonlinear neutral differential equation: 𝑥(𝑡)−31++4𝑡𝑡^𝑛𝑛𝑥√𝑡^(𝑛)+1+√3+2𝑡𝑥⁵(3𝑡+sin𝑡)+𝑥³(𝑡−1/𝑡)+𝑡𝑥(3𝑡−ln𝑡)𝑥₄1+𝑡^𝑛+31+|||𝑥⁸3𝑡²−2𝑥²¹𝑡−√𝑡−1|||𝑡²−𝑡𝑥⁶(𝑡−2)+5𝑡𝑥𝑡(1+1/𝑡)^𝑡1+3𝑡^3𝑛+11+||𝑥³4𝑡−cos³𝑡−4𝑥⁴(𝑡−1)||=12sin𝑡+𝑛2𝜋,𝑡≥2,(4.1) where 𝑡₀=2,𝑚=2, and 𝑛∈ℕ. Put 𝑘₁=4,𝑘₂=6,𝛽=0,𝑟₀=𝑟₁=1/2,𝑝₀=3,𝑝₁=4, 𝑝(𝑡)=3+4𝑡^𝑛1+𝑡^𝑛,𝑞₁(𝑡)=1+√3+2𝑡1+𝑡^𝑛+3,𝑞₂(𝑡)=𝑡1+3𝑡^3𝑛+1𝑔(𝑡)=1,2sin𝑡+𝑛2𝜋,𝑟(𝑡)=12sin𝑡,𝜏(𝑡)=√𝜎𝑡,𝜎₁₁(𝑡)=3𝑡+sin𝑡,₁₂(𝑡)=𝑡−1𝑡,𝜎¹³𝜎(𝑡)=3𝑡²,𝜎₁₄(𝑡)=𝑡−√𝑡−1,𝜎₂₁(𝑡)=3𝑡−ln𝑡,₂₂(𝑡)=𝑡²−𝑡,𝜎₂₃(𝑡)=𝑡−2,𝜎₂₄(𝑡)=𝑡1+1𝑡^𝑡𝜎,𝜎₂₅(𝑡)=4𝑡−cos³𝑡,₂₆(𝑡)=𝑡−1,𝑓₁(𝑢,𝑣,𝑤,𝑧)=𝑢⁵+𝑣³𝑓₂1+||𝑤⁸−2𝑧²¹||,(𝑢,𝑣,𝑤,𝑧,𝑦,𝑠)=𝑢𝑣⁴𝑤⁶+5𝑧1+||𝑦³−4𝑠⁴||,∀(𝑡,𝑢,𝑣,𝑤,𝑧,𝑦,𝑠)∈𝑡⁰,+∞×ℝ⁶.(4.2) Clearly (𝐴1), (𝐴2), (𝐴3), and (3.1)–(3.3) hold.
Let 𝑀 and 𝑁 be arbitrarily positive constants satisfying 𝑀>(3/2)𝑁+7/12. It is easy to verify that (3.4) holds. It follows from Theorem 3.1 that (4.1) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀.
Let 𝑀 and 𝑁 be arbitrarily positive constants satisfying 𝑁>3𝑀/2+7/12. It is easy to verify that (3.6) holds. It follows from Theorem 3.1 that (4.1) has uncountably many bounded negative solutions 𝑥∈𝐴(−𝑁,−𝑀) with −𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤−𝑀.

Example 4.4. Consider the 𝑛th-order forced nonlinear neutral differential equation: 𝑥(𝑡)+82++1𝑡0⁵𝑡⁵𝑥√𝑡²+1−1^(𝑛)+𝑡²+3𝑡³𝑥⁷3𝑡²𝑥(𝑡−1)𝑥³(𝑡ln𝑡)+3+𝑡₂2+sin³𝑡²+𝑡^𝑛+51+𝑥²(𝑡−1)𝑥⁴(𝑡ln𝑡)𝑥⁵𝑡²+1+7𝑥³𝑡⁴−2+𝑥⁹𝑡+√𝑡𝑥⁸(𝑡−4)√𝑡+1+𝑡^𝑛+31+𝑥⁵𝑡+√𝑡−4𝑥⁴(𝑡−4)−3⁶=(−1)^𝑛𝑛!𝑡^𝑛+1+𝑛!(1−𝑡)^𝑛+1,𝑡≥3,(4.3) where 𝑡₀=3,𝑚=2, and 𝑛∈ℕ. Put 𝑘₁=3,𝑘₂=4,𝛽=−1,𝑟₀=1/2,𝑟₁=0,𝑝₀=4,𝑝₁=10, 𝑝(𝑡)=−8+10𝑡⁵2+𝑡⁵,𝑞₁(𝑡)=𝑡²+3𝑡³2+sin³𝑡²+𝑡^𝑛+5,𝑞₂(𝑡)=3+𝑡²√𝑡+1+𝑡^𝑛+3𝑔(𝑡)=(−1)_𝑛,𝑛!𝑡^𝑛+1+𝑛!(1−𝑡)^𝑛+1,𝑟(𝑡)=1𝑡(1−𝑡),𝜏(𝑡)=√𝜎𝑡²+1−1,₁₁𝜎(𝑡)=3𝑡²,𝜎₁₂(𝑡)=𝑡−1,𝜎₁₃(𝑡)=𝑡ln𝑡,𝜎₂₁(𝑡)=𝑡²+1,₂₂(𝑡)=𝑡⁴−2,𝜎₂₃(𝑡)=𝑡+√𝑡,𝜎₂₄(𝑡)=𝑡−4,𝑓₁(𝑢,𝑣,𝑤)=𝑢⁷𝑣𝑤³1+𝑣²𝑤⁴𝑓₂,(𝑢,𝑣,𝑤,𝑧)=𝑢⁵+7𝑣³+𝑤⁹𝑧⁸1+𝑤⁵−4𝑧⁴−3⁶,∀(𝑡,𝑢,𝑣,𝑤,𝑧)∈𝑡₀,+∞×ℝ⁴.(4.4) Clearly (𝐴1), (𝐴2), (𝐴3), (3.2), (3.3), and (3.36) hold.
Let 𝑀 and 𝑁 be arbitrarily positive constants satisfying 𝑀>(32/5)𝑁+5/6. It is easy to verify that (3.37) holds. It follows from Theorem 3.2 that (4.3) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀.
Let 𝑀 and 𝑁 be arbitrarily positive constants satisfying 𝑁>(32/5)𝑀+1/3. It is easy to verify that (3.39) holds. It follows from Theorem 3.2 that (4.3) has uncountably many bounded negative solutions 𝑥∈𝐴(−𝑁,−𝑀) with −𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤−𝑀.

Example 4.5. Consider the 𝑛th-order forced nonlinear neutral differential equation: ⎡⎢⎢⎣𝑥(𝑡)−2sin𝑡²−√𝑡5+sin𝑡²−√𝑡𝑥(𝑡−5)²⎤⎥⎥⎦^(𝑛)+3√𝑡−1+𝑡⁵𝑥³(𝑡−4)+1−√√𝑡²+1+𝑡^𝑛+62+cos⁵𝑥√𝑡+1−3𝑡+1ln²𝑡+𝑡⁴𝑥⁹2𝑡+sin𝑡²+11−2𝑡³+3𝑡⁴+𝑡^𝑛+5ln2+𝑥²𝑡²√1+2𝑡=(−1)^𝑛cos𝑡+𝑛𝜋2,𝑡≥1,(4.5) where 𝑡₀=1,𝑚=2, and 𝑛∈ℕ. Put 𝑘₁=𝑘₂=2,𝛽=−4,𝑟₀=𝑟₁=1,𝑝₀=1/2,𝑝₁=1/3, 𝑝(𝑡)=2sin𝑡²−√𝑡5+sin𝑡²−√𝑡,𝑞¹(𝑡)=3√𝑡−1+𝑡⁵√𝑡²+1+𝑡^𝑛+6,𝑞₂(𝑡)=1−√𝑡+1ln²𝑡+𝑡⁴1−2𝑡³+3𝑡⁴+𝑡^𝑛+5𝑔(𝑡)=(−1)_𝑛,cos𝑡+𝑛𝜋2,𝑟(𝑡)=(−1)^𝑛𝜎cos𝑡,𝜏(𝑡)=(𝑡−4)²𝜎₁₁(𝑡)=𝑡−5,₁₂𝑓(𝑡)=√𝑡+1−3,𝜎₂₁(𝑡)=2𝑡+sin𝑡²+1,𝜎₂₂(𝑡)=𝑡²√1+2𝑡,₁(𝑢,𝑣)=𝑢³2+cos⁵𝑣,𝑓²(𝑢,𝑣)=𝑢⁹ln2+𝑣²,∀(𝑡,𝑢,𝑣)∈𝑡⁰,+∞×ℝ².(4.6) Clearly (𝐴1), (𝐴3), (3.2), (3.3), and (3.47) hold.
Let 𝑀 and 𝑁 be arbitrarily positive constants satisfying 𝑀>6𝑁+12. It is easy to verify that (3.48) holds. It follows from Theorem 3.3 that (4.5) has uncountably many bounded positive solutions 𝑥∈𝐴(𝑁,𝑀) with 𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀.
Let 𝑀 and 𝑁 be arbitrarily positive constants satisfying 𝑁>6𝑀+12. It is easy to verify that (3.50) holds. It follows from Theorem 3.3 that (4.5) has uncountably many bounded negative solutions 𝑥∈𝐴(−𝑁,−𝑀) with −𝑁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤−𝑀.

Example 4.6. Consider the 𝑛th-order forced nonlinear neutral differential equation: 𝑥(𝑡)−(−1)^𝑛5+9ln²𝑡1+ln²𝑡𝑥√+𝑡₂𝑡−1^(𝑛)+𝑡⁸+9𝑡⁵+32𝑥³𝑡ln²𝑡+5𝑥⁷(𝑡−16)2+sin𝑡³−5𝑡𝑥⁹𝑡−si𝑡n𝑡𝑥⁴(𝑡−cos𝑡)+4𝑥⁶1+𝑡+𝑡²+𝑡³1+𝑡+𝑡²²+(𝑡+1)²−√𝑡𝑥⁵𝑡arctan𝑡³+1/1+√𝑡+1ln1+𝑥⁶(𝑡+1)/1+𝑥²(𝑡−2)=(−1)_𝑛2𝑡^𝑛+3+√𝑡sin3𝑡⁵−11+𝑥²𝑡²−𝑡𝑥⁴𝑡²+𝑡𝑛!ln𝑡−∑^𝑛𝑖=1(1/𝑖)𝑡^𝑛+1,𝑡≥4,(4.7) where 𝑡₀=4,𝑚=3, and 𝑛∈ℕ. Put 𝑘₁=2,𝑘₂=3,𝑘₃=5,𝛽=−12,𝑝₀=5,𝑝₁=9, 𝑝(𝑡)=(−1)^𝑛5+9ln²𝑡1+ln²𝑡,𝑞¹𝑞(𝑡)=𝑡⁸+9𝑡⁵+3,𝑞₂(𝑡)=𝑡²2+sin𝑡³−5𝑡₃,=(𝑡+1)²−√𝑡2𝑡^𝑛+3+√𝑡sin3𝑡⁵−1,𝑔(𝑡)=(−1)^𝑛𝑛!ln𝑡−∑^𝑛𝑖=11/𝑖𝑡^𝑛+1,𝑟(𝑡)=ln𝑡𝜏(𝑡)=√𝑡,𝑡−1,𝜎₁₁(𝑡)=𝑡ln²𝑡,𝜎₁₂(𝑡)=𝑡−16,𝜎₂₁(𝑡)=𝑡−sin𝑡𝜎₂₂𝑡,(𝑡)=𝑡−cos𝑡,𝜎₂₃(𝑡)=1+𝑡+𝑡²+𝑡³1+𝑡+𝑡²,𝜎₃₁(𝑡)=𝑡arctan𝑡³+1𝜎1+√𝑡+1₃₂,(𝑡)=𝑡+𝑓1,𝜎₃₃(𝑡)=𝑡−2,𝜎₃₄(𝑡)=𝑡²−𝑡,𝜎₃₅(𝑡)=𝑡²+𝑡,₁𝑓₃(𝑢,𝑣)=2𝑢³+5𝑣⁷,𝑓₂(𝑢,𝑣,𝑤)=𝑢⁹𝑣⁴+4𝑤⁶²,(𝑢,𝑣,𝑤,𝑦,𝑧)=𝑢⁵ln1+𝑣⁶/1+𝑤²1+𝑦²𝑧⁴,∀(𝑡,𝑢,𝑣,𝑤,𝑦,𝑧)∈𝑡₀,+∞×ℝ⁵.(4.8) Clearly (𝐴1), (𝐴2), (𝐴3), (2.24), (3.59), and (3.60) hold. It follows from Theorem 3.4 that each bounded solution of (4.7) either oscillates or tends to 0 as 𝑡→+∞.

Example 4.7. Consider the 𝑛th-order forced nonlinear neutral differential equation: 𝑥(𝑡)−(−1)^𝑛cos³(3𝑡−1)4+cos³(3𝑡−1)𝑥(𝑡−sin𝑡)^(𝑛)+𝑡³+2𝑡²−√𝑡+1𝑥⁵√𝑡−2−1+√1+𝑥²√𝑡−2−1𝑡²−1𝑥³(𝑡−1/𝑡)+5𝑥⁷(𝑡−1/𝑡)ln2+𝑥⁶(𝑡−1/𝑡)𝑡^2𝑛+1+2𝑡^𝑛ln³1+𝑡²+1=2^𝑛sin√2𝑡+𝑛𝜋/4𝑒^√2𝑡,𝑡≥6,(4.9) where 𝑡₀=6,𝑚=2, and 𝑛∈ℕ. Put 𝑘₁=𝑘₂=1,𝛽=1,𝑝₀=1/3, 𝑝(𝑡)=(−1)^𝑛cos³(3𝑡−1)4+cos³(3𝑡−1),𝑞¹𝑞(𝑡)=𝑡³+2𝑡²−√𝑡+1,₂(𝑡)=√𝑡²−1𝑡^2𝑛+1+2𝑡^𝑛ln³1+𝑡²+1,𝑔(𝑡)=2^𝑛sin√2𝑡+𝑛𝜋/4𝑒^√2𝑡,𝑟(𝑡)=sin√2𝑡𝑒^√2𝑡𝜏(𝑡)=𝑡−sin𝑡,𝜎₁,(𝑡)=√𝑡−2−1𝜎₂(𝑡)=𝑡−1𝑡,𝑓¹(𝑢)=𝑢⁵1+𝑢²𝑓₂,(𝑢)=𝑢³+5𝑢⁷ln2+𝑢⁶,∀(𝑡,𝑢)∈𝑡₀,+∞×ℝ.(4.10) Clearly (𝐴1), (𝐴3), (2.29), (3.59), and (3.60) hold. It follows from Theorem 3.5 that each bounded solution of (4.9) either oscillates or tends to 0 as 𝑡→+∞.

Next, we prove that the necessary part of Theorem  2.1 in [26] does not hold by means of (4.9). It is easy to verify that the conditions of Theorem  2.1 in [26] are fulfilled. Suppose that the necessary part of Theorem  2.1 in [26] is true. Because each bounded solution of (4.9) either oscillates or tends to 0 as 𝑡→+∞, it follows that the necessary part of Theorem  2.1 in [26] gives that^𝑡+∞₀𝑠^𝑛−1𝑞_𝑖(𝑠)𝑑𝑠=+∞,𝑖∈{1,2},(4.11) which yields that+∞=^𝑡+∞₀𝑠^𝑛−1𝑞₂(𝑠)𝑑𝑠=^𝑡+∞₀𝑠^𝑛−1√𝑠²−1𝑠^2𝑛+1+2𝑠^𝑛ln³1+𝑠²+1𝑑𝑠≤^𝑡+∞₀1𝑠^𝑛+1𝑑𝑠<+∞,(4.12) which is a contradiction.

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This study was supported by research funds from Dong-A University.