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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 546302, 33 pages
http://dx.doi.org/10.1155/2012/546302
Research Article

Bounded Oscillation of a Forced Nonlinear Neutral Differential Equation

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Mathematics, Kunming University, Kunming, Yunnan 650214, China
3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
4Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 20 December 2011; Accepted 5 March 2012

Academic Editor: Miroslava RůžičkovÑ

Copyright Β© 2012 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the 𝑛th-order forced nonlinear neutral differential equation [π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑))](𝑛)+βˆ‘π‘šπ‘–=1π‘žπ‘–(𝑑)𝑓𝑖(π‘₯(πœŽπ‘–1(𝑑)),π‘₯(πœŽπ‘–2(𝑑)),…,π‘₯(πœŽπ‘–π‘˜π‘–(𝑑)))=𝑔(𝑑),𝑑β‰₯𝑑0. 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:[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑))(𝑛)+π‘šξ“π‘–=1π‘žπ‘–(𝑑)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑑),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑑),…,π‘₯π‘–π‘˜π‘–(𝑑)ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.1) where 𝑑0βˆˆβ„ and 𝑛,π‘š,π‘˜π‘–βˆˆβ„• are constants for 1β‰€π‘–β‰€π‘š. In what follows, we assume that(𝐴1)  𝑝,𝑔,𝜏,πœŽπ‘–π‘—βˆˆπΆ([𝑑0,+∞),ℝ) and π‘žπ‘–βˆˆπΆ([𝑑0,+∞),ℝ+) satisfy that lim𝑑→+∞𝜏(𝑑)=lim𝑑→+βˆžπœŽπ‘–π‘—(𝑑)=+∞,1β‰€π‘—β‰€π‘˜π‘–,1β‰€π‘–β‰€π‘š,(1.2) and there exists 1≀𝑖0β‰€π‘š such that π‘žπ‘–0 is positive eventually:(𝐴2) β€‰πœ is strictly increasing and 𝜏(𝑑)<𝑑 in [𝑑0,+∞);(𝐴3) β€‰π‘“π‘–βˆˆπΆ(β„π‘˜π‘–,ℝ) satisfies that 𝑓𝑖𝑒1,𝑒2,…,π‘’π‘˜π‘–ξ€Έξ€·π‘’>0,βˆ€1,𝑒2,…,π‘’π‘˜π‘–ξ€Έβˆˆξ€·β„+⧡{0}π‘˜π‘–,𝑓𝑖𝑒1,𝑒2,…,π‘’π‘˜π‘–ξ€Έξ€·π‘’<0,βˆ€1,𝑒2,…,π‘’π‘˜π‘–ξ€Έβˆˆξ€·β„βˆ’ξ€Έβ§΅{0}π‘˜π‘–(1.3) for 1β‰€π‘–β‰€π‘š.

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:[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(π‘‘βˆ’πœ)ξ…ž+π‘ž(𝑑)π‘₯(π‘‘βˆ’πœŽ)βˆ’π‘Ÿ(𝑑)π‘₯(π‘‘βˆ’π›Ώ)=0,𝑑β‰₯𝑑0,(1.4) where 𝑝,π‘ž,π‘ŸβˆˆπΆ([𝑑0,+∞),ℝ+),𝜏>0, and 𝜎β‰₯𝛿β‰₯0. Das and Misra [7] studied the nonhomogeneous neutral delay differential equation:[]π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’πœ)ξ…ž+π‘ž(𝑑)𝑓(π‘₯(π‘‘βˆ’πœŽ))=𝑔(𝑑),𝑑β‰₯𝑑0,(1.5) where π‘ž,π‘”βˆˆπΆ([𝑇,+∞),ℝ+⧡{0}),𝜎>0,𝜏>0,π‘βˆˆ[0,1),π‘“βˆΆβ„β†’β„, 𝑑𝑓(𝑑)>0 for 𝑑≠0, 𝑓 is nondecreasing, Lipschitzian, and satisfies βˆ«π‘˜0(1/𝑓(𝑑))𝑑𝑑<+∞ for every π‘˜>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:[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(π‘‘βˆ’πœ)ξ…ž+π‘ž(𝑑)𝑓(π‘₯(π‘‘βˆ’πœŽ))=𝑔(𝑑),𝑑β‰₯0,(1.6) where π‘βˆˆπΆ(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 𝑑→+∞. 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:[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(π‘‘βˆ’πœ)ξ…ž+𝑛𝑖=1𝑓𝑖(𝑑)π‘₯π‘‘βˆ’πœŽπ‘–ξ€Έ=0,𝑑β‰₯𝑑0,(1.7) where π‘βˆˆπΆ([𝑑0,+∞),ℝ),𝜏>0,πœŽπ‘–βˆˆβ„+, and π‘“π‘–βˆˆπΆ([𝑑0,+∞),ℝ) for 1β‰€π‘–β‰€π‘š. Γ‡akmak and Tiryaki [6] showed several sufficient conditions for the oscillation of the forced second-order nonlinear differential equations with delayed argument in the form:π‘₯ξ…žξ…ž(𝑑)+𝑝(𝑑)𝑓(π‘₯(𝛼(𝑑)))=𝑔(𝑑),𝑑β‰₯𝑑0β‰₯0,(1.8) where 𝑝,𝛼,π‘”βˆˆπΆ([𝑑0,+∞),ℝ), 𝛼(𝑑)≀𝑑,lim𝑑→+βˆžπ›Ό(𝑑)=+∞, and π‘“βˆˆπΆ(ℝ,ℝ). Travis [20] investigated the oscillatory behavior of the second-order differential equation with functional argument:π‘₯ξ…žξ…ž(𝑑)+𝑝(𝑑)𝑓(π‘₯(𝑑),π‘₯(𝛼(𝑑)))=0,𝑑β‰₯𝑑0,(1.9) where 𝑝,π›ΌβˆˆπΆ([𝑑0,+∞),ℝ) and π‘“βˆˆπΆ(ℝ2,ℝ) satisfies that 𝑓(𝑠,𝑑) has the same sign of 𝑠 and 𝑑 when they have the same sign. Lin [12] got some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral differential equation:[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(π‘‘βˆ’πœ)ξ…žξ…ž+π‘ž(𝑑)𝑓(π‘₯(π‘‘βˆ’πœŽ))=0,𝑑β‰₯0,(1.10) where 𝑝,π‘žβˆˆπΆ(ℝ+,ℝ),π‘βˆˆ[0,1) with 0≀𝑝(𝑑)≀𝑝 eventually, π‘“βˆˆπΆ(ℝ,ℝ), 𝑓 is nondecreasing and 𝑑𝑓(𝑑)>0 for 𝑑≠0. KulenoviΔ‡ and HadΕΎiomerspahiΔ‡ [9] deduced the existence of a nonoscillatory solution for the neutral delay differential equation of second order with positive and negative coefficients:[π‘₯](𝑑)+𝑐π‘₯(π‘‘βˆ’πœ)ξ…žξ…ž+π‘ž1ξ€·(𝑑)π‘₯π‘‘βˆ’πœŽ1ξ€Έβˆ’π‘ž2ξ€·(𝑑)π‘₯π‘‘βˆ’πœŽ2ξ€Έ=0,𝑑β‰₯𝑑0,(1.11) where 𝑐≠±1, 𝜏>0,πœŽπ‘–βˆˆβ„+,π‘žπ‘–βˆˆπΆ([𝑑0,+∞),ℝ+), and βˆ«π‘‘+∞0π‘žπ‘–(𝑑)𝑑𝑑<+∞ for π‘–βˆˆ{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:[]π‘₯(𝑑)+𝑐π‘₯(π‘‘βˆ’πœ)(𝑛)+(βˆ’1)𝑛+1[]𝑃(𝑑)π‘₯(π‘‘βˆ’πœŽ)βˆ’π‘„(𝑑)π‘₯(π‘‘βˆ’π›Ώ)=0,𝑑β‰₯𝑑0,[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(π‘‘βˆ’πœ)(𝑛)+π‘šξ“π‘–=1π‘žπ‘–(𝑑)𝑓𝑖π‘₯ξ€·π‘‘βˆ’πœŽπ‘–ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.12) where π‘βˆˆβ„β§΅{Β±1},𝜏,𝜎,𝛿,πœŽπ‘–βˆˆβ„+,𝑃,π‘„βˆˆπΆ([𝑑0,+∞),ℝ+), and 𝑝,𝑔,π‘“π‘–βˆˆπΆ([𝑑0,+∞),ℝ) for 1β‰€π‘–β‰€π‘š. 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:[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(π‘‘βˆ’πœ)(𝑛)||||βˆ’π‘ž(𝑑)π‘₯(π‘‘βˆ’πœŽ)π›Όβˆ’1π‘₯(π‘‘βˆ’πœŽ)=0,𝑑β‰₯𝑑0,(1.13) where 𝜏>0,𝜎>0,𝛼β‰₯1, and 𝑝,π‘žβˆˆπΆ([𝑑0,+∞),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:[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))(𝑛)+π‘ž(𝑑)π‘₯(𝜎(𝑑))=0,𝑑β‰₯𝑑0,(1.14) where 𝑛 is even, 𝑝,π‘ž,𝜏,𝜎∈𝐢([𝑑0,+∞),ℝ+),𝑝(𝑑)<1,𝜏(𝑑)≀𝑑 and 𝜎(𝑑)≀𝑑 for π‘‘βˆˆ[𝑑0,+∞), and lim𝑑→+∞𝜏(𝑑)=lim𝑑→+∞𝜎(𝑑)=+∞. 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:[]π‘₯(𝑑)+𝑐π‘₯(𝜏(𝑑))(𝑛)+π‘ž(𝑑)𝑓(π‘₯(𝜎(𝑑)))=0,𝑑β‰₯𝑑0,[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))(𝑛)+π‘ž(𝑑)𝑓(π‘₯(𝜎(𝑑)))=𝑔(𝑑),𝑑β‰₯𝑑0,(1.15) where π‘βˆˆβ„β§΅{Β±1}, 𝜏,𝜎∈𝐢([𝑑0,+∞),ℝ+),𝑝,π‘ž,π‘”βˆˆπΆ([𝑑0,+∞),ℝ), and π‘“βˆˆπΆ(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:[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))(𝑛)+π‘šξ“π‘–=1π‘žπ‘–(𝑑)𝑓𝑖π‘₯ξ€·πœŽπ‘–(𝑑)ξ€Έξ€Έ=0,𝑑β‰₯𝑑0,[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))(𝑛)+π‘šξ“π‘–=1π‘žπ‘–(𝑑)𝑓𝑖π‘₯ξ€·πœŽπ‘–(𝑑)ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.16) where π‘›βˆˆβ„•β§΅{1},π‘šβˆˆβ„•,𝑝,𝑓𝑖,𝑔,𝜏,πœŽπ‘–βˆˆπΆ([𝑑0,+∞),ℝ), 𝑓𝑖 is nondecreasing and 𝑒𝑓𝑖(𝑒)>0 for 𝑒≠0,πœŽπ‘–βˆˆπΆ1([𝑑0,+∞),ℝ),πœŽξ…žπ‘–(𝑑)>0,πœŽπ‘–(𝑑)≀𝑑 for π‘‘βˆˆ[𝑑0,+∞), lim𝑑→+∞𝜏(𝑑)=lim𝑑→+βˆžπœŽπ‘–(𝑑)=+∞ for 1β‰€π‘–β‰€π‘š, 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:[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑))(𝑛)+π‘šξ“π‘–=1π‘žπ‘–(𝑑)𝑓𝑖π‘₯ξ€·πœŽπ‘–(𝑑)ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.17) where π‘›βˆˆβ„•β§΅{1},π‘šβˆˆβ„•, and(𝐢1)  𝑝,π‘žπ‘–,𝜏,π‘”βˆˆπΆ([𝑑0,+∞),ℝ) for 𝑖=1,2,…,π‘š and lim𝑑→+∞𝜏(𝑑)=+∞;(𝐢2)  𝑝 and 𝑔 are oscillating functions;(𝐢3) β€‰πœŽπ‘–βˆˆπΆ([𝑑0,+∞),ℝ),πœŽξ…žπ‘–(𝑑)>0,πœŽπ‘–(𝑑)≀𝑑 and lim𝑑→+βˆžπœŽπ‘–(𝑑)=+∞ for 𝑖=1,2,…,π‘š;(𝐢4) β€‰π‘“π‘–βˆˆπΆ(ℝ,ℝ) is nondecreasing function, 𝑒𝑓𝑖(𝑒)>0 for 𝑒≠0 and 𝑖=1,2,…,π‘š.

That is, they claimed the following result.

Theorem 1.1 (see [26, Theorem 2.1]). Assume that(𝐢5) there is an oscillating function π‘ŸβˆˆπΆ([𝑑0,+∞),ℝ) such that π‘Ÿ(𝑛)(𝑑)=𝑔(𝑑) and lim𝑑→+βˆžπ‘Ÿ(𝑑)=0;(𝐢6)  𝑝 is an oscillating function and |𝑝(𝑑)|≀𝑝0<1/2;(𝐢7)β€‰β€‰π‘žπ‘–(𝑑)β‰₯0, 𝑖=1,2,…,π‘š.Then, every bounded solution of (1.17) either oscillates or tends to zero if and only if ξ€œπ‘‘+∞0π‘ π‘›βˆ’1π‘žπ‘–(𝑠)𝑑𝑠=+∞,𝑖=1,2,…,π‘š.(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 𝑓𝑖. 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 𝑑→+∞ 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=(βˆ’βˆž,+∞),ℝ+=[0,+∞),β„βˆ’=(βˆ’βˆž,0] and𝑑𝛽=min0ξ€½πœ,inf(𝑑),πœŽπ‘–π‘—ξ€Ίπ‘‘(𝑑)βˆΆπ‘‘βˆˆ0ξ€Έ,+∞,1β‰€π‘—β‰€π‘–π‘˜,1β‰€π‘–β‰€π‘šξ€Ύξ€Ύ.(2.1) By a solution of (1.1), we mean a function π‘₯∈𝐢([𝛽,+∞),ℝ) for some 𝑇β‰₯𝑑0+𝛽, such that π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑)) is 𝑛 times continuously differentiable in [𝑇,+∞) and such that (1.1) is satisfied for 𝑑β‰₯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 𝐡𝐢([𝛽,+∞),ℝ) stand for the Banach space of all bounded continuous functions in [𝛽,+∞) with the norm β€–π‘₯β€–=sup𝑑β‰₯𝛽|π‘₯(𝑑)| for each π‘₯∈𝐡𝐢([𝛽,+∞),ℝ) and[𝐴(𝑁,𝑀)={π‘₯∈𝐡𝐢(𝛽,+∞),ℝ)βˆΆπ‘β‰€π‘₯(𝑑)≀𝑀,𝑑β‰₯𝛽}for𝑀,π‘βˆˆβ„with𝑀>𝑁.(2.2) It is easy to see that 𝐴(𝑁,𝑀) is a bounded closed and convex subset of the Banach space 𝐡𝐢([𝛽,+∞),ℝ).

Lemma 2.1. Let π‘›βˆˆβ„• and π‘₯βˆˆπΆπ‘›([𝑑0,+∞),ℝ) be bounded. If π‘₯(𝑛)(𝑑)≀0 eventually, then(a)lim𝑑→+∞π‘₯(𝑑) exists and lim𝑑→+∞π‘₯(𝑖)(𝑑)=0 for 1β‰€π‘–β‰€π‘›βˆ’1; furthermore, there exists πœƒ=0 for 𝑛 odd and πœƒ=1 for 𝑛 even such that(b)(βˆ’1)πœƒ+𝑖π‘₯(𝑖)(𝑑)β‰₯0 eventually for 1≀𝑖≀𝑛;(c)(βˆ’1)πœƒ+𝑖π‘₯(𝑖) is nonincreasing eventually for 0β‰€π‘–β‰€π‘›βˆ’1.

Proof. Now, we consider two possible cases below.
Case 1. Assume that 𝑛=1. Let πœƒ=0. Note that π‘₯β€²(𝑑)≀0 eventually. It follows that there exists a constant 𝑑1>𝑑0 satisfying π‘₯ξ…ž(𝑑)≀0, for all 𝑑β‰₯𝑑1, which yields that π‘₯ is nonincreasing in [𝑑1,+∞). Since π‘₯ is bounded in [𝑑0,+∞), it follows that lim𝑑→+∞π‘₯(𝑑) exists.
Case 2. Assume that 𝑛β‰₯2. Notice that πœƒ+𝑛 is odd. It follows that (βˆ’1)πœƒ+𝑛π‘₯(𝑛)(𝑑)β‰₯0 eventually, which implies that there exists a constant 𝑑1>𝑑0 satisfying (βˆ’1)πœƒ+𝑛π‘₯(𝑛)(𝑑)β‰₯0,βˆ€π‘‘β‰₯𝑑1,(2.3) which means that (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)𝑑(𝑑)isnonincreasingin1ξ€Έ,+∞.(2.4)
Suppose that there exists a constant 𝑑2β‰₯𝑑1 satisfying (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)(𝑑2)<0, which together with (2.4) gives that (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)(𝑑)≀(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)𝑑2ξ€Έ<0,βˆ€π‘‘β‰₯𝑑2,(2.5) which guarantees that (βˆ’1)πœƒ+π‘›βˆ’2π‘₯(π‘›βˆ’2)(𝑑) is increasing in [𝑑2,+∞) and (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’2)(𝑑)βˆ’(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’2)𝑑2ξ€Έ=ξ€œπ‘‘π‘‘2(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)(𝑠)𝑑𝑠≀(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)𝑑2ξ€Έξ€·π‘‘βˆ’π‘‘2ξ€ΈβŸΆβˆ’βˆžasπ‘‘βŸΆ+∞,(2.6) that is, lim𝑑→+∞π‘₯(π‘›βˆ’2)(𝑑)=βˆ’βˆž,(2.7) which means that lim𝑑→+∞π‘₯(π‘›βˆ’3)(𝑑)=lim𝑑→+∞π‘₯(π‘›βˆ’4)(𝑑)=β‹…β‹…β‹…=lim𝑑→+∞π‘₯ξ…ž(𝑑)=lim𝑑→+∞π‘₯(𝑑)=βˆ’βˆž,(2.8) which contradicts the boundedness of π‘₯. Consequently, we have (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)(𝑑)β‰₯0,βˆ€π‘‘β‰₯𝑑1.(2.9) Combining (2.4) and (2.9), we conclude easily that there exists a constant 𝐿β‰₯0 with lim𝑑→+∞(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)(𝑑)=𝐿.(2.10)
Next, we claim that 𝐿=0. Otherwise, there exists a constant 𝑏>𝑑1 satisfying (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)𝐿(𝑑)β‰₯2>0,βˆ€π‘‘β‰₯𝑏,(2.11) which yields that (βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’2)(𝑑)βˆ’(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’2)=ξ€œ(𝑏)𝑑𝑏(βˆ’1)πœƒ+π‘›βˆ’1π‘₯(π‘›βˆ’1)(𝑠)𝑑𝑠β‰₯𝐿(π‘‘βˆ’π‘)2⟢+∞asπ‘‘βŸΆ+∞,(2.12) which gives that lim𝑑→+∞π‘₯(π‘›βˆ’2)(𝑑)=+∞,(2.13) which means that lim𝑑→+∞π‘₯(π‘›βˆ’3)(𝑑)=lim𝑑→+∞π‘₯(π‘›βˆ’4)(𝑑)=β‹…β‹…β‹…=lim𝑑→+∞π‘₯ξ…ž(𝑑)=lim𝑑→+∞π‘₯(𝑑)=+∞,(2.14) which contradicts the boundedness of π‘₯ in [𝑑0,+∞). Hence, 𝐿=0, that is, lim𝑑→+∞π‘₯(π‘›βˆ’1)(𝑑)=0.(2.15)
Repeating the proof of (2.3)–(2.15), we deduce similarly that (βˆ’1)πœƒ+𝑗π‘₯(𝑗)𝑑isnonincreasingandnonnegativein1ξ€Έ,,+∞lim𝑑→+∞π‘₯(𝑗)(𝑑)=0,1β‰€π‘—β‰€π‘›βˆ’1,(2.16) which together with the boundedness of π‘₯ implies that (βˆ’1)πœƒπ‘₯ is nonincreasing in [𝑑1,+∞) and lim𝑑→+∞π‘₯(𝑑) exists.
Thus, (2.3) and (2.16) yield (a)–(c). This completes the proof.

Lemma 2.2. Let π‘₯,𝑝,𝜏,π‘Ÿ,π‘¦βˆˆπΆ([𝑑0,+∞),ℝ) satisfy (𝐴2) and 𝑦(𝑑)=π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑))βˆ’π‘Ÿ(𝑑),βˆ€π‘‘β‰₯𝑑0;(2.17)π‘₯isboundedandlim𝑑→+∞𝜏(𝑑)=+∞;(2.18)lim𝑑→+βˆžπ‘¦(𝑑)=lim𝑑→+∞||||π‘Ÿ(𝑑)=0,𝑝(𝑑)β‰₯𝑝0>1eventually,(2.19) where 𝑝0 is a fixed constant. Then, lim𝑑→+∞π‘₯(𝑑)=0.

Proof. Since 𝜏 is a strictly increasing continuous function, 𝜏(𝑑)<𝑑 in [𝑑0,+∞) and lim𝑑→+∞𝜏(𝑑)=+∞, it follows that the inverse function πœβˆ’1 of 𝜏 is also strictly increasing continuous, πœβˆ’1(𝑑)>𝑑 in [𝜏(𝑑0),+∞) and limπ‘—β†’βˆžπœβˆ’π‘—(𝑑)=+∞, where πœβˆ’π‘—=πœβˆ’(π‘—βˆ’1)(πœβˆ’1) for all π‘—βˆˆβ„•. Equation (2.18) implies that there exists a constant 𝐡>0 with ||||π‘₯(𝑑)≀𝐡,βˆ€π‘‘β‰₯𝑑0.(2.20) Using (2.18) and (2.19), we deduce that, for any πœ€>0, there exist sufficiently large numbers 𝑇>1+|𝑑0| and πΎβˆˆβ„• satisfying 𝐡𝑝𝐾0<πœ€4ξ€½||||,||||ξ€Ύ<πœ€ξ€·π‘,max𝑦(𝑑)π‘Ÿ(𝑑)0ξ€Έβˆ’14,||||𝑝(𝑑)β‰₯𝑝0,βˆ€π‘‘β‰₯𝑇.(2.21) In view of (2.17), (2.20), and (2.21), we infer that for all 𝑑β‰₯𝑇||π‘₯||=||π‘₯ξ€·πœ(𝑑)βˆ’1ξ€Έξ€·πœ(𝑑)βˆ’π‘¦βˆ’1ξ€Έξ€·πœ(𝑑)βˆ’π‘Ÿβˆ’1ξ€Έ||(𝑑)||π‘ξ€·πœβˆ’1ξ€Έ||≀||π‘₯ξ€·πœ(𝑑)βˆ’1(ξ€Έ||+||π‘¦ξ€·πœπ‘‘)βˆ’1(ξ€Έ||+||π‘Ÿξ€·πœπ‘‘)βˆ’1(ξ€Έ||𝑑)||π‘ξ€·πœβˆ’1ξ€Έ||<1(𝑑)𝑝0||π‘₯ξ€·πœβˆ’1ξ€Έ||+πœ€ξ€·π‘(𝑑)0ξ€Έβˆ’12𝑝0≀1𝑝01𝑝0||π‘₯ξ€·πœβˆ’2ξ€Έ||+πœ€ξ€·π‘(𝑑)0ξ€Έβˆ’12𝑝0ξƒ­+πœ€ξ€·π‘0ξ€Έβˆ’12𝑝0=1𝑝20||π‘₯ξ€·πœβˆ’2ξ€Έ||+πœ€ξ€·π‘(𝑑)0ξ€Έβˆ’12𝑝0ξ‚΅11+𝑝0≀1≀⋅⋅⋅𝑝𝐾0||π‘₯ξ€·πœβˆ’πΎξ€Έ||+πœ€ξ€·π‘(𝑑)0ξ€Έβˆ’12𝑝011+𝑝01+β‹…β‹…β‹…+𝑝0πΎβˆ’1ξƒͺ≀𝐡𝑝𝐾0+πœ€ξ€·π‘0ξ€Έβˆ’12𝑝0β‹…11βˆ’1/𝑝0<πœ€,(2.22) which gives that lim𝑑→+∞π‘₯(𝑑)=0. This completes the proof.

Lemma 2.3. Let π‘₯,𝑝,𝜏,π‘Ÿ, and 𝑦 be in 𝐢([𝑑0,+∞),ℝ) satisfying (𝐴2), (2.17), (2.18), and lim𝑑→+∞||||𝑦(𝑑)=𝑑>0,lim𝑑→+βˆžπ‘π‘Ÿ(𝑑)=0;(2.23)1β‰₯||||𝑝(𝑑)β‰₯𝑝0>1eventually,𝑝20>𝑝0+𝑝1,(2.24) where 𝑑,𝑝0, and 𝑝1 are constants. Then, there exists 𝐿>0 such that |π‘₯(𝑑)|β‰₯𝐿 eventually.

Proof. Obviously, (2.20) holds. It follows from (2.18), (2.23), and (2.24) that for πœ€=𝑑[𝑝0(𝑝0βˆ’1)βˆ’π‘1]/(𝑝0(𝑝0βˆ’1)+𝑝1)>0, there exist πΎβˆˆβ„• and 𝑇>1+|𝑑0| satisfying 𝐡𝑝𝐾0<πœ€4𝑝1πœ€,π‘‘βˆ’4<||𝑦||πœ€(𝑑)<𝑑+4,||π‘Ÿ||<πœ€(𝑑)4𝑝0,𝑝1β‰₯||𝑝||(𝑑)β‰₯𝑝0,βˆ€π‘‘β‰₯𝑇.(2.25) Put 𝐿=𝑑[𝑝0(𝑝0βˆ’1)βˆ’π‘1]/2𝑝1𝑝0(𝑝0βˆ’1). In light of (2.17), we conclude that for each 𝑑β‰₯𝑇π‘₯π‘₯ξ€·πœ(𝑑)=βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘¦ξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ=1(𝑑)π‘ξ€·πœβˆ’1π‘₯ξ€·πœ(𝑑)βˆ’2ξ€Έ(𝑑)π‘ξ€·πœβˆ’2ξ€Έβˆ’π‘¦ξ€·πœ(𝑑)βˆ’2ξ€Έ(𝑑)π‘ξ€·πœβˆ’2ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’2ξ€Έ(𝑑)π‘ξ€·πœβˆ’2ξ€Έξƒ­βˆ’π‘¦ξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ=π‘₯ξ€·πœ(𝑑)βˆ’2ξ€Έ(𝑑)Ξ 2𝑖=1𝑝(πœβˆ’π‘–βˆ’(𝑑))2𝑗=1π‘¦ξ€·πœβˆ’π‘—ξ€Έ(𝑑)Π𝑗𝑖=1𝑝(πœβˆ’π‘–(βˆ’π‘‘))2𝑗=1π‘Ÿξ€·πœβˆ’π‘—ξ€Έ(𝑑)Π𝑗𝑖=1𝑝(πœβˆ’π‘–(=π‘₯ξ€·πœπ‘‘))=β‹…β‹…β‹…βˆ’πΎξ€Έ(𝑑)Π𝐾𝑖=1𝑝(πœβˆ’π‘–βˆ’(𝑑))𝐾𝑗=1π‘¦ξ€·πœβˆ’π‘—ξ€Έ(𝑑)Π𝑗𝑖=1𝑝(πœβˆ’π‘–βˆ’(𝑑))𝐾𝑗=1π‘Ÿξ€·πœβˆ’π‘—ξ€Έ(𝑑)Π𝑗𝑖=1𝑝(πœβˆ’π‘–,(𝑑))(2.26) which together with (2.20) and (2.25) yields that for any 𝑑β‰₯𝑇||||β‰₯||π‘¦ξ€·πœπ‘₯(𝑑)βˆ’1ξ€Έ||(𝑑)||π‘ξ€·πœβˆ’1ξ€Έ||βˆ’||π‘₯ξ€·πœ(𝑑)βˆ’πΎξ€Έ||(𝑑)Π𝐾𝑖=1||𝑝(πœβˆ’π‘–||βˆ’(𝑑))𝐾𝑗=2||π‘¦ξ€·πœβˆ’π‘—ξ€Έ||(𝑑)Π𝑗𝑖=1||𝑝(πœβˆ’π‘–||βˆ’(𝑑))𝐾𝑗=1||π‘Ÿξ€·πœβˆ’π‘—ξ€Έ||(𝑑)Π𝑗𝑖=1||𝑝(πœβˆ’π‘–||β‰₯(𝑑))π‘‘βˆ’πœ€/4𝑝1βˆ’π΅π‘πΎ0βˆ’ξ‚€πœ€π‘‘+4𝐾𝑗=21𝑝𝑗0βˆ’πœ€4𝑝0𝐾𝑗=11𝑝𝑗0β‰₯π‘‘βˆ’πœ€/4𝑝1βˆ’πœ€4𝑝1βˆ’ξ‚€πœ€π‘‘+4⋅1/𝑝201βˆ’1/𝑝0βˆ’πœ€4𝑝0β‹…1/𝑝01βˆ’1/𝑝0=π‘‘βˆ’πœ€/2𝑝1βˆ’π‘‘+πœ€/2𝑝0𝑝0ξ€Έ=π‘‘ξ€Ίπ‘βˆ’10𝑝0ξ€Έβˆ’1βˆ’π‘1ξ€»βˆ’ξ€Ίπ‘(πœ€/2)0𝑝0ξ€Έβˆ’1+𝑝1𝑝1𝑝0𝑝0ξ€Έβˆ’1=𝐿.(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 π‘₯,𝑝,𝜏,π‘Ÿ, and 𝑦 be in 𝐢([𝑑0,+∞),ℝ) satisfying (𝐴2), (2.17), (2.18), and lim𝑑→+βˆžπ‘¦(𝑑)=lim𝑑→+∞||||π‘Ÿ(𝑑)=0;(2.28)𝑝(𝑑)≀𝑝0<12eventually,(2.29) where 𝑝0 is a constant. Then, lim𝑑→+∞π‘₯(𝑑)=0.

Lemma 2.5. Let π‘₯, 𝑝, 𝜏, π‘Ÿ, and 𝑦 be in 𝐢([𝑑0,+∞),ℝ) satisfying (𝐴2), (2.17), (2.18), (2.23), and (2.29). Then, there exists 𝐿>0 such that |π‘₯(𝑑)|β‰₯𝐿 eventually.

Lemma 2.6 (Krasnoselskii’s fixed point theorem). Let 𝑋 be a Banach space, let π‘Œ be a nonempty bounded closed convex subset of 𝑋, and let 𝑓, 𝑔 be mappings of π‘Œ into 𝑋 such that 𝑓π‘₯+π‘”π‘¦βˆˆπ‘Œ for every pair π‘₯,π‘¦βˆˆπ‘Œ. If 𝑓 is a contraction mapping and 𝑔 is completely continuous, then the mapping 𝑓+𝑔 has a fixed point in π‘Œ.

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,𝑝1βˆˆβ„+⧡{0},π‘Ÿ0,π‘Ÿ1βˆˆβ„+, and π‘ŸβˆˆπΆπ‘›([𝑑0,+∞),ℝ) satisfying 𝑝1β‰₯𝑝(𝑑)β‰₯𝑝0>1eventually,𝑝20>𝑝0+𝑝1;π‘Ÿ(3.1)(𝑛)(𝑑)=𝑔(𝑑),βˆ’π‘Ÿ0β‰€π‘Ÿ(𝑑)β‰€π‘Ÿ1ξ€œeventually;(3.2)𝑑+∞0π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑑𝑠<+∞.(3.3) Then, the following hold:(a)for arbitrarily positive constants 𝑀 and 𝑁 with 𝑝0ξ€Έξ€·π‘βˆ’1𝑀>1ξ€Έπ‘βˆ’1𝑁+1π‘Ÿ1𝑝0+π‘Ÿ0,(3.4) equation (1.1) has uncountably many bounded positive solutions π‘₯∈𝐴(𝑁,𝑀) with 𝑁≀liminf𝑑→+∞π‘₯(𝑑)≀limsup𝑑→+∞π‘₯(𝑑)≀𝑀;(3.5)(b)for arbitrarily positive constants 𝑀 and 𝑁 with 𝑝0ξ€Έξ€·π‘βˆ’1𝑁>1ξ€Έπ‘βˆ’1𝑀+1π‘Ÿ0𝑝0+π‘Ÿ1,(3.6) equation (1.1) has uncountably many bounded negative solutions π‘₯∈𝐴(βˆ’π‘,βˆ’π‘€) with βˆ’π‘β‰€liminf𝑑→+∞π‘₯(𝑑)≀limsup𝑑→+∞π‘₯(𝑑)β‰€βˆ’π‘€.(3.7)

Proof. It follows from (3.1) and (3.2) that there exists an enough large constant 𝑇0 with πœβˆ’1(𝑇0)>1+|𝑑0|+|𝛽| satisfying 𝑝0≀𝑝(𝑑)≀𝑝1,π‘Ÿ(𝑛)(𝑑)=𝑔(𝑑),βˆ’π‘Ÿ0β‰€π‘Ÿ(𝑑)β‰€π‘Ÿ1,βˆ€π‘‘β‰₯𝑇0.(3.8)
(a) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.4). Let 𝐷∈((𝑝1βˆ’1)𝑁+(𝑝1π‘Ÿ1/𝑝0),(𝑝0βˆ’1)π‘€βˆ’π‘Ÿ0). First of all, we prove that there exist two mappings 𝐹𝐷,𝐺𝐷∢𝐴(𝑁,𝑀)→𝐡𝐢([𝛽,+∞),ℝ) and a constant 𝑇𝐷>πœβˆ’1(𝑇0) such that 𝐹𝐷+𝐺𝐷 has a fixed point π‘₯∈𝐴(𝑁,𝑀), which is also a bounded positive solution of (1.1) with 𝑁≀liminf𝑑→+∞π‘₯(𝑑)≀limsup𝑑→+∞π‘₯(𝑑)≀𝑀. Put ξ€½||𝑓𝐡=max𝑖𝑒1,𝑒2,…,π‘’π‘˜π‘–ξ€Έ||βˆΆπ‘’π‘—βˆˆ[]𝑁,𝑀,1β‰€π‘—β‰€π‘˜π‘–ξ€Ύ,1β‰€π‘–β‰€π‘š.(3.9) In light of (3.3), (3.9), and (𝐴2), we infer that there exists a sufficiently large number 𝑇𝐷>πœβˆ’1(𝑇0) satisfying 𝐡𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–ξ‚»(𝑠)𝑑𝑠<minπ‘€βˆ’π·+𝑀+π‘Ÿ0𝑝0,𝐷+𝑁𝑝1βˆ’π‘Ÿ1𝑝0ξ‚Όβˆ’π‘.(3.10) Define two mappings 𝐹𝐷,𝐺𝐷∢𝐴(𝑁,𝑀)→𝐢([𝛽,+∞),ℝ) by 𝐹𝐷π‘₯ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩𝐷(𝑑)=π‘ξ€·πœβˆ’1ξ€Έ+π‘₯ξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ(𝑑),𝑑β‰₯𝑇𝐷𝐹𝐷π‘₯𝑇𝐷,𝛽≀𝑑<𝑇𝐷,𝐺(3.11)𝐷π‘₯ξ€ΈβŽ§βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺ⎩(𝑑)=(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘›βˆ’1Γ—π‘šξ“π‘–=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–(𝑠)𝑑𝑠,𝑑β‰₯𝑇𝐷,𝐺𝐷π‘₯𝑇𝐷,𝛽≀𝑑<𝑇𝐷,(3.12) for each π‘₯∈𝐴(𝑁,𝑀). In view of (3.1), (3.8), and (3.10)–(3.12), we conclude that for any π‘₯,π‘’βˆˆπ΄(𝑁,𝑀) and 𝑑β‰₯𝑇𝐷||𝐹𝐷π‘₯𝐹(𝑑)βˆ’π·π‘’ξ€Έ||=||||π‘₯ξ€·πœ(𝑑)βˆ’1ξ€Έξ€·πœ(𝑑)βˆ’π‘’βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ||||≀1(𝑑)𝑝0𝐹‖π‘₯βˆ’π‘’β€–,𝐷π‘₯𝐺(𝑑)+𝐷𝑒=𝐷(𝑑)π‘ξ€·πœβˆ’1(ξ€Έ+π‘₯ξ€·πœπ‘‘)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1(ξ€Έβˆ’π‘Ÿξ€·πœπ‘‘)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1(ξ€Έ+𝑑)(βˆ’1)π‘›π‘ξ€·πœβˆ’1(ξ€Έ(Γ—ξ€œπ‘‘)π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)π‘“π‘–ξ€·π‘’ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),𝑒𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘’π‘–π‘˜π‘–β‰€π·(𝑠)𝑑𝑠𝑝0+𝑀𝑝0+π‘Ÿ0𝑝0+𝐡𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–<(𝑠)𝑑𝑠𝐷+𝑀+π‘Ÿ0𝑝0ξ‚»+minπ‘€βˆ’π·+𝑀+π‘Ÿ0𝑝0,𝐷+𝑁𝑝1βˆ’π‘Ÿ1𝑝0ξ‚Όξ€·πΉβˆ’π‘β‰€π‘€,𝐷π‘₯𝐺(𝑑)+𝐷𝑒=𝐷(𝑑)π‘ξ€·πœβˆ’1ξ€Έ+π‘₯ξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ+(𝑑)(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1(𝑑)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)π‘“π‘–ξ€·π‘’ξ€·πœŽπ‘–1(ξ€Έξ€·πœŽπ‘ ),𝑒𝑖2(ξ€Έξ€·πœŽπ‘ ),…,π‘’π‘–π‘˜π‘–(β‰₯𝐷𝑠)𝑑𝑠𝑝1+𝑁𝑝1βˆ’π‘Ÿ1𝑝0βˆ’π΅π‘0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–>(𝑠)𝑑𝑠𝐷+𝑁𝑝1βˆ’π‘Ÿ1𝑝0ξ‚»βˆ’minπ‘€βˆ’π·+𝑀+π‘Ÿ0𝑝0,𝐷+𝑁𝑝1βˆ’π‘Ÿ1𝑝0ξ‚Ό||ξ€·πΊβˆ’π‘β‰₯𝑁,𝐷𝑒||=||||(𝑑)(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)π‘“π‘–ξ€·π‘’ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),𝑒𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘’π‘–π‘˜π‘–|||||≀𝐡(𝑠)𝑑𝑠𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–ξ‚»(𝑠)𝑑𝑠<minπ‘€βˆ’π·+𝑀+π‘Ÿ0𝑝0,𝐷+𝑁𝑝1βˆ’π‘Ÿ1𝑝0ξ‚Όβˆ’π‘<𝑀,(3.13) which ensures that ‖‖𝐹𝐷π‘₯βˆ’πΉπ·π‘’β€–β€–=sup𝑑β‰₯𝑇𝐷||𝐹𝐷π‘₯𝐹(𝑑)βˆ’π·π‘’ξ€Έ||≀1(𝑑)𝑝0𝐹‖π‘₯βˆ’π‘’β€–,βˆ€π‘₯,π‘’βˆˆπ΄(𝑁,𝑀),(3.14)𝐷π‘₯+πΊπ·β€–β€–πΊπ‘’βˆˆπ΄(𝑁,𝑀),βˆ€π‘₯,π‘’βˆˆπ΄(𝑁,𝑀),(3.15)𝐷𝑒‖‖≀𝑀,βˆ€π‘’βˆˆπ΄(𝑁,𝑀).(3.16) It follows from (3.11), (3.12), (3.15), and (3.16) that 𝐹𝐷 and 𝐺𝐷 map 𝐴(𝑁,𝑀) into 𝐡𝐢([𝛽,+∞),ℝ), respectively.
Now, we show that 𝐺𝐷 is continuous in 𝐴(𝑁,𝑀). Let {π‘₯𝑙}π‘™βˆˆβ„•βŠ‚π΄(𝑁,𝑀) and π‘₯∈𝐴(𝑁,𝑀) with limπ‘™β†’βˆžπ‘₯𝑙=π‘₯, given πœ€>0. It follows from the uniform continuity of 𝑓𝑖 in [𝑁,𝑀]π‘˜π‘– for 1β‰€π‘–β‰€π‘š and limπ‘™β†’βˆžπ‘₯𝑙=π‘₯ that there exist 𝛿>0 and πΎβˆˆβ„• satisfying ||𝑓𝑖𝑒𝑖1,𝑒𝑖2,…,π‘’π‘–π‘˜π‘–ξ€Έβˆ’π‘“π‘–ξ€·π‘£π‘–1,𝑣𝑖2,…,π‘£π‘–π‘˜π‘–ξ€Έ||<πœ€ξ€·1+1/𝑝0ξ€Έβˆ«(π‘›βˆ’1)!𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘›βˆ’1βˆ‘π‘šπ‘–=1π‘žπ‘–(𝑠)𝑑𝑠,βˆ€π‘’π‘–π‘—,π‘£π‘–π‘—βˆˆ[],||𝑒𝑁,π‘€π‘–π‘—βˆ’π‘£π‘–π‘—||<𝛿,1β‰€π‘—β‰€π‘˜π‘–β€–β€–π‘₯,1β‰€π‘–β‰€π‘š,π‘™β€–β€–βˆ’π‘₯<𝛿,βˆ€π‘™β‰₯𝐾.(3.17) In view of (3.8), (3.12), (3.17), we arrive at ‖𝐺𝐷π‘₯π‘™βˆ’πΊπ·π‘₯β€–=sup𝑑β‰₯𝑇𝐷||||(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–ξ€Ίπ‘“(𝑠)𝑖π‘₯π‘™ξ€·πœŽπ‘–1ξ€Έ(𝑠),π‘₯π‘™ξ€·πœŽπ‘–2ξ€Έ(𝑠),…,π‘₯π‘™ξ€·πœŽπ‘–π‘˜π‘–(𝑠)ξ€Έξ€Έβˆ’π‘“π‘–ξ€·π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–|||||(𝑠)𝑑𝑠≀sup𝑑β‰₯𝑇𝐷1𝑝0Γ—ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–||𝑓(𝑠)𝑖π‘₯π‘™ξ€·πœŽπ‘–1ξ€Έ(𝑠),π‘₯π‘™ξ€·πœŽπ‘–2ξ€Έ(𝑠),…,π‘₯π‘™ξ€·πœŽπ‘–π‘˜π‘–(𝑠)ξ€Έξ€Έβˆ’π‘“π‘–ξ€·π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–||≀1(𝑠)𝑑𝑠𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–πœ€(𝑠)𝑑𝑠⋅1+1/𝑝0∫(π‘›βˆ’1)!𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘›βˆ’1βˆ‘π‘šπ‘–=1π‘žπ‘–(𝑠)𝑑𝑠<πœ€,βˆ€π‘™β‰₯𝐾,(3.18) which means that 𝐺𝐷 is continuous in 𝐴(𝑁,𝑀).
Next, we show that 𝐺𝐷(𝐴(𝑁,𝑀)) is equicontinuous in [𝛽,+∞). Let πœ€>0. Taking into account (3.3) and (𝐴2), we know that there exists π‘‡βˆ—>𝑇𝐷 satisfying 1𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1(π‘‡βˆ—)π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–πœ€(𝑠)𝑑𝑠<4.(3.19) Put 𝐡1𝑠=maxπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)βˆΆπœβˆ’1ξ€·π‘‡π·ξ€Έβ‰€π‘ β‰€πœβˆ’1ξ€·π‘‡βˆ—ξ€Έξƒ°.(3.20) It follows from the uniform continuity of π‘πœβˆ’1 and πœβˆ’1 in [𝑇𝐷,π‘‡βˆ—] that there exists 𝛿>0 satisfying ||π‘ξ€·πœβˆ’1𝑑1ξ€·πœξ€Έξ€Έβˆ’π‘βˆ’1𝑑2||<ξ€Έξ€Έπœ€π‘20(π‘›βˆ’1)!4ξ‚ƒβˆ«1+𝐡𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘›βˆ’1βˆ‘π‘šπ‘–=1π‘žπ‘–ξ‚„,(𝑠)π‘‘π‘ βˆ€π‘‘1,𝑑2βˆˆξ€Ίπ‘‡π·,π‘‡βˆ—ξ€»||𝑑with1βˆ’π‘‘2||||𝜏<𝛿;βˆ’1𝑑1ξ€Έβˆ’πœβˆ’1𝑑2ξ€Έ||<πœ€π‘0(π‘›βˆ’1)!4𝐡1+𝐡1+∫(π‘›βˆ’1)𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘’π‘›βˆ’1βˆ‘π‘šπ‘–=1π‘žπ‘–ξ‚„,(𝑠)π‘‘π‘ βˆ€π‘‘1,𝑑2βˆˆξ€Ίπ‘‡π·,π‘‡βˆ—ξ€»||𝑑with1βˆ’π‘‘2||<𝛿.(3.21) Let π‘₯∈𝐴(𝑁,𝑀) and 𝑑1,𝑑2∈[𝛽,+∞) with |𝑑1βˆ’π‘‘2|<𝛿. We consider three possible cases.
Case 1. Let 𝑑1,𝑑2∈[π‘‡βˆ—,+∞). In view of (3.8), (3.9), (3.12), and (3.19), we conclude that ||𝐺𝐷π‘₯𝑑1ξ€Έβˆ’ξ€·πΊπ·π‘₯𝑑2ξ€Έ||=1||||1(π‘›βˆ’1)!π‘ξ€·πœβˆ’1𝑑1Γ—ξ€œξ€Έξ€Έπœ+βˆžβˆ’1𝑑1ξ€Έξ€·π‘ βˆ’πœβˆ’1𝑑1ξ€Έξ€Έπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–βˆ’1(𝑠)ξ€Έξ€Έπ‘‘π‘ π‘ξ€·πœβˆ’1𝑑2Γ—ξ€œξ€Έξ€Έπœ+βˆžβˆ’1𝑑2ξ€Έξ€·π‘ βˆ’πœβˆ’1𝑑2ξ€Έξ€Έπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–|||||≀𝐡(𝑠)𝑑𝑠𝑝0ξƒ¬ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1𝑑1ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–ξ€œ(𝑠)𝑑𝑠+𝜏+βˆžβˆ’1𝑑2ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–ξƒ­<πœ€(𝑠)𝑑𝑠2.(3.22)
Case 2. Let 𝑑1,𝑑2∈[𝑇𝐷,π‘‡βˆ—]. In terms of (3.8), (3.9), (3.12), (3.21), we arrive at ||𝐺𝐷π‘₯𝑑1ξ€Έβˆ’ξ€·πΊπ·π‘₯𝑑2ξ€Έ||=1||||1(π‘›βˆ’1)!π‘ξ€·πœβˆ’1𝑑1Γ—ξ€œξ€Έξ€Έπœ+βˆžβˆ’1𝑑1ξ€Έξ€·π‘ βˆ’πœβˆ’1𝑑1ξ€Έξ€Έπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–βˆ’1(𝑠)ξ€Έξ€Έπ‘‘π‘ π‘ξ€·πœβˆ’1𝑑2Γ—ξ€œξ€Έξ€Έπœ+βˆžβˆ’1𝑑2ξ€Έξ€·π‘ βˆ’πœβˆ’1𝑑2ξ€Έξ€Έπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–|||||≀1(𝑠)𝑑𝑠||||1(π‘›βˆ’1)!π‘ξ€·πœβˆ’1𝑑1βˆ’1ξ€Έξ€Έπ‘ξ€·πœβˆ’1𝑑2||||Γ—ξ€œξ€Έξ€Έπœ+βˆžβˆ’1𝑑1ξ€Έξ€·π‘ βˆ’πœβˆ’1𝑑1ξ€Έξ€Έπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–+1(𝑠)ξ€Έξ€Έπ‘‘π‘ π‘ξ€·πœβˆ’1𝑑2Γ—βŽ‘βŽ’βŽ’βŽ£|||||ξ€œξ€Έξ€Έπœβˆ’1(𝑑2)πœβˆ’1𝑑1ξ€Έξ€·π‘ βˆ’πœβˆ’1𝑑1ξ€Έξ€Έπ‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–|||||+ξ€œ(𝑠)ξ€Έξ€Έπ‘‘π‘ πœ+βˆžβˆ’1(𝑑2)|||ξ€·π‘ βˆ’πœβˆ’1𝑑1ξ€Έξ€Έπ‘›βˆ’1βˆ’ξ€·π‘ βˆ’πœβˆ’1𝑑2ξ€Έξ€Έπ‘›βˆ’1|||Γ—π‘šξ“π‘–=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–β‰€π΅(𝑠)𝑑𝑠||π‘ξ€·πœ(π‘›βˆ’1)!βˆ’1𝑑1ξ€·πœξ€Έξ€Έβˆ’π‘βˆ’1𝑑2||ξ€Έξ€Έπ‘ξ€·πœβˆ’1𝑑1π‘ξ€·πœξ€Έξ€Έβˆ’1𝑑2ξ€œξ€Έξ€Έπœ+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(1𝑠)𝑑𝑠+𝑝0Γ—βŽ‘βŽ’βŽ’βŽ£|||||ξ€œπœβˆ’1(𝑑2)πœβˆ’1𝑑1ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–|||||+ξ€œ(𝑠)π‘‘π‘ πœ+βˆžβˆ’1𝑑2ξ€Έ(π‘›βˆ’1)𝑠max{π‘›βˆ’2,0}||πœβˆ’1𝑑1ξ€Έβˆ’πœβˆ’1𝑑2ξ€Έ||π‘šξ“π‘–=1π‘žπ‘–β‰€π΅(𝑠)𝑑𝑠𝑝20||π‘ξ€·πœ(π‘›βˆ’1)!βˆ’1𝑑1ξ€·πœξ€Έξ€Έβˆ’π‘βˆ’1𝑑2||ξ€œξ€Έξ€Έπœ+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–+𝐡(𝑠)𝑑𝑠𝑝0𝐡(π‘›βˆ’1)!1ξ€œ+(π‘›βˆ’1)𝜏+βˆžβˆ’1ξ€·π‘‡π·ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–ξƒ­||𝜏(𝑠)π‘‘π‘ βˆ’1𝑑1ξ€Έβˆ’πœβˆ’1𝑑2ξ€Έ||<πœ€2.(3.23)
Case 3. Let 𝑑1,𝑑2∈[𝛽,𝑇𝐷]. By (3.12), we have ||𝐺𝐷π‘₯𝑑1ξ€Έβˆ’ξ€·πΊπ·π‘₯𝑑2ξ€Έ||=||𝐺𝐷π‘₯π‘‡ξ€Έξ€·π·ξ€Έβˆ’ξ€·πΊπ·π‘₯𝑇𝐷||=0<πœ€.(3.24) Thus, 𝐺𝐷(𝐴(𝑁,𝑀)) is equicontinuous in [𝛽,+∞). Consequently, 𝐺𝐷(𝐴(𝑁,𝑀)) is relatively compact by (3.16) and the continuity of 𝐺𝐷. By means of (3.14), (3.15), and Lemma 2.6, we infer that 𝐹𝐷+𝐺𝐷 possesses a fixed point π‘₯∈𝐴(𝑁,𝑀), that is, π‘₯𝐷(𝑑)=π‘ξ€·πœβˆ’1ξ€Έ+π‘₯ξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ+(𝑑)(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–(𝑠)𝑑𝑠,βˆ€π‘‘β‰₯𝑇𝐷,(3.25) which gives that π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑))=βˆ’π·+π‘Ÿ(𝑑)+(βˆ’1)π‘›βˆ’1Γ—ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑠),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑠),…,π‘₯π‘–π‘˜π‘–(𝑠)𝑑𝑠,βˆ€π‘‘β‰₯πœβˆ’1𝑇𝐷,[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(𝜏(𝑑))(𝑛)=𝑔(𝑑)βˆ’π‘šξ“π‘–=1π‘žπ‘–(𝑑)𝑓𝑖π‘₯ξ€·πœŽπ‘–1ξ€Έξ€·πœŽ(𝑑),π‘₯𝑖2ξ€Έξ€·πœŽ(𝑑),…,π‘₯π‘–π‘˜π‘–(𝑑)ξ€Έξ€Έ,βˆ€π‘‘β‰₯πœβˆ’1𝑇𝐷,(3.26) which mean that π‘₯∈𝐴(𝑁,𝑀) is a bounded positive solution of (1.1) with 𝑁≀liminf𝑑→+∞π‘₯(𝑑)≀limsup𝑑→+∞π‘₯(𝑑)≀𝑀.(3.27)
Let 𝐷1 and 𝐷2 be two arbitrarily different numbers in ((𝑝1βˆ’1)𝑁+(𝑝1π‘Ÿ1/𝑝0),(𝑝0βˆ’1)π‘€βˆ’π‘Ÿ0). Similarly, we conclude that for each π‘™βˆˆ{1,2} there exist two mappings 𝐹𝐷𝑗,πΊπ·π‘—βˆΆπ΄(𝑁,𝑀)→𝐡𝐢([𝛽,+∞),ℝ) and a sufficiently large number 𝑇𝐷𝑙>πœβˆ’1(𝑇0) satisfying (3.8)–(3.12), where 𝐷,𝑇𝐷,𝐹𝐷, and 𝐺𝐷 are replaced by 𝐷𝑙,𝑇𝐷𝑙,𝐹𝐷𝑙, and 𝐺𝐷𝑙, respectively, and 𝐹𝐷𝑙+𝐺𝐷𝑙 has a fixed point π‘₯π‘™βˆˆπ΄(𝑁,𝑀), which is also a bounded positive solution with 𝑁≀liminf𝑑→+∞π‘₯𝑙(𝑑)≀limsup𝑑→+∞π‘₯𝑙(𝑑)≀𝑀, that is, π‘₯𝑙𝐷(𝑑)=π‘™π‘ξ€·πœβˆ’1ξ€Έ+π‘₯(𝑑)π‘™ξ€·πœβˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’π‘Ÿξ€·πœ(𝑑)βˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ+(𝑑)(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–(𝑠)𝑓𝑖π‘₯π‘™ξ€·πœŽπ‘–1ξ€Έ(𝑠),π‘₯π‘™ξ€·πœŽπ‘–2ξ€Έ(𝑠),…,π‘₯π‘™ξ€·πœŽπ‘–π‘˜π‘–(𝑠)𝑑𝑠,βˆ€π‘‘β‰₯𝑇𝐷𝑙.(3.28) It follows from (3.3) that there exists 𝑇3>max{𝑇𝐷1,𝑇𝐷2} satisfying 𝐡𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1𝑇3ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–||𝐷(𝑠)𝑑𝑠<1βˆ’π·2||4𝑝1.(3.29) Combining (3.8), (3.28), and (3.29), we conclude easily that ||π‘₯1(𝑑)βˆ’π‘₯2||=||||𝐷(𝑑)1βˆ’π·2π‘ξ€·πœβˆ’1ξ€Έ+π‘₯(𝑑)1ξ€·πœβˆ’1ξ€Έ(𝑑)βˆ’π‘₯2ξ€·πœβˆ’1ξ€Έ(𝑑)π‘ξ€·πœβˆ’1ξ€Έ+(𝑑)(βˆ’1)π‘›π‘ξ€·πœβˆ’1ξ€ΈΓ—ξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)ξ€·π‘ βˆ’πœβˆ’1ξ€Έ(𝑑)π‘›βˆ’1Γ—π‘šξ“π‘–=1π‘žπ‘–ξ€Ίπ‘“(𝑠)𝑖π‘₯1ξ€·πœŽπ‘–1ξ€Έ(𝑠),π‘₯1ξ€·πœŽπ‘–2ξ€Έ(𝑠),…,π‘₯1ξ€·πœŽπ‘–π‘˜π‘–(𝑠)ξ€Έξ€Έβˆ’π‘“π‘–ξ€·π‘₯2ξ€·πœŽπ‘–1ξ€Έ(𝑠),π‘₯2ξ€·πœŽπ‘–2ξ€Έ(𝑠),…,π‘₯2ξ€·πœŽπ‘–π‘˜π‘–||||β‰₯||𝐷(𝑠)𝑑𝑠1βˆ’π·2||π‘ξ€·πœβˆ’1ξ€Έβˆ’||π‘₯(𝑑)1ξ€·πœβˆ’1ξ€Έ(𝑑)βˆ’π‘₯2ξ€·πœβˆ’1ξ€Έ||(𝑑)π‘ξ€·πœβˆ’1ξ€Έβˆ’(𝑑)2π΅π‘ξ€·πœβˆ’1ξ€Έξ€œ(𝑑)(π‘›βˆ’1)!𝜏+βˆžβˆ’1(𝑑)π‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–β‰₯||𝐷(𝑠)𝑑𝑠1βˆ’π·2||𝑝1βˆ’β€–β€–π‘₯1βˆ’π‘₯2‖‖𝑝0βˆ’2𝐡𝑝0ξ€œ(π‘›βˆ’1)!𝜏+βˆžβˆ’1𝑇3ξ€Έπ‘ π‘šπ‘›βˆ’1𝑖=1π‘žπ‘–>||𝐷(𝑠)𝑑𝑠1βˆ’π·2||𝑝1βˆ’β€–β€–π‘₯1βˆ’π‘₯2‖‖𝑝0βˆ’||𝐷1βˆ’π·2||2𝑝1=||𝐷1βˆ’π·2||2𝑝1βˆ’β€–β€–π‘₯1βˆ’π‘₯2‖‖𝑝0,βˆ€π‘‘β‰₯𝑇3,(3.30) which guarantees that β€–β€–π‘₯1βˆ’π‘₯2β€–β€–β‰₯𝑝0||𝐷1βˆ’π·2||2𝑝1ξ€·1+𝑝0ξ€Έ>0,(3.31) that is, π‘₯1β‰ π‘₯2. Hence, (1.1) has uncountably many bounded positive solutions π‘₯∈𝐴(𝑁,𝑀) with 𝑁≀liminf𝑑→+∞π‘₯(𝑑)≀limsup𝑑→+∞π‘₯(𝑑)≀𝑀.

(b) Assume that 𝑀 and 𝑁 are arbitrary positive constants satisfying (3.6) and put 𝐡2ξ€½||𝑓=max𝑖𝑒1,𝑒2,…,π‘’π‘˜π‘–ξ€Έ||βˆΆπ‘’π‘—βˆˆ[]βˆ’π‘,βˆ’π‘€,1β‰€π‘—β‰€π‘˜π‘–ξ€Ύ,1β‰€π‘–β‰€π‘š.(3.32) Let 𝐷∈((1βˆ’π‘0)𝑁+π‘Ÿ1,(1βˆ’π‘1)π‘€βˆ’(𝑝1π‘Ÿ0/𝑝0)). It follows from (3.3), (3.8), (3.32), and (𝐴2) that there exists 𝑇𝐷>πœβˆ’1(𝑇0) satisfying 𝐡2𝑝0ξ€œ(π‘›βˆ’