Unbounded Positive Solutions and Mann Iterative Schemes of a Second-Order Nonlinear Neutral Delay Difference Equation

Abstract

This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation Δ²(𝑥_𝑛+𝑎_𝑛𝑥_𝑛−𝜏)+Δℎ(𝑛,𝑥_ℎ1𝑛,𝑥_ℎ2𝑛,…,𝑥_ℎ𝑘𝑛)+𝑓(𝑛,𝑥_𝑓1𝑛,𝑥_𝑓2𝑛,…,𝑥_𝑓𝑘𝑛)=𝑏_𝑛,∀𝑛≥𝑛₀. Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.

1. Introduction and Preliminaries

Recently, the oscillation, nonoscillation, asymptotic behavior, and existence of solutions of different classes of linear and nonlinear second-order difference equations have been studied by many authors; see, for example, [1–26] and the references cited therein. Using the Banach fixed point theorem, Jinfa [5] discussed the existence of a bounded nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients: Δ²𝑥_𝑛+𝑝𝑥_𝑛−𝑚+𝑝_𝑛𝑥_𝑛−𝑘−𝑞_𝑛𝑥_𝑛−𝑙=0,∀𝑛≥𝑛₀ (1) under the condition 𝑝≠−1. Luo and Bainov [13] and M. Migda and J. Migda [16] considered the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima: Δ²𝑥_𝑛+𝑝_𝑛𝑥_𝑛−𝑘+𝑞_𝑛max𝑥_𝑠∶𝑛−𝑙≤𝑠≤𝑛=0,∀𝑛≥1 (2) and the second-order neutral difference equation: Δ²𝑥_𝑛+𝑝𝑥_𝑛−𝑘+𝑓𝑛,𝑥_𝑛=0,∀𝑛≥1, (3) respectively. Meng and Yan [15] studied the existence of bounded nonoscillatory solutions for the second-order nonlinear nonautonomous neutral delay difference equation: Δ²𝑥_𝑛−𝑝𝑥_𝑛−𝜏=^𝑚_𝑛=1𝑞_𝑖𝑥_{𝑛−𝜎𝑖}+𝑓𝑛,𝑥_{𝑛−𝜂1𝑛},…,𝑥_{𝑛−𝜂𝑙𝑛},∀𝑛≥𝑛₀. (4) Applying the cone compression and expansion theorem in Fréchet spaces, Tian and Ge [21] established the existence of multiple positive solutions of the second-order discrete equation on the half-line: Δ²𝑥_𝑛−1−𝑝Δ𝑥_𝑛−1−𝑞𝑥_𝑛−1+𝑓𝑛,𝑥_𝑛=0,∀𝑛≥1 (5) with certain boundary value conditions. But to the best of our knowledge, results on multiplicity of unbounded solutions for neutral delay difference equations are very scarce in the literature. Nothing has been done with the existence of uncountably many unbounded positive solutions for (1)~(5) and any other second-order neutral delay difference equations:

Inspired and motivated by the results in [1–26], in this paper we introduce and study the second-order nonlinear neutral delay difference equation: Δ²𝑥_𝑛+𝑎_𝑛𝑥_𝑛−𝜏+Δℎ𝑛,𝑥_ℎ1𝑛,𝑥_ℎ2𝑛,…,𝑥_ℎ𝑘𝑛+𝑓𝑛,𝑥_𝑓1𝑛,𝑥_𝑓2𝑛,…,𝑥_𝑓𝑘𝑛=𝑏_𝑛,∀𝑛≥𝑛₀, (6) where 𝜏,𝑘,𝑛₀∈ℕ, {𝑎_𝑛}_{𝑛∈ℕ𝑛0},{𝑏_𝑛}_{𝑛∈ℕ𝑛0}⊂ℝ, ℎ,𝑓∈𝐶(ℕ_𝑛0×ℝ^𝑘,ℝ), {ℎ_𝑙𝑛}_{𝑛∈ℕ𝑛0},{𝑓_𝑙𝑛}_{𝑛∈ℕ𝑛0}⊆ℕ, and lim_𝑛→∞ℎ_𝑙𝑛=lim_𝑛→∞𝑓_𝑙𝑛=+∞,𝑙∈{1,2,…,𝑘}. (7) By means of the Banach fixed point theorem and some new techniques, we establish sufficient conditions for the existence of uncountably many unbounded positive solutions of (6), suggest a few Mann iterative schemes with errors for approximating these unbounded positive solutions, and prove their convergence and the error estimates. The results obtained in this paper extend the result in [5]. Four nontrivial examples are interested in the text to illustrate the importance of our results.

Throughout this paper, we assume that Δ is the forward difference operator defined by Δ𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛, ℝ=(−∞,+∞), ℝ⁺=[0,+∞), ℤ, ℕ₀, and ℕ denote the sets of all integers, nonnegative integers, and positive integers, respectively, ℕ_𝑡={}𝑛𝑛∶𝑛∈ℕwith𝑛≥𝑡,∀𝑡∈ℕ,𝛽=min₀−𝜏,infℎ_𝑙𝑛,𝑓_𝑙𝑛∶1≤𝑙≤𝑘,𝑛∈ℕ_𝑛0𝐻∈ℕ,_𝑛=maxℎ_𝑙𝑛{,𝐹∶𝑙∈1,2,…,𝑘}_𝑛=max𝑓_𝑙𝑛∶𝑙∈{1,2,…,𝑘},∀𝑛∈ℕ_𝑛0, (8)𝑙^∞_𝛽 represents the Banach space of all real sequences on ℕ_𝛽 with norm ‖𝑥‖=sup_{𝑛∈ℕ𝛽}𝑥_𝑛𝑛<+∞foreach𝑥=𝑥_{𝑛𝑛∈ℕ𝛽}∈𝑙^∞_𝛽,𝐴(𝑁,𝑀)=𝑥=𝑥_{𝑛𝑛∈ℕ𝛽}∈𝑙^∞_𝛽𝑥∶𝑁≤^𝑛𝑛≤𝑀,𝑛∈ℕ_𝛽forany𝑀>𝑁>0. (9) It is clear that 𝐴(𝑁,𝑀) is a closed and convex subset of   𝑙^∞_𝛽. By a solution of (6), we mean a sequence {𝑥_𝑛}_{𝑛∈ℕ𝛽} with a positive integer 𝑇≥𝑛₀+𝜏+𝛽 such that (6) holds for all 𝑛≥𝑇.

The following lemmas play important roles in this paper.

Lemma 1 (see [27]). Let {𝛼_𝑛}_𝑛∈ℕ0,{𝛽_𝑛}_𝑛∈ℕ0,{𝛾_𝑛}_𝑛∈ℕ0, and {𝑡_𝑛}_𝑛∈ℕ0 be four nonnegative sequences satisfying the inequality 𝛼_𝑛+1≤1−𝑡_𝑛𝛼_𝑛+𝑡_𝑛𝛽_𝑛+𝛾_𝑛,∀𝑛∈ℕ₀, (10) where {𝑡_𝑛}_𝑛∈ℕ0⊂[0,1], ∑^∞_𝑛=0𝑡_𝑛=+∞, lim_𝑛→∞𝛽_𝑛=0, and ∑^∞_𝑛=0𝛾_𝑛<+∞. Then lim_𝑛→∞𝛼_𝑛=0.

Lemma 2 (see [11]). Let 𝜏,𝑛₀∈ℕ and {𝑏_𝑛}_{𝑛∈ℕ𝑛0} be a nonnegative sequence. Then(a)∑^∞_𝑖=0∑^∞_{𝑠=𝑛0+𝑖𝜏}𝑏_𝑠<+∞⇔∑^∞_𝑠=𝑛0𝑠𝑏_𝑠<+∞; (b)∑^∞_𝑖=0∑^∞_{𝑠=𝑛0+𝑖𝜏}∑^∞_𝑡=𝑠𝑏_𝑡<+∞⇔∑^∞_𝑠=𝑛0∑^∞_𝑡=𝑠𝑠𝑏_𝑡<+∞.

2. Existence of Uncountably Many Unbounded Positive Solutions

Using the Banach fixed point theorem and the Mann iterative schemes with errors, we next discuss the existence of uncountably many unbounded positive solutions of (6), prove that the Mann iterative schemes with errors converge to these unbounded positive solutions, and compute the error estimates between the Mann iterative schemes with errors and the unbounded positive solutions.

Theorem 3. Assume that there exist two constants 𝑀 and 𝑁 with 𝑀>𝑁>0 and four nonnegative sequences {𝑃_𝑛}_{𝑛∈ℕ𝑛0},{𝑄_𝑛}_{𝑛∈ℕ𝑛0},{𝑅_𝑛}_{𝑛∈ℕ𝑛0}, and {𝑊_𝑛}_{𝑛∈ℕ𝑛0} satisfying 𝑓𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘−𝑓𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘≤𝑃_𝑛max𝑢_𝑙−𝑢_𝑙,ℎ∶1≤𝑙≤𝑘𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘−ℎ𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘≤𝑅_𝑛max𝑢_𝑙−𝑢_𝑙,∀∶1≤𝑙≤𝑘𝑛,𝑢_𝑙,𝑢_𝑙∈ℕ_𝑛0×ℝ⁺⧵{0}²𝑓,1≤𝑙≤𝑘,𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘≤𝑄_𝑛,ℎ𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘≤𝑊_𝑛,∀𝑛,𝑢_𝑙∈ℕ_𝑛0×ℝ⁺⧵{0},1≤𝑙≤𝑘,lim_𝑛→∞1𝑛^∞∞_{𝑖=0𝑠=𝑛+𝑖𝜏}max𝑅_𝑠𝐻_𝑠,𝑊_𝑠=0,lim_𝑛→∞1𝑛^∞∞_𝑖=0^∞_{𝑠=𝑛+𝑖𝜏𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡𝑎=0,_𝑛=−1𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦. (11)(12)(13)(14)(15) Then
(a) for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that for each 𝑥₀={𝑥_0𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {𝑥_𝑚}_𝑚∈ℕ0={𝑥_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} generated by the scheme 𝑥_𝑚+1𝑛=1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑛+𝛼_𝑚−𝑛𝐿_∞^∞∑_𝑖=1∑_{𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑛,𝑛≥𝑇,𝑚≥0,1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑇+𝛼_𝑚−𝑇𝐿_∞^∞∑_𝑖=1∑_{𝑠=𝑇+𝑖𝜏}ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑇,𝛽≤𝑛<𝑇,𝑚≥0 (16) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (6) and has the following error estimate: 𝑥_𝑚+1−𝑥≤1−(1−𝜃)𝛼_𝑚𝑥_𝑚−𝑥+2𝑀𝛽_𝑚,∀𝑚∈ℕ₀, (17) where {𝛾_𝑚}_𝑚∈ℕ0={𝛾_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} is an arbitrary sequence in 𝐴(𝑁,𝑀) and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] such that ^∞_𝑚=0𝛼_𝑚=+∞,_∞𝑚=0𝛽_𝑚𝜉<+∞or𝑡ℎ𝑒𝑟𝑒𝑒𝑥𝑖𝑠𝑡𝑠𝑎𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒_{𝑚𝑚∈ℕ0}⊂[)𝛽0,+∞𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔_𝑚=𝜉_𝑚𝛼_𝑚,𝑚∈ℕ₀,lim_𝑚→∞𝜉_𝑚=0; (18) (b) equation (6) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀).

Proof. First of all we show that (a) holds. Set 𝐿∈(𝑁,𝑀). It follows from (13), (14), and (15) that there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 satisfying 1𝜃=𝑇^∞∞_{𝑖=1𝑠=𝑇+𝑖𝜏}𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,1𝑇^∞∞_{𝑖=1𝑠=𝑇+𝑖𝜏}𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+^∞_𝑡=𝑠𝑏_𝑡{},𝑎<min𝑀−𝐿,𝐿−𝑁_𝑛=−1,∀𝑛≥𝑇. (19)(20)(21) Define a mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛽 by 𝑆_𝐿𝑥_𝑛=−𝑛𝐿_∞^∞∑_𝑖=1∑_{𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡,𝑆𝑛≥𝑇,_𝐿𝑥_𝑇,𝛽≤𝑛<𝑇 (22) for each 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀). In view of (11), (12), (19), (20), and (22), we deduce that for each 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽},𝑦={𝑦_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀) and for all 𝑛≥𝑇𝑆_𝐿𝑥_𝑛𝑛−𝑆_𝐿𝑦_𝑛𝑛≤1𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−ℎ𝑠,𝑦_ℎ1𝑠,𝑦_ℎ2𝑠,…,𝑦_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑓𝑡,𝑦_𝑓1𝑡,𝑦_𝑓2𝑡,…,𝑦_𝑓𝑘𝑡≤1𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}𝑅_𝑠max𝑥_ℎ𝑙𝑠−𝑦_ℎ𝑙𝑠+∶1≤𝑙≤𝑘_∞𝑡=𝑠𝑃_𝑡max𝑥_𝑓𝑙𝑡−𝑦_𝑓𝑙𝑡≤∶1≤𝑙≤𝑘𝑥−𝑦𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}𝑅_𝑠maxℎ_𝑙𝑠+∶1≤𝑙≤𝑘_∞𝑡=𝑠𝑃_𝑡max𝑓_𝑙𝑡≤∶1≤𝑙≤𝑘𝑥−𝑦𝑇^∞∞_{𝑖=1𝑠=𝑇+𝑖𝜏}𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,𝑆=𝜃𝑥−𝑦_𝐿𝑥_𝑛𝑛=1−𝐿𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡≤1𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡+𝑏_𝑡≤1𝑇^∞∞_{𝑖=1𝑠=𝑇+𝑖𝜏}𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡<min{𝑀−𝐿,𝐿−𝑁}, (23) which yield that 𝑆_𝐿(𝐴(𝑁,𝑀))⊆𝐴(𝑁,𝑀),𝑆_𝐿𝑥−𝑆_𝐿𝑦(),≤𝜃𝑥−𝑦∀𝑥,𝑦∈𝐴𝑁,𝑀 (24) which means that 𝑆_𝐿 is a contraction in 𝐴(𝑁,𝑀). It follows from the Banach fixed point theorem that 𝑆_𝐿 has a unique fixed point 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), that is, 𝑥_𝑛−=𝑛𝐿_{∞∞𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡,𝑥∀𝑛≥𝑇,_𝑛−𝜏=()𝐿−𝑛−𝜏_{∞∞𝑖=1()𝜏𝑠=𝑛+𝑖−1}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡,∀𝑛≥𝑇+𝜏, (25) which imply that 𝑥_𝑛−𝑥_𝑛−𝜏=𝜏𝐿+^∞_𝑠=𝑛ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡,∀𝑛≥𝑇+𝜏, (26) which yields that Δ𝑥_𝑛−𝑥_𝑛−𝜏=−ℎ𝑛,𝑥_ℎ1𝑛,𝑥_ℎ2𝑛,…,𝑥_ℎ𝑘𝑛+^∞_𝑡=𝑛𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡,Δ∀𝑛≥𝑇+𝜏,₂𝑥_𝑛−𝑥_𝑛−𝜏=−Δℎ𝑛,𝑥_ℎ1𝑛,𝑥_ℎ2𝑛,…,𝑥_ℎ𝑘𝑛−𝑓𝑛,𝑥_𝑓1𝑛,𝑥_𝑓2𝑛,…,𝑥_𝑓𝑘𝑛+𝑏_𝑛,∀𝑛≥𝑇+𝜏, (27) which together with (21) gives that 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽} is an unbounded positive solution of (6) in 𝐴(𝑁,𝑀). It follows from (16), (19), (21), (22), and (24) that for any 𝑚∈ℕ₀ and 𝑛≥𝑇𝑥_𝑚+1𝑛𝑛−𝑥_𝑛𝑛=1𝑛1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑛+𝛼_𝑚−𝑛𝐿_{∞∞𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑛−𝑥_𝑛≤1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑛−𝑥_𝑛𝑛+𝛼_𝑚𝑆_𝐿𝑥_𝑚𝑛−𝑆_𝐿𝑥_𝑛𝑛+𝛽_𝑚𝛾_𝑚𝑛−𝑥_𝑛𝑛≤1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚−𝑥+𝜃𝛼_𝑚𝑥_𝑚−𝑥+2𝑀𝛽_𝑚≤1−(1−𝜃)𝛼_𝑚𝑥_𝑚−𝑥+2𝑀𝛽_𝑚, (28) which implies that 𝑥_𝑚+1−𝑥≤1−(1−𝜃)𝛼_𝑚𝑥_𝑚−𝑥+2𝑀𝛽_𝑚,∀𝑚∈ℕ₀. (29) That is, (17) holds. Thus, Lemma 1 and (17) and (18) guarantee that lim_𝑚→∞𝑥_𝑚=𝑥.
Next we show that (b) holds. Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. As in the proof of (a), we deduce similarly that for each 𝑐∈{1,2} there exist constants 𝜃_𝑐∈(0,1), 𝑇_𝑐≥𝑛₀+𝜏+𝛽, and a mapping 𝑆_𝐿𝑐 satisfying (19)~(24), where 𝜃,𝐿, and 𝑇 are replaced by 𝜃_𝑐,𝐿_𝑐 and 𝑇_𝑐, respectively, and the mapping 𝑆_𝐿𝑐 has a fixed point 𝑧^𝑐={𝑧^𝑐_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), which is an unbounded positive solution of (6) in 𝐴(𝑁,𝑀), that is, 𝑧^𝑐_𝑛=𝑛𝐿_𝑐−^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑧^𝑐_ℎ1𝑠,𝑧^𝑐_ℎ2𝑠,…,𝑧^𝑐_ℎ𝑘𝑠−^∞_𝑡=𝑠𝑓𝑡,𝑧^𝑐_𝑓1𝑡,𝑧^𝑐_𝑓2𝑡,…,𝑧^𝑐_𝑓𝑘𝑡−𝑏_𝑡,∀𝑛≥𝑇_𝑐, (30) which together with (11) and (17) implies that for 𝑛≥max{𝑇₁,𝑇₂}𝑧¹_𝑛𝑛−𝑧²_𝑛𝑛≥𝐿₁−𝐿₂−1𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}ℎ𝑠,𝑧¹_ℎ1𝑠,𝑧¹_ℎ2𝑠,…,𝑧¹_ℎ𝑘𝑠−ℎ𝑠,𝑧²_ℎ1𝑠,𝑧²_ℎ2𝑠,…,𝑧²_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑧¹_𝑓1𝑡,𝑧¹_𝑓2𝑡,…,𝑧¹_𝑓𝑘𝑡−𝑓𝑡,𝑧²_𝑓1𝑡,𝑧²_𝑓2𝑡,…,𝑧²_𝑓𝑘𝑡≥𝐿₁−𝐿₂−1𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}𝑅_𝑠max𝑧¹_ℎ𝑙𝑠−𝑧²_ℎ𝑙𝑠+∶1≤𝑙≤𝑘_∞𝑡=𝑠𝑃_𝑡max𝑧¹_𝑓𝑙𝑡−𝑧²_𝑓𝑙𝑡≥𝐿∶1≤𝑙≤𝑘₁−𝐿₂−𝑧¹−𝑧²𝑛^∞∞_{𝑖=1𝑠=𝑛+𝑖𝜏}𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡≥𝐿₁−𝐿₂−𝑧¹−𝑧²max𝑇₁,𝑇₂×^∞∞_{𝑖=1𝑠=max{𝑇1,𝑇2}+𝑖𝜏}𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡≥𝐿₁−𝐿₂−max𝜃₁,𝜃₂𝑧¹−𝑧², (31) which yields that 𝑧¹−𝑧²≥𝐿₁−𝐿₂1+max𝜃₁,𝜃₂>0, (32) that is, 𝑧¹≠𝑧². This completes the proof.

Theorem 4. Assume that there exist two constants 𝑀 and 𝑁 with 𝑀>𝑁>0 and four nonnegative sequences {𝑃_𝑛}_{𝑛∈ℕ𝑛0}, {𝑄_𝑛}_{𝑛∈ℕ𝑛0}, {𝑅_𝑛}_{𝑛∈ℕ𝑛0}, and {𝑊_𝑛}_{𝑛∈ℕ𝑛0} satisfying (11), (12), lim_𝑛→∞1𝑛^∞_𝑠=𝑛max𝑅_𝑠𝐻_𝑠,𝑊_𝑠=0,lim_𝑛→∞1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡𝑎=0,_𝑛=1𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦. (33)(34)(35) Then
(a) for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that for each 𝑥₀={𝑥_0𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {𝑥_𝑚}_𝑚∈ℕ0={𝑥_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} generated by the scheme 𝑥_𝑚+1𝑛=1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑛+𝛼_𝑚+𝑛𝐿_∞∑∑_𝑖=1^{𝑛+2𝑖𝜏−1}_{()𝜏𝑠=𝑛+2𝑖−1}ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑛,𝑛≥𝑇,𝑚≥0,1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑇+𝛼_𝑚+𝑇𝐿_∞∑∑_𝑖=1^{𝑇+2𝑖𝜏−1}_{()𝜏𝑠=𝑇+2𝑖−1}ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑇,𝛽≤𝑛<𝑇,𝑚≥0 (36) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (6) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0={𝛾_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} is an arbitrary sequence in 𝐴(𝑁,𝑀), and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18);
(b) equation (6) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀).

Proof. Let 𝐿∈(𝑁,𝑀). It follows from (33)~(35) that there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 satisfying 1𝜃=𝑇^∞_𝑠=𝑇𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,1𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡{},𝑎<min𝑀−𝐿,𝐿−𝑁_𝑛=1,∀𝑛≥𝑇. (37)(38)(39) Define a mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛽 by 𝑆_𝐿𝑥_𝑛=+𝑛𝐿_∞∑∑_𝑖=1^{𝑛+2𝑖𝜏−1}_{()𝜏𝑠=𝑛+2𝑖−1}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡,𝑆𝑛≥𝑇,_𝐿𝑥_𝑇,𝛽≤𝑛<𝑇 (40) for each 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀). Using (11), (12), (37), (38), and (40), we get that for each 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽},𝑦={𝑦_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀) and 𝑛≥𝑇𝑆_𝐿𝑥_𝑛𝑛−𝑆_𝐿𝑦_𝑛𝑛≤1𝑛^∞_𝑖=1^{𝑛+2𝑖𝜏−1}_{()𝜏𝑠=𝑛+2𝑖−1}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−ℎ𝑠,𝑦_ℎ1𝑠,𝑦_ℎ2𝑠,…,𝑦_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑓𝑡,𝑦_𝑓1𝑡,𝑦_𝑓2𝑡,…,𝑦_𝑓𝑘𝑡≤𝑥−𝑦𝑛^∞_𝑖=1^{𝑛+2𝑖𝜏−1}_{()𝜏𝑠=𝑛+2𝑖−1}𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡≤𝑥−𝑦𝑇^∞_𝑠=𝑇𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,𝑆=𝜃𝑥−𝑦_𝐿𝑥_𝑛𝑛≤1−𝐿𝑛×^∞_𝑖=1^{𝑛+2𝑖𝜏−1}_{()𝜏𝑠=𝑛+2𝑖−1}ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡+𝑏_𝑡≤1𝑇^∞_𝑖=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡<min{𝑀−𝐿,𝐿−𝑁}, (41) which imply (24). Consequently, (24) means that 𝑆_𝐿 is a contraction in 𝐴(𝑁,𝑀) and has a unique fixed point 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), which is also an unbounded positive solution of (6) in 𝐴(𝑁,𝑀). The rest of the proof is similar to the proof of Theorem 3 and is omitted. This completes the proof.

Theorem 5. Assume that there exist three constants 𝑎,𝑀, and 𝑁 with (1−𝑎)𝑀>𝑁>0 and four nonnegative sequences {𝑃_𝑛}_{𝑛∈ℕ𝑛0}, {𝑄_𝑛}_{𝑛∈ℕ𝑛0}, {𝑅_𝑛}_{𝑛∈ℕ𝑛0}, and {𝑊_𝑛}_{𝑛∈ℕ𝑛0} satisfying (11), (12), (33), (34) and 0≤𝑎_𝑛≤𝑎<1𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦. (42) Then
(a) for any 𝐿∈(𝑎𝑀+𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that for any 𝑥₀={𝑥_0𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {𝑥_𝑚}_𝑚∈ℕ0={𝑥_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} generated by the scheme 𝑥_𝑚+1𝑛=1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑛+𝛼_𝑚𝑛𝐿−𝑎_𝑛𝑥_{𝑚𝑛−𝜏}+∑^∞_𝑠=𝑛ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑛,𝑛≥𝑇,𝑚≥0,1−𝛼_𝑚−𝛽_𝑚𝑥_𝑚𝑇+𝛼_𝑚𝑇𝐿−𝑎_𝑇𝑥_{𝑚𝑇−𝜏}+∑^∞_𝑠=𝑇ℎ𝑠,𝑥_𝑚ℎ1𝑠,𝑥_𝑚ℎ2𝑠,…,𝑥_{𝑚ℎ𝑘𝑠}−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_{𝑚𝑓1𝑡},𝑥_{𝑚𝑓2𝑡}𝑥,…,_{𝑚𝑓𝑘𝑡}−𝑏_𝑡+𝛽_𝑚𝛾_𝑚𝑇,𝛽≤𝑛<𝑇,𝑚≥0 (43) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (6) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0={𝛾_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} is an arbitrary sequence in 𝐴(𝑁,𝑀) and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18);
(b) equation (6) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀).

Proof. Put 𝐿∈(𝑎𝑀+𝑁,𝑀). It follows from (33), (34), and (42) that there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 satisfying 1𝜃=𝑎+𝑇^∞_𝑠=𝑇𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,1𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡<min{𝑀−𝐿,𝐿−𝑎𝑀−𝑁},0≤𝑎_𝑛≤𝑎<1,∀𝑛≥𝑇. (44) Define a mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛽 by 𝑆_𝐿𝑥_𝑛=𝑛𝐿−𝑎_𝑛𝑥_𝑛−𝜏+∑^∞_𝑠=𝑛ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−∑^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑏_𝑡𝑆,𝑛≥𝑇,_𝐿𝑥_𝑇,𝛽≤𝑛<𝑇, (45) for each 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀). In view of (11), (12), and (44) and (45), we obtain that for each 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽},𝑦={𝑦_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀) and 𝑛≥𝑇, 𝑆_𝐿𝑥_𝑛𝑛−𝑆_𝐿𝑦_𝑛𝑛≤𝑎_𝑛𝑥_𝑛−𝜏−𝑦_𝑛−𝜏𝑛+1𝑛^∞_𝑠=𝑛ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠−ℎ𝑠,𝑦_ℎ1𝑠,𝑦_ℎ2𝑠,…,𝑦_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡−𝑓𝑡,𝑦_𝑓1𝑡,𝑦_𝑓2𝑡,…,𝑦_𝑓𝑘𝑡≤1𝑎+𝑇^∞_𝑠=𝑇𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,𝑆𝑥−𝑦=𝜃𝑥−𝑦_𝐿𝑥_𝑛𝑛+1≤𝐿𝑛^∞_𝑠=𝑛ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡+𝑏_𝑡1≤𝐿+𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡{}𝑆<𝐿+min𝑀−𝐿,𝐿−𝑎𝑀−𝑁≤𝑀,_𝐿𝑥_𝑛𝑛−1≥𝐿−𝑎𝑀𝑛^∞_𝑠=𝑛ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡+𝑏_𝑡1≥𝐿−𝑎𝑀−𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡>𝐿−𝑎𝑀−min{𝑀−𝐿,𝐿−𝑎𝑀−𝑁}≥𝑁, (46) which give (24), in turn, which implies that 𝑆_𝐿 is a contraction in 𝐴(𝑁,𝑀) and possesses a unique fixed point 𝑥={𝑥_𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), which is an unbounded positive solution of (6) in 𝐴(𝑁,𝑀). The rest of the proof is similar to that of Theorem 3 and is omitted. This completes the proof.

Theorem 6. Assume that there exist constants 𝑎,𝑀, and 𝑁 with (1+𝑎)𝑀>𝑁>0 and four nonnegative sequences {𝑃_𝑛}_{𝑛∈ℕ𝑛0}, {𝑄_𝑛}_{𝑛∈ℕ𝑛0}, {𝑅_𝑛}_{𝑛∈ℕ𝑛0}, and {𝑊_𝑛}_{𝑛∈ℕ𝑛0} satisfying (11), (12), (33), (34), and −1<𝑎≤𝑎_𝑛≤0𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦. (47) Then
(a) for any 𝐿∈(𝑁,(1+𝑎)𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that for any 𝑥₀={𝑥_0𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀) and the Mann iterative sequence with errors {𝑥_𝑚}_𝑚∈ℕ0={𝑥_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} generated by (43) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (6) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0={𝛾_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} is an arbitrary sequence in 𝐴(𝑁,𝑀), {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18);
(b) equation (6) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀).

Proof. Put 𝐿∈(𝑁,(1+𝑎)𝑀). It follows from (33), (34), and (47) that there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 satisfying 1𝜃=−𝑎+𝑇^∞_𝑠=𝑇𝑅_𝑠𝐻_𝑠+^∞_𝑡=𝑠𝑃_𝑡𝐹_𝑡,1𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡<min{(1+𝑎)𝑀−𝐿,𝐿−𝑁},−1<𝑎≤𝑎_𝑛≤0,∀𝑛≥𝑇. (48)(49)(50) Define a mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛽 by (45). By virtue of (12), (45), (48), and (50), we easily verify that 𝑆_𝐿𝑥_𝑛𝑛+1≤𝐿−𝑎𝑀𝑛^∞_𝑠=𝑛ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡+𝑏_𝑡1≤𝐿−𝑎𝑀+𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡)}𝑆<𝐿+min{(1+𝑎𝑀−𝐿,𝐿−𝑁≤𝑀,_𝐿𝑥_𝑛𝑛−1≥𝐿𝑛^𝑛−1_𝑠=𝑇ℎ𝑠,𝑥_ℎ1𝑠,𝑥_ℎ2𝑠,…,𝑥_ℎ𝑘𝑠+^∞_𝑡=𝑠𝑓𝑡,𝑥_𝑓1𝑡,𝑥_𝑓2𝑡,…,𝑥_𝑓𝑘𝑡+𝑏_𝑡1≥𝐿−𝑇^∞_𝑠=𝑇𝑊_𝑠+^∞_𝑡=𝑠𝑄_𝑡+𝑏_𝑡>𝐿−min{(1+𝑎)𝑀−𝐿,𝐿−𝑁}≥𝑁, (51) which yield that 𝑆_𝐿(𝐴(𝑁,𝑀))⊆𝐴(𝑁,𝑀). The rest of the proof is similar to that of Theorem 5 and is omitted. This completes the proof.

Remark 7. Theorems 3~6 extend and improve Theorem 1 in [5].

3. Examples

In this section we suggest four examples to explain the results presented in Section 2. Note that Theorem 1 in [5] is useless for all these examples.

Example 8. Consider the second-order nonlinear neutral delay difference equation: Δ²𝑥_𝑛−𝑥_𝑛−𝜏+Δsin²𝑥_𝑛−2𝑛⁴+1𝑛⁶+𝑛³+21+𝑥_𝑛2³=𝑛²−3𝑛+1𝑛⁷+𝑛⁴+1,𝑛≥3, (52) where 𝜏∈ℕ is fixed. Let 𝑛₀=3, 𝑘=1, 𝛽=min{3−𝜏,1}, 𝑀 and 𝑁 two positive constants with 𝑀>𝑁 and 𝑎_𝑛=−1,𝑏_𝑛=𝑛²−3𝑛+1𝑛⁷+𝑛⁴,𝑓()=1+1𝑛,𝑢𝑛⁶+𝑛³+21+|𝑢|³,ℎ(𝑛,𝑢)=sin²𝑢𝑛⁴,𝐹_𝑛=𝑓_1𝑛=𝑛²,𝐻_𝑛=ℎ_1𝑛𝑃=𝑛−2,_𝑛=3𝑀²1+𝑁³²𝑛⁶,𝑄_𝑛=1𝑛⁶,𝑅_𝑛=2𝑛⁴,𝑊_𝑛=1𝑛⁴,∀(𝑛,𝑢)∈ℕ_𝑛0×ℝ. (53) It is easy to see that (11), (12), and (15) are satisfied. Note that 𝑠max𝑅^∞_{𝑠=𝑛𝑠}𝐻_𝑠,𝑊_𝑠=^∞_𝑠=𝑛2()𝑠max𝑠−2𝑠⁴,1𝑠⁴=^∞_𝑠=𝑛2(𝑠−2)𝑠³<+∞,∀𝑛∈ℕ_𝑛0,_{∞∞𝑠=𝑛𝑡=𝑠}𝑠max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡=^∞∞_{𝑠=𝑛𝑡=𝑠}𝑠max3𝑀²1+𝑁³²𝑡⁴,1𝑡⁶,𝑡²−3𝑡+1𝑡⁷+𝑡⁴+1≤max1,3𝑀²1+𝑁^{32∞∞𝑠=𝑛}_𝑡=𝑠𝑠𝑡⁴≤max1,3𝑀²1+𝑁^{32∞∞𝑠=𝑛}_𝑡=𝑠1𝑡³=max1,3𝑀²1+𝑁^32∞_𝑡=𝑛𝑡−𝑛+1𝑡³≤max1,3𝑀²1+𝑁^32∞_𝑡=𝑛1𝑡²<+∞,∀𝑛∈ℕ_𝑛0, (54)(55) which together with Lemma 2 yield that (13) and (14) hold. It follows from Theorem 3 that (52) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀). On the other hand, for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that for each 𝑥₀={𝑥_0𝑛}_{𝑛∈ℕ𝛽}∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {𝑥_𝑚}_𝑚∈ℕ0={𝑥_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} generated by (16) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (52) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0={𝛾_𝑚𝑛}_{(𝑚,𝑛)∈ℕ0×ℕ𝛽} is an arbitrary sequence in 𝐴(𝑁,𝑀) and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18).

Example 9. Consider the second-order nonlinear neutral delay difference equation: Δ²𝑥_𝑛+𝑥_𝑛−𝜏+Δsin²𝑥_3𝑛3+1𝑛³𝑛²+21+𝑥⁴_2𝑛2−3+(−1)^𝑛𝑛³𝑥_{𝑛2−𝑛−1}+𝑥_{(𝑛+1)(𝑛+2)}𝑛¹¹+𝑛⁵+11+𝑥²_{𝑛2−𝑛−1}+𝑥²_{()𝑛+1)(𝑛+2}=𝑛²−ln𝑛𝑛⁶+𝑛⁵+1,𝑛≥5, (56) where 𝜏∈ℕ is fixed. Let 𝑛₀=5, 𝑘=2, 𝛽=5−𝜏, 𝑀 and 𝑁 two positive constants with 𝑀>𝑁 and 𝑎_𝑛=1,𝑏_𝑛=𝑛²−ln𝑛𝑛⁶+𝑛⁵,𝑓()=()+1𝑛,𝑢,𝑣−1^𝑛𝑛³(𝑢+𝑣)𝑛¹¹+𝑛⁵+11+𝑢²+𝑣²,ℎ(𝑛,𝑢,𝑣)=sin²𝑣𝑛³𝑛²+21+𝑢⁴,𝑓_1𝑛=𝑛²−𝑛−1,𝐹_𝑛=𝑓_2𝑛=(),ℎ𝑛+1)(𝑛+2_1𝑛=2𝑛²−3,𝐻_𝑛=ℎ_2𝑛=3𝑛³𝑃+1,_𝑛=𝑄_𝑛=4𝑛⁸,𝑅_𝑛=𝑊_𝑛=10𝑛⁵,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (57) It is clear that (11), (12), and (35) are fulfilled. Note that lim_𝑛→∞1𝑛^∞_𝑠=𝑛max𝑅_𝑠𝐻_𝑠,𝑊_𝑠=lim_𝑛→∞1𝑛^∞_𝑠=𝑛max103𝑠³+1𝑠⁵,10𝑠⁵1=0,𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡=1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}4()max𝑡+1)(𝑡+2𝑡⁸,4𝑡⁸,𝑡²−ln𝑡𝑡⁶+𝑡⁵≤4+1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}1𝑡⁴≤4𝑛^∞_𝑡=𝑛1𝑡³→0as𝑛→∞, (58) which yields that lim_𝑛→∞1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡=0. (59) Thus, Theorem 4 guarantees that (56) possesses uncountably unbounded positive solutions in 𝐴(𝑁,𝑀). On the other hand, for any 𝐿∈(𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝜏+𝑛₀+𝛽 such that the Mann iterative sequence with error {𝑥_𝑚}_𝑚∈ℕ0 generated by (36) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (56) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0 is an arbitrary sequence in 𝐴(𝑁,𝑀) and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18).

Example 10. Consider the second-order nonlinear neutral delay difference equation: Δ²𝑥_𝑛+3𝑛³−14𝑛³𝑥+2^𝑛−𝜏+Δsinln1+𝑛²𝑥_3𝑛2−1𝑛⁹−√−𝑛𝑛−4₂−(−1)^{𝑛(𝑛−1)/2}𝑛⁷+3𝑛⁵−12^{|𝑥34𝑛+1|}+(−1)^𝑛𝑛⁶1+𝑥²_𝑛−2−1𝑛⁵+11+𝑥_𝑛+4=(−1)^𝑛𝑛³−1𝑛⁸ln³𝑛+1,𝑛≥7, (60) where 𝜏∈ℕ is fixed. Let 𝑛₀=7, 𝑘=2, 𝑎=3/4, 𝛽=min{7−𝜏,5}, 𝑀 and 𝑁 two positive constants with 𝑀>4𝑁 and 𝑎_𝑛=3𝑛³−14𝑛³+2,𝑏_𝑛=(−1)^𝑛𝑛³−1𝑛⁸ln³,𝑓()=()𝑛+1𝑛,𝑢,𝑣−1^𝑛𝑛⁶1+𝑢²−1𝑛⁵+1|𝑣|,ℎ()=1+𝑛,𝑢,𝑣sinln1+𝑛²|𝑢|𝑛⁹−√−𝑛𝑛−4²−(−1)^{𝑛(𝑛−1)/2}𝑛⁷+3𝑛⁵−12^|𝑣|,𝑓_1𝑛𝐹=𝑛−2,_𝑛=𝑓_2𝑛=𝑛+4,ℎ_1𝑛=3𝑛²𝐻−1,_𝑛=ℎ_2𝑛=4𝑛³𝑃+1,_𝑛=𝑄_𝑛=2𝑛⁴,𝑅_𝑛=𝑊_𝑛=2𝑛⁵,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (61) It is not difficult to verify that (11), (12), and (42) are fulfilled. Note that lim_𝑛→∞1𝑛^∞_𝑠=𝑛max𝑅_𝑠𝐻_𝑠,𝑊_𝑠=lim_𝑛→∞1𝑛^∞_𝑠=𝑛2max4𝑠³+1𝑠⁵,2𝑠⁵=lim_𝑛→∞1𝑛^∞_𝑠=𝑛24𝑠³+1𝑠⁵=0,lim_𝑛→∞1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡=lim_𝑛→∞1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max2𝑡+8𝑡⁴,2𝑡⁴,(−1)^𝑡𝑡³−1𝑡⁸ln³𝑡+1=lim_𝑛→∞1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}2𝑡+8𝑡⁴=0. (62) That is, (33) and (34) are satisfied. Consequently Theorem 5 implies that (60) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀). On the other hand, for any 𝐿∈((3/4)𝑀+𝑁,𝑀), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that the Mann iterative sequence with error {𝑥_𝑚}_𝑚∈ℕ0 generated by (43) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (60) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0 is an arbitrary sequence in 𝐴(𝑁,𝑀) and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18).

Example 11. Consider the second-order nonlinear neutral delay difference equation: Δ²𝑥_𝑛+1−2𝑛²2+3𝑛²𝑥_𝑛−𝜏𝑛+Δ²−1𝑛⁵+2𝑛³−11+𝑥²_2𝑛−15+sin²𝑛³𝑥_5𝑛2−2(𝑛+2)⁸=(−1)^𝑛𝑛⁴+𝑛³+3𝑛²−1𝑛⁹+𝑛⁷+3𝑛⁶+𝑛⁴+1,𝑛≥11, (63) where 𝜏∈ℕ is fixed. Let 𝑛₀=11, 𝑘=1, 𝑎=−4/5, 𝛽=min{11−𝜏,7}, 𝑀 and 𝑁 two positive constants with 𝑀>5𝑁 and 𝑎_𝑛=1−2𝑛²2+3𝑛²,𝑏_𝑛=(−1)^𝑛𝑛⁴+𝑛³+3𝑛²−1𝑛⁹+𝑛⁷+3𝑛⁶+𝑛⁴,𝑓()=+1𝑛,𝑢sin²𝑛³𝑢(𝑛+2)⁸,ℎ()=𝑛𝑛,𝑢²−1𝑛⁵+2𝑛³−11+𝑢²,𝐹_𝑛=𝑓_1𝑛=5𝑛²𝐻−2,_𝑛=ℎ_1𝑛=2𝑛−15,𝑃_𝑛=2𝑛⁵,𝑄_𝑛=1𝑛⁸,𝑅_𝑛=2𝑛³,𝑊_𝑛=1𝑛³,∀(𝑛,𝑢)∈ℕ_𝑛0×ℝ. (64) Obviously, (11), (12), and (50) are satisfied. Note that lim_𝑛→∞1𝑛^∞_𝑠=𝑛max𝑅_𝑠𝐻_𝑠,𝑊_𝑠=lim_𝑛→∞1𝑛^∞_𝑠=𝑛max4𝑠−30𝑠³,1𝑠³1=0,𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡=1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}2max5𝑡²−2𝑡⁵,1𝑡⁸,(−1)^𝑡𝑡⁴+𝑡³+3𝑡²−1𝑡⁹+𝑡⁷+3𝑡⁶+𝑡⁴≤+110𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}1𝑡³→0as𝑛→∞, (65) which gives that lim_𝑛→∞1𝑛^∞∞_{𝑠=𝑛𝑡=𝑠}max𝑃_𝑡𝐹_𝑡,𝑄_𝑡,𝑏_𝑡=0. (66) That is, (33) and (34) hold. Thus, Theorem 6 shows that (63) possesses uncountably many unbounded positive solutions in 𝐴(𝑁,𝑀). On the other hand, for any 𝐿∈(𝑁,𝑀/5), there exist 𝜃∈(0,1) and 𝑇≥𝑛₀+𝜏+𝛽 such that the Mann iterative sequence with error {𝑥_𝑚}_𝑚∈ℕ0 generated by (43) converges to an unbounded positive solution 𝑥∈𝐴(𝑁,𝑀) of (63) and has the error estimate (17), where {𝛾_𝑚}_𝑚∈ℕ0 is an arbitrary sequence in 𝐴(𝑁,𝑀) and {𝛼_𝑚}_𝑚∈ℕ0 and {𝛽_𝑚}_𝑚∈ℕ0 are any sequences in [0,1] satisfying (18).

Acknowledgments

This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2002165).