Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 328956 | https://doi.org/10.1155/2011/328956

Zeqing Liu, Ming Jia, Jeong Sheok Ume, Shin Min Kang, "Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation", Abstract and Applied Analysis, vol. 2011, Article ID 328956, 16 pages, 2011. https://doi.org/10.1155/2011/328956

Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation

Academic Editor: Sergey V. Zelik
Received23 May 2011
Accepted31 Jul 2011
Published11 Oct 2011

Abstract

This paper deals with the higher-order nonlinear neutral delay differential equation (𝑑𝑛/𝑑𝑑𝑛)[π‘₯(𝑑)+βˆ‘π‘šπ‘–=1𝑝𝑖(𝑑)π‘₯(𝑇𝑖(𝑑))]+(π‘‘π‘›βˆ’1/π‘‘π‘‘π‘›βˆ’1)𝑓(𝑑,π‘₯(𝛼1(𝑑)),…,π‘₯(π›Όπ‘˜(𝑑)))+β„Ž(𝑑,π‘₯(𝛽1(𝑑)),…,π‘₯(π›½π‘˜(𝑑)))=𝑔(𝑑), 𝑑β‰₯π‘‘π‘œ, where 𝑛,π‘š,π‘˜βˆˆβ„•, 𝑝𝑖,πœπ‘–,𝛽𝑗,π‘”βˆˆπΆ([π‘‘π‘œ,+∞),ℝ), π›Όπ‘—βˆˆπΆπ‘›βˆ’1([π‘‘π‘œ,+∞),ℝ), π‘“βˆˆπΆπ‘›βˆ’1([π‘‘π‘œ,+∞)Γ—β„π‘˜,ℝ), β„ŽβˆˆπΆ([π‘‘π‘œ,+∞)Γ—β„π‘˜,ℝ), and lim𝑑→+βˆžπœπ‘–(𝑑)=lim𝑑→+βˆžπ›Όπ‘—(𝑑)=lim𝑑→+βˆžπ›½π‘—(𝑑)=+∞, π‘–βˆˆ{1,2,…,π‘š},π‘—βˆˆ{1,2,…,π‘˜}. By making use of the Leray-Schauder nonlinear alterative theorem, we establish the existence of uncountably many bounded positive solutions for the above equation. Our results improve and generalize some corresponding results in the field. Three examples are given which illustrate the advantages of the results presented in this paper.

1. Introduction and Preliminaries

This paper is concerned with the higher-order nonlinear neutral delay differential equation: 𝑑𝑛𝑑𝑑𝑛π‘₯(𝑑)+π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘₯𝑖+𝑑(𝑑)π‘›βˆ’1π‘‘π‘‘π‘›βˆ’1𝑓𝛼𝑑,π‘₯1𝛼(𝑑),…,π‘₯π‘˜ξ€·ξ€·π›½(𝑑)ξ€Έξ€Έ+β„Žπ‘‘,π‘₯1𝛽(𝑑),…,π‘₯π‘˜(𝑑)ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.1) where 𝑛,π‘š,π‘˜βˆˆβ„•,𝑝𝑖,πœπ‘–,𝛽𝑗,π‘”βˆˆπΆ([𝑑0,+∞),ℝ),π›Όπ‘—βˆˆπΆπ‘›βˆ’1([𝑑0,+∞),ℝ),π‘“βˆˆπΆπ‘›βˆ’1([𝑑0,+∞)Γ—β„π‘˜,ℝ),β„ŽβˆˆπΆ([𝑑0,+∞)Γ—β„π‘˜,ℝ), and limπ‘‘βŸΆ+βˆžπœπ‘–(𝑑)=limπ‘‘βŸΆ+βˆžπ›Όπ‘—(𝑑)=limπ‘‘βŸΆ+βˆžπ›½π‘—(𝑑)=+∞,π‘–βˆˆ{1,2,…,π‘š},π‘—βˆˆ{1,2,…,π‘˜}.(1.2)

Theory of neutral delay differential equations has undergone a rapid development in the last over thirty years. We refer the readers to [1–8] and the references therein for a wealth of reference materials on the subject. The authors [1–8] and others discussed the oscillation, nonoscillation, and existence of a nonoscillatoy solution for some special cases of (1.1) under various conditions. By using the Banach fixed point theorem, Zhang et al. [4] and KulenoviΔ‡ and HadΕΎiomerspahiΔ‡ [1] studied, respectively, the existence of a nonoscillatory solution for the first-order neutral delay differential equation: 𝑑[]𝑑𝑑π‘₯(𝑑)+𝑝(𝑑)π‘₯(π‘‘βˆ’πœ)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœŽ)βˆ’π‘„(𝑑)π‘₯(π‘‘βˆ’π›Ώ)=0,𝑑β‰₯𝑑0,(1.3) where 𝜏>0, 𝜎,π›Ώβˆˆβ„+, 𝑃,π‘„βˆˆπΆ([𝑑0,+∞),ℝ+), and π‘βˆˆπΆ([𝑑0,+∞),ℝ), and the second-order neutral delay differential equation with positive and negative coefficients: 𝑑2𝑑𝑑2[]π‘₯(𝑑)+𝑝π‘₯(π‘‘βˆ’πœ)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœŽ)βˆ’π‘„(𝑑)π‘₯(π‘‘βˆ’π›Ώ)=0,𝑑β‰₯𝑑0,(1.4) where π‘βˆˆβ„β§΅{Β±1}, 𝜎,π›Ώβˆˆβ„+ and 𝑃,π‘„βˆˆπΆ([𝑑0,+∞),ℝ+). Zhang et al. [6] considered the second-order nonlinear neutral differential equation with positive and negative terms: 𝑑2𝑑𝑑2[]π‘₯(𝑑)βˆ’π‘π‘₯(𝜏(𝑑))+𝑓1ξ€·ξ€·πœŽπ‘‘,π‘₯1(𝑑)ξ€Έξ€Έβˆ’π‘“2ξ€·ξ€·πœŽπ‘‘,π‘₯2(𝑑)ξ€Έξ€Έ=0,𝑑β‰₯𝑑0(1.5) and its corresponding equation with forced term: 𝑑2𝑑𝑑2[]π‘₯(𝑑)βˆ’π‘π‘₯(𝜏(𝑑))+𝑓1ξ€·ξ€·πœŽπ‘‘,π‘₯1(𝑑)ξ€Έξ€Έβˆ’π‘“2ξ€·ξ€·πœŽπ‘‘,π‘₯2(𝑑)ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.6) where 𝑑β‰₯𝑑0, 𝑝,𝜏,πœŽπ‘–βˆˆπΆ([𝑑0,∞),ℝ), π‘“π‘–βˆˆπΆ([𝑑0,∞)×ℝ,ℝ), and lim𝑑→+∞𝜏(𝑑)=lim𝑑→+βˆžπœŽπ‘–(𝑑)=+∞ for π‘–βˆˆ{1,2}. Lin [2] investigated sufficient conditions of oscillation and nonoscillation for the second-order nonlinear neutral differential equation: 𝑑2𝑑𝑑2[]π‘₯(𝑑)βˆ’π‘(𝑑)π‘₯(π‘‘βˆ’πœ)+π‘ž(𝑑)𝑓(π‘₯(π‘‘βˆ’πœŽ))=0,𝑑β‰₯0,(1.7) where 𝜏>0,𝜎>0, 𝑝,π‘žβˆˆπΆ(ℝ+,ℝ+), π‘“βˆˆπΆ(ℝ,ℝ) with π‘₯𝑓(π‘₯)>0 for all π‘₯β‰ 0. Liu and Huang [3] used the coincidence degree theory to establish the existence and uniqueness of 𝑇-periodic solutions for the second-order neutral functional differential equation of the form 𝑑2𝑑𝑑2[]π‘₯(𝑑)+𝐡π‘₯(π‘‘βˆ’π›Ώ)+𝐢𝑑π‘₯(𝑑)𝑑𝑑+𝑔(π‘₯(π‘‘βˆ’πœ(𝑑)))=𝑝(𝑑),𝑑β‰₯0,(1.8) where 𝜏,𝑝,π‘”βˆΆβ„β†’β„ are continuous functions, 𝐡, 𝛿, 𝐢 are constants, 𝜏 and 𝑝 are 𝑇-periodic, 𝐢≠0,|𝐡|β‰ 1, and 𝑇>0. Zhou and Zhang [8] extended the results in [1] to the higher-order neutral functional differential equation with positive and negative coefficients: 𝑑𝑛𝑑𝑑𝑛[]π‘₯(𝑑)+𝑝π‘₯(π‘‘βˆ’πœ)+(βˆ’1)𝑛+1[]𝑃(𝑑)π‘₯(π‘‘βˆ’πœŽ)βˆ’π‘„(𝑑)π‘₯(π‘‘βˆ’π›Ώ)=0,𝑑β‰₯𝑑0,(1.9) where π‘βˆˆβ„β§΅{Β±1}, 𝜏,𝜎,π›Ώβˆˆβ„+ and 𝑃,π‘„βˆˆπΆ([𝑑0,+∞),ℝ+). Zhou et al. [7] used the Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation: 𝑑𝑛𝑑𝑑𝑛[]+π‘₯(𝑑)+𝑝(𝑑)π‘₯(π‘‘βˆ’πœ)π‘šξ“π‘–=1π‘žπ‘–ξ€·π‘₯ξ€·(𝑑)π‘“π‘‘βˆ’πœŽπ‘–ξ€Έξ€Έ=𝑔(𝑑),𝑑β‰₯𝑑0,(1.10) where 𝜏,πœŽπ‘–βˆˆβ„+, 𝑝,π‘žπ‘–,π‘”βˆˆπΆ([𝑑0,+∞),ℝ) for π‘–βˆˆ{1,2,…,π‘š} and π‘“βˆˆπΆ(ℝ,ℝ). Zhang et al. [5] obtained some sufficient conditions for the oscillation of all solutions of the even order nonlinear neutral differential equations with variable coefficients: 𝑑𝑛𝑑𝑑𝑛[]π‘₯(𝑑)+𝑝(𝑑)π‘₯(𝜏(𝑑))+π‘ž(𝑑)𝑓(π‘₯(𝜎(𝑑)))=0,𝑑β‰₯𝑑0,(1.11) where 𝑛 is an even number, 𝑝,π‘ž,𝜎,𝜏∈𝐢([𝑑0,+∞),ℝ+) with 0≀𝑝(𝑑)<1, for all 𝑑β‰₯𝑑0, lim𝑑→+∞𝜏(𝑑)=lim𝑑→+∞,πœŽπ‘–(𝑑)=+∞ and π‘“βˆˆπΆ([𝑑0,+∞),ℝ).

The purpose of this paper is to investigate the solvability of (1.1). By constructing appropriate mappings and using the Laray-Schauder nonlinear alternative theorem, we establish a few sufficient conditions which ensure the existence of uncountably many bounded positive solutions for (1.1). Our results improve and generalize some corresponding results in [1, 2, 4, 6–8]. Three examples are given to illustrate the advantages of the results presented in this paper.

Throughout this paper, we assume that ℝ, ℝ+, and β„• denote the sets of all real numbers, nonnegative numbers, and positive integers, respectively, and ξ€½πœπœˆ=inf𝑖(𝑑),𝛼𝑗(𝑑),𝛽𝑗𝑑(𝑑)βˆΆπ‘‘βˆˆ0ξ€Έξ€Ύ,+∞,π‘–βˆˆ{1,2,…,π‘š},π‘—βˆˆ{1,2,…,π‘˜}.(1.12) Let CB([𝜈,+∞),ℝ) stand for the Banach space of all continuous and bounded functions in [𝜈,+∞) with norm β€–π‘₯β€–=sup𝑑β‰₯𝜈|π‘₯(𝑑)| for all π‘₯∈CB([𝜈,+∞),ℝ) and [π‘ˆπΈ(𝑁)={π‘₯∈CB(𝜈,+∞),ℝ)∢π‘₯(𝑑)β‰₯𝑁for𝑑β‰₯𝜈},(𝑀)={π‘₯∈𝐸(𝑁)βˆΆβ€–π‘₯β€–<𝑀},(1.13) where 𝑀,π‘βˆˆβ„+ with 𝑀>𝑁>0. Clearly, 𝐸(𝑁) is a nonempty closed convex subset of CB([𝜈,+∞),ℝ) and π‘ˆ(𝑀) is an open subset of 𝐸(𝑁).

By a solution of (1.1), we mean a function π‘₯∈𝐢([𝜈,+∞),ℝ) with some 𝑇β‰₯𝑑0+|𝜈| such that βˆ‘π‘₯(𝑑)+π‘šπ‘–=1𝑝𝑖(𝑑)π‘₯(πœπ‘–(𝑑)) is 𝑛 times continuously differentiable in [𝑇,+∞) and 𝑓(𝑑,π‘₯(𝛼1(𝑑)),…,π‘₯(π›Όπ‘˜(𝑑))) is π‘›βˆ’1 times continuously differentiable in [𝑇,+∞) and (1.1) holds for 𝑑β‰₯𝑇.

Lemma 1.1 (the Leray-Schauder nonlinear alterative theorem [9]). Let 𝐸 be a closed convex subset of a Banach space 𝑋 and let π‘ˆ be an open subset of 𝐸 with π‘β‹†βˆˆπ‘ˆ. Also, πΊβˆΆπ‘ˆβ†’πΈ is a continuous, condensing mapping with 𝐺(π‘ˆ) bounded, where π‘ˆ denotes the closure of π‘ˆ Then,
(𝐴1)𝐺 has a fixed point in π‘ˆ, or(𝐴2) there are π‘₯βˆˆπœ•π‘ˆ and πœ†βˆˆ(0,1) with π‘₯=(1βˆ’πœ†)𝑝⋆+πœ†πΊπ‘₯.

2. Main Results

Now, we apply the Leray-Schauder nonlinear alterative theorem to investigate the existence of uncountably many bounded positive solutions of (1.1) under certain conditions.

Theorem 2.1. Assume that there exist constants 𝑀,𝑁,𝑝0,𝑑1 and functions 𝐹,𝐻∈𝐢([𝑑0,+∞),ℝ+) satisfying ||𝑓𝑑,𝑒1,…,π‘’π‘˜ξ€Έ||≀𝐹(𝑑),βˆ€π‘‘,𝑒1,…,π‘’π‘˜ξ€Έβˆˆξ€Ίπ‘‘0ξ€ΈΓ—[],+βˆžπ‘,π‘€π‘˜,||β„Žξ€·(2.1)𝑑,𝑣1,…,π‘£π‘˜ξ€Έ||≀𝐻(𝑑),βˆ€π‘‘,𝑣1,…,π‘£π‘˜ξ€Έβˆˆξ€Ίπ‘‘0ξ€ΈΓ—[],+βˆžπ‘,π‘€π‘˜ξ‚»ξ€œ,(2.2)max𝑑+∞0ξ€œπΉ(𝑠)𝑑𝑠,𝑑+∞0π‘ π‘›βˆ’1ξ€½||||ξ€Ύξ‚Όξ€·max𝑔(𝑠),𝐻(𝑠)𝑑𝑠<+∞,(2.3)0<𝑁<1βˆ’2𝑝0𝑀,π‘šξ“π‘–=1||𝑝𝑖||(𝑑)≀𝑝0<12,βˆ€π‘‘β‰₯𝑑1β‰₯𝑑0.(2.4) Then, (1.1) has uncountably many bounded positive solutions in π‘ˆ(𝑀).

Proof. Let 𝐿∈(𝑝0𝑀+𝑁,(1βˆ’π‘0)𝑀). It follows from (2.3) and (2.4) that there exists a constant 𝑇>1+|𝑑0|+|𝑑1|+|𝜈| satisfying ξ€œπ‘‡+βˆžπΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžπ‘ π‘›βˆ’1ξ€Ί||𝑔||(𝑠)+𝐻(𝑠)𝑑𝑠<minπΏβˆ’π‘0ξ€·π‘€βˆ’π‘,1βˆ’π‘0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2.(2.5) Choose πœ–0∈(0,min{πΏβˆ’π‘0π‘€βˆ’π‘,(1βˆ’π‘0)π‘€βˆ’πΏ,(π‘€βˆ’π‘/2)}) with ξ€œπ‘‡+βˆžπΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžπ‘ π‘›βˆ’1ξ€Ί||𝑔||(𝑠)+𝐻(𝑠)𝑑𝑠<minπΏβˆ’π‘0ξ€·π‘€βˆ’π‘,1βˆ’π‘0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2ξ‚‡βˆ’πœ–0.(2.6) Put 𝑝⋆=π‘€βˆ’πœ–0. Clearly, π‘β‹†βˆˆπ‘ˆ(𝑀). Define two mappings 𝐴𝐿,π΅πΏβˆΆπ‘ˆ(𝑀)→𝐢𝐡([𝜈,+∞),ℝ) by 𝐴𝐿π‘₯ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩(𝑑)=πΏβˆ’π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘₯𝑖+(𝑑)(βˆ’1)π‘›ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1𝐴𝑔(𝑠)𝑑𝑠,𝑑β‰₯𝑇,𝐿π‘₯𝐡(𝑇),πœˆβ‰€π‘‘<𝑇,(2.7)𝐿π‘₯ξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ€œ(𝑑)=𝑑+βˆžπ‘“ξ€·ξ€·π›Όπ‘ ,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(+𝑠)𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜ξ€·π΅(𝑠)𝑑𝑠,𝑑β‰₯𝑇,𝐿π‘₯ξ€Έ(𝑇),πœˆβ‰€π‘‘<𝑇(2.8) for all π‘₯βˆˆπ‘ˆ(𝑀). It is clear to see that 𝐴𝐿π‘₯ and 𝐡𝐿π‘₯ are continuous for each π‘₯βˆˆπ‘ˆ(𝑀). Let 𝐷𝐿=𝐴𝐿+𝐡𝐿. In view of (2.1), (2.2), and (2.4)–(2.8), we get that 𝐴𝐿π‘₯𝐡(𝑑)+𝐿π‘₯ξ€Έ(𝑑)=πΏβˆ’π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘₯𝑖+((𝑑)βˆ’1)π‘›ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1+ξ€œπ‘”(𝑠)𝑑𝑠𝑑+βˆžπ‘“ξ€·ξ€·π›Όπ‘ ,π‘₯1𝛼(𝑠),…,π‘₯π‘˜+((𝑠)ξ€Έξ€Έπ‘‘π‘ βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜(𝑠)𝑑𝑠β‰₯πΏβˆ’π‘0ξ€œπ‘€βˆ’π‘‡+βˆžξ€œπΉ(𝑠)π‘‘π‘ βˆ’π‘‡+βˆžπ‘ π‘›βˆ’1ξ€Ί||||𝑔(𝑠)+𝐻(𝑠)𝑑𝑠β‰₯πΏβˆ’π‘0ξ‚†π‘€βˆ’minπΏβˆ’π‘0ξ€·π‘€βˆ’π‘,1βˆ’π‘0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2+πœ€0[>𝑁,βˆ€(𝑑,π‘₯)βˆˆπ‘‡,+∞)Γ—π‘ˆ(𝑀),(2.9) which gives that π·πΏβˆΆπ‘ˆ(𝑀)→𝐸(𝑁).
Now, we show that π΅πΏβˆΆπ‘ˆ(𝑀)β†’CB([𝜈,+∞),ℝ) is continuous and compact. Let {π‘₯π‘š}π‘šβˆˆβ„•βŠ†π‘ˆ(𝑀) be an arbitrary sequence and π‘₯∈𝐢([𝜈,+∞),ℝ) with β€–β€–π‘₯π‘šβ€–β€–βˆ’π‘₯⟢asπ‘šβŸΆβˆž.(2.10) Since π‘ˆ(𝑀) is closed, it follows that π‘₯βˆˆπ‘ˆ(𝑀). For any (𝑠,π‘š)∈[𝑇,+∞)Γ—β„•, put πΉπ‘š||𝑓(𝑠)=𝑠,π‘₯π‘šξ€·π›Ό1ξ€Έ(𝑠),…,π‘₯π‘šξ€·π›Όπ‘˜ξ€·ξ€·π›Ό(𝑠)ξ€Έξ€Έβˆ’π‘“π‘ ,π‘₯1𝛼(𝑠),…,π‘₯π‘˜||,𝐻(𝑠)ξ€Έξ€Έπ‘š(||β„Žξ€·π‘ )=𝑠,π‘₯π‘šξ€·π›½1(𝑠),…,π‘₯π‘šξ€·π›½π‘˜(𝛽𝑠)ξ€Έξ€Έβˆ’β„Žπ‘ ,π‘₯1(𝛽𝑠),…,π‘₯π‘˜(||.𝑠)ξ€Έξ€Έ(2.11) It follows from (2.1), (2.2), and (2.11) that ||πΉπ‘š||||𝐻(𝑠)≀2𝐹(𝑠),π‘š||[(𝑠)≀2𝐻(𝑠),βˆ€(𝑠,π‘š)βˆˆπ‘‡,+∞)Γ—β„•,(2.12) which together with (2.8)–(2.11), the continuity of 𝑓,β„Ž,𝛼𝑗,𝛽𝑗 for π‘—βˆˆ{1,2,…,π‘˜}, and the Lebesgue dominated convergence theorem yields that ||𝐡𝐿π‘₯π‘šξ€Έξ€·π΅(𝑑)βˆ’πΏπ‘₯ξ€Έ||β‰€ξ€œ(𝑑)𝑑+∞||𝑓𝑠,π‘₯π‘šξ€·π›Ό1(𝑠),…,π‘₯π‘šξ€·π›Όπ‘˜(𝛼𝑠)ξ€Έξ€Έβˆ’π‘“π‘ ,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(||+1𝑠)ξ€Έξ€Έπ‘‘π‘ Γ—ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1||β„Žξ€·π‘ ,π‘₯π‘šξ€·π›½1ξ€Έ(𝑠),…,π‘₯π‘šξ€·π›½π‘˜ξ€·ξ€·π›½(𝑠)ξ€Έξ€Έβˆ’β„Žπ‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜||β‰€ξ€œ(𝑠)𝑑𝑠𝑇+βˆžπΉπ‘š1(𝑠)𝑑𝑠+ξ€œ(π‘›βˆ’1)!𝑇+βˆžπ‘ π‘›βˆ’1π»π‘š(𝑠)𝑑𝑠,βˆ€π‘‘β‰₯𝑇,limsupπ‘šβ†’βˆžβ€–β€–π΅πΏπ‘₯π‘šβˆ’π΅πΏπ‘₯‖‖≀limsupπ‘šβ†’βˆžξ‚΅ξ€œπ‘‡+βˆžπΉπ‘š1(𝑠)𝑑𝑠+ξ€œ(π‘›βˆ’1)!𝑇+βˆžπ‘ π‘›βˆ’1π»π‘šξ‚Ά(𝑠)𝑑𝑠=0,(2.13) which means that 𝐡𝐿 is continuous in π‘ˆ(𝑀). It follow from (2.1), (2.2), (2.6), and (2.8) that ‖‖𝐡𝐿π‘₯β€–β€–=sup𝑑β‰₯𝜈||𝐡𝐿π‘₯ξ€Έ||β‰€ξ€œ(𝑑)𝑇+∞1𝐹(𝑠)𝑑𝑠+ξ€œ(π‘›βˆ’1)!𝑇+βˆžπ‘ π‘›βˆ’1𝐻(𝑠)𝑑𝑠<minπΏβˆ’π‘0ξ€·π‘€βˆ’π‘,1βˆ’π‘0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2ξ‚‡βˆ’πœ–0<𝑀,βˆ€π‘₯βˆˆπ‘ˆ(𝑀),(2.14) which yields that 𝐡𝐿(π‘ˆ(𝑀)) is uniformly bounded in [𝜈,+∞).
Let πœ€ be an arbitrary positive number. Equation (2.3) ensures that there exists π‘‡βˆ—>𝑇 satisfying ξ€œπ‘‡+βˆžβˆ—πΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžβˆ—π‘ π‘›βˆ’1π»πœ€(𝑠)𝑑𝑠<2.(2.15) Set πœ€π›Ώ=ξ€Ί1+4𝑄+𝑀+𝑄(π‘‡βˆ—βˆ’π‘‡)π‘›βˆ’1𝐹,𝑄=max(𝑑),𝐻(𝑑)βˆΆπ‘‘βˆˆπ‘‡,π‘‡βˆ—ξ€»ξ€Ύ.(2.16) For any π‘₯βˆˆπ‘ˆ(𝑀) and 𝑑1,𝑑2∈[𝜈,+∞) with |𝑑1βˆ’π‘‘2|<𝛿, we consider the following three cases.Case 1 (π‘‡βˆ—β‰€π‘‘1<𝑑2). In view of (2.1), (2.2), (2.8), and (2.15), we deduce that ||𝐡𝐿π‘₯𝑑2ξ€Έβˆ’ξ€·π΅πΏπ‘₯𝑑1ξ€Έ||=||||ξ€œπ‘‘+∞2𝑓𝛼𝑠,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(+𝑠)𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞2ξ€·π‘ βˆ’π‘‘2ξ€Έπ‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜βˆ’ξ€œ(𝑠)𝑑𝑠𝑑+∞1𝑓𝛼𝑠,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(βˆ’π‘ )𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞1ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜||||ξ€œ(𝑠)𝑑𝑠≀2𝑇+βˆžβˆ—2𝐹(𝑠)𝑑𝑠+ξ€œ(π‘›βˆ’1)!𝑇+βˆžβˆ—π‘ π‘›βˆ’1𝐻(𝑠)𝑑𝑠<πœ€.(2.17)Case 2 (𝑇≀𝑑1<𝑑2β‰€π‘‡βˆ—). Suppose that 𝑛=1. It follows from (2.1), (2.2), (2.6), and (2.8) that ||𝐡𝐿π‘₯𝑑2ξ€Έβˆ’ξ€·π΅πΏπ‘₯𝑑1ξ€Έ||=||||ξ€œπ‘‘+∞2𝑓𝛼𝑠,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(ξ€œπ‘ )𝑑𝑠+𝑑+∞2β„Žξ€·ξ€·π›½π‘ ,π‘₯1(𝛽𝑠),…,π‘₯π‘˜(βˆ’ξ€œπ‘ )𝑑𝑠𝑑+∞1𝑓𝛼𝑠,π‘₯1𝛼(𝑠),…,π‘₯π‘˜ξ€œ(𝑠)ξ€Έξ€Έπ‘‘π‘ βˆ’π‘‘+∞1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜||||β‰€ξ€œ(𝑠)𝑑𝑠𝑑2𝑑1||𝑑(𝐹(𝑠)+𝐻(𝑠))𝑑𝑠≀2𝑄1βˆ’π‘‘2||<πœ€.(2.18)Suppose that π‘›βˆˆπ‘β§΅{1}. It follows from the mean value theorem that, for each π‘ βˆˆ(𝑑2,+∞), there exists 𝜁∈(π‘ βˆ’π‘‘2,π‘ βˆ’π‘‘1) satisfying |||ξ€·π‘ βˆ’π‘‘2ξ€Έπ‘›βˆ’1βˆ’ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’1|||=(π‘›βˆ’1)πœπ‘›βˆ’2||𝑑1βˆ’π‘‘2||≀(π‘›βˆ’1)π‘ π‘›βˆ’1||𝑑1βˆ’π‘‘2||,(2.19) which together with (2.1), (2.2), (2.6), and (2.8) yields that ||𝐡𝐿π‘₯𝑑2ξ€Έβˆ’ξ€·π΅πΏπ‘₯𝑑1ξ€Έ||=||||ξ€œπ‘‘+∞2𝑓𝛼𝑠,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(+𝑠)𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞2ξ€·π‘ βˆ’π‘‘2ξ€Έπ‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜βˆ’ξ€œ(𝑠)𝑑𝑠𝑑+∞1𝑓𝛼𝑠,π‘₯1(𝛼𝑠),…,π‘₯π‘˜(βˆ’π‘ )𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞1ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜||||β‰€ξ€œ(𝑠)𝑑𝑠𝑑2𝑑11𝐹(𝑠)𝑑𝑠+ξ‚΅ξ€œ(π‘›βˆ’1)!𝑑+∞2|||ξ€·π‘ βˆ’π‘‘2ξ€Έπ‘›βˆ’1βˆ’ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’1|||ξ€œπ»(𝑠)𝑑𝑠+𝑑2𝑑1ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’1ξ‚Ά||𝑑𝐻(𝑠)𝑑𝑠≀𝑄1βˆ’π‘‘2||+||𝑑(π‘›βˆ’1)1βˆ’π‘‘2||ξ€œ(π‘›βˆ’1)!𝑑+∞2π‘ π‘›βˆ’1𝑇𝐻(𝑠)𝑑𝑠+βˆ—ξ€Έβˆ’π‘‡π‘›βˆ’1𝑄||𝑑1βˆ’π‘‘2||≀𝑇𝑄+𝑀+βˆ—ξ€Έβˆ’π‘‡π‘›βˆ’1𝑄||𝑑1βˆ’π‘‘2||<πœ€.(2.20)Case 3 (πœˆβ‰€π‘‘1<𝑑2≀𝑇). Equation (2.8) gives that ||𝐡𝐿π‘₯𝑑2ξ€Έβˆ’ξ€·π΅πΏπ‘₯𝑑1ξ€Έ||=||𝐡𝐿π‘₯𝐡(𝑇)βˆ’πΏπ‘₯ξ€Έ||(𝑇)=0.(2.21) Thus, 𝐡𝐿(π‘ˆ(𝑀)) is equicontinuous in [𝜈,+∞). Hence, 𝐡𝐿(π‘ˆ(𝑀)) is a relatively compact subset of 𝐢([𝜈,+∞),ℝ). That is, 𝐡𝐿 is a compact mapping.
Note that for any π‘₯,π‘¦βˆˆπ‘ˆ(𝑀) and 𝑑β‰₯𝑇||𝐴𝐿π‘₯𝐴(𝑑)βˆ’πΏπ‘¦ξ€Έ||=|||||(𝑑)πΏβˆ’π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘₯𝑖+(𝑑)(βˆ’1)π‘›ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1𝑔(𝑠)π‘‘π‘ βˆ’πΏ+π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘¦π‘–ξ€Έβˆ’(𝑑)(βˆ’1)π‘›ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1|||||≀𝑔(𝑠)π‘‘π‘ π‘šξ“π‘–=1||𝑝𝑖||(𝑑)β€–π‘₯βˆ’π‘¦β€–β‰€π‘0β€–π‘₯βˆ’π‘¦β€–,(2.22)which implies that ‖‖𝐴𝐿π‘₯βˆ’π΄πΏπ‘¦β€–β€–β‰€π‘0β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπ‘ˆ(𝑀),(2.23) which together with (2.4) gives that 𝐴𝐿 is a contraction mapping. It follows that π·πΏβˆΆπ‘ˆ(𝑀)→𝐸(𝑁) is a continuous and condensing mapping. Let π‘₯0(𝑑)=𝑁 for all π‘‘βˆˆ[𝜈,+∞). Notice that π‘₯0βˆˆπ‘ˆ(𝑀). Thus, (2.4)–(2.7), (2.14), and (2.23) yield that ‖‖𝐷𝐿π‘₯‖‖≀‖‖𝐴𝐿π‘₯β€–β€–+‖‖𝐡𝐿π‘₯‖‖≀‖‖𝐴𝐿π‘₯βˆ’π΄πΏπ‘₯0β€–β€–+‖‖𝐴𝐿π‘₯0β€–β€–+𝑀≀𝑝0β€–β€–π‘₯βˆ’π‘₯0β€–β€–+𝑀+sup𝑑β‰₯𝑇|||||πΏβˆ’π‘šξ“π‘–=1𝑝𝑖(𝑑)π‘₯0ξ€·πœπ‘–ξ€Έ+((𝑑)βˆ’1)π‘›ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1|||||𝑔(𝑠)𝑑𝑠≀𝑝0(𝑀+𝑁)+𝑀+𝐿+𝑝0ξ€œπ‘+𝑇+βˆžπ‘ π‘›βˆ’1||||≀𝑔(𝑠)𝑑𝑠2+𝑝0𝑀+2𝑝0𝑁+𝐿,βˆ€π‘₯βˆˆπ‘ˆ(𝑀),(2.24) that is, 𝐷𝐿(π‘ˆ(𝑀)) is uniformly bounded in [𝛾,+∞).
Put 𝑆1[𝑆={π‘₯∈CB(𝜈,+∞),ℝ)βˆΆπ‘β‰€π‘₯(𝑑)≀𝑀,βˆ€π‘‘β‰₯𝜈,β€–π‘₯β€–=𝑀},2=[{π‘₯∈CB(𝜈,+∞),ℝ)βˆΆπ‘β‰€π‘₯(𝑑)≀𝑀,βˆ€π‘‘β‰₯𝜈andthereexistsπ‘‘βˆ—ξ€·π‘‘β‰₯𝜈satisfyingπ‘₯βˆ—ξ€Έξ€Ύ.=𝑁(2.25) It is easy to verify that πœ•π‘ˆ(𝑀)=𝑆1βˆͺ𝑆2.
Next, we show that (𝐴2) in Lemma 1.1 does not hold. Otherwise, there exist π‘₯βˆˆπœ•π‘ˆ(𝑀) and πœ†βˆˆ(0,1) satisfying π‘₯=(1βˆ’πœ†)𝑝⋆+πœ†π·πΏπ‘₯. We have to discuss the following possible cases. Case 1. Let π‘₯βˆˆπ‘†1. By means of (2.1), (2.2), and (2.4)–(2.8), we get that, for 𝑑β‰₯𝑇, π‘₯(𝑑)=(1βˆ’πœ†)𝑝⋆𝐴+πœ†ξ€Ίξ€·πΏπ‘₯𝐡(𝑑)+𝐿π‘₯ξ€Έξ€»ξ€·(𝑑)≀(1βˆ’πœ†)π‘€βˆ’πœ–0+πœ†πΏ+π‘šξ“π‘–=1||𝑝𝑖||π‘₯ξ€·πœ(𝑑)𝑖+1(𝑑)ξ€œ(π‘›βˆ’1)!𝑑+βˆžπ‘ π‘›βˆ’1||||+ξ€œπ‘”(𝑠)𝑑𝑠𝑑+∞||𝑓𝛼𝑠,π‘₯1𝛼(𝑠),…,π‘₯π‘˜||+1(𝑠)ξ€Έξ€Έπ‘‘π‘ ξ€œ(π‘›βˆ’1)!𝑑+βˆžπ‘ π‘›βˆ’1||β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜||ξ‚Ήξ€·(𝑠)𝑑𝑠≀(1βˆ’πœ†)π‘€βˆ’πœ–0ξ€Έξ‚Έ+πœ†πΏ+𝑝0ξ€œπ‘€+𝑑+∞1𝐹(𝑠)𝑑𝑠+ξ€œ(π‘›βˆ’1)!𝑑+βˆžπ‘ π‘›βˆ’1ξ€Ί||||𝑔(𝑠)+𝐻(𝑠)𝑑𝑠<(1βˆ’πœ†)π‘€βˆ’πœ–0+πœ†πΏ+𝑝0𝑀+minπΏβˆ’π‘0ξ€·π‘€βˆ’π‘,1βˆ’π‘0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2ξ‚‡βˆ’πœ–0ξ‚„β‰€π‘€βˆ’πœ–0,(2.26) which implies that 𝑀=β€–π‘₯β€–=sup𝑑β‰₯𝜈||||π‘₯(𝑑)β‰€π‘€βˆ’πœ–0<𝑀,(2.27) which is a contradiction.Case 2. Let π‘₯βˆˆπ‘†2. It follows from (2.1), (2.2), and (2.4)–(2.8) that 𝑑𝑁=π‘₯βˆ—ξ€Έ=(1βˆ’πœ†)𝑝⋆𝐴+πœ†ξ€Ίξ€·πΏπ‘₯π‘‘ξ€Έξ€·βˆ—ξ€Έ+𝐡𝐿π‘₯π‘‘ξ€Έξ€·βˆ—=ξ€·ξ€Έξ€»(1βˆ’πœ†)π‘€βˆ’πœ–0𝐴+πœ†ξ€Ίξ€·πΏπ‘₯𝑑maxβˆ—+𝐡,𝑇𝐿π‘₯𝑑maxβˆ—ξ€·,𝑇β‰₯(1βˆ’πœ†)π‘€βˆ’πœ–0+πœ†πΏβˆ’π‘šξ“π‘–=1||𝑝𝑖𝑑maxβˆ—||π‘₯ξ€·πœ,𝑇𝑖𝑑maxβˆ—βˆ’1,π‘‡ξ€Ύξ€Έξ€Έξ€œ(π‘›βˆ’1)!+∞max{π‘‘βˆ—,𝑇}π‘ π‘›βˆ’1||||ξ€œπ‘”(𝑠)π‘‘π‘ βˆ’+∞max{π‘‘βˆ—,𝑇}||𝑓𝛼𝑠,π‘₯1𝛼(𝑠),…,π‘₯π‘˜||βˆ’1(𝑠)𝑑𝑠(ξ€œπ‘›βˆ’1)!+∞max{π‘‘βˆ—,𝑇}π‘ π‘›βˆ’1||β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜||ξ‚Ήξ€·(𝑠)𝑑𝑠β‰₯(1βˆ’πœ†)π‘€βˆ’πœ–0ξ€Έξ‚Έ+πœ†πΏβˆ’π‘0ξ€œπ‘€βˆ’+∞max{π‘‘βˆ—,𝑇}1𝐹(𝑠)π‘‘π‘ βˆ’ξ€œ(π‘›βˆ’1)!+∞max{π‘‘βˆ—,𝑇}π‘ π‘›βˆ’1ξ€Ί||||𝑔(𝑠)+𝐻(𝑠)𝑑𝑠β‰₯(1βˆ’πœ†)π‘€βˆ’πœ–0+πœ†πΏβˆ’π‘0ξ‚†π‘€βˆ’minπΏβˆ’π‘0ξ€·π‘€βˆ’π‘,1βˆ’π‘0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2+πœ–0ξ‚„β‰₯ξ€·(1βˆ’πœ†)π‘€βˆ’πœ–0ξ€Έξ€·+πœ†π‘+πœ–0ξ€Έξ€½β‰₯minπ‘€βˆ’πœ–0,𝑁+πœ–0ξ€Ύ=𝑁+πœ–0,(2.28) which is absurd.
Thus, Lemma 1.1 ensures that 𝐷𝐿 has a fixed point π‘₯βˆˆπ‘ˆ(𝑀); that is, π‘₯(𝑑)=πΏβˆ’π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘₯𝑖+(𝑑)(βˆ’1)π‘›ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1+ξ€œπ‘”(𝑠)𝑑𝑠𝑑+βˆžπ‘“ξ€·ξ€·π›Όπ‘ ,π‘₯1𝛼(𝑠),…,π‘₯π‘˜+(𝑠)𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1β„Žξ€·ξ€·π›½π‘ ,π‘₯1𝛽(𝑠),…,π‘₯π‘˜(𝑠)𝑑𝑠,βˆ€π‘‘β‰₯𝑇,(2.29) which yields that 𝑑𝑛𝑑𝑑𝑛π‘₯(𝑑)+π‘šξ“π‘–=1π‘π‘–ξ€·πœ(𝑑)π‘₯𝑖+𝑑(𝑑)π‘›βˆ’1π‘‘π‘‘π‘›βˆ’1𝑓𝛼𝑑,π‘₯1𝛼(𝑑),…,π‘₯π‘˜ξ€·ξ€·π›½(𝑑)ξ€Έξ€Έ+β„Žπ‘‘,π‘₯1𝛽(𝑑),…,π‘₯π‘˜(𝑑)ξ€Έξ€Έ=𝑔(𝑑),βˆ€π‘‘β‰₯𝑇,(2.30) which means that π‘₯βˆˆπ‘ˆ(𝑀) is a bounded positive solution of (1.1).
Let 𝐿1,𝐿2∈(𝑝0𝑀+𝑁,(1βˆ’π‘0)𝑀) with 𝐿1≠𝐿2. Similarly, we can prove that, for each π‘Ÿβˆˆ{1,2}, there exist a constant π‘‡π‘Ÿ>1+|𝑑0|+|𝑑1|+|𝜈| and two mappings π΄πΏπ‘Ÿ,π΅πΏπ‘ŸβˆΆπ‘ˆ(𝑀)β†’CB([𝜈,+∞),ℝ) satisfying (2.6)–(2.8), where 𝑇,𝐿,𝐴𝐿, and 𝐡𝐿 are replaced by π‘‡π‘Ÿ,πΏπ‘Ÿ,π΄πΏπ‘Ÿ, and π΅πΏπ‘Ÿ, respectively, and π΄πΏπ‘Ÿ+π΅πΏπ‘Ÿ has a fixed point π‘§π‘Ÿβˆˆπ‘ˆ(𝑀), which is a bounded positive solution of (1.1) in π‘ˆ(𝑀). In order to prove that (1.1) possesses uncountably many bounded positive solutions in π‘ˆ(𝑀), we need only to prove that 𝑧1≠𝑧2. By means of (2.1)–(2.3), we know that there exists 𝑇3>max{𝑇1,𝑇2} satisfying ξ€œπ‘‡+∞3ξ€œπΉ(𝑠)𝑑𝑠+𝑇+∞3π‘ π‘›βˆ’1||𝐿𝐻(𝑠)𝑑𝑠<1βˆ’πΏ2||4.(2.31) It follows from (2.1), (2.2), (2.4), (2.7), (2.8), and (2.31) that for 𝑑β‰₯𝑇3||𝑧1(𝑑)βˆ’π‘§2||=|||||𝐿(𝑑)1βˆ’πΏ2βˆ’π‘šξ“π‘–=1𝑝𝑖(𝑑)𝑧1ξ€·πœπ‘–ξ€Έ+(𝑑)π‘šξ“π‘–=1𝑝𝑖(𝑑)𝑧2ξ€·πœπ‘–ξ€Έ+ξ€œ(𝑑)𝑑+βˆžπ‘“ξ€·π‘ ,𝑧1𝛼1(𝑠),…,𝑧1ξ€·π›Όπ‘˜(ξ€œπ‘ )ξ€Έξ€Έπ‘‘π‘ βˆ’π‘‘+βˆžπ‘“ξ€·π‘ ,𝑧2𝛼1(𝑠),…,𝑧2ξ€·π›Όπ‘˜(+𝑠)𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1β„Žξ€·π‘ ,𝑧1𝛽1ξ€Έ(𝑠),…,𝑧1ξ€·π›½π‘˜βˆ’(𝑠)𝑑𝑠(βˆ’1)π‘›βˆ’1ξ€œ(π‘›βˆ’1)!𝑑+∞(π‘ βˆ’π‘‘)π‘›βˆ’1β„Žξ€·π‘ ,𝑧2𝛽1ξ€Έ(𝑠),…,𝑧2ξ€·π›½π‘˜||||β‰₯||𝐿(𝑠)𝑑𝑠1βˆ’πΏ2||βˆ’π‘0‖‖𝑧1βˆ’π‘§2β€–β€–ξ€œβˆ’2𝑇+∞3ξ€œπΉ(𝑠)π‘‘π‘ βˆ’2𝑇+∞3π‘ π‘›βˆ’1β‰₯||𝐿𝐻(𝑠)𝑑𝑠1βˆ’πΏ2||βˆ’π‘0‖‖𝑧1βˆ’π‘§2β€–β€–βˆ’||𝐿1βˆ’πΏ2||2,(2.32)which implies that ‖‖𝑧1βˆ’π‘§2β€–β€–β‰₯||𝐿1βˆ’πΏ2||2ξ€·1+𝑝0ξ€Έ>0,(2.33) that is, 𝑧1≠𝑧2. This completes the proof.

Theorem 2.2. Assume that there exist constants 𝑀,𝑁,𝑝0,𝑑1 and functions 𝐹,𝐻∈𝐢([𝑑0,+∞),ℝ+) satisfying (2.1)–(2.3) and 𝑁<1+𝑝0𝑝𝑀,max𝑖(𝑑)∢1β‰€π‘–β‰€π‘šβ‰€0,π‘šξ“π‘–=1𝑝𝑖(𝑑)β‰₯𝑝0>βˆ’1,βˆ€π‘‘β‰₯𝑑1β‰₯𝑑0.(2.34) Then, (1.1) has uncountably many bounded positive solutions in π‘ˆ(𝑀).

Proof. Let 𝐿∈(𝑁,(1+𝑝0)𝑀). It follows from (2.3) and (2.34) that there exists a constant 𝑇>1+|𝑑0|+|𝑑1|+|𝜈| satisfying ξ€œπ‘‡+βˆžπΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžπ‘ π‘›βˆ’1ξ€Ί||𝑔||(𝑠)+𝐻(𝑠)𝑑𝑠<minπΏβˆ’π‘,1+𝑝0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2.(2.35) Take πœ–0∈(0,min{πΏβˆ’π‘,(1+𝑝0)π‘€βˆ’πΏ,(π‘€βˆ’π‘/2)}) such that ξ€œπ‘‡+βˆžπΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžπ‘ π‘›βˆ’1ξ€Ί||𝑔||(𝑠)+𝐻(𝑠)𝑑𝑠≀minπΏβˆ’π‘,1+𝑝0ξ€Έπ‘€βˆ’πΏ,π‘€βˆ’π‘2ξ‚‡βˆ’πœ–0.(2.36) The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.

Theorem 2.3. Assume that there exist constants 𝑀,𝑁,𝑝0,𝑑1 and functions 𝐹,𝐻∈𝐢([𝑑0,+∞),ℝ+) satisfying (2.1)–(2.3) and 𝑁<1βˆ’π‘0𝑝𝑀,min𝑖(𝑑)∢1β‰€π‘–β‰€π‘šβ‰₯0,π‘šξ“π‘–=1𝑝𝑖(𝑑)≀𝑝0<1,βˆ€π‘‘β‰₯𝑑1β‰₯𝑑0.(2.37) Then, (1.1) has uncountably many bounded positive solutions in π‘ˆ(𝑀).

Proof. Let 𝐿∈(𝑝0𝑀+𝑁,𝑀). It follows from (2.3) and (2.37) that there exists a constant 𝑇>1+|𝑑0|+|𝑑1|+|𝜈| satisfying ξ€œπ‘‡+βˆžπΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžπ‘ π‘›βˆ’1ξ€Ί||𝑔||(𝑠)+𝐻(𝑠)𝑑𝑠<minπΏβˆ’π‘0π‘€βˆ’π‘,π‘€βˆ’πΏ,π‘€βˆ’π‘2.(2.38) Choose πœ–0∈(0,min{πΏβˆ’π‘0π‘€βˆ’π‘,π‘€βˆ’πΏ,(π‘€βˆ’π‘/2)}) such that ξ€œπ‘‡+βˆžπΉξ€œ(𝑠)𝑑𝑠+𝑇+βˆžπ‘ π‘›βˆ’1ξ€Ί||𝑔||(𝑠)+𝐻(𝑠)𝑑𝑠≀minπΏβˆ’π‘0π‘€βˆ’π‘,π‘€βˆ’πΏ,π‘€βˆ’π‘2ξ‚‡βˆ’πœ–0.(2.39) The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.

Remark 2.4. Theorems 2.1–2.3 extend, improve, and unify the theorem in [1], Theorem  2.2 in [2], Theorem  1 in [4], Theorems  2.1 and 2.3 in [6], Theorems  1 and 3 in [7], and Theorems  1 and 3 in [8].

3. Examples and Applications

Now, we construct three nontrivial examples to show the superiority and applications of Theorems 2.1–2.3, respectively.

Example 3.1. Consider the higher-order nonlinear neutral delay differential equation: 𝑑𝑛𝑑𝑑𝑛𝑑π‘₯(𝑑)βˆ’2cos𝑑1+6𝑑2π‘₯(𝑑+2)+(βˆ’1)𝑛𝑑sin1βˆ’π‘‘3ξ€Έπ‘₯𝑑1+4𝑑3ξ€Έξƒ­+π‘‘βˆ’π‘‘π‘›βˆ’1π‘‘π‘‘π‘›βˆ’11+𝑑2π‘₯4(π‘‘βˆ’1)βˆ’π‘‘π‘₯2𝑑2ξ€Έ+21+𝑑5+𝑑π‘₯2𝑑2ξ€Έ+2sin2ξ€·π‘‘βˆ’π‘‘3π‘₯2ξ€Έ(π‘‘βˆ’2)1+π‘₯2(π‘‘βˆ’2)+𝑑3ξƒ­+𝑑3+(1/𝑑)1+𝑑𝑛+4cos3ξ€·π‘₯𝑑2+ξ€·ln𝑑ln1+π‘₯2ξ€·2𝑑+𝑑2ξ€·π‘₯𝑑sin2ξ€Έξ€Έ1+𝑑𝑛+3+𝑑2=1βˆ’π‘‘3𝑑𝑛+4ξ€·ln1+𝑑3ξ€Έ,𝑑β‰₯3.(3.1) Let 𝑑0=𝑑1=3,𝑝0=5/12,π‘š=2,π‘˜=3,𝑀=36,𝑁=3,𝜈=1, 𝑝1𝑑(𝑑)=βˆ’2cos𝑑1+6𝑑2,𝑝2(𝑑)=(βˆ’1)𝑛𝑑sin1βˆ’π‘‘3ξ€Έ1+4𝑑,𝜏1(𝑑)=𝑑+2,𝜏2(𝑑)=𝑑3π›Όβˆ’π‘‘,1(𝑑)=π‘‘βˆ’1,𝛼2(𝑑)=𝑑2+2,𝛼3(𝑑)=π‘‘βˆ’2,𝛽1(𝑑)=𝑑2ln𝑑,𝛽2(𝑑)=2𝑑,𝛽3(𝑑)=𝑑2,𝑓(𝑑,𝑒,𝑣,𝑀)=1+𝑑2𝑒4βˆ’π‘‘π‘£21+𝑑5+𝑑𝑣2sin2ξ€·π‘‘βˆ’π‘‘3𝑀2ξ€Έ1+𝑀2+𝑑3,π‘‘β„Ž(𝑑,𝑒,𝑣,𝑀)=3+(1/𝑑)1+𝑑𝑛+4cos3𝑒+ln1+𝑣2ξ€Έ+𝑑2sin𝑀1+𝑑𝑛+3+𝑑2,𝐹(𝑑)=1+𝑑2𝑀4+𝑑𝑀21+𝑑5+𝑑𝑀21+𝑁2+𝑑3𝑑,𝐻(𝑑)=3+(