Bounded Positive Solutions for a Third Order Discrete Equation

Abstract

This paper studies the following third order neutral delay discrete equation Δ(𝑎_𝑛Δ²(𝑥_𝑛+𝑝_𝑛𝑥_𝑛−𝜏))+𝑓(𝑛,𝑥_{𝑛−𝑑1𝑛},…,𝑥_{𝑛−𝑑𝑙𝑛})=𝑔_𝑛,𝑛≥𝑛₀, where 𝜏,𝑙∈ℕ, 𝑛₀∈ℕ∪{0}, {𝑎_𝑛}_{𝑛∈ℕ𝑛0}, {𝑝_𝑛}_{𝑛∈ℕ𝑛0}, {𝑔_𝑛}_{𝑛∈ℕ𝑛0} are real sequences with 𝑎_𝑛≠0 for 𝑛≥𝑛₀, {𝑑_𝑖𝑛}_{𝑛∈ℕ𝑛0}⊆ℤ with lim_𝑛→∞(𝑛−𝑑_𝑖𝑛)=+∞ for 𝑖∈{1,2,…,𝑙} and 𝑓∈𝐶(ℕ_𝑛0×ℝ^𝑙,ℝ). By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

1. Introduction and Preliminaries

The oscillatory, nonoscillatory and asymptotic behaviors and existence of solutions for various difference equations have received more and more attentions in recent years. For details, we refer the reader to [1–11] and the references therein.

In 2005, M. Migda and J. Migda [10] studied the asymptotic behavior of solutions for the second order neutral difference equation Δ²𝑥_𝑛+𝑝𝑥_𝑛−𝑘+𝑓𝑛,𝑥_𝑛=0,𝑛≥1, (1.1) where 𝑝∈ℝ, 𝑘 is a nonnegative integer and 𝑓∶ℕ×ℝ→ℝ. In 2008, Cheng and Chu [7] established sufficient and necessary conditions of oscillation for the second order difference equation Δ𝑟_𝑛−1Δ𝑥_𝑛−1+𝑝_𝑛𝑥^𝛾_𝑛=0,𝑛≥1, (1.2) where 𝛾 is the quotient of two odd positive integers and 𝑝_𝑛,𝑟_𝑛∈(0,+∞) for 𝑛∈ℕ. In 2000, Li et al. [9] gave several necessary and/or sufficient conditions of the existence of unbounded positive solution for the nonlinear difference equation Δ𝑟_𝑛Δ𝑥_𝑛+𝑓𝑛,𝑥_𝑛=0,𝑛≥𝑛₀, (1.3) where 𝑛₀ is a fixed nonnegative integer, 𝑟∶ℕ_𝑛0→(0,+∞) and 𝑓∶ℕ_𝑛0×ℝ→ℝ. In 2003, using the Leray-Schauder's nonlinear alternative theorem, Agarwal et al. [1] presented the existence of nonoscillatory solutions for the discrete equation Δ𝑎_𝑛Δ𝑥_𝑛+𝑝𝑥_𝑛−𝜏+𝐹𝑛+1,𝑥_{𝑛+1−𝜎}=0,𝑛≥1, (1.4) where 𝜏,𝜎 are fixed nonnegative integers, 𝑝∈ℝ,𝑎∶ℕ→(0,+∞) and 𝐹∶ℕ×(0,+∞)→[0,+∞) is continuous. In 1995, Yan and Liu [11] proved the existence of a bounded nonoscillatory solution for the third order difference equation Δ³𝑥_𝑛+𝑓𝑛,𝑥_𝑛,𝑥_𝑛−𝑟=0,𝑛≥𝑛₀ (1.5) by utilizing the Schauder's fixed point theorem. In 2005, Andruch-Sobiło and Migda [2] studied the third order linear difference equations of neutral type Δ³𝑥_𝑛−𝑝_𝑛𝑥_𝜎𝑛±𝑞_𝑛𝑥_𝜏𝑛=0,𝑛≥𝑛₀ (1.6) and obtained sufficient conditions under which all solutions of (1.6) are oscillatory.

The aim of this paper is to study the following third order neutral delay discrete equation Δ𝑎_𝑛Δ²𝑥_𝑛+𝑝_𝑛𝑥_𝑛−𝜏+𝑓𝑛,𝑥_{𝑛−𝑑1𝑛},…,𝑥_{𝑛−𝑑𝑙𝑛}=𝑔_𝑛,𝑛≥𝑛₀, (1.7) where 𝜏,𝑙∈ℕ,𝑛₀∈ℕ∪{0},{𝑎_𝑛}_{𝑛∈ℕ𝑛0},{𝑝_𝑛}_{𝑛∈ℕ𝑛0},{𝑔_𝑛}_{𝑛∈ℕ𝑛0} are real sequences with 𝑎_𝑛≠0 for 𝑛≥𝑛₀, {𝑑_𝑖𝑛}_{𝑛∈ℕ𝑛0}⊆ℤ with lim_𝑛→∞(𝑛−𝑑_𝑖𝑛)=+∞ for 𝑖∈{1,2,…,𝑙} and 𝑓∈𝐶(ℕ_𝑛0×ℝ^𝑙,ℝ). By making use of the Leray-Schauder's nonlinear alternative theorem, we establish the existence results of bounded positive solutions for (1.7), which extend substantially Theorem 2 in [11]. Three nontrivial examples are given to illustrate the superiority and applications of the results presented in this paper.

Let us recall and introduce the below concepts, signs and lemmas. Let ℝ,ℤ and ℕ denote the sets of all real numbers, integers and positive integers, respectively, ℕ_𝑛0=𝑛∶𝑛∈ℕwith𝑛≥𝑛₀,ℤ_𝛽=,𝑛𝑛∶𝑛∈ℤwith𝑛≥𝛽𝛽=min₀−𝜏,inf𝑛−𝑑_𝑖𝑛∶1≤𝑖≤𝑙,𝑛∈ℕ_𝑛0 (1.8) and 𝑙^∞_𝛽 stand for the Banach space of all bounded sequences on ℤ_𝛽 with norm ‖𝑥‖=sup_{𝑛∈ℤ𝛽}𝑥_𝑛for𝑥=𝑥_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽. (1.9) For any constants 𝑀>𝑁>0, put 𝐸(𝑁)=𝑥=𝑥_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽∶𝑥_𝑛≥𝑁for𝑛∈ℤ_𝛽,𝑈(𝑀)={𝑥∈𝐸(𝑁)∶‖𝑥‖<𝑀}. (1.10) It is easy to verify that 𝐸(𝑁) is a nonempty closed convex subset of 𝑙^∞_𝛽 and 𝑈(𝑀) is a nonempty open subset of 𝐸(𝑁).

By a solution of (1.7), we mean a sequence {𝑥_𝑛}_{𝑛∈ℤ𝛽} with a positive integer 𝑇≥𝜏+|𝛽| such that (1.7) holds for all 𝑛≥𝑇.

For any subset 𝑈 of a Banach space 𝑋, let 𝑈 and 𝜕𝑈 denote the closure and boundary of 𝑈 in 𝑋, respectively.

Lemma 1.1 (see [8]). A bounded, uniformly Cauchy subset 𝐷 of 𝑙^∞_𝛽 is relatively compact.

Lemma 1.2 (Leray-Schauder's Nonlinear Alternative Theorem [1]). Let 𝐸 be a nonempty closed convex subset of a Banach space 𝑋 and 𝑈 be an open subset of 𝐸 with 𝑝^∗∈𝑈. Also 𝐺∶𝑈→𝐸 is a continuous, condensing mapping with 𝐺(𝑈) bounded. Then either(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. Existence of Bounded Positive Solutions

Now we investigate sufficient conditions of the existence of bounded positive solutions for (1.7) by using the Leray-Schauder's Nonlinear Alternative Theorem.

Theorem 2.1. Assume that there exist constants 𝑘₀∈ℕ_𝑛0 and 𝑀,𝑁,𝑝 and 𝑝 satisfying ^∞_𝑠=𝑘0^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[𝑁,𝑀],1≤𝑖≤𝑙<+∞;_{∞𝑠=𝑘0∞𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗<+∞;0<𝑁<1−𝑝−𝑝𝑀,min𝑝,𝑝≥0,𝑝+𝑝<1−𝑝≤𝑝_𝑛≤𝑝,𝑛≥𝑘₀. (2.1)(2.2)(2.3)(2.4) Then (1.7) possesses a bounded positive solution in 𝑈(𝑀).

Proof. Let 𝐿∈(𝑝𝑀+𝑁,𝑀(1−𝑝)). It follows from (2.1)–(2.3) that there exists a positive integer 𝑇>1+𝜏+𝑘₀+|𝛽| sufficiently large satisfying ^∞∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[𝑁,𝑀],1≤𝑖≤𝑙<min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿. (2.5) Choose 𝑝^∗=𝑀−𝜀₀ with 𝜀₀∈(0,min{𝐿−𝑝𝑀−𝑁,𝑀(1−𝑝)−𝐿}) and ^∞∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[𝑁,𝑀],1≤𝑖≤𝑙≤min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿−𝜀₀. (2.6) Note that 𝑀>𝑀−𝜀₀=𝑝^∗>𝑀−min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿≥𝑁+𝑀1+𝑝−𝐿>𝑁, (2.7) which implies that 𝑝^∗={𝑝^∗}_{𝑛∈ℤ𝛽}∈𝑈(𝑀). Define two mappings 𝐴_𝐿,𝐵_𝐿∶𝑈(𝑀)→𝑙^∞_𝛽 by 𝐴_𝐿𝑥_𝑛=𝐿−𝑝_𝑛𝑥_𝑛−𝜏,𝑛≥𝑇+1𝐿−𝑝_𝑇𝑥_𝑇𝐵,𝛽≤𝑛≤𝑇;_𝐿𝑥_𝑛=−^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗},𝑛≥𝑇+10,𝛽≤𝑛≤𝑇 (2.8)(2.9) for all 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀).
We now show that 𝐷_𝐿=𝐴_𝐿+𝐵_𝐿∶𝑈(𝑀)⟶𝐸(𝑁). (2.10) For each 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀), by (2.4)–(2.9), we have 𝐴_𝐿𝑥_𝑛+𝐵_𝐿𝑥_𝑛=𝐿−𝑝_𝑛𝑥_𝑛−𝜏−^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≥𝐿−𝑝𝑀−^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≥𝐿−𝑝𝑀−min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿+𝜀₀≥𝑁+𝜀₀𝐴>𝑁,𝑛≥𝑇+1_𝐿𝑥_𝑛+𝐵_𝐿𝑥_𝑛=𝐿−𝑝_𝑇𝑥_𝑇≥𝐿−𝑝𝑀>𝑁,𝛽≤𝑛≤𝑇, (2.11)

We next assert that 𝐵_𝐿∶𝑈(𝑀)⟶𝑙^∞_𝛽isacontinuous,compactmapping. (2.12) Let {𝑥^𝛼}_𝛼∈ℕ⊆𝑈(𝑀) be an arbitrary sequence and 𝑥⁰∈𝑙^∞_𝛽 with 𝑥^𝛼−𝑥⁰⟶0as𝛼⟶∞. (2.13) Since 𝑈(𝑀) is closed, it follows that 𝑥⁰∈𝑈(𝑀). Given 𝜀>0. Using (2.1), (2.13) and the continuity of 𝑓, we infer that there exists 𝑇^∗∗,𝑇^∗∈ℕ with 𝑇^∗>𝑇+1 satisfying ^∞_𝑠=𝑇∗^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]<𝜀𝑁,𝑀,1≤𝑖≤𝑙;16_{∞𝑘=𝑇∗}1𝑎_𝑘^∞_𝑗=𝑘sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]<𝜀𝑁,𝑀,1≤𝑖≤𝑙16(𝑇^∗);−𝑇_{𝑇∗−1𝑇𝑠=𝑇∗−1𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑇∗sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]<𝜀𝑁,𝑀,1≤𝑖≤𝑙;16_{𝑇∗−1𝑠=T𝑇∗−1𝑘=𝑠}1𝑎_𝑘^𝑇∗−1_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}−𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}<𝜀2,𝛼≥𝑇^∗∗. (2.14)(2.15)(2.16)(2.17)

Combining (2.9) and (2.14)–(2.17), we conclude that 𝐵_𝐿𝑥^𝛼−𝐵_𝐿𝑥⁰=maxsup_{𝛽≤𝑛≤𝑇}𝐵_𝐿𝑥^𝛼_𝑛−𝐵_𝐿𝑥⁰_𝑛,sup_{𝑛≥𝑇+1}𝐵_𝐿𝑥^𝛼_𝑛−𝐵_𝐿𝑥⁰_𝑛≤max0,sup_{𝑛≥𝑇+1}^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}−𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}≤^{𝑇∗−1∞}_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}−𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}+^∞_𝑠=𝑇∗^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}+𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}≤^{𝑇∗−1𝑇}_𝑠=𝑇^∗−1_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}−𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}+^{𝑇∗−1∞}_{𝑠=𝑇𝑘=𝑇∗}1𝑎_𝑘^∞_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}+𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}+2^∞_𝑠=𝑇∗^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]≤𝑁,𝑀,1≤𝑖≤𝑙_{𝑇∗−1𝑇𝑠=𝑇∗−1𝑘=𝑠}1𝑎_𝑘^𝑇∗−1_𝑗=𝑘𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{𝑗−𝑑𝑙𝑗}−𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}+^{𝑇∗−1𝑇}_𝑠=𝑇^∗−1_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑇∗𝑓𝑗,𝑥^𝛼_{𝑗−𝑑1𝑗},…,𝑥^𝛼_{j−𝑑𝑙𝑗}+𝑓𝑗,𝑥⁰_{𝑗−𝑑1𝑗},…,𝑥⁰_{𝑗−𝑑𝑙𝑗}+2𝑇^∗−𝑇^∞_𝑘=𝑇∗1𝑎_𝑘^∞_𝑗=𝑘sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]𝜀𝑁,𝑀,1≤𝑖≤𝑙+2⋅<𝜀162𝜀+2⋅𝑇16+2^∗⋅𝜀−𝑇16(𝑇^∗)+𝜀−𝑇8<𝜀,𝛼≥𝑇^∗∗, (2.18) which means that 𝐵_𝐿 is continuous in 𝑈(𝑀). On the other hand, in light of (2.6) and (2.9), we get that for each 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀)𝐵_𝐿𝑥=maxsup_{𝛽≤𝑛≤𝑇}𝐵_𝐿𝑥_𝑛,sup_{𝑛≥𝑇+1}𝐵_𝐿𝑥_𝑛≤sup_{𝑛≥𝑇+1}^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≤^∞∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[𝑁,𝑀],1≤𝑖≤𝑙≤min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿−𝜀₀≤𝑀, (2.19) which yields that 𝐵_𝐿(𝑈(𝑀)) is a bounded subset of 𝑙^∞_𝛽. By virtue of (2.1) and (2.2), we deduce that for any 𝜀>0, there exists 𝑇₀>𝑇 satisfying ^∞_𝑠=𝑇0^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[𝑁,𝑀],1≤𝑖≤𝑙<𝜀, (2.20) which together with (2.9) gives that for any 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀)𝐵_𝐿𝑥_𝑛−𝐵_𝐿𝑥_𝑚=^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}−^𝑚−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≤^𝑚−1∞_{𝑠=𝑛𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}<^∞_𝑠=𝑇0^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[𝑁,𝑀],1≤𝑖≤𝑙<𝜀,𝑚>𝑛≥𝑇₀, (2.21) which means that 𝐵_𝐿(𝑈(𝑀)) is uniformly Cauchy. Thus Lemma 1.1 ensures that 𝐵_𝐿(𝑈(𝑀)) is a relatively compact subset of 𝑙^∞_𝛽.

Let 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽},𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀). In view of (2.3), (2.4) and (2.8), we know that 𝐴_𝐿𝑥_𝑛−𝐴_𝐿𝑦_𝑛=𝐿−𝑝_𝑛𝑥_𝑛−𝜏−𝐿+𝑝_𝑛𝑦_𝑛−𝜏≤𝑝_𝑛𝑥−𝑦≤𝑝+𝑝𝐴𝑥−𝑦,𝑛≥𝑇+1,_𝐿𝑥_𝑛−𝐴_𝐿𝑦_𝑛=𝐿−𝑝_𝑇𝑥_𝑇−𝜏−𝐿+𝑝_𝑇𝑦_𝑇−𝜏≤𝑝_𝑇𝑥−𝑦≤𝑝+𝑝𝑥−𝑦,𝛽≤𝑛≤𝑇, (2.22) which implies that 𝐴𝑥−𝐴𝑦≤𝑝+𝑝𝑥−𝑦, (2.23) which together with (2.10) and (2.12) guarantees that 𝐷_𝐿∶𝑈(𝑀)→𝐸(𝑁) is a continuous, condensing mapping.

In order to show the existence of a fixed point of 𝐷_𝐿, we need to prove that (A₂) in Lemma 1.2 does not hold. Otherwise there exist 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}∈𝜕𝑈(𝑀) and 𝜆∈(0,1) such that 𝑥=(1−𝜆)𝑝^∗+𝜆𝐷_𝐿𝑥. Let 𝑆₁=𝑥∈𝑙^∞_𝛽∶𝑁≤𝑥_𝑛‖𝑥‖,𝑆≤𝑀,∀𝑛≥𝛽,=𝑀₂=𝑥∈𝑙^∞_𝛽∶𝑁≤𝑥_𝑛≤𝑀,∀𝑛≥𝛽,andthereexists𝑛^∗≥𝛽satisfying𝑥_𝑛∗=𝑁. (2.24) It is easy to verify that 𝜕𝑈(𝑀)=𝑆₁∪𝑆₂. Now we have to discuss two possible cases as follows:

Case  1. Let 𝑥∈𝑆₁. It follows from (2.3), (2.4), (2.8) and (2.9) that 𝑥_𝑛=(1−𝜆)𝑝^∗+𝜆𝐴_𝐿𝑥_𝑛+𝐵_𝐿𝑥_𝑛=(1−𝜆)𝑝^∗+𝜆𝐿−𝑝_𝑛𝑥_𝑛−𝜏−^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≤(1−𝜆)𝑝^∗+𝜆𝐿+𝑝𝑀+^𝑛−1∞_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≤(1−𝜆)𝑀−𝜀₀+𝜆𝐿+𝑝𝑀+min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿−𝜀₀≤(1−𝜆)𝑀−𝜀₀+𝜆𝑀−𝜀₀=𝑀−𝜀₀𝑥,𝑛≥𝑇+1,_𝑛=(1−𝜆)𝑝^∗+𝜆𝐿−𝑝_𝑇𝑥_𝑇≤(1−𝜆)𝑝^∗+𝜆𝐿+𝑝𝑀≤(1−𝜆)𝑀−𝜀₀+𝜆𝑀−𝜀₀=𝑀−𝜀₀,𝛽≤𝑛≤𝑇, (2.25)(2.26) which yield that 𝑀=‖𝑥‖≤𝑀−𝜀₀<𝑀, (2.27) which is a contradiction;

Case  2. Let 𝑥∈𝑆₂. If 𝑛^∗≥𝑇+1, by (2.3), (2.4), (2.8) and (2.9), we deduce that 𝑁=𝑥_𝑛∗=(1−𝜆)𝑝^∗+𝜆𝐴_𝐿𝑥_𝑛∗+𝐵_𝐿𝑥_𝑛∗=(1−𝜆)𝑝^∗+𝜆𝐿−𝑝_𝑛∗𝑥_{𝑛∗−𝜏}−^{𝑛∗−1∞}_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}≥(1−𝜆)𝑀−𝜀₀+𝜆𝐿−𝑝𝑀−^{𝑛∗−1∞}_{𝑠=𝑇𝑘=𝑠}1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+𝑓𝑗,𝑥_{𝑗−𝑑1𝑗},…,𝑥_{𝑗−𝑑𝑙𝑗}>(1−𝜆)𝑁+𝜆𝐿−𝑝𝑀−min𝐿−𝑝𝑀−𝑁,𝑀1−𝑝−𝐿+𝜀₀≥(1−𝜆)𝑁+𝜆𝑁+𝜀₀=𝑁+𝜀₀>𝑁, (2.28) which is impossible; if 𝑛^∗≤𝑇, by (2.3), (2.4), (2.8) and (2.9), we arrive at 𝑁=𝑥_𝑛∗=(1−𝜆)𝑝^∗+𝜆𝐿−𝑝_𝑇𝑥_𝑇≥(1−𝜆)𝑝^∗+𝜆𝐿−≥()𝑝𝑀1−𝜆N+𝜆𝑁+𝜀₀=𝑁+𝜀₀>𝑁, (2.29) which is absurd.

Consequently Lemma 1.2 ensures that there is 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) such that 𝐷_𝐿𝑥=𝐴_𝐿𝑥+𝐵_𝐿𝑥=𝑥, which is a bounded positive solution of (1.7). This completes the proof.

Remark 2.2. Under the conditions of Theorem 2.1 we prove also that (1.7) has uncountably many bounded positive solutions in 𝑈(𝑀).
In fact, as in the proof of Theorem 2.1, for any different 𝐿₁,𝐿₂∈(𝑝𝑀+𝑁,𝑀(1−𝑝)) we conclude that for each 𝑟∈{1,2}, there exist a constant 𝑇_𝑟>1+𝜏+𝑘₀+|𝛽| and two mappings 𝐴_𝑟,𝐵_𝑟∶𝑈(𝑀)→𝑙^∞_𝛽 satisfying (2.6)–(2.9) and ^∞_{𝑠=min{𝑇1,𝑇2}}^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]≤𝐿𝑁,𝑀,1≤𝑖≤𝑙¹−𝐿₂4, (2.30) where 𝑇,𝐿,𝐴_𝐿 and 𝐵_𝐿 are replaced by 𝑇_𝑟,𝑟,𝐴_𝑟 and 𝐵_𝑟, respectively, and 𝐴_𝑟+𝐵_𝑟 has a fixed point 𝑧^𝑟={𝑧^𝑟_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀), which is a bounded positive solution of (1.7). In order to prove that (1.7) possesses uncountably many bounded positive solutions in 𝑈(𝑀), we need only to prove that 𝑧¹≠𝑧². It follows from (2.8), (2.9) and (2.30) that for 𝑛≥min{𝑇₁,𝑇₂}𝑧¹_𝑛−𝑧²_𝑛=𝐿₁−𝑝_𝑛𝑧¹_𝑛−𝜏−^𝑛−1_𝑠=𝑇1^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑧¹_{𝑗−𝑑1𝑗},…,𝑧¹_{𝑗−𝑑𝑙𝑗}−𝐿₂+𝑝_𝑛𝑧²_n−𝜏+^𝑛−1_𝑠=𝑇2^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘−𝑔_𝑗+𝑓𝑗,𝑧²_{𝑗−𝑑1𝑗},…,𝑧²_{𝑗−𝑑𝑙𝑗}≥𝐿₁−𝐿₂−𝑝+𝑝𝑧¹−𝑧²−2^∞_{𝑇𝑠=min1,𝑇2}^∞_𝑘=𝑠1𝑎_𝑘^∞_𝑗=𝑘𝑔_𝑗+sup𝑓𝑗,𝑤₁,…,𝑤_𝑙∶𝑤_𝑖∈[]≥𝐿𝑁,𝑀,1≤𝑖≤𝑙₁−𝐿₂−𝑝+𝑝𝑧¹−𝑧²−𝐿₁−𝐿₂2, (2.31) which implies that 𝑧¹−𝑧²≥𝐿₁−𝐿₂21+𝑝+𝑝>0, (2.32) which yields that 𝑧¹≠𝑧².

Remark 2.3. If either 𝑝=0 or 𝑝=0, then Theorem 2.1 reduces to the below results, respectively.

Theorem 2.4. Assume that there exist constants 𝑘₀∈ℕ_𝑛0 and 𝑀,𝑁 and 𝑝 satisfying (2.1), (2.2) and 0<𝑁<1−𝑝𝑀,0≤𝑝_𝑛≤𝑝<1,𝑛≥𝑘₀. (2.33) Then (1.7) possesses a bounded positive solution in 𝑈(𝑀).

Theorem 2.5. Assume that there exist constants 𝑘₀∈ℕ_𝑛0 and 𝑀,𝑁 and 𝑝 satisfying (2.1), (2.2) and 0<𝑁<1−𝑝𝑀,−1<−𝑝≤𝑝_𝑛≤0,𝑛≥𝑘₀. (2.34) Then (1.7) possesses a bounded positive solution in 𝑈(𝑀).

Remark 2.6. Theorems 2.1–2.5 include Theorem 2 in [11] as special cases. Examples 3.1–3.3 in Section 3 explain that Theorems 2.1–2.5 are genuine generalizations of Theorem 2 in [11].

3. Examples and Applications

Now we construct three nontrivial examples to explain the superiority and applications of Theorems 2.1–2.5, respectively.

Example 3.1. Consider the third order neutral delay discrete equation Δ(−1)^𝑛𝑛³Δ²𝑥_𝑛+𝑛sin𝑛²𝑥3𝑛+1^𝑛−𝜏+𝑛³𝑥_𝑛2−2−√𝑛𝑥³_𝑛3−6𝑛⁵+5𝑛+𝑛𝑥²_2𝑛+5=cos𝑛ln𝑛²+1√𝑛³+1,𝑛≥1, (3.1) where 𝜏∈ℕ is fixed. Let 𝑙=3,𝑛₀=1,𝑘₀=2,𝑝=1/2,𝑝=1/3,𝑀=7,𝑁=1, 𝑎_𝑛=(−1)^𝑛𝑛³,𝑝_𝑛=𝑛sin𝑛²,𝑔3𝑛+1_𝑛=cos𝑛ln𝑛²+1√𝑛³+1,𝑑_1𝑛=−𝑛²𝑑+𝑛+2,_3𝑛()=𝑛=−𝑛−5,𝑓𝑛,𝑢,𝑣,𝑤³𝑢−√𝑛𝑣³𝑛⁵+5𝑛+𝑛𝑤²,∀(𝑛,𝑢,𝑣,𝑤)∈ℕ×ℝ³. (3.2) It is easy to verify that (2.1)–(2.4) hold. It follows from Theorem 2.1 that (3.1) has a bounded positive solution in 𝑈(𝑀). However Theorem 2 in [11] is useless for (3.1).

Example 3.2. Consider the third order neutral delay discrete equation Δ(𝑛+1)²ln³(𝑛+2)Δ²𝑥_𝑛+3𝑛−4𝑥4𝑛+2^𝑛−𝜏+𝑥²_{𝑛()𝑛+1/2}𝑥³_{𝑛()𝑛−1/2}√𝑛³=()+1−1^𝑛𝑛²,𝑛≥1, (3.3) where 𝜏∈ℕ is fixed. Let 𝑙=2,𝑛₀=1,𝑘₀=2,𝑝=3/4,𝑀=40,𝑁=8, 𝑎_𝑛=(𝑛+1)²ln³(𝑛+2),𝑝_𝑛=3𝑛−44𝑛+2,𝑑_1𝑛=𝑛(1−𝑛)2,𝑑_2𝑛=𝑛(3−𝑛)2,𝑔_𝑛=(−1)^𝑛𝑛²()=𝑢,𝑓𝑛,𝑢,𝑣²𝑣³√𝑛³()+1,∀𝑛,𝑢,𝑣∈ℕ×ℝ². (3.4) It is clear that (2.1), (2.2) and (2.33) hold. Consequently Theorem 2.4 guarantees that (3.3) has a bounded positive solution in 𝑈(𝑀). But Theorem 2 in [11] is inapplicable for (3.3).

Example 3.3. Consider the third order neutral delay discrete equation Δ√𝑛⁵+1Δ²𝑥_𝑛−𝑛³+13𝑛³𝑥+4^𝑛−𝜏+√𝑛𝑥⁶_{𝑛2−2𝑛}𝑛⁴+𝑥+𝑛+1𝑛^32𝑛+32=𝑛+2sin²−𝑛𝑛²+1,𝑛≥1, (3.5) where 𝜏∈ℕ is fixed. Let 𝑙=2,𝑛₀=1,𝑘₀=3,𝑝=1/3,𝑀=30,𝑁=19, 𝑎_𝑛=√𝑛⁵+1,𝑝_𝑛𝑛=−³+13𝑛³+4,𝑑_1𝑛=𝑛(3−𝑛),𝑑_2𝑛𝑔=−𝑛−3,_𝑛=sin𝑛²−𝑛𝑛²()=𝑢+1,𝑓𝑛,𝑢,𝑣⁶√𝑛𝑛⁴+𝑣+𝑛+1𝑛³²()+2,∀𝑛,𝑢,𝑣∈ℕ×ℝ². (3.6) Obviously, (2.1), (2.2) and (2.34) hold. Thus Theorem 2.5 ensures that (3.5) has a bounded positive solution in 𝑈(𝑀). While Theorem 2 in [11] is unfit for (3.5)

Acknowledgment

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