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

Abstract

This paper deals with the higher-order nonlinear neutral delay differential equation (𝑑^𝑛/𝑑𝑡^𝑛)[𝑥(𝑡)+∑^𝑚𝑖=1𝑝_𝑖(𝑡)𝑥(𝑇_𝑖(𝑡))]+(𝑑^𝑛−1/𝑑𝑡^𝑛−1)𝑓(𝑡,𝑥(𝛼₁(𝑡)),…,𝑥(𝛼_𝑘(𝑡)))+ℎ(𝑡,𝑥(𝛽₁(𝑡)),…,𝑥(𝛽_𝑘(𝑡)))=𝑔(𝑡), 𝑡≥𝑡_𝑜, where 𝑛,𝑚,𝑘∈ℕ, 𝑝_𝑖,𝜏_𝑖,𝛽_𝑗,𝑔∈𝐶([𝑡_𝑜,+∞),ℝ), 𝛼_𝑗∈𝐶^𝑛−1([𝑡_𝑜,+∞),ℝ), 𝑓∈𝐶^𝑛−1([𝑡_𝑜,+∞)×ℝ^𝑘,ℝ), ℎ∈𝐶([𝑡_𝑜,+∞)×ℝ^𝑘,ℝ), and lim_𝑡→+∞𝜏_𝑖(𝑡)=lim_𝑡→+∞𝛼_𝑗(𝑡)=lim_𝑡→+∞𝛽_𝑗(𝑡)=+∞, 𝑖∈{1,2,…,𝑚},𝑗∈{1,2,…,𝑘}. 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: 𝑑^𝑛𝑑𝑡^𝑛𝑥(𝑡)+^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)+𝑑^𝑛−1𝑑𝑡^𝑛−1+ℎ𝑡,𝑥𝛽₁𝑓𝑡,𝑥𝛼₁(𝑡),…,𝑥𝛼_𝑘(𝑡)(𝑡),…,𝑥𝛽_𝑘(𝑡)=𝑔(𝑡),𝑡≥𝑡₀,(1.1) where 𝑛,𝑚,𝑘∈ℕ,𝑝_𝑖,𝜏_𝑖,𝛽_𝑗,𝑔∈𝐶([𝑡₀,+∞),ℝ),𝛼_𝑗∈𝐶^𝑛−1([𝑡₀,+∞),ℝ),𝑓∈𝐶^𝑛−1([𝑡₀,+∞)×ℝ^𝑘,ℝ),ℎ∈𝐶([𝑡₀,+∞)×ℝ^𝑘,ℝ), and lim_𝑡⟶+∞𝜏_𝑖(𝑡)=lim_𝑡⟶+∞𝛼_𝑗(𝑡)=lim_𝑡⟶+∞𝛽_𝑗(𝑡)=+∞,𝑖∈{1,2,…,𝑚},𝑗∈{1,2,…,𝑘}.(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 Kulenović and Hadžiomerspahić [1] studied, respectively, the existence of a nonoscillatory solution for the first-order neutral delay differential equation: 𝑑𝑑𝑡[𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏)]+𝑃(𝑡)𝑥(𝑡−𝜎)−𝑄(𝑡)𝑥(𝑡−𝛿)=0,𝑡≥𝑡⁰,(1.3) where 𝜏>0, 𝜎,𝛿∈ℝ⁺, 𝑃,𝑄∈𝐶([𝑡₀,+∞),ℝ⁺), and 𝑝∈𝐶([𝑡₀,+∞),ℝ), and the second-order neutral delay differential equation with positive and negative coefficients: 𝑑²𝑑𝑡²[𝑥(𝑡)+𝑝𝑥(𝑡−𝜏)]+𝑃(𝑡)𝑥(𝑡−𝜎)−𝑄(𝑡)𝑥(𝑡−𝛿)=0,𝑡≥𝑡₀,(1.4) where 𝑝∈ℝ⧵{±1}, 𝜎,𝛿∈ℝ⁺ and 𝑃,𝑄∈𝐶([𝑡₀,+∞),ℝ⁺). Zhang et al. [6] considered the second-order nonlinear neutral differential equation with positive and negative terms: 𝑑²𝑑𝑡²[𝑥(𝑡)−𝑝𝑥(𝜏(𝑡))]+𝑓₁𝑡,𝑥𝜎₁(𝑡)−𝑓₂𝑡,𝑥𝜎₂(𝑡)=0,𝑡≥𝑡₀(1.5) and its corresponding equation with forced term: 𝑑²𝑑𝑡²[𝑥(𝑡)−𝑝𝑥(𝜏(𝑡))]+𝑓₁𝑡,𝑥𝜎₁(𝑡)−𝑓₂𝑡,𝑥𝜎₂(𝑡)=𝑔(𝑡),𝑡≥𝑡₀,(1.6) where 𝑡≥𝑡₀, 𝑝,𝜏,𝜎_𝑖∈𝐶([𝑡₀,∞),ℝ), 𝑓_𝑖∈𝐶([𝑡₀,∞)×ℝ,ℝ), and lim_𝑡→+∞𝜏(𝑡)=lim_𝑡→+∞𝜎_𝑖(𝑡)=+∞ for 𝑖∈{1,2}. Lin [2] investigated sufficient conditions of oscillation and nonoscillation for the second-order nonlinear neutral differential equation: 𝑑²𝑑𝑡²[𝑥(𝑡)−𝑝(𝑡)𝑥(𝑡−𝜏)]+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=0,𝑡≥0,(1.7) where 𝜏>0,𝜎>0, 𝑝,𝑞∈𝐶(ℝ⁺,ℝ⁺), 𝑓∈𝐶(ℝ,ℝ) with 𝑥𝑓(𝑥)>0 for all 𝑥≠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 𝑑²𝑑𝑡²[𝑥(𝑡)+𝐵𝑥(𝑡−𝛿)]+𝐶𝑑𝑥(𝑡)𝑑𝑡+𝑔(𝑥(𝑡−𝜏(𝑡)))=𝑝(𝑡),𝑡≥0,(1.8) where 𝜏,𝑝,𝑔∶ℝ→ℝ are continuous functions, 𝐵, 𝛿, 𝐶 are constants, 𝜏 and 𝑝 are 𝑇-periodic, 𝐶≠0,|𝐵|≠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: 𝑑^𝑛𝑑𝑡^𝑛[𝑥(𝑡)+𝑝𝑥(𝑡−𝜏)]+(−1)^𝑛+1[𝑃(𝑡)𝑥(𝑡−𝜎)−𝑄(𝑡)𝑥(𝑡−𝛿)]=0,𝑡≥𝑡₀,(1.9) where 𝑝∈ℝ⧵{±1}, 𝜏,𝜎,𝛿∈ℝ⁺ and 𝑃,𝑄∈𝐶([𝑡₀,+∞),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: 𝑑^𝑛𝑑𝑡^𝑛[𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏)]+^𝑚_𝑖=1𝑞_𝑖(𝑡)𝑓𝑥𝑡−𝜎_𝑖=𝑔(𝑡),𝑡≥𝑡₀,(1.10) where 𝜏,𝜎_𝑖∈ℝ⁺, 𝑝,𝑞_𝑖,𝑔∈𝐶([𝑡₀,+∞),ℝ) for 𝑖∈{1,2,…,𝑚} and 𝑓∈𝐶(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: 𝑑^𝑛𝑑𝑡^𝑛[𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))]+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0,𝑡≥𝑡₀,(1.11) where 𝑛 is an even number, 𝑝,𝑞,𝜎,𝜏∈𝐶([𝑡₀,+∞),ℝ⁺) with 0≤𝑝(𝑡)<1, for all 𝑡≥𝑡₀, lim_𝑡→+∞𝜏(𝑡)=lim_𝑡→+∞,𝜎_𝑖(𝑡)=+∞ and 𝑓∈𝐶([𝑡₀,+∞),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 ℕ denote the sets of all real numbers, nonnegative numbers, and positive integers, respectively, and 𝜈=inf𝜏_𝑖(𝑡),𝛼_𝑗(𝑡),𝛽_𝑗(𝑡)∶𝑡∈𝑡₀,+∞,𝑖∈{1,2,…,𝑚},𝑗∈{1,2,…,𝑘}.(1.12) Let CB([𝜈,+∞),ℝ) stand for the Banach space of all continuous and bounded functions in [𝜈,+∞) with norm ‖𝑥‖=sup_𝑡≥𝜈|𝑥(𝑡)| for all 𝑥∈CB([𝜈,+∞),ℝ) and 𝐸(𝑁)={𝑈𝑥∈(𝑀C)B=([{𝜈𝑥,+∈∞𝐸)(,𝑁ℝ))∶∶‖𝑥𝑥‖(𝑡)<≥𝑀𝑁}f,or𝑡≥𝜈},(1.13) where 𝑀,𝑁∈ℝ⁺ with 𝑀>𝑁>0. Clearly, 𝐸(𝑁) is a nonempty closed convex subset of CB([𝜈,+∞),ℝ) and 𝑈(𝑀) is an open subset of 𝐸(𝑁).

By a solution of (1.1), we mean a function 𝑥∈𝐶([𝜈,+∞),ℝ) with some 𝑇≥𝑡₀+|𝜈| such that 𝑥(𝑡)+∑^𝑚𝑖=1𝑝_𝑖(𝑡)𝑥(𝜏_𝑖(𝑡)) is 𝑛 times continuously differentiable in [𝑇,+∞) and 𝑓(𝑡,𝑥(𝛼₁(𝑡)),…,𝑥(𝛼_𝑘(𝑡))) is 𝑛−1 times continuously differentiable in [𝑇,+∞) and (1.1) holds for 𝑡≥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 𝑈 be an open subset of 𝐸 with 𝑝^⋆∈𝑈. Also, 𝐺∶𝑈→𝐸 is a continuous, condensing mapping with 𝐺(𝑈) bounded, where 𝑈 denotes the closure of 𝑈 Then,
(𝐴₁)𝐺 has a fixed point in 𝑈, or(𝐴₂) there are 𝑥∈𝜕𝑈 and 𝜆∈(0,1) with 𝑥=(1−𝜆)𝑝^⋆+𝜆𝐺𝑥.

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,𝑡₁ and functions 𝐹,𝐻∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying ||||ℎ𝑓𝑡𝑡,,𝑣𝑢₁₁,…,𝑢_𝑘||≤𝐹(𝑡),∀𝑡,𝑢₁,…,𝑢_𝑘∈𝑡₀,+∞×[𝑁,𝑀]^𝑘,(2.1)max,…_𝑡+∞0,𝑣_𝑘||≤𝐻(𝑡),∀𝑡,𝑣₁,…,𝑣_𝑘∈𝑡₀,+∞×[𝑁,𝑀]^𝑘,(2.2)0<𝑁<1𝐹−(𝑠2)𝑝𝑑₀𝑠,^𝑡+∞₀𝑠^𝑛−1max||𝑔(𝑠)||,𝐻(𝑠)𝑑𝑠<+∞,(2.3)𝑀,^𝑚_𝑖=1||𝑝_𝑖(𝑡)||≤𝑝₀<12,∀𝑡≥𝑡¹≥𝑡₀.(2.4) Then, (1.1) has uncountably many bounded positive solutions in 𝑈(𝑀).

Proof. Let 𝐿∈(𝑝₀𝑀+𝑁,(1−𝑝₀)𝑀). It follows from (2.3) and (2.4) that there exists a constant 𝑇>1+|𝑡₀|+|𝑡₁|+|𝜈| satisfying ^𝑇+∞𝐹(𝑠)𝑑𝑠+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠<min𝐿−𝑝₀𝑀−𝑁,1−𝑝₀𝑀−𝐿,𝑀−𝑁2.(2.5) Choose 𝜖₀∈(0,min{𝐿−𝑝₀𝑀−𝑁,(1−𝑝₀)𝑀−𝐿,(𝑀−𝑁/2)}) with ^𝑇+∞𝐹(𝑠)𝑑𝑠+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠<min𝐿−𝑝₀𝑀−𝑁,1−𝑝₀𝑀−𝐿,𝑀−𝑁2−𝜖⁰.(2.6) Put 𝑝^⋆=𝑀−𝜖₀. Clearly, 𝑝^⋆∈𝑈(𝑀). Define two mappings 𝐴_𝐿,𝐵_𝐿∶𝑈(𝑀)→𝐶𝐵([𝜈,+∞),ℝ) by 𝐴_𝐿𝑥(𝑡)=⎧⎪⎨⎪⎩𝐿−^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)+(−1)^𝑛(𝑛−1)!^𝑡+∞𝐴(𝑠−𝑡)^𝑛−1𝑔(𝑠)𝑑𝑠,𝑡≥𝑇,_𝐿𝐵𝑥(𝑇),𝜈≤𝑡<𝑇,(2.7)_𝐿𝑥(𝑡)=⎧⎪⎪⎨⎪⎪⎩^𝑡+∞+(𝑓−1𝑠),𝑥_𝑛−1𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞𝐵(𝑠−𝑡)^𝑛−1ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠,𝑡≥𝑇,_𝐿𝑥(𝑇),𝜈≤𝑡<𝑇(2.8) for all 𝑥∈𝑈(𝑀). It is clear to see that 𝐴_𝐿𝑥 and 𝐵_𝐿𝑥 are continuous for each 𝑥∈𝑈(𝑀). Let 𝐷_𝐿=𝐴_𝐿+𝐵_𝐿. In view of (2.1), (2.2), and (2.4)–(2.8), we get that 𝐴=_𝐿𝑥𝐿−(𝑡)_𝑚+𝐵_𝐿𝑥(𝑡)_𝑖=1𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)+(−1)^𝑛(𝑛−1)!^𝑡+∞+(𝑠−𝑡)^𝑛−1𝑔(𝑠)𝑑𝑠_𝑡+∞+(−1)𝑓_𝑛−1𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞≥𝐿−𝑝₀(𝑠−𝑡)^𝑛−1ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠≥𝐿−𝑝₀𝑀−^𝑇+∞𝐹(𝑠)𝑑𝑠−^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠𝑀−min𝐿−𝑝₀𝑀−𝑁,1−𝑝₀𝑀−𝐿,𝑀−𝑁2+𝜀⁰>𝑁,∀(𝑡,𝑥)∈[𝑇,+∞)×𝑈(𝑀),(2.9) which gives that 𝐷_𝐿∶𝑈(𝑀)→𝐸(𝑁).
Now, we show that 𝐵_𝐿∶𝑈(𝑀)→CB([𝜈,+∞),ℝ) is continuous and compact. Let {𝑥_𝑚}_𝑚∈ℕ⊆𝑈(𝑀) be an arbitrary sequence and 𝑥∈𝐶([𝜈,+∞),ℝ) with ‖‖𝑥_𝑚−𝑥‖‖⟶as𝑚⟶∞.(2.10) Since 𝑈(𝑀) is closed, it follows that 𝑥∈𝑈(𝑀). For any (𝑠,𝑚)∈[𝑇,+∞)×ℕ, put 𝐻𝐹_𝑚(𝑠)=||𝑓𝑠,𝑥_𝑚𝛼₁(𝑠),…,𝑥_𝑚𝛼_𝑘(𝑠)−𝑓𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)_𝑚||,(𝑠)=||ℎ𝑠,𝑥_𝑚𝛽₁(𝑠),…,𝑥_𝑚𝛽_𝑘(𝑠)−ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)||.(2.11) It follows from (2.1), (2.2), and (2.11) that ||𝐹_𝑚(𝑠)||≤2𝐹(𝑠),||𝐻_𝑚(𝑠)||≤2𝐻(𝑠),∀(𝑠,𝑚)∈[𝑇,+∞)×ℕ,(2.12) which together with (2.8)–(2.11), the continuity of 𝑓,ℎ,𝛼_𝑗,𝛽_𝑗 for 𝑗∈{1,2,…,𝑘}, and the Lebesgue dominated convergence theorem yields that ||𝐵≤_𝐿𝑥_𝑚(𝑡)−𝐵_𝐿𝑥(𝑡)_𝑡+∞||+||1𝑓𝑠,𝑥_𝑚𝛼₁(𝑠),…,𝑥_𝑚𝛼_𝑘(𝑠)−𝑓𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)||𝑑𝑠×(𝑛−1)!_𝑡+∞≤(𝑠−𝑡)^𝑛−1||ℎ𝑠,𝑥_𝑚𝛽₁(𝑠),…,𝑥_𝑚𝛽_𝑘(𝑠)−ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)||𝑑𝑠_𝑇+∞𝐹_𝑚(𝑠)𝑑𝑠+1(𝑛−1)!^𝑇+∞limsup_𝑚→∞𝑠^𝑛−1𝐻_𝑚(𝑠)𝑑𝑠,∀𝑡≥𝑇,‖‖𝐵_𝐿𝑥_𝑚−𝐵_𝐿𝑥‖‖≤limsup_𝑚→∞^𝑇+∞𝐹_𝑚(𝑠)𝑑𝑠+1(𝑛−1)!^𝑇+∞𝑠^𝑛−1𝐻_𝑚(𝑠)𝑑𝑠=0,(2.13) which means that 𝐵_𝐿 is continuous in 𝑈(𝑀). It follow from (2.1), (2.2), (2.6), and (2.8) that ‖‖𝐵_𝐿𝑥‖‖=≤sup_𝑡≥𝜈||𝐵_𝐿𝑥(𝑡)_𝑇+∞||𝐹(𝑠)𝑑𝑠+1(𝑛−1)!^𝑇+∞<min𝐿−𝑝₀𝑠^𝑛−1𝐻(𝑠)𝑑𝑠𝑀−𝑁,1−𝑝₀𝑀−𝐿,𝑀−𝑁2−𝜖⁰<𝑀,∀𝑥∈𝑈(𝑀),(2.14) which yields that 𝐵_𝐿(𝑈(𝑀)) is uniformly bounded in [𝜈,+∞).
Let 𝜀 be an arbitrary positive number. Equation (2.3) ensures that there exists 𝑇^∗>𝑇 satisfying ^𝑇+∞_∗𝐹(𝑠)𝑑𝑠+^𝑇+∞_∗𝑠^𝑛−1𝐻(𝑠)𝑑𝑠<𝜀2.(2.15) Set 𝛿=𝜀1+4𝑄+𝑀+𝑄(𝑇^∗−𝑇)^𝑛−1,𝑄=max𝐹(𝑡),𝐻(𝑡)∶𝑡∈𝑇,𝑇^∗.(2.16) For any 𝑥∈𝑈(𝑀) and 𝑡₁,𝑡₂∈[𝜈,+∞) with |𝑡₁−𝑡₂|<𝛿, we consider the following three cases.Case 1 (𝑇^∗≤𝑡₁<𝑡₂). In view of (2.1), (2.2), (2.8), and (2.15), we deduce that ||𝐵=_𝐿||||𝑥_𝑡+∞2𝑡₂−𝐵_𝐿𝑥𝑡₁||+(−1𝑓)𝑠_𝑛−1,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞₂𝑠−𝑡₂^𝑛−1−ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠_𝑡+∞1−(−1)𝑓_𝑛−1𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞₁𝑠−𝑡₁^𝑛−1≤2_𝑇+∞∗ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠||||𝐹(𝑠)𝑑𝑠+2(𝑛−1)!^𝑇+∞∗<𝜀.𝑠^𝑛−1𝐻(𝑠)𝑑𝑠(2.17)Case 2 (𝑇≤𝑡₁<𝑡₂≤𝑇^∗). Suppose that 𝑛=1. It follows from (2.1), (2.2), (2.6), and (2.8) that ||𝐵=_𝐿||||𝑥_𝑡+∞2𝑡₂−𝐵_𝐿𝑥𝑡₁||−𝑓𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠+^𝑡+∞₂ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠_𝑡+∞1≤𝑓𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠−^𝑡+∞₁ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠_𝑡2𝑡1||||≤2𝑄(||𝑡𝐹₁(𝑠)+𝐻(𝑠))𝑑𝑠<𝜀.−𝑡₂||(2.18)Suppose that 𝑛∈𝑁⧵{1}. It follows from the mean value theorem that, for each 𝑠∈(𝑡₂,+∞), there exists 𝜁∈(𝑠−𝑡₂,𝑠−𝑡₁) satisfying |||𝑠−𝑡₂^𝑛−1−𝑠−𝑡₁^𝑛−1|||=(𝑛−1)𝜁_𝑛−2||𝑡₁−𝑡₂||≤(𝑛−1)𝑠^𝑛−1||𝑡₁−𝑡₂||,(2.19) which together with (2.1), (2.2), (2.6), and (2.8) yields that ||𝐵=_𝐿||||𝑥_𝑡+∞2𝑡₂−𝐵_𝐿𝑥𝑡₁||+(−1𝑓)𝑠_𝑛−1,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞₂𝑠−𝑡₂^𝑛−1−ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠_𝑡+∞1−(−1)𝑓_𝑛−1𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞₁𝑠−𝑡₁^𝑛−1≤ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠_𝑡2𝑡1||||𝐹(𝑠)𝑑𝑠+1(𝑛−1)!^𝑡+∞₂|||𝑠−𝑡₂^𝑛−1−𝑠−𝑡₁^𝑛−1|||𝐻(𝑠)𝑑𝑠+^𝑡2_𝑡1≤𝑄||𝑡₁𝑠−𝑡₁^𝑛−1𝐻(𝑠)𝑑𝑠−𝑡₂||+(𝑛−1)||𝑡¹−𝑡₂||(𝑛−1)!^𝑡+∞₂≤𝑄+𝑀+𝑇_∗𝑠^𝑛−1𝐻(𝑠)𝑑𝑠+𝑇^∗−𝑇^𝑛−1𝑄||𝑡₁−𝑡₂||<𝜀.−𝑇^𝑛−1𝑄||𝑡₁−𝑡₂||(2.20)Case 3 (𝜈≤𝑡₁<𝑡₂≤𝑇). Equation (2.8) gives that ||𝐵_𝐿𝑥𝑡₂−𝐵_𝐿𝑥𝑡₁||=||𝐵_𝐿𝑥(𝑇)−𝐵_𝐿𝑥(𝑇)||=0.(2.21) Thus, 𝐵_𝐿(𝑈(𝑀)) is equicontinuous in [𝜈,+∞). Hence, 𝐵_𝐿(𝑈(𝑀)) is a relatively compact subset of 𝐶([𝜈,+∞),ℝ). That is, 𝐵_𝐿 is a compact mapping.
Note that for any 𝑥,𝑦∈𝑈(𝑀) and 𝑡≥𝑇||𝐴=_𝐿|||||𝑥𝐿−(𝑡)_𝑚−𝐴_𝐿𝑦(𝑡)||_𝑖=1𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)+(−1)^𝑛(𝑛−1)!^𝑡+∞−𝐿+_𝑚(𝑠−𝑡)^𝑛−1𝑔(𝑠)𝑑𝑠_𝑖=1𝑝_𝑖(𝑡)𝑦𝜏_𝑖(𝑡)−(−1)^𝑛(𝑛−1)!^𝑡+∞≤(𝑠−𝑡)^𝑛−1𝑔(𝑠)𝑑𝑠_𝑚|||||_𝑖=1≤𝑝₀||𝑝_𝑖(𝑡)||‖𝑥−𝑦‖‖𝑥−𝑦‖,(2.22)which implies that ‖‖𝐴_𝐿𝑥−𝐴_𝐿𝑦‖‖≤𝑝₀‖𝑥−𝑦‖,∀𝑥,𝑦∈𝑈(𝑀),(2.23) which together with (2.4) gives that 𝐴_𝐿 is a contraction mapping. It follows that 𝐷_𝐿∶𝑈(𝑀)→𝐸(𝑁) is a continuous and condensing mapping. Let 𝑥₀(𝑡)=𝑁 for all 𝑡∈[𝜈,+∞). Notice that 𝑥₀∈𝑈(𝑀). Thus, (2.4)–(2.7), (2.14), and (2.23) yield that ‖‖𝐷_𝐿𝑥‖‖≤≤‖‖‖‖𝐴𝐴_𝐿𝐿𝑥‖‖+‖‖𝐵_𝐿𝑥‖‖≤𝑝₀𝑥−𝐴_𝐿𝑥₀‖‖+‖‖𝐴_𝐿𝑥₀‖‖+𝑀‖‖𝑥−𝑥₀‖‖+𝑀+sup_𝑡≥𝑇|||||𝐿−_𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥₀𝜏_𝑖(𝑡)+(−1)^𝑛(𝑛−1)!^𝑡+∞≤𝑝₀(𝑠−𝑡)^𝑛−1𝑔(𝑠)𝑑𝑠|||||≤2(+𝑀𝑝+₀𝑁)+𝑀+𝐿+𝑝₀𝑁+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||𝑑𝑠𝑀+2𝑝₀𝑁+𝐿,∀𝑥∈𝑈(𝑀),(2.24) that is, 𝐷_𝐿(𝑈(𝑀)) is uniformly bounded in [𝛾,+∞).
Put 𝑆₁={𝑥∈C𝑆B([𝜈,+∞),ℝ)∶𝑁≤𝑥(𝑡)≤𝑀,∀𝑡≥𝜈,‖𝑥‖=𝑀},₂={∀𝑥𝑡≥∈𝜈CBan(d[𝜈t,h+er∞e)e,xℝis)ts∶𝑡𝑁_∗≤𝑥(𝑡)≤𝑀,≥𝜈satisfying𝑥𝑡^∗=𝑁.(2.25) It is easy to verify that 𝜕𝑈(𝑀)=𝑆₁∪𝑆₂.
Next, we show that (𝐴₂) in Lemma 1.1 does not hold. Otherwise, there exist 𝑥∈𝜕𝑈(𝑀) and 𝜆∈(0,1) satisfying 𝑥=(1−𝜆)𝑝^⋆+𝜆𝐷_𝐿𝑥. We have to discuss the following possible cases. Case 1. Let 𝑥∈𝑆₁. By means of (2.1), (2.2), and (2.4)–(2.8), we get that, for 𝑡≥𝑇, 𝑥(𝑡)=≤((11−−𝜆𝜆))𝑝^⋆𝑀+−𝜆𝜖₀𝐴_𝐿𝑥(𝑡)+𝐵_𝐿𝑥(𝑡)+𝜆𝐿+_𝑚_𝑖=1||𝑝_𝑖(𝑡)||𝑥𝜏_𝑖(𝑡)+1(𝑛−1)!^𝑡+∞+𝑠^𝑛−1||𝑔(𝑠)||𝑑𝑠_𝑡+∞+1||𝑓𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)||𝑑𝑠(𝑛−1)!^𝑡+∞≤(1−𝜆)𝑀−𝜖₀𝑠^𝑛−1||ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)||𝑑𝑠<(1−𝜆)𝑀−𝜖₀+𝜆𝐿+𝑝₀𝑀+^𝑡+∞𝐹(𝑠)𝑑𝑠+(𝑛−11)!^𝑡+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠≤𝑀−𝜖₀+𝜆𝐿+𝑝₀𝑀+min𝐿−𝑝₀𝑀−𝑁,1−𝑝₀𝑀−𝐿,𝑀−2𝑁−𝜖⁰,(2.26) which implies that 𝑀=‖𝑥‖=sup_𝑡≥𝜈||𝑥(𝑡)||≤𝑀−𝜖₀<𝑀,(2.27) which is a contradiction.Case 2. Let 𝑥∈𝑆₂. It follows from (2.1), (2.2), and (2.4)–(2.8) that 𝑁==(𝑥1−𝑡^∗𝜆)𝑝_⋆=(1−𝜆)𝑀+−𝜆𝜖₀𝐴_𝐿𝑥𝑡^∗+𝐵_𝐿𝑥𝑡^∗≥(1−𝜆)𝑀−𝜖₀+𝜆𝐴_𝐿𝑥max𝑡^∗,𝑇+𝐵_𝐿𝑥max𝑡^∗,𝑇−1+𝜆𝐿−^𝑚_𝑖=1||𝑝_𝑖max𝑡^∗,𝑇||𝑥𝜏_𝑖max𝑡^∗,𝑇(𝑛−1)!^+∞max{𝑡_∗,𝑇}−1𝑠^𝑛−1||𝑔(𝑠)||𝑑𝑠−^+∞max{𝑡_∗,𝑇}||𝑓𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)||𝑑𝑠(𝑛−1)!^+∞max{𝑡_∗,𝑇}≥(1−𝜆)𝑀−𝜖₀𝑠^𝑛−1||ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)||𝑑𝑠+𝜆𝐿−𝑝₀𝑀−^+∞max{𝑡_∗,𝑇}𝐹(𝑠)𝑑𝑠−1(𝑛−1)!^+∞max{𝑡_∗,𝑇}≥(1−𝜆)𝑀−𝜖₀𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠≥(1−𝜆)𝑀−𝜖₀+𝜆𝐿−𝑝₀𝑀−min𝐿−𝑝₀𝑀−𝑁,1−𝑝₀𝑀−𝐿,𝑀−2𝑁+𝜖⁰≥min𝑀−𝜖₀+𝜆𝑁+𝜖₀=𝑁+𝜖₀,𝑁+𝜖₀,(2.28) which is absurd.
Thus, Lemma 1.1 ensures that 𝐷_𝐿 has a fixed point 𝑥∈𝑈(𝑀); that is, 𝑥(𝑡)=𝐿−^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)+(−1)^𝑛(𝑛−1)!^𝑡+∞+(𝑠−𝑡)^𝑛−1𝑔(𝑠)𝑑𝑠_𝑡+∞+(−1)𝑓_𝑛−1𝑠,𝑥𝛼₁(𝑠),…,𝑥𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞(𝑠−𝑡)^𝑛−1ℎ𝑠,𝑥𝛽₁(𝑠),…,𝑥𝛽_𝑘(𝑠)𝑑𝑠,∀𝑡≥𝑇,(2.29) which yields that 𝑑^𝑛𝑑𝑡^𝑛𝑥(𝑡)+^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑥𝜏_𝑖(𝑡)+𝑑^𝑛−1𝑑𝑡^𝑛−1+ℎ𝑡,𝑥𝛽₁𝑓𝑡,𝑥𝛼₁(𝑡),…,𝑥𝛼_𝑘(𝑡)(𝑡),…,𝑥𝛽_𝑘(𝑡)=𝑔(𝑡),∀𝑡≥𝑇,(2.30) which means that 𝑥∈𝑈(𝑀) is a bounded positive solution of (1.1).
Let 𝐿₁,𝐿₂∈(𝑝₀𝑀+𝑁,(1−𝑝₀)𝑀) with 𝐿₁≠𝐿₂. Similarly, we can prove that, for each 𝑟∈{1,2}, there exist a constant 𝑇_𝑟>1+|𝑡₀|+|𝑡₁|+|𝜈| and two mappings 𝐴_𝐿𝑟,𝐵_𝐿𝑟∶𝑈(𝑀)→CB([𝜈,+∞),ℝ) satisfying (2.6)–(2.8), where 𝑇,𝐿,𝐴_𝐿, and 𝐵_𝐿 are replaced by 𝑇_𝑟,𝐿_𝑟,𝐴_𝐿𝑟, and 𝐵_𝐿𝑟, respectively, and 𝐴_𝐿𝑟+𝐵_𝐿𝑟 has a fixed point 𝑧_𝑟∈𝑈(𝑀), which is a bounded positive solution of (1.1) in 𝑈(𝑀). In order to prove that (1.1) possesses uncountably many bounded positive solutions in 𝑈(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 ^𝑇+∞₃𝐹(𝑠)𝑑𝑠+^𝑇+∞₃𝑠^𝑛−1𝐻(𝑠)𝑑𝑠<||𝐿¹−𝐿₂||4.(2.31) It follows from (2.1), (2.2), (2.4), (2.7), (2.8), and (2.31) that for 𝑡≥𝑇₃||𝑧₁=(𝑡|||||)𝐿−𝑧₂(𝑡)₁||−𝐿₂−^𝑚_𝑖=1𝑝_𝑖(𝑡)𝑧₁𝜏_𝑖(𝑡)+^𝑚_𝑖=1+𝑝_𝑖(𝑡)𝑧₂𝜏_𝑖(𝑡)_𝑡+∞+(−1)𝑓_𝑛−1𝑠,𝑧₁𝛼₁(𝑠),…,𝑧₁𝛼_𝑘(𝑠)𝑑𝑠−^𝑡+∞𝑓𝑠,𝑧₂𝛼₁(𝑠),…,𝑧₂𝛼_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞−(−1)_𝑛−1(𝑠−𝑡)^𝑛−1ℎ𝑠,𝑧₁𝛽₁(𝑠),…,𝑧₁𝛽_𝑘(𝑠)𝑑𝑠(𝑛−1)!^𝑡+∞≥||𝐿(𝑠−𝑡)^𝑛−1ℎ𝑠,𝑧₂𝛽₁(𝑠),…,𝑧₂𝛽_𝑘(𝑠)𝑑𝑠₁||||≥||𝐿−𝐿₂||−𝑝₀‖‖𝑧₁−𝑧₂‖‖−2^𝑇+∞₃𝐹(𝑠)𝑑𝑠−2^𝑇+∞₃𝑠^𝑛−1𝐻(𝑠)𝑑𝑠₁−𝐿₂||−𝑝₀‖‖𝑧₁−𝑧₂‖‖−||𝐿¹−𝐿₂||2,(2.32)which implies that ‖‖𝑧₁−𝑧₂‖‖≥||𝐿¹−𝐿₂||21+𝑝₀>0,(2.33) that is, 𝑧₁≠𝑧₂. This completes the proof.

Theorem 2.2. Assume that there exist constants 𝑀,𝑁,𝑝₀,𝑡₁ and functions 𝐹,𝐻∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (2.1)–(2.3) and 𝑁<1+𝑝₀𝑀,max𝑝_𝑖(𝑡)∶1≤𝑖≤𝑚≤0,^𝑚_𝑖=1𝑝_𝑖(𝑡)≥𝑝₀>−1,∀𝑡≥𝑡₁≥𝑡₀.(2.34) Then, (1.1) has uncountably many bounded positive solutions in 𝑈(𝑀).

Proof. Let 𝐿∈(𝑁,(1+𝑝₀)𝑀). It follows from (2.3) and (2.34) that there exists a constant 𝑇>1+|𝑡₀|+|𝑡₁|+|𝜈| satisfying ^𝑇+∞𝐹(𝑠)𝑑𝑠+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠<min𝐿−𝑁,1+𝑝₀𝑀−𝐿,𝑀−𝑁2.(2.35) Take 𝜖₀∈(0,min{𝐿−𝑁,(1+𝑝₀)𝑀−𝐿,(𝑀−𝑁/2)}) such that ^𝑇+∞𝐹(𝑠)𝑑𝑠+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠≤min𝐿−𝑁,1+𝑝₀𝑀−𝐿,𝑀−𝑁2−𝜖⁰.(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,𝑡₁ and functions 𝐹,𝐻∈𝐶([𝑡₀,+∞),ℝ⁺) satisfying (2.1)–(2.3) and 𝑁<1−𝑝₀𝑀,min𝑝_𝑖(𝑡)∶1≤𝑖≤𝑚≥0,^𝑚_𝑖=1𝑝_𝑖(𝑡)≤𝑝₀<1,∀𝑡≥𝑡₁≥𝑡₀.(2.37) Then, (1.1) has uncountably many bounded positive solutions in 𝑈(𝑀).

Proof. Let 𝐿∈(𝑝₀𝑀+𝑁,𝑀). It follows from (2.3) and (2.37) that there exists a constant 𝑇>1+|𝑡₀|+|𝑡₁|+|𝜈| satisfying ^𝑇+∞𝐹(𝑠)𝑑𝑠+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠<min𝐿−𝑝₀𝑀−𝑁,𝑀−𝐿,𝑀−𝑁2.(2.38) Choose 𝜖₀∈(0,min{𝐿−𝑝₀𝑀−𝑁,𝑀−𝐿,(𝑀−𝑁/2)}) such that ^𝑇+∞𝐹(𝑠)𝑑𝑠+^𝑇+∞𝑠^𝑛−1||𝑔(𝑠)||+𝐻(𝑠)𝑑𝑠≤min𝐿−𝑝₀𝑀−𝑁,𝑀−𝐿,𝑀−𝑁2−𝜖⁰.(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: 𝑑^𝑛𝑑𝑡^𝑛𝑥(𝑡)−𝑡²cos𝑡1+6𝑡²𝑥(𝑡+2)+(−1)^𝑛𝑡sin1−𝑡³1+4𝑡𝑥𝑡³+𝑑−𝑡_𝑛−1𝑑𝑡^𝑛−11+𝑡²𝑥⁴(𝑡−1)−𝑡𝑥²𝑡²+21+𝑡⁵+𝑡𝑥²𝑡²+2sin²𝑡−𝑡³𝑥²(𝑡−2)+𝑡₃1+𝑥²(𝑡−2)+𝑡³+(1/𝑡)1+𝑡^𝑛+4cos³𝑥𝑡²ln𝑡+ln1+𝑥²2^𝑡+𝑡²sin𝑥𝑡²1+𝑡^𝑛+3+𝑡²=1−𝑡³𝑡^𝑛+4ln1+𝑡³,𝑡≥3.(3.1) Let 𝑡₀=𝑡₁=3,𝑝₀=5/12,𝑚=2,𝑘=3,𝑀=36,𝑁=3,𝜈=1, 𝑝₁(𝑡)=−𝑡²cos𝑡1+6𝑡²,𝑝₂(𝑡)=(−1)^𝑛𝑡sin1−𝑡³1+4𝑡,𝜏¹𝛼(𝑡)=𝑡+2,𝜏₂(𝑡)=𝑡³−𝑡,₁(𝑡)=𝑡𝛽−₃1,𝛼₂(𝑡)=𝑡²+2,𝛼₃(𝑡)=𝑡−2,𝛽₁(𝑡)=𝑡²ln𝑡,𝛽₂(𝑡)=2^𝑡,(𝑡)=𝑡²,𝑓(𝑡,𝑢,𝑣,𝑤)=1+𝑡²𝑢⁴−𝑡𝑣²1+𝑡⁵+𝑡𝑣²sin²𝑡−𝑡³𝑤²1+𝑤²+𝑡³ℎ(𝑡,𝑢,𝑣,𝑤)=𝑡₃,+(1/𝑡)1+𝑡^𝑛+4cos³𝑢+ln1+𝑣²+𝑡²sin𝑤1+𝑡^𝑛+3+𝑡²𝐹(𝑡)=1+𝑡₂,𝑀⁴+𝑡𝑀²1+𝑡⁵+𝑡𝑀²1+𝑁²+𝑡³,𝐻(𝑡)=𝑡³+(1/𝑡)1+𝑡^𝑛+4+ln1+𝑀²+𝑡²1+𝑡^𝑛+3+𝑡²𝑔(𝑡)=1−𝑡₃,𝑡^𝑛+4ln1+𝑡³,∀(𝑡,𝑢,𝑣,𝑤)∈𝑡⁰,+∞×ℝ³.(3.2) It is clear that (2.1)–(2.4) hold. Consequently, Theorem 2.1 ensures that (3.1) has uncountably many bounded positive solutions in 𝑈(𝑀). But Theorem in [1], Theorems  2.1 and 2.3 in [6], Theorems  1 and 3 in [7], and Theorems  1 and 3 in [8] are null for (3.1).

Example 3.2. Consider the higher-order nonlinear neutral delay differential equation: 𝑑^𝑛𝑑𝑡^𝑛𝑥(𝑡)−1+𝑡²3𝑡²𝑥𝑡+𝑡²−2𝑡²1+7𝑡²+𝑑𝑥𝑡²−4𝑡_𝑛−1𝑑𝑡^𝑛−12+𝑡⁴𝑥²(𝑡−(1/𝑡))1+𝑡⁶+𝑡²𝑥²(𝑡−(1/𝑡))+𝑡1+𝑡⁴1+𝑡𝑥²2𝑡²−𝑡₃−𝑥²𝑡²−𝑡−𝑡²𝑥³(𝑡ln(1+𝑡))=1+1√+𝑡^𝑛+42+sin²𝑡³𝑥𝑡²−𝑡𝑥⁴(𝑡ln(1+𝑡))𝑡1+𝑡^𝑛+41−𝑡+𝑡²ln(1+2|𝑡|),𝑡≥1.(3.3) Let 𝑡₀=𝑡₁=1,𝑝₀=−13/21,𝑚=2,𝑘=2,𝑀=400,𝑁=100,𝜈=−4, 𝑝₁(𝑡)=−𝑡²1+3𝑡²,𝑝₂(𝑡)=−2𝑡²1+7𝑡²𝛼,𝜏₁(𝑡)=𝑡+𝑡²,𝜏₂(𝑡)=𝑡²−4𝑡,₁(𝑡)=𝑡−1𝑡,𝛼²𝑓(𝑡,𝑢,𝑣)=2+𝑡₄(𝑡)=2𝑡²−𝑡,𝛽₁(𝑡)=𝑡²−𝑡,𝛽₂(𝑡)=𝑡ln(1+𝑡),𝑢²1+𝑡⁶+𝑡²𝑢²1+𝑡⁴1+𝑡𝑣²,ℎ(𝑡,𝑢,𝑣)=𝑡³−𝑢²−𝑡²𝑣³𝐹(𝑡)=2+𝑡₄1+𝑡^𝑛+42+sin²𝑡³𝑢𝑣⁴𝑀²1+𝑡⁶+𝑡²𝑀²1+𝑡⁴1+𝑡𝑁²,𝐻(𝑡)=𝑡³+𝑀²+𝑡²𝑀³2+2𝑡^𝑛+4𝑔(𝑡)=1+√,𝑡1+𝑡^𝑛+41−𝑡+𝑡²ln(1+2|𝑡|),∀(𝑡,𝑢,𝑣)∈𝑡₀,+∞×ℝ².(3.4) It is easy to verify that (2.1)–(2.3) and (2.34) hold. Consequently, Theorem 2.2 guarantees that (3.3) has uncountably many bounded positive solutions in 𝑈(𝑀). But Theorem in [1], Theorem  1 in [4], Theorem  2.3 in [6], Theorem  3 in [7], and Theorem  3 in [8] are useless for (3.3).

Example 3.3. Consider the higher-order nonlinear neutral delay differential equation: 𝑑^𝑛𝑑𝑡^𝑛𝑥(𝑡)+1+𝑡²𝑡⁴+4𝑡⁴𝑥𝑡⁴+1+2ln1+𝑡²1+3ln1+𝑡²𝑥2+ln²+𝑑_𝑛−1𝑡𝑑𝑡^𝑛−1𝑡+𝑥³𝑡³+2𝑡²−(−1)^𝑛𝑡²sin𝑥𝑡²+𝑡₃1+𝑡⁵+𝑡²𝑥⁴𝑡²−𝑥²(𝑡−2)+𝑡²𝑥⁵(𝑡−2)1+𝑡^𝑛+51+𝑡𝑥²𝑡²ln(1+𝑡)=𝑡−(−1)^𝑛ln1+√1+𝑡²𝑡^𝑛+(5/2)+√1+cos²𝑡,𝑡≥2.(3.5) Let 𝑡₀=𝑡₁=2,𝑝₀=11/12,𝑚=2,𝑘=2,𝑀=24,𝑁=1,𝜈=0, 𝑝₁(𝑡)=𝑡⁴1+𝑡²+4𝑡⁴,𝑝₂(𝑡)=2ln1+𝑡²1+3ln1+𝑡²,𝜏¹𝛼(𝑡)=𝑡⁴+1,𝜏₂(𝑡)=2+ln²𝑡,₁𝑓(𝑡()𝑡,=𝑢,𝑡𝑣³)+=2𝑡𝑡²+,𝑢₃𝛼₂(𝑡)=𝑡²,𝛽₁(𝑡)=𝑡−2,𝛽₂(𝑡)=𝑡²ln(1+𝑡),−(−1)^𝑛𝑡²sin𝑣1+𝑡⁵+𝑡²𝑣⁴,ℎ(𝑡,𝑢,𝑣)=𝑡³−𝑢²+𝑡²𝑢⁵𝐹(𝑡)=𝑡+𝑀₃1+𝑡^𝑛+51+𝑡𝑣²,+𝑡²1+𝑡⁵+𝑡²𝑁⁴,𝐻(𝑡)=𝑡³+𝑀²+𝑡²𝑀⁵𝑔(𝑡)=𝑡−(−1)_𝑛1+𝑡^𝑛+51+𝑡𝑁²,ln1+√1+𝑡²𝑡^𝑛+(5/2)+√1+cos²𝑡,∀(𝑡,𝑢,𝑣)∈𝑡⁰,+∞×ℝ².(3.6) Obviously, (2.1)–(2.3) and (2.37) hold. It follows from Theorem 2.3 that (3.5) has uncountably many bounded positive solutions in 𝑈(𝑀). But Theorem in [1], Theorem  2.2 in [2], Theorem  2.1 in [4], Theorem  2.1 in [6], Theorem  1 in [7], and Theorem  1 in [8] are inapplicable for (3.5).

Acknowledgments

This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2011) and Changwon National University in 2011.