Positive Solutions and Iterative Approximations for a Nonlinear Two-Dimensional Difference System with Multiple Delays

Abstract

This paper studies the nonlinear two-dimensional difference system with multiple delays Δ(𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1)+𝑓₁(𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛)=𝑞_1𝑛,Δ(𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2)+𝑓₂(𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛)=𝑞_2𝑛,𝑛≥𝑛₀. Using the Banach fixed point theorem and a few new analysis techniques, we show the existence of uncountably many bounded positive solutions for the system, suggest Mann iterative algorithms with errors, and discuss the error estimates between the positive solutions and iterative sequences generated by the Mann iterative algorithms. Examples to illustrates the results are included.

1. Introduction and Preliminaries

In recent years, there has been an increasing interest in the study of oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, positive solutions, nonoscillatory solutions, and periodic solutions, respectively, for various difference equations and systems, see for example, [1–34] and the references cited therein.

Jiang and Tang [18] and Graef and Thandapani [14] studied the oscillation of the linear two-dimensional difference system Δ𝑥_𝑛=𝑝_𝑛𝑦_𝑛,Δ𝑦_𝑛−1=−𝑞_𝑛𝑥_𝑛,𝑛≥𝑛₀, (1.1) and nonlinear two-dimensional difference system Δ𝑥_𝑛=𝑏_𝑛𝑔𝑦_𝑛,Δ𝑦_𝑛−1=−𝑎_𝑛𝑓𝑥_𝑛,𝑛≥𝑛₀, (1.2) where 𝑛₀∈ℕ, {𝑝_𝑛}_𝑛≥𝑛0,{𝑞_𝑛}_𝑛≥𝑛0,{𝑎_𝑛}_𝑛≥𝑛0, and {𝑏_𝑛}_𝑛≥𝑛0 are nonnegative sequences, 𝑓,𝑔∈𝐶(ℝ,ℝ) with 𝑢𝑓(𝑢)>0 and 𝑢𝑔(𝑢)>0 for all 𝑢≠0. Jiang and Tang [19] also gave some necessary and sufficient conditions for all solutions of System (1.2) to be oscillatory. Agarwal et al. [2] discussed the two-dimensional nonlinear difference system of the form Δ𝑥_𝑛=𝑎_𝑛𝑓𝑦_𝑛,Δ𝑦_𝑛−1=𝑏_𝑛𝑔𝑥_𝑛,𝑛≥𝑛₀, (1.3) where 𝑛₀∈ℕ, {𝑎_𝑛}_𝑛≥𝑛0 and {𝑏_𝑛}_𝑛≥𝑛0 are positive sequences, 𝑓,𝑔∈𝐶(ℝ,ℝ) are increasing with 𝑢𝑓(𝑢)>0 and 𝑢𝑔(𝑢)>0 for all 𝑢≠0, and they provided a classification scheme of positive solutions for System (1.3) and established conditions for the existence of solutions with designated asymptotic behavior. Li [21] introduced the two-dimensional nonlinear difference system: Δ𝑥_𝑛=𝑎_𝑛𝑓𝑦_𝑛,Δ𝑦_𝑛=−𝑏_𝑛𝑔𝑥_𝑛,𝑛≥𝑛₀, (1.4) where 𝑛₀∈ℕ, {𝑎_𝑛}_𝑛≥𝑛0 and {𝑏_𝑛}_𝑛≥𝑛0 are nonnegative sequences, 𝑓,𝑔∈𝐶(ℝ,ℝ) with 𝑢𝑓(𝑢)>0, and 𝑢𝑔(𝑢)>0 for all 𝑢≠0, he showed both classification schemes for nonoscillatory solutions of System (1.4) and gave necessary and sufficient conditions for the existence of these solutions. Huo and Li [15, 16] considered the nonlinear two-dimensional difference system Δ𝑥_𝑛=𝑏_𝑛𝑔𝑦_𝑛,Δ𝑦_𝑛=−𝑎_𝑛𝑓𝑥_𝑛+𝑟_𝑛,𝑛≥𝑛₀, (1.5) and the Emden-Fowler difference system Δ𝑥_𝑛=𝑏_𝑛𝑔𝑦_𝑛,Δ𝑦_𝑛−1=−𝑎_𝑛𝑓𝑥_𝑛+𝑟_𝑛,𝑛≥1, (1.6) where 𝑛₀∈ℕ, {𝑎_𝑛}_𝑛≥𝑛0 is a real sequence, {𝑏_𝑛}_𝑛≥𝑛0 is a nonnegative sequence, ∑^∞_𝑖=1|𝑟_𝑖|<+∞, 𝑓,𝑔∈𝐶(ℝ,ℝ) with 𝑢𝑓(𝑢)>0, and 𝑢𝑔(𝑢)>0 for all 𝑢≠0, they proved some oscillation results for Systems (1.5) and (1.6). Jiang and Li [17] investigated the nonlinear two-dimensional difference system: Δ𝑥_𝑛=𝑎_𝑛𝑔𝑦_𝑛,Δ𝑦_𝑛−1=𝑓𝑛,𝑥_𝑛,𝑛≥𝑛₀, (1.7) where 𝑛₀∈ℕ, {𝑎_𝑛}_𝑛≥𝑛0 is a nonnegative sequence, 𝑓∈𝐶(ℕ_𝑛0×ℝ,ℝ),𝑔∈𝐶(ℝ,ℝ) with 𝑢𝑓(𝑛,𝑢)>0 for all 𝑛∈ℕ_𝑛0, and 𝑢≠0 and 𝑢𝑔(𝑢)>0 for all 𝑢≠0, they obtained some necessary and sufficient conditions for all solutions of System (1.7) to be oscillatory. Thandapani and Kumar [31] studied the oscillation for the nonlinear two-dimensional difference system of the neutral type Δ𝑥_𝑛−𝑎_𝑛𝑥_𝜎𝑛=𝑝_𝑛𝑔𝑦_𝑛,Δ𝑦_𝑛=𝛿𝑞_𝑛𝑓𝑥_𝜏𝑛,𝑛≥𝑛₀, (1.8) where 𝛿=±1,𝑛₀∈ℕ, {𝑎_𝑛}_𝑛≥𝑛0 is a positive sequence, {𝑝_𝑛}_𝑛≥𝑛0 and {𝑞_𝑛}_𝑛≥𝑛0 are nonnegative sequences with ∑^∞_𝑖=𝑛0𝑝_𝑖=+∞, lim_𝑛→∞𝜎_𝑛=lim_𝑛→∞𝜏_𝑛=+∞, 𝑓,𝑔∈𝐶(ℝ,ℝ) are nondecreasing with 𝑢𝑓(𝑢)>0 and 𝑢𝑔(𝑢)>0 for all 𝑢≠0. Wu and Liu [33] established the existence and multiplicity of periodic solutions for the first-order neutral difference system Δ𝑥_𝑛−𝑐𝑥_𝑛−𝜏=𝑎_1𝑛𝑔₁𝑥_𝑛𝑥_𝑛−𝜆𝑏_1𝑛𝑓₁𝑥_{𝑛−𝜏1𝑛},𝑦_{𝑛−𝜌1𝑛},Δ𝑦_𝑛−𝑐𝑦_𝑛−𝜏=𝑎_2𝑛𝑔₂𝑦_𝑛𝑦_𝑛−𝜇𝑏_2𝑛𝑓₂𝑥_{𝑛−𝜏2𝑛},𝑦_{𝑛−𝜌2𝑛}, (1.9) where 𝜏∈ℕ,𝜆,𝜇∈ℝ⁺⧵{0}, 𝑐∈ℝ with |𝑐|≠1, {𝑎_1𝑛},{𝑎_2𝑛},{𝑏_1𝑛}, and {𝑏_2𝑛} are positive 𝑇-periodic sequences, {𝜏_1𝑛},{𝜏_2𝑛},{𝜌_1𝑛}, and {𝜌_2𝑛} are positive 𝑇-periodic integer sequences. Tang [30] proved the existence results of a bounded nonoscillatory solution for the second-order linear delay difference equation Δ²𝑥_𝑛=𝑝_𝑛𝑥_𝑛−𝑘,𝑛≥0, (1.10) where 𝑘∈ℕ and {𝑝_𝑛}_𝑛≥0 is a nonnegative sequence. Cheng [20] utilized the Banach fixed point theorem to discuss the existence of a 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, (1.11) where 𝑚,𝑘,𝑙∈ℕ, 𝑝∈ℝ⧵{−1}, {𝑝_𝑛}_𝑛≥0 and {𝑞_𝑛}_𝑛≥0 are nonnegative sequences with ∑^∞_𝑛=𝑛0𝑛max{𝑝_𝑛,𝑞_𝑛}<+∞. M. Migda and J. Migda [29] obtained the asymptotic behavior of the second-order neutral difference equation: Δ²𝑥_𝑛+𝑝𝑥_𝑛−𝑘+𝑓𝑛,𝑥_𝑛=0,𝑛≥1, (1.12) where 𝑝∈ℝ, 𝑘∈ℕ₀, and 𝑓∈𝐶(ℕ×ℝ,ℝ). Liu et al. [27] studied the global existence of uncountably many bounded nonoscillatory solutions for the second-order nonlinear neutral delay difference equation: Δ𝑎_𝑛Δ𝑥_𝑛+𝑏𝑥_𝑛−𝜏+𝑓𝑛,𝑥_{𝑛−𝑑1𝑛},𝑥_{𝑛−𝑑2𝑛},…,𝑥_{𝑛−𝑑𝑘𝑛}=𝑐_𝑛,𝑛≥𝑛₀, (1.13) where 𝑏∈ℝ,𝜏,𝑘∈ℕ,𝑛₀∈ℕ₀,{𝑎_𝑛}_{𝑛∈ℕ𝑛0} is a positive sequence, {𝑐_𝑛}_{𝑛∈ℕ𝑛0} is a real sequence, ⋃^𝑘_𝑙=1{𝑑_𝑙𝑛}_{𝑛∈ℕ𝑛0}⊆ℤ with lim_𝑛→∞(𝑛−𝑑_𝑙𝑛)=+∞ for 𝑙∈{1,2,…,𝑘}, and 𝑓∶ℕ_𝑛0×ℝ^𝑘→ℝ is a mapping.

The purpose of this paper is to study the below nonlinear two-dimensional difference system with multiple delays Δ𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1+𝑓₁𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛=𝑞_1𝑛,Δ𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2+𝑓₂𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛=𝑞_2𝑛,𝑛≥𝑛₀, (1.14) where ℎ,𝑘,𝜏_𝑖∈ℕ, 𝑛₀∈ℕ₀,{𝑝_𝑖𝑛}_{𝑛∈ℕ𝑛0},{𝑞_𝑖𝑛}_{𝑛∈ℕ𝑛0}⊂ℝ, and 𝑓_𝑖∈𝐶(ℕ_𝑛0×ℝ^ℎ+𝑘,ℝ) for 𝑖∈Λ₂, {𝑎_𝑙𝑛}_{𝑛∈ℕ𝑛0},{𝑐_𝑙𝑛}_{𝑛∈ℕ𝑛0}⊂ℤ with lim_𝑛→∞𝑎_𝑙𝑛=lim_𝑛→∞𝑐_𝑙𝑛=+∞ for 𝑙∈Λ_ℎ and {𝑏_𝑗𝑛}_{𝑛∈ℕ𝑛0},{𝑑_𝑗𝑛}_{𝑛∈ℕ𝑛0}⊂ℤ with lim_𝑛→∞𝑏_𝑗𝑛=lim_𝑛→∞𝑑_𝑗𝑛=+∞ for 𝑗∈Λ_𝑘. It is easy to see that the system (1.14) includes Systems (1.1)–(1.9), (1.10)–(1.13), and a lot of the first- and second-order nonlinear, half-linear, and quasilinear difference equations as special cases. Using the Banach fixed point theorem and some analysis techniques, we prove the existence of uncountably many bounded positive solutions for System (1.14), establish their iterative approximations, and discuss the error estimates between the positive solutions and iterative approximations. Our results sharp, and improve [14, Theorem 1] and [27, Theorems 2.1–2.7]. To illustrate our results, fifteen examples are also included.

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,ℕ, and ℕ₀ stand for the sets of all integers, positive integers, and nonnegative integers, respectively, ℕ_𝑛0=𝑛∶𝑛∈ℕ₀with𝑛≥𝑛₀,Λ_𝑛={}𝑛1,…,𝑛,∀𝑛∈ℕ,𝛼=inf₀−𝜏₁,𝑛₀−𝜏₂,𝑎_𝑙𝑛,𝑐_𝑙𝑛,𝑏_𝑗𝑛,𝑑_𝑗𝑛∶𝑛∈ℕ_𝑛0,𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,ℤ_𝛼={𝑛∶𝑛∈ℤwith𝑛≥𝛼}, (1.15)𝑙^∞_𝛼 denotes the Banach space of all bounded sequences on ℤ_𝛼 with norm ‖𝑥‖=sup_{𝑛∈ℤ𝛼}𝑥_𝑛for𝑥=𝑥_{𝑛𝑛∈ℤ𝛼}∈𝑙^∞_𝛼. (1.16) Let 𝐴()==𝑥𝑁,𝑀𝑥,𝑦_{𝑛𝑛∈ℤ𝛼},𝑦_{𝑛𝑛∈ℤ𝛼}∈𝑙^∞_𝛼×𝑙^∞_𝛼∶𝑁≤𝑥_𝑛≤𝑀,𝑁≤𝑦_𝑛≤𝑀,𝑛∈ℤ_𝛼,𝐴𝑁,𝑀,𝑁₀,𝑀₀=𝑥,𝑦=𝑥_{𝑛𝑛∈ℤ𝛼},𝑦_{𝑛𝑛∈ℤ𝛼}∈𝑙^∞_𝛼×𝑙^∞_𝛼∶𝑁≤𝑥_𝑛≤𝑀,𝑁₀≤𝑦_𝑛≤𝑀₀,𝑛∈ℤ_𝛼, (1.17) for any 𝑀>𝑁>0 and 𝑀₀>𝑁₀>0. It is easy to see that 𝐴(𝑁,𝑀) and 𝐴(𝑁,𝑀,𝑁₀,𝑀₀) are bounded closed and convex subsets of the Banach space 𝑙^∞_𝛼×𝑙^∞_𝛼 with norm 𝑥,𝑦₁=max‖𝑥‖,𝑦for𝑥,𝑦∈𝑙^∞_𝛼×𝑙^∞_𝛼. (1.18)

By a solution of System (1.14), we mean a sequence ({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼}) with a positive integer 𝑇≥𝑛₀+𝜏₁+𝜏₂+|𝛼| such that System (1.14) is satisfied for all 𝑛≥𝑇. A solution ({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼}) of System (1.14) is said to be positive if both components are positive.

Lemma 1.1 (see [35]). Let {𝐴_𝑛}_𝑛≥0,{𝐵_𝑛}_𝑛≥0,{𝐶_𝑛}_𝑛≥0, and {𝐷_𝑛}_𝑛≥0 be four nonnegative real sequences satisfying the inequality A_𝑛+1≤1−𝐷_𝑛𝐴_𝑛+𝐷_𝑛𝐵_𝑛+𝐶_𝑛,∀𝑛≥0, (1.19) where {𝐷_𝑛}_𝑛≥0⊂[0,1],∑^∞_𝑛=0𝐷_𝑛=+∞,lim_𝑛→∞𝐵_𝑛=0, and ∑^∞_𝑛=0𝐶_𝑛<+∞. Then lim_𝑛→∞𝐴_𝑛=0.

Lemma 1.2 (see [36]). Let 𝜏∈ℕ,𝑛₀∈ℕ₀, and {𝐵_𝑛}_{𝑛∈ℕ𝑛0} be a nonnegative sequence. Then ^∞∞_{𝑖=0𝑛=𝑛0+𝑖𝜏}𝐵_𝑛<+∞⟺^∞_𝑛=𝑛0𝑛𝐵_𝑛<+∞. (1.20)

2. Main Results

In this section, we investigate the existence of uncountably many bounded positive solutions for System (1.14) and their iterative approximations and the error estimates between the positive solutions and iterative approximations.

Theorem 2.1. Assume that there exist constants 𝑀,𝑁∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0, and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying 𝑓_𝑖𝑛,𝑢₁,…,𝑢_ℎ,𝑣₁,…,𝑣_𝑘≤𝑟_𝑖𝑛,∀𝑛,𝑢_𝑙,𝑣_𝑗∈ℕ_𝑛1×[𝑁,𝑀]²,𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,𝑖∈Λ₂,𝑓_𝑖𝑛,𝑢₁,…,𝑢_ℎ,𝑣₁,…,𝑣_𝑘−𝑓_𝑖𝑛,𝑤₁,…,𝑤_ℎ,𝑧₁,…,𝑧_𝑘≤𝑡_𝑖𝑛max𝑢_𝑙−𝑤_𝑙,𝑣_𝑗−𝑧_𝑗∶𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,∀𝑛,𝑢_𝑙,𝑤_𝑙,𝑣_𝑗,𝑧_𝑗∈ℕ_𝑛1×[𝑁,𝑀]⁴,𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,𝑖∈Λ₂,_{∞𝑛=𝑛0}𝑛max𝑟_𝑖𝑛,𝑡_𝑖𝑛,𝑞_𝑖𝑛∶𝑖∈Λ₂<+∞,𝑁<𝑀,𝑝_𝑖𝑛=−1,∀𝑛≥𝑛₁,𝑖∈Λ₂. (2.1)(2.2)(2.3)(2.4) Then,(a) for each 𝐿∈(𝑁,𝑀), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by the schemes 𝑥_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛+𝛽_𝜇𝐿−∑^∞∞_𝑙=0∑_{()𝜏𝑚=𝑛+𝑙+11}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑇𝐿+𝛽_𝜇𝐿−∑^∞∞_𝑙=0∑_{𝑚=𝑇𝐿+()𝜏𝑙+11}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿,𝑦_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛+𝛽_𝜇𝐿−∑^∞∞_𝑙=0∑_{()𝜏𝑚=𝑛+𝑙+12}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑇𝐿+𝛽_𝜇𝐿−∑^∞∞_𝑙=0∑_{𝑚=𝑇𝐿+()𝜏𝑙+12}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿 (2.5)(2.6) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate: 𝑥^𝜇+1,𝑦^𝜇+1−𝑥,𝑦₁≤1−𝛽_𝜇1−𝜃_𝐿𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇,∀𝜇≥0, (2.7) where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1] and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with ^∞_𝜇=0𝛽_𝜇=+∞,_∞𝜇=0𝛾_𝜇<+∞orthereexistsasequence𝜂_{𝜇𝜇∈ℕ0}⊂[)𝛾0,+∞satisfying_𝜇=𝛽_𝜇𝜂_𝜇,∀𝜇∈ℕ₀,lim_𝜇→∞𝜂_𝜇=0, (2.8)(2.9)(b) System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Proof. (a) Let 𝐿∈(𝑁,𝑀). Now we construct a contraction mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) and prove that its fixed point is a bounded positive solution of System (1.14). It follows from (2.3) and (2.4) that there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| satisfying 𝜃_𝐿=^∞_{𝑚=𝑇𝐿}(1+𝑚)max𝑡_𝑖𝑚∶𝑖∈Λ₂,_{∞𝑚=𝑇𝐿}(1+𝑚)max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min{𝑀−𝐿,𝐿−𝑁}. (2.10)(2.11) Define three mappings 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼×𝑙^∞_𝛼,𝑆_1𝐿,𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼 by 𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿−∑^∞∞_𝑙=0∑_{()𝜏𝑚=𝑛+𝑙+11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛=𝐿−∑^∞∞_𝑙=0∑_{()𝜏𝑚=𝑛+𝑙+12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿,𝑆_2𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿,𝑆_𝐿𝑥_𝑛,𝑦_𝑛=𝑆_1𝐿𝑥_𝑛,𝑦_𝑛,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛,𝑛∈ℤ_𝛼, (2.12)(2.13)(2.14) for all (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀). It follows from (2.1), (2.3), (2.12), (2.13), and Lemma 1.2 that 𝑆_1𝐿 and 𝑆_2𝐿 are well defined.
In view of (2.1), (2.2), and (2.10)–(2.14), we know that for any (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀) and (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), 𝑆_𝐿𝑥,𝑦−𝑆_𝐿(𝑢,𝑣)₁=max𝑆_𝑖𝐿𝑥,𝑦−𝑆_𝑖𝐿(𝑢,𝑣)∶𝑖∈Λ₂=maxsup_{𝑛≥𝑇𝐿}𝑆_𝑖𝐿𝑥_𝑛,𝑦_𝑛−𝑆_𝑖𝐿𝑢_𝑛,𝑣_𝑛∶𝑖∈Λ₂≤maxsup_{𝑛≥𝑇𝐿}^∞∞_{𝑙=0𝑚=𝑛+(𝑙+1)𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚,sup_{𝑛≥𝑇𝐿}^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_{𝑑𝑏1𝑚},…,𝑦_𝑑𝑘𝑚−𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚≤max^∞∞_{𝑙=0𝑚=𝑇𝐿+()𝜏𝑙+11}𝑡_1𝑚max𝑥_𝑎𝑖𝑚−𝑢_𝑎𝑖𝑚,𝑦_𝑏𝑗𝑚−𝑣_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘,_{∞∞𝑙=0𝑚=𝑇𝐿+(𝑙+1)𝜏2}𝑡_2𝑚max𝑥_𝑐𝑖𝑚−𝑢_𝑐𝑖𝑚,𝑦_𝑑𝑗𝑚−𝑣_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘≤max^∞∞_{𝑙=0𝑚=𝑇𝐿+()𝜏𝑙+11}𝑡_1𝑚,^∞∞_{𝑙=0𝑚=𝑇𝐿+()𝜏𝑙+12}𝑡_2𝑚max‖𝑥−𝑢‖,𝑦−𝑣≤max^∞_{𝑚=𝑇𝐿}𝑚1+𝜏₁𝑡_1𝑚,^∞_{𝑚=𝑇𝐿}𝑚1+𝜏₂𝑡_2𝑚𝑥,𝑦−(𝑢,𝑣)₁≤^∞_{𝑚=𝑇𝐿}(1+𝑚)max𝑡_1𝑚,𝑡_2𝑚𝑥,𝑦−(𝑢,𝑣)₁=𝜃_𝐿𝑥,𝑦−(𝑢,𝑣)₁,max𝑆_1𝐿𝑥_𝑛,𝑦_𝑛−𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛−𝐿≤max^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,_{∞∞𝑙=0𝑚=𝑛+(𝑙+1)𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚≤max^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+11}𝑟_1𝑚+𝑞_1𝑚,^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+12}𝑟_2𝑚+𝑞_2𝑚≤max^∞_{𝑚=𝑇𝐿}𝑚1+𝜏₁𝑟_1𝑚+𝑞_1𝑚,^∞_{𝑚=𝑇𝐿}𝑚1+𝜏₂𝑟_2𝑚+𝑞_2𝑚≤^∞_{𝑚=𝑇𝐿}(1+𝑚)max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min{𝑀−𝐿,𝐿−𝑁},∀𝑛≥𝑇_𝐿, (2.15) which give that 𝑆_𝐿𝑥,𝑦−𝑆_𝐿(𝑢,𝑣)₁≤𝜃_𝐿𝑥,𝑦−(𝑢,𝑣)₁,∀𝑥,𝑦,(𝑢,𝑣)∈𝐴(𝑁,𝑀),𝑁≤𝑆_1𝐿𝑥_𝑛,𝑦_𝑛≤𝑀,𝑁≤𝑆_2𝐿𝑥_𝑛,𝑦_𝑛≤𝑀,∀𝑥_{𝑛𝑛∈ℤ𝛼},𝑦_{𝑛𝑛∈ℤ𝛼}∈𝐴(𝑁,𝑀),𝑛≥𝑇_𝐿. (2.16)(2.17) Observe that 𝜃_𝐿∈(0,1). It is easy to see that (2.12)–(2.17) give that 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) is a contraction. Consequently, 𝑆_𝐿 possesses a unique fixed point (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), which implies that 𝑥_𝑛=𝐿−^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑦_𝑛=𝐿−^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_L, (2.18) which yield that 𝑥_𝑛−𝑥_𝑛−𝜏1=𝐿−^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚−𝐿+^∞∞_{𝑙=0𝑚=𝑛+𝑙𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚=^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,∀𝑛≥𝑇_𝐿+𝜏₁,𝑦_𝑛−𝑦_𝑛−𝜏2=𝐿−^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚−𝐿+^∞∞_{𝑙=0𝑚=𝑛+𝑙𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚=^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,∀𝑛≥𝑇_𝐿+𝜏₂, (2.19) which give that Δ𝑥_𝑛−𝑥_𝑛−𝜏1=−𝑓₁𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛+𝑞_1𝑛,∀𝑛≥𝑇_𝐿+𝜏₁,Δ𝑦_𝑛−𝑦_𝑛−𝜏2=−𝑓₂𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛+𝑞_2𝑛,∀𝑛≥𝑇_𝐿+𝜏₂, (2.20) that is, (𝑥,𝑦) is a bounded positive solution of System (1.14) in 𝐴(𝑁,𝑀).
In light of (2.5), (2.6), and (2.11)–(2.16), we arrive at 𝑥^𝜇+1,𝑦^𝜇+1−𝑥,𝑦₁=maxsup_{𝑛≥𝑇𝐿}𝑥_𝑛^𝜇+1−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}𝑦_𝑛^𝜇+1−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇𝐿−^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+11}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚−𝑥_𝑛+𝛾_𝜇𝛿^𝜇_1𝑛−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇𝐿−^∞∞_{𝑙=0()𝜏𝑚=𝑛+𝑙+12}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚−𝑦_𝑛+𝛾_𝜇𝛿^𝜇_2𝑛−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇𝑆_1𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_1𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇𝑆_2𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_2𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇≤max1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇−𝑥+𝜃_𝐿𝛽_𝜇𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇,1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇−𝑦+𝜃_𝐿𝛽_𝜇𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇≤1−𝛽_𝜇1−𝜃_𝐿𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇,∀𝜇≥0, (2.21) that is, (2.7) holds. It follows from (2.7)–(2.9) and Lemma 1.1 that lim_𝜇→∞‖(𝑥^𝜇,𝑦^𝜇)−(𝑥,𝑦)‖₁=0.
(b) Let 𝐿₁,𝐿₂∈(𝑁,𝑀), and 𝐿₁≠𝐿₂. As in the proof of (a), we similarly infer that for each 𝑖∈Λ₂, there exist a constant 𝜃_𝐿𝑖∈(0,1), a positive integer 𝑇_𝐿𝑖≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼|, and mappings 𝑆_𝐿𝑖,𝑆_1𝐿𝑖,𝑆_2𝐿𝑖 satisfying (2.10)–(2.14), where 𝐿,𝜃_𝐿 and 𝑇_𝐿 are replaced by 𝐿_𝑖,𝜃_𝐿𝑖 and 𝑇_𝐿𝑖, respectively, and the contraction mappings 𝑆_𝐿1 and 𝑆_𝐿2 have the unique fixed points (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼}),(𝑤,𝑧)=({𝑤_𝑛}_{𝑛∈ℤ𝛼},{𝑧_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), respectively, (𝑢,𝑣) and (𝑤,𝑧) are bounded positive solutions of System (1.14) in 𝐴(𝑁,𝑀). In order to show that System (1.14) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀), we prove only that (𝑢,𝑣)≠(𝑤,𝑧). By means of (2.10) and (2.12)–(2.16), we conclude that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁=max{‖𝑢−𝑤‖,‖𝑣−𝑧‖}≥maxsup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑢_𝑛−𝑤_𝑛,sup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑣_𝑛−𝑧_𝑛≥max𝐿₁−𝐿₂−^∞∞_{𝑙=0𝑇𝑚=max𝐿1,𝑇𝐿2+()𝜏𝑙+11}𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘m−𝑓₁𝑚,𝑤_𝑎1𝑚,…,𝑤_𝑎ℎ𝑚,𝑧_𝑏1𝑚,…,𝑧_𝑏𝑘𝑚,𝐿₁−𝐿₂−^∞∞_{𝑙=0𝑚=max{𝑇𝐿1,𝑇𝐿2}+(𝑙+1)𝜏2}𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚−𝑓₂𝑚,𝑤_𝑐1𝑚,…,𝑤_𝑐ℎ𝑚,𝑧_𝑑1𝑚,…,z_𝑑𝑘𝑚≥max𝐿₁−𝐿₂−^∞∞_{𝑙=0𝑇𝑚=max𝐿1,𝑇𝐿2+()𝜏𝑙+11}𝑡_1𝑚max𝑢_𝑎𝑖𝑚−𝑤_𝑎𝑖𝑚,𝑣_𝑏𝑗𝑚−𝑧_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾,𝐿₁−𝐿₂−^∞∞_{𝑙=0𝑇𝑚=max𝐿1,𝑇𝐿2+()𝜏𝑙+12}𝑡_2𝑚×max𝑢_𝑐𝑖𝑚−𝑤_𝑐𝑖𝑚,𝑣_𝑑𝑗𝑚−𝑧_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾≥max𝐿₁−𝐿₂−^∞∞_{𝑙=0𝑇𝑚=max𝐿1,𝑇𝐿2+()𝜏𝑙+11}𝑡_1𝑚‖,‖,𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂−^∞∞_{𝑙=0𝑇𝑚=max𝐿1,𝑇𝐿2+()𝜏𝑙+12}𝑡_2𝑚‖,‖≥𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}(1+𝑚)max𝑡_𝑖𝑚∶𝑖∈Λ₂‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝐿₁−𝐿₂−max𝜃_𝐿1,𝜃_𝐿2‖(𝑢,𝑣)−(𝑤,𝑧)‖₁, (2.22) which implies that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝐿₁−𝐿₂1+max𝜃_𝐿1,𝜃_𝐿2>0, (2.23) that is, (𝑢,𝑣)≠(𝑤,𝑧). This completes the proof.

Theorem 2.2. Assume that there exist constants 𝑀,𝑁∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0, and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), max𝑟^∞_{𝑛=𝑛0𝑖𝑛},𝑡_𝑖𝑛,𝑞_𝑖𝑛∶𝑖∈Λ₂<+∞,𝑁<𝑀,𝑝_𝑖𝑛=1,∀𝑛≥𝑛₁,𝑖∈Λ₂. (2.24)(2.25) Then,(a) for each 𝐿∈(𝑁,𝑀), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by the schemes 𝑥_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛+𝛽_𝜇𝐿+∑^∞∑_𝑙=1^{𝑛+2𝑙𝜏1−1}_{()𝜏𝑚=𝑛+2𝑙−11}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑇𝐿+𝛽_𝜇𝐿+∑^∞∑_𝑙=1^{𝑇+2𝑙𝜏1−1}_{𝑚=𝑇𝐿+()𝜏2𝑙−11}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿,𝑦_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛+𝛽_𝜇𝐿+∑^∞∑_𝑙=1^{𝑛+2𝑙𝜏2−1}_{()𝜏𝑚=𝑛+2𝑙−12}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑇𝐿+𝛽_𝜇𝐿+∑^∞∑^𝑇_𝑙=1^{𝐿+2𝑙𝜏2−1}_{𝑚=𝑇𝐿+()𝜏2𝑙−12}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿 (2.26)(2.27) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Proof. (a) Let 𝐿∈(𝑁,𝑀). Now we construct a contraction mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) and prove that its fixed point is a bounded positive solution of System (1.14). Note that (2.24) and (2.25) guarantee that there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| satisfying 𝜃_𝐿=^∞_{𝑚=𝑇𝐿}max𝑡_𝑖𝑚∶𝑖∈Λ₂,_{∞𝑚=𝑇𝐿}max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min{𝑀−𝐿,𝐿−𝑁}. (2.28)(2.29) Define three mappings 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼×𝑙^∞_𝛼,𝑆_1𝐿,𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼 by (2.14), 𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿+∑^∞∑_𝑙=1^{𝑛+2𝑙𝜏1−1}_{𝑚=𝑛+(2𝑙−1)𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛=𝐿+∑^∞∑_𝑙=1^{𝑛+2𝑙𝜏2−1}_{𝑚=𝑛+(2𝑙−1)𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿,𝑆_2𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿, (2.30)(2.31) for all (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀).
Using (2.1), (2.2), (2.14), and (2.28)–(2.31), we deduce that for any (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼}) and (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), 𝑆_𝐿𝑥,𝑦−𝑆_𝐿(𝑢,𝑣)₁=max𝑆_𝑖𝐿𝑥,𝑦−𝑆_𝑖𝐿(𝑢,𝑣)∶𝑖∈Λ₂=maxsup_{𝑛≥𝑇𝐿}𝑆_𝑖𝐿𝑥_𝑛,𝑦_𝑛−𝑆_𝑖𝐿𝑢_𝑛,𝑣_𝑛∶𝑖∈Λ₂≤maxsup_{𝑛≥𝑇𝐿}^∞_𝑙=1^{𝑚=𝑛+2𝑙𝜏1−1}_{𝑚=𝑛+(2𝑙−1)𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚,sup_{𝑛≥𝑇𝐿}^∞_𝑙=1^{𝑚=𝑛+2𝑙𝜏2−1}_{𝑚=𝑛+(2𝑙−1)𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚≤max^∞_{𝑚=𝑇𝐿+𝜏1}𝑡_1𝑚max𝑥_𝑎𝑖𝑚−𝑢_𝑎𝑖𝑚,𝑦_𝑏𝑗𝑚−𝑣_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘,_{∞𝑚=𝑇𝐿+𝜏2}𝑡_2𝑚max𝑥_𝑐𝑖𝑚−𝑢_𝑐𝑖𝑚,𝑦_𝑑𝑗𝑚−𝑣_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘≤max^∞_{𝑚=𝑇𝐿+𝜏1}𝑡_1𝑚,^∞_{𝑚=𝑇𝐿+𝜏2}𝑡_2𝑚‖,≤max‖𝑥−𝑢𝑦−𝑣_{∞𝑚=𝑇𝐿}max𝑡_𝑖𝑚∶𝑖∈Λ₂𝑥,𝑦−(𝑢,𝑣)₁=𝜃_𝐿𝑥,𝑦−(𝑢,𝑣)₁,max𝑆_1𝐿𝑥_𝑛,𝑦_𝑛−𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛−𝐿≤max^∞_𝑙=1^{𝑛+2𝑙𝜏1−1}_{()𝜏𝑚=𝑛+2𝑙−11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,_{∞𝑙=1𝑛+2𝑙𝜏2−1()𝜏𝑚=𝑛+2𝑙−12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚≤max^∞_𝑙=1^{𝑛+2𝑙𝜏1−1}_{()𝜏𝑚=𝑛+2𝑙−11}𝑟_1𝑚+𝑞_1𝑚,^∞_𝑙=1^{𝑛+2𝑙𝜏2−1}_{()𝜏𝑚=𝑛+2𝑙−12}𝑟_2𝑚+𝑞_2𝑚≤max^∞_{𝑚=𝑇𝐿+𝜏1}𝑟_1𝑚+𝑞_1𝑚,^∞_{𝑚=𝑇𝐿+𝜏2}𝑟_2𝑚+𝑞_2𝑚≤^∞_{𝑚=𝑇𝐿}max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min{𝑀−𝐿,𝐿−𝑁},∀𝑛≥𝑇_𝐿, (2.32) which imply (2.16) and (2.17) and which ensure that 𝑆_𝐿 is a self-mapping from 𝐴(𝑁,𝑀) into itself and is a contraction by 𝜃_𝐿∈(0,1). The Banach fixed point theorem means that 𝑆_𝐿 has a unique fixed point (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), that is, 𝑥_𝑛=𝐿+^∞_𝑙=1^{𝑛+2𝑙𝜏1−1}_{𝑚=𝑛+(2𝑙−1)𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑦_𝑛=𝐿+^∞_𝑙=1^{𝑛+2𝑙𝜏2−1}_{𝑚=𝑛+(2𝑙−1)𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿, (2.33) which reveal that 𝑥_𝑛+𝑥_𝑛−𝜏1=𝐿+^∞_𝑙=1^{𝑛+2𝑙𝜏1−1}_{()𝜏𝑚=𝑛+2𝑙−11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚+𝐿+^∞_𝑙=1^{𝑛+(2𝑙−1)𝜏1−1}_{()𝜏𝑚=𝑛+2𝑙−11}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚=2𝐿+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_a1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,∀𝑛≥𝑇_𝐿+𝜏₁,𝑦_𝑛+𝑦_𝑛−𝜏2=𝐿+^∞_𝑙=1^{𝑛+2𝑙𝜏2−1}_{()𝜏𝑚=𝑛+2𝑙−12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚+𝐿+^∞_𝑙=1^{𝑛+(2𝑙−1)𝜏2−1}_{()𝜏𝑚=𝑛+2𝑙−12}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚=2𝐿+^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,∀𝑛≥𝑇_𝐿+𝜏₂, (2.34) which yield that Δ𝑥_𝑛−𝑥_𝑛−𝜏1=−𝑓₁𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛+𝑞_1𝑛,∀𝑛≥𝑇_𝐿+𝜏₁,Δ𝑦_𝑛−𝑦_𝑛−𝜏2=−𝑓₂𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛+𝑞_2𝑛,∀𝑛≥𝑇_𝐿+𝜏₂, (2.35) that is, (𝑥,𝑦) is a bounded positive solution of System (1.14) in 𝐴(𝑁,𝑀).
It follows from (2.14), (2.16), (2.26)–(2.28), (2.30), and (2.31) that 𝑥^𝜇+1,𝑦^𝜇+1−𝑥,𝑦₁=maxsup_{𝑛≥𝑇𝐿}𝑥_𝑛^𝜇+1−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}𝑦_𝑛^𝜇+1−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇𝐿+^∞_𝑙=1^{𝑛+2𝑙𝜏1−1}_{()𝜏𝑚=𝑛+2𝑙−11}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚−𝑥_𝑛+𝛾_𝜇𝛿^𝜇_1𝑛−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇𝐿+^∞_𝑙=1^{𝑛+2𝑙𝜏2−1}_{()𝜏𝑚=𝑛+2𝑙−12}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚−𝑦_𝑛+𝛾_𝜇𝛿^𝜇_2𝑛−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇𝑆_1𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_1𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇𝑆_2𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_2𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇≤1−𝛽_𝜇1−𝜃_𝐿(𝑥^𝜇,𝑦^𝜇)−𝑥,𝑦₁+2𝑀𝛾_𝜇,∀𝜇≥0, (2.36) which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) mean that lim_𝜇→∞‖(𝑥^𝜇,𝑦^𝜇)−(𝑥,𝑦)‖₁=0.
(b) Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. As in the proof of (a), we conclude that for each 𝑖∈Λ₂, there exist a constant 𝜃_𝐿𝑖∈(0,1), a positive integer 𝑇_𝐿𝑖≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼|, and mappings 𝑆_𝐿𝑖,𝑆_1𝐿𝑖,𝑆_2𝐿𝑖 satisfying (2.14) and (2.28)–(2.31), where 𝐿,𝜃_𝐿, and 𝑇_𝐿 are replaced by 𝐿_𝑖,𝜃_𝐿𝑖, and 𝑇_𝐿𝑖, respectively, and the contraction mappings 𝑆_𝐿1 and 𝑆_𝐿2 have the unique fixed points (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼}),(𝑤,𝑧)=({𝑤_𝑛}_{𝑛∈ℤ𝛼},{𝑧_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), respectively, (𝑢,𝑣) and (𝑤,𝑧) are bounded positive solutions of the system (1.14) in 𝐴(𝑁,𝑀). In order to show that System (1.14) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀), we prove only that (𝑢,𝑣)≠(𝑤,𝑧). By virtue of (2.14) and (2.28)–(2.31), we infer that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁=max{‖𝑢−𝑤‖,‖𝑣−𝑧‖}≥maxsup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑢_𝑛−𝑤_𝑛,sup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑣_𝑛−𝑧_𝑛≥max𝐿₁−𝐿₂−^∞_𝑙=1^{max{𝑇𝐿1,𝑇𝐿2}+2𝑙𝜏1−1}_{𝑚=max{𝑇𝐿1,𝑇𝐿2}+(2𝑙−1)𝜏1}𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚−𝑓₁𝑚,𝑤_𝑎1𝑚,…,𝑤_𝑎ℎ𝑚,𝑧_𝑏1𝑚,…,𝑧_𝑏𝑘𝑚,𝐿₁−𝐿₂−^∞_𝑙=1^{max{𝑇𝐿1,𝑇𝐿2}+2𝑙𝜏2−1}_{𝑚=max{𝑇𝐿1,𝑇𝐿2}+(2𝑙−1)𝜏2}𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚−𝑓₂𝑚,𝑤_𝑐1𝑚,…,𝑤_𝑐ℎ𝑚,𝑧_𝑑1𝑚,…,𝑧_𝑑𝑘𝑚≥max𝐿₁−𝐿₂−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2+𝜏1}𝑡_1𝑚max𝑢_𝑎𝑖𝑚−𝑤_𝑎𝑖𝑚,𝑣_𝑏𝑗𝑚−𝑧_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾,𝐿₁−𝐿₂−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2+𝜏2}𝑡_2𝑚max𝑢_𝑐𝑖𝑚−𝑤_𝑐𝑖𝑚,𝑣_𝑑𝑗𝑚−𝑧_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾≥max𝐿₁−𝐿₂−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2+𝜏1}𝑡_1𝑚‖,‖,𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2+𝜏2}𝑡_2𝑚‖,‖≥𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}max𝑡_𝑖𝑚∶𝑖∈Λ₂‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝐿₁−𝐿₂−max𝜃_𝐿1,𝜃_𝐿2‖(𝑢,𝑣)−(𝑤,𝑧)‖₁, (2.37) which yields that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝐿₁−𝐿₂1+max𝜃_𝐿1,𝜃_𝐿2>0, (2.38) that is, (𝑢,𝑣)≠(𝑤,𝑧). This completes the proof.

Theorem 2.3. Assume that there exist constants 𝑀,𝑁∈ℝ⁺⧵{0},𝑝,𝑝∈ℝ⁺, 𝑛₁∈ℕ_𝑛0, and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.24), and 𝑁<1−𝑝−𝑝𝑀,𝑝+𝑝<1,−𝑝≤𝑝_𝑖𝑛≤𝑝,∀𝑛≥𝑛₁,𝑖∈Λ₂. (2.39) Then,(a) for each 𝐿∈(𝑁+𝑝𝑀,𝑀(1−𝑝)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by the schemes 𝑥_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛+𝛽_𝜇𝐿−𝑝_1𝜇𝑥^𝜇_𝑛−𝜏1+∑^∞_𝑚=𝑛𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑇𝐿+𝛽_𝜇𝐿−𝑝_1𝜇𝑥^𝜇_{𝑇𝐿−𝜏1}+∑^∞_{𝑚=𝑇𝐿}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿,𝑦_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛+𝛽_𝜇𝐿−𝑝_2𝜇𝑦^𝜇_𝑛−𝜏2+∑^∞_𝑚=𝑛𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑇𝐿+𝛽_𝜇𝐿−𝑝_2𝜇𝑦^𝜇_{𝑇𝐿−𝜏2}+∑^∞_{𝑚=𝑇𝐿}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿 (2.40)(2.41) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Proof. (a) Let 𝐿∈(𝑁+𝑝𝑀,𝑀(1−𝑝)). Now we construct a contraction mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) and prove that its fixed point is a bounded positive solution of the system (1.14). Notice that (2.24) and (2.39) mean that there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| satisfying 𝜃_𝐿=𝑝+𝑝+^∞_{𝑚=𝑇𝐿}max𝑡_𝑖𝑚∶𝑖∈Λ₂,_{∞𝑚=𝑇𝐿}max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min𝑀1−𝑝−𝐿,𝐿−𝑁−𝑝𝑀. (2.42)(2.43) Define three mappings 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼×𝑙^∞_𝛼,𝑆_1𝐿,𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼 by (2.14), 𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿−𝑝_1𝑛𝑥_𝑛−𝜏1+∑^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛=𝐿−𝑝_2𝑛𝑦_𝑛−𝜏2+∑^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿,𝑆_2𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿, (2.44)(2.45) for all (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀).
It follows from (2.1), (2.2), (2.14), and (2.42)–(2.45) that for any (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼}) and (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), 𝑆_𝐿𝑥,𝑦−𝑆_𝐿(𝑢,𝑣)₁=max𝑆_𝑖𝐿𝑥,𝑦−𝑆_𝑖𝐿(𝑢,𝑣)∶𝑖∈Λ₂=maxsup_{𝑛≥𝑇𝐿}𝑆_𝑖𝐿𝑥_𝑛,𝑦_𝑛−𝑆_𝑖𝐿𝑢_𝑛,𝑣_𝑛∶𝑖∈Λ₂≤maxsup_{𝑛≥𝑇𝐿}𝑝_1𝑛𝑥_𝑛−𝜏1−𝑢_𝑛−𝜏1+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚,sup_{𝑛≥𝑇𝐿}𝑝_2𝑛𝑦_𝑛−𝜏2−𝑣_𝑛−𝜏2+^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚≤max𝑝+𝑝‖𝑥−𝑢‖+^∞_{𝑚=𝑇𝐿}𝑡_1𝑚max𝑥_𝑎𝑙𝑚−𝑢_𝑎𝑙𝑚,𝑦_𝑏𝑗𝑚−𝑣_𝑏𝑗𝑚∶𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,𝑝+𝑝𝑦−𝑣+^∞_{𝑚=𝑇𝐿}𝑡_2𝑚max𝑥_𝑐𝑙𝑚−𝑢_𝑐𝑙𝑚,𝑦_𝑑𝑗𝑚−𝑣_𝑑𝑗𝑚∶𝑙∈Λ_ℎ,𝑗∈Λ_𝑘≤max𝑝+𝑝‖𝑥−𝑢‖+^∞_{𝑚=𝑇𝐿}𝑡_1𝑚‖,,𝑝max‖𝑥−𝑢𝑦−𝑣+𝑝𝑦−𝑣+^∞_{𝑚=𝑇𝐿}𝑡_2𝑚‖,≤𝑝max‖𝑥−𝑢𝑦−𝑣+𝑝+^∞_{𝑚=𝑇𝐿}max𝑡_𝑖𝑚∶𝑖∈Λ₂𝑥,𝑦−(𝑢,𝑣)₁=𝜃_𝐿𝑥,𝑦−(𝑢,𝑣)₁,𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿−𝑝_1𝑛𝑥_𝑛−𝜏1+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚≤𝐿+𝑝𝑀+^∞_{𝑚=𝑇𝐿}𝑟_1𝑚+𝑞_1𝑚≤𝐿+𝑝𝑀+min𝑀1−𝑝−𝐿,𝐿−𝑁−𝑝𝑀≤𝑀,∀𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿−𝑝_1𝑛𝑥_𝑛−𝜏1+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚≥𝐿−𝑝𝑀−^∞_{𝑚=𝑇𝐿}𝑟_1𝑚+𝑞_1𝑚≥𝐿−𝑝𝑀−min𝑀1−𝑝−𝐿,𝐿−𝑁−𝑝𝑀≥𝑁,∀𝑛≥𝑇_𝐿,𝑁≤𝑆_2𝐿𝑥_𝑛,𝑦_𝑛≤𝑀,∀𝑛≥𝑇_𝐿, (2.46) which give (2.16) and (2.17), which together with 𝜃_𝐿∈(0,1) guarantee that 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) is a contraction. Thus the Banach fixed point theorem ensures that 𝑆_𝐿 has a unique fixed point (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), that is, 𝑥_𝑛=𝐿−𝑝_1𝑛𝑥_𝑛−𝜏1+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑦_𝑛=𝐿−𝑝_2𝑛𝑦_𝑛−𝜏2+^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿, (2.47) which yield that Δ𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1=−𝑓₁𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛+𝑞_1𝑛,Δ𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2=𝑓₂𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛+𝑞_2𝑛,∀𝑛≥𝑇_𝐿, (2.48) that is, (𝑥,𝑦) is a bounded positive solution of System (1.14) in 𝐴(𝑁,𝑀).
In light of (2.14), (2.16), (2.40), (2.41), (2.44), and (2.45), we deduce that 𝑥^𝜇+1,𝑦^𝜇+1−𝑥,𝑦₁=maxsup_{𝑛≥𝑇𝐿}𝑥_𝑛^𝜇+1−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}𝑦_𝑛^𝜇+1−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇𝐿−𝑝_1𝑛𝑥_𝑛−𝜏1+^∞_𝑚=𝑛𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚−𝑥_𝑛+𝛾_𝜇𝛿^𝜇_1𝑛−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇𝐿−𝑝_2𝑛𝑦_𝑛−𝜏2+^∞_𝑚=𝑛𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚−𝑦_𝑛+𝛾_𝜇𝛿^𝜇_2𝑛−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇−𝑥+𝛽_𝜇𝑆_1𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_1𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇−𝑦+𝛽_𝜇𝑆_2𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_2𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇≤1−𝛽_𝜇1−𝜃_𝐿𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇,∀𝜇≥0, (2.49) which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) imply that lim_𝜇→∞‖(𝑥^𝜇,𝑦^𝜇)−(𝑥,𝑦)‖₁=0.
(b) Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. Similar to the proof of (a), we know that for each 𝑖∈Λ₂, there exist a constant 𝜃_𝐿𝑖∈(0,1), a positive integer 𝑇_𝐿𝑖≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| and mappings 𝑆_𝐿𝑖,𝑆_1𝐿𝑖,𝑆_2𝐿𝑖 satisfying (2.14) and (2.42)–(2.45), where 𝐿,𝜃_𝐿, and 𝑇_𝐿 are replaced by 𝐿_𝑖,𝜃_𝐿𝑖 and 𝑇_𝐿𝑖, respectively, and the contraction mappings 𝑆_𝐿1 and 𝑆_𝐿2 have the unique fixed points (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼}),(𝑤,𝑧)=({𝑤_𝑛}_{𝑛∈ℤ𝛼},{𝑧_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), which are bounded positive solutions of System (1.14) in 𝐴(𝑁,𝑀), respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀), we prove only that (𝑢,𝑣)≠(𝑤,𝑧). In terms of (2.14) and (2.42)–(2.45), we get that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁=max{‖𝑢−𝑤‖,‖𝑣−𝑧‖}≥maxsup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑢_𝑛−𝑤_𝑛,sup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑣_𝑛−𝑧_𝑛≥maxsup_{𝑛≥max{𝑇𝐿1,𝑇𝐿2}}𝐿₁−𝐿₂−𝑝_1𝑛𝑢_𝑛−𝜏1−𝑤_𝑛−𝜏1−^∞_𝑚=𝑛𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚−𝑓₁𝑚,𝑤_𝑎1𝑚,…,𝑤_𝑎ℎ𝑚,𝑧_𝑏1𝑚,…,𝑧_𝑏𝑘𝑚,sup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝐿₁−𝐿₂−𝑝_2𝑛𝑣_𝑛−𝜏2−𝑧_𝑛−𝜏2−^∞_𝑚=𝑛𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚−𝑓₂𝑚,𝑤_𝑐1𝑚,…,𝑤_𝑐ℎ𝑚,𝑧_𝑑1𝑚,…,𝑧_𝑑𝑘𝑚≥max𝐿₁−𝐿₂−𝑝+𝑝‖‖−𝑢−𝑤_{∞𝑇𝑚=max𝐿1,𝑇L2}𝑡_1𝑚max𝑢_𝑎𝑖𝑚−𝑤_𝑎𝑖𝑚,𝑣_𝑏𝑗𝑚−𝑧_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾,𝐿₁−𝐿₂−𝑝+𝑝‖‖−𝑣−𝑧_{∞𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_2𝑚max𝑢_𝑐𝑖𝑚−𝑤_𝑐𝑖𝑚,𝑣_𝑑𝑗𝑚−𝑧_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾≥max𝐿₁−𝐿₂−𝑝+𝑝‖𝑢−𝑤‖−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_1𝑚‖,‖,𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂−𝑝+𝑝‖𝑣−𝑧‖−^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_2𝑚‖,‖≥𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂−𝑝+𝑝+^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}max𝑡_𝑖𝑚∶𝑖∈Λ₂‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝐿₁−𝐿₂−max𝜃_𝐿1,𝜃_𝐿2‖(𝑢,𝑣)−(𝑤,𝑧)‖₁, (2.50) which yields that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝐿₁−𝐿₂1+max𝜃_𝐿1,𝜃_𝐿2>0, (2.51) that is, (𝑢,𝑣)≠(𝑤,𝑧). This completes the proof.

Theorem 2.4. Assume that there exist constants 𝑀,𝑁,𝑝,𝑝∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.24), and 0<𝑁𝑝−1<𝑀𝑝−1,−𝑝≤𝑝_𝑖𝑛≤−𝑝,∀𝑛≥𝑛₁,𝑖∈Λ₂. (2.52) Then,(a) for each 𝐿∈(𝑁(𝑝−1),𝑀(𝑝−1)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by the schemes 𝑥_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛+𝛽_𝜇−𝐿𝑝_1𝑛+𝜏1−𝑥^𝜇_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1×∑^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑇𝐿+𝛽_𝜇−𝐿𝑝_{1𝑇𝐿+𝜏1}−𝑥^𝜇_{𝑇𝐿+𝜏1}𝑝_{1𝑇𝐿+𝜏1}+1𝑝_{1𝑇𝐿+𝜏1}×∑^∞_{𝑚=𝑇𝐿+𝜏1}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿,𝑦_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛+𝛽_𝜇−𝐿𝑝_2𝑛+𝜏2−𝑦^𝜇_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2×∑^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑇𝐿+𝛽_𝜇−𝐿𝑝_{2𝑇𝐿+𝜏2}−𝑦^𝜇_{𝑇𝐿+𝜏2}𝑝_{2𝑇𝐿+𝜏2}+1𝑝_{2𝑇𝐿+𝜏2}×∑^∞_{𝑚=𝑇𝐿+𝜏2}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿 (2.53)(2.54) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1] and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Proof. (a) Let 𝐿∈(𝑁(𝑝−1),𝑀(𝑝−1)). Now we construct a contraction mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) and prove that its fixed point is a bounded positive solution of the system (1.14). Observe that (2.24) and (2.52) guarantee that there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| satisfying 𝜃_𝐿=1𝑝1+^∞_𝑚=T𝐿max𝑡_𝑖𝑚∶𝑖∈Λ₂,_{∞𝑚=𝑇𝐿}max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min𝑀𝑝−1−𝐿,𝑝𝐿+𝑁𝑝−𝑁. (2.55)(2.56) Define three mappings 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼×𝑙^∞_𝛼,𝑆_1𝐿,𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼 by (2.14), 𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=−𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_𝑚=∑_𝑛+𝜏1𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛=−𝐿𝑝_2𝑛+𝜏2−𝑦_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2^∞_𝑚=∑_𝑛+𝜏2𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿,𝑆_2𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿, (2.57)(2.58) for all (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀).
It follows from (2.1), (2.2), (2.14), and (2.55)–(2.58) that for any (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼}) and (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀)𝑆_𝐿𝑥,𝑦−𝑆_𝐿(𝑢,𝑣)₁=max𝑆_𝑖𝐿𝑥,𝑦−𝑆_𝑖𝐿(𝑢,𝑣)∶𝑖∈Λ₂=maxsup_{𝑛≥𝑇𝐿}𝑆_𝑖𝐿𝑥_𝑛,𝑦_𝑛−𝑆_𝑖𝐿𝑢_𝑛,𝑣_𝑛∶𝑖∈Λ₂≤maxsup_{𝑛≥𝑇𝐿}1𝑝_1𝑛+𝜏1𝑥_𝑛+𝜏1−𝑢_𝑛+𝜏1+^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚,sup_{𝑛≥𝑇𝐿}1𝑝_2𝑛+𝜏2𝑦_𝑛+𝜏2−𝑣_𝑛+𝜏2+^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚≤1𝑝max‖𝑥−𝑢‖+^∞_{𝑚=𝑇𝐿}𝑡_1𝑚max𝑥_𝑎𝑖𝑚−𝑢_𝑎𝑖𝑚,𝑦_𝑏𝑗𝑚−𝑣_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘,𝑦−𝑣+^∞_{𝑚=𝑇𝐿}𝑡_2𝑚max𝑥_𝑐𝑖𝑚−𝑢_𝑐𝑖𝑚,𝑦_𝑑𝑗𝑚−𝑣_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘≤1𝑝max‖𝑥−𝑢‖+^∞_{𝑚=𝑇𝐿}𝑡_1𝑚‖,,+max‖𝑥−𝑢𝑦−𝑣𝑦−𝑣^∞_{𝑚=𝑇𝐿}𝑡_2𝑚‖,≤1max‖𝑥−𝑢𝑦−𝑣𝑝1+^∞_{𝑚=𝑇𝐿}max𝑡_𝑖𝑚∶𝑖∈Λ₂𝑥,𝑦−(𝑢,𝑣)₁=𝜃_𝐿𝑥,𝑦−(𝑢,𝑣)₁,𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=−𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚≤𝐿𝑝+𝑀𝑝+1𝑝^∞_{𝑚=𝑇𝐿+𝜏1}𝑟_1𝑚+𝑞_1𝑚≤𝐿𝑝+𝑀𝑝+1𝑝min𝑀𝑝−1−𝐿,𝑝𝐿+𝑁𝑝−𝑁≤𝑀,∀𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=−𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚≥𝐿𝑝+𝑁𝑝−1𝑝^∞_{𝑚=𝑇𝐿+𝜏1}𝑟_1𝑚+𝑞_1𝑚≥𝐿𝑝+𝑁𝑝−1𝑝min𝑀𝑝−1−𝐿,𝑝𝐿+𝑁𝑝−𝑁≥𝑁,∀𝑛≥𝑇_𝐿,𝑁≤𝑆_2𝐿𝑥_𝑛,𝑦_𝑛≤𝑀,∀𝑛≥𝑇_𝐿, (2.59) which yield (2.16) and (2.17), which together with 𝜃_𝐿∈(0,1) ensure that 𝑆_𝐿 is a contraction in 𝐴(𝑁,𝑀). By the Banach fixed point theorem, we deduce that 𝑆_𝐿 has a unique fixed point (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), that is, 𝑥_𝑛=−𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑦_𝑛=−𝐿𝑝_2𝑛+𝜏2−𝑦_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿, (2.60) which yield that 𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1=−𝐿+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿+𝜏₁,𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2=−𝐿+^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿+𝜏₂,Δ𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1=−𝑓₁𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛+𝑞_1𝑛,∀𝑛≥𝑇_𝐿+𝜏₁,Δ𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2=−𝑓₂𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛+𝑞_2𝑛,∀𝑛≥𝑇_𝐿+𝜏₂, (2.61) that is, (𝑥,𝑦) is a bounded positive solution of the system (1.14) in 𝐴(𝑁,𝑀).
In light of (2.14), (2.16), (2.53)–(2.55), (2.57), and (2.58), we get that 𝑥^𝜇+1,𝑦^𝜇+1−𝑥,𝑦₁=maxsup_{𝑛≥𝑇𝐿}𝑥_𝑛^𝜇+1−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}𝑦_𝑛^𝜇+1−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇×−𝐿𝑝_1𝑛+𝜏1−𝑥^𝜇_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_𝑚=𝑛𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚−𝑥_𝑛+𝛾_𝜇𝛿^𝜇_1𝑛−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇×−𝐿𝑝_2𝑛+𝜏2−𝑦^𝜇_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2^∞_𝑚=𝑛𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚−𝑦_𝑛+𝛾_𝜇𝛿^𝜇_2𝑛−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇−𝑥+𝛽_𝜇𝑆_1𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_1𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇−𝑦+𝛽_𝜇𝑆_2𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_2𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇≤1−𝛽_𝜇1−𝜃_𝐿𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇,∀𝜇≥0, (2.62) which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) imply that lim_𝜇→∞‖(𝑥^𝜇,𝑦^𝜇)−(𝑥,𝑦)‖₁=0.
(b) Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. Similar to the proof of (a), we conclude that for each 𝑖∈Λ₂, there exist a constant 𝜃_𝐿𝑖∈(0,1), a positive integer 𝑇_𝐿𝑖≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼|, and mappings 𝑆_𝐿𝑖,𝑆_1𝐿𝑖,𝑆_2𝐿𝑖 satisfying (2.14) and (2.55)–(2.58), where 𝐿,𝜃_𝐿, and 𝑇_𝐿 are replaced by 𝐿_𝑖,𝜃_𝐿𝑖, and 𝑇_𝐿𝑖, respectively, and the contraction mappings 𝑆_𝐿1 and 𝑆_𝐿2 have the unique fixed points (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼}),(𝑤,𝑧)=({𝑤_𝑛}_{𝑛∈ℤ𝛼},{𝑧_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), which are bounded positive solutions of System (1.14) in 𝐴(𝑁,𝑀), respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀), we prove only that (𝑢,𝑣)≠(𝑤,𝑧). By virtue of (2.14) and (2.55)–(2.58), we get that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁=max{‖𝑢−𝑤‖,‖𝑣−𝑧‖}≥maxsup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑢_𝑛−𝑤_𝑛,sup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑣_𝑛−𝑧_𝑛≥maxsup_{𝑛≥max{𝑇𝐿1,𝑇𝐿2}}𝐿₁−𝐿₂𝑝_1𝑛+𝜏1−𝑢_𝑛+𝜏1−𝑤_𝑛+𝜏1𝑝_1𝑛+𝜏1−1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚−𝑓₁𝑚,𝑤_𝑎1𝑚,…,𝑤_𝑎ℎ𝑚,𝑧_𝑏1𝑚,…,𝑧_𝑏𝑘𝑚,sup_{𝑛≥max{𝑇𝐿1,𝑇𝐿2}}𝐿₁−𝐿₂𝑝_2𝑛+𝜏2−𝑢_𝑛+𝜏2−𝑤_𝑛+𝜏2𝑝_2𝑛+𝜏2−1𝑝_2𝑛+𝜏2×^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚−𝑓₂𝑚,𝑤_𝑐1𝑚,…,𝑤_𝑐ℎ𝑚,𝑧_𝑑1𝑚,…,𝑧_𝑑𝑘𝑚𝐿≥max¹−𝐿₂𝑝−‖𝑢−𝑤‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_1𝑚max𝑢_𝑎𝑖𝑚−𝑤_𝑎𝑖𝑚,𝑣_𝑏𝑗𝑚−𝑧_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾,𝐿₁−𝐿₂𝑝−‖𝑣−𝑧‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_2𝑚max𝑢_𝑐𝑖𝑚−𝑤_𝑐𝑖𝑚,𝑣_𝑑𝑗𝑚−𝑧_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝐾𝐿≥max¹−𝐿₂𝑝−‖𝑢−𝑤‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_1𝑚‖,‖,𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂𝑝−‖𝑣−𝑧‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_2𝑚‖,‖≥𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂𝑝−1𝑝1+max𝜃_𝐿1,𝜃_𝐿2‖(𝑢,𝑣)−(𝑤,𝑧)‖₁, (2.63) which yields that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝑝𝐿₁−𝐿₂𝑝𝑝+max𝜃_𝐿1,𝜃_𝐿2>0, (2.64) that is, (𝑢,𝑣)≠(𝑤,𝑧). This completes the proof.

Theorem 2.5. Assume that there exist constants 𝑀,𝑁,𝑝,𝑝∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.24), and 𝑁𝑝²−𝑝𝑝<𝑀𝑝²−𝑝𝑝,1<𝑝≤𝑝_𝑖𝑛≤𝑝<𝑝²,∀𝑛≥𝑛₁,𝑖∈Λ₂. (2.65) Then,(a) for each 𝐿∈(𝑝(𝑀/𝑝+𝑁),𝑝(𝑁/𝑝+𝑀)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by the schemes 𝑥_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛+𝛽_𝜇𝐿𝑝_1𝑛+𝜏1−𝑥^𝜇_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1×∑^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑇𝐿+𝛽_𝜇𝐿𝑝_{1𝑇𝐿+𝜏1}−𝑥^𝜇_{𝑇𝐿+𝜏1}𝑝_{1𝑇𝐿+𝜏1}+1𝑝_{1𝑇𝐿+𝜏1}×∑^∞_{𝑚=𝑇𝐿+𝜏1}𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚+𝛾_𝜇𝛿^𝜇_1𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿,𝑦_𝑛^𝜇+1=1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛+𝛽_𝜇𝐿𝑝_2𝑛+𝜏2−𝑦^𝜇_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2×∑^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑛,𝜇≥0,𝑛≥𝑇_𝐿,1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑇𝐿+𝛽_𝜇𝐿𝑝_{2𝑇𝐿+𝜏2}−𝑦^𝜇_{𝑇𝐿+𝜏2}𝑝_{2𝑇𝐿+𝜏2}+1𝑝_{2𝑇𝐿+𝜏2}×∑^∞_{𝑚=𝑇𝐿+𝜏2}𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚+𝛾_𝜇𝛿^𝜇_2𝑇𝐿,𝜇≥0,𝛼≤𝑛<𝑇_𝐿 (2.66)(2.67) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Proof. (a) Let 𝐿∈(𝑝(𝑀/𝑝+𝑁),𝑝(𝑁/𝑝+𝑀)). Now we construct a contraction mapping 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝐴(𝑁,𝑀) and prove that its fixed point is a bounded positive solution of System (1.14). Observe that (2.24) and (2.65) imply that there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| satisfying 𝜃_𝐿=1𝑝1+^∞_{𝑚=𝑇𝐿}max𝑡_𝑖𝑚∶𝑖∈Λ₂,_{∞𝑚=𝑇𝐿}max𝑟_𝑖𝑚+𝑞_𝑖𝑚∶𝑖∈Λ₂<min𝑁𝑝𝑝+𝑀𝑝−𝐿,𝐿𝑝𝑝−𝑀−𝑁𝑝. (2.68)(2.69) Define three mappings 𝑆_𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼×𝑙^∞_𝛼,𝑆_1𝐿,𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛼 by (2.14), 𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_𝑚=∑_𝑛+𝜏1𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿,𝑆_2𝐿𝑥_𝑛,𝑦_𝑛=𝐿𝑝_2𝑛+𝜏2−𝑦_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2^∞_𝑚=∑_𝑛+𝜏2𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿,𝑆_2𝐿𝑥_𝑇𝐿,𝑦_𝑇𝐿,𝛼≤𝑛<𝑇_𝐿, (2.70)(2.71) for all (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀).
Using (2.1), (2.2), (2.14), (2.65), and (2.68)–(2.71), we deduce that for any (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼}) and (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), 𝑆_𝐿𝑥,𝑦−𝑆_𝐿(𝑢,𝑣)₁=max𝑆_𝑖𝐿𝑥,𝑦−𝑆_𝑖𝐿(𝑢,𝑣)∶𝑖∈Λ₂=maxsup_{𝑛≥𝑇𝐿}𝑆_𝑖𝐿𝑥_𝑛,𝑦_𝑛−𝑆_𝑖𝐿𝑢_𝑛,𝑣_𝑛∶𝑖∈Λ₂≤maxsup_{𝑛≥𝑇𝐿}1𝑝_1𝑛+𝜏1𝑥_𝑛+𝜏1−𝑢_𝑛+𝜏1+^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚,sup_{𝑛≥𝑇𝐿}1𝑝_2𝑛+𝜏2𝑦_𝑛+𝜏2−𝑣_𝑛+𝜏2+^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚≤1𝑝max‖𝑥−𝑢‖+^∞_{𝑚=𝑇𝐿}𝑡_1𝑚max𝑥_𝑎𝑖𝑚−𝑢_𝑎𝑖𝑚,𝑦_𝑏𝑗𝑚−𝑣_𝑏𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘,𝑦−𝑣+^∞_{𝑚=𝑇𝐿}𝑡_2𝑚max𝑥_𝑐𝑖𝑚−𝑢_𝑐𝑖𝑚,𝑦_𝑑𝑗𝑚−𝑣_𝑑𝑗𝑚∶𝑖∈Λ_ℎ,𝑗∈Λ_𝑘≤1𝑝max‖𝑥−𝑢‖+^∞_{𝑚=𝑇𝐿}𝑡_1𝑚max‖𝑥−𝑢‖,𝑦−𝑣,𝑦−𝑣+^∞_{𝑚=𝑇𝐿}𝑡_2𝑚‖,≤1max‖𝑥−𝑢𝑦−𝑣𝑝1+^∞_{𝑚=𝑇𝐿}max𝑡_1𝑚,𝑡_2𝑚𝑥,𝑦−(𝑢,𝑣)₁=𝜃_𝐿𝑥,𝑦−(𝑢,𝑣)₁,𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚≤𝐿𝑝−𝑁𝑝+1𝑝^∞_{𝑚=𝑇𝐿+𝜏1}𝑟_1𝑚+𝑞_1𝑚≤𝐿𝑝−𝑁𝑝+1𝑝min𝑁𝑝𝑝+𝑀𝑝−𝐿,𝐿𝑝𝑝−𝑀−𝑁𝑝≤𝑀,∀𝑛≥𝑇_𝐿,𝑆_1𝐿𝑥_𝑛,𝑦_𝑛=𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚≥𝐿𝑝−𝑀𝑝−1𝑝^∞_{𝑚=𝑇𝐿+𝜏1}𝑟_1𝑚+𝑞_1𝑚≥𝐿𝑝−𝑀𝑝−1𝑝min𝑁𝑝𝑝+𝑀𝑝−𝐿,𝐿𝑝𝑝−𝑀−𝑁𝑝≥𝑁,∀𝑛≥𝑇_𝐿,𝑁≤𝑆_2𝐿𝑥_𝑛,𝑦_𝑛≤𝑀,∀𝑛≥𝑇_𝐿, (2.72) which yield (2.16) and (2.17), and which guarantee that 𝑆_𝐿 is a self-mapping from 𝐴(𝑁,𝑀) into itself and is a contraction by 𝜃_𝐿∈(0,1). Thus the Banach fixed point theorem means that 𝑆_𝐿 has a unique fixed point (𝑥,𝑦)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), that is, 𝑥_𝑛=𝐿𝑝_1𝑛+𝜏1−𝑥_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿,𝑦_𝑛=𝐿𝑝_2𝑛+𝜏2−𝑦_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿, (2.73) which yield that 𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1=𝐿+^∞_𝑚=𝑛𝑓₁𝑚,𝑥_𝑎1𝑚,…,𝑥_𝑎ℎ𝑚,𝑦_𝑏1𝑚,…,𝑦_𝑏𝑘𝑚−𝑞_1𝑚,𝑛≥𝑇_𝐿+𝜏₁,𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2=𝐿+^∞_𝑚=𝑛𝑓₂𝑚,𝑥_𝑐1𝑚,…,𝑥_𝑐ℎ𝑚,𝑦_𝑑1𝑚,…,𝑦_𝑑𝑘𝑚−𝑞_2𝑚,𝑛≥𝑇_𝐿+𝜏₂,Δ𝑥_𝑛+𝑝_1𝑛𝑥_𝑛−𝜏1=−𝑓₁𝑛,𝑥_𝑎1𝑛,…,𝑥_𝑎ℎ𝑛,𝑦_𝑏1𝑛,…,𝑦_𝑏𝑘𝑛+𝑞_1𝑛,∀𝑛≥𝑇_𝐿+𝜏₁,Δ𝑦_𝑛+𝑝_2𝑛𝑦_𝑛−𝜏2=−𝑓₂𝑛,𝑥_𝑐1𝑛,…,𝑥_𝑐ℎ𝑛,𝑦_𝑑1𝑛,…,𝑦_𝑑𝑘𝑛+𝑞_2𝑛,∀𝑛≥𝑇_𝐿+𝜏₂, (2.74) that is, (𝑥,𝑦) is a bounded positive solution of the system (1.14) in 𝐴(𝑁,𝑀).
In view of (2.14), (2.16), (2.65)–(2.67), (2.70), and (2.71), we gain that 𝑥^𝜇+1,𝑦^𝜇+1−𝑥,𝑦₁=maxsup_{𝑛≥𝑇𝐿}𝑥_𝑛^𝜇+1−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}𝑦_𝑛^𝜇+1−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇_𝑛−𝑥_𝑛+𝛽_𝜇×𝐿𝑝_1𝑛+𝜏1−𝑥^𝜇_𝑛+𝜏1𝑝_1𝑛+𝜏1+1𝑝_1𝑛+𝜏1^∞_𝑚=𝑛𝑓₁𝑚,𝑥^𝜇_𝑎1𝑚,…,𝑥^𝜇_𝑎ℎ𝑚,𝑦^𝜇_𝑏1𝑚,…,𝑦^𝜇_𝑏𝑘𝑚−𝑞_1𝑚−𝑥_𝑛+𝛾_𝜇𝛿^𝜇_1𝑛−𝑥_𝑛,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇_𝑛−𝑦_𝑛+𝛽_𝜇×𝐿𝑝_2𝑛+𝜏2−𝑦^𝜇_𝑛+𝜏2𝑝_2𝑛+𝜏2+1𝑝_2𝑛+𝜏2^∞_𝑚=𝑛𝑓₂𝑚,𝑥^𝜇_𝑐1𝑚,…,𝑥^𝜇_𝑐ℎ𝑚,𝑦^𝜇_𝑑1𝑚,…,𝑦^𝜇_𝑑𝑘𝑚−𝑞_2𝑚−𝑦_𝑛+𝛾_𝜇𝛿^𝜇_2𝑛−𝑦_𝑛≤maxsup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑥^𝜇−𝑥+𝛽_𝜇𝑆_1𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_1𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇,sup_{𝑛≥𝑇𝐿}1−𝛽_𝜇−𝛾_𝜇𝑦^𝜇−𝑦+𝛽_𝜇𝑆_2𝐿𝑥^𝜇_𝑛,𝑦^𝜇_𝑛−𝑆_2𝐿𝑥_𝑛,𝑦_𝑛+2𝑀𝛾_𝜇≤1−𝛽_𝜇1−𝜃_𝐿𝑥^𝜇,𝑦^𝜇−𝑥,𝑦₁+2𝑀𝛾_𝜇,∀𝜇≥0, (2.75) which implies (2.7). Thus, Lemma 1.1 and (2.7)–(2.9) imply that lim_𝜇→∞‖(𝑥^𝜇,𝑦^𝜇)−(𝑥,𝑦)‖₁=0.
(b) Let 𝐿₁,𝐿₂∈(𝑁,𝑀) and 𝐿₁≠𝐿₂. Similar to the proof of (a), we conclude that for each 𝑖∈Λ₂, there exist a constant 𝜃_𝐿𝑖∈(0,1), a positive integer 𝑇_𝐿𝑖≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼|, and mappings 𝑆_𝐿𝑖,𝑆_1𝐿𝑖,𝑆_2𝐿𝑖 satisfying (2.14) and (2.68)–(2.71), where 𝐿,𝜃_𝐿, and 𝑇_𝐿 are replaced by 𝐿_𝑖,𝜃_𝐿𝑖, and 𝑇_𝐿𝑖, respectively, and the contraction mappings 𝑆_𝐿1 and 𝑆_𝐿2 have the unique fixed points (𝑢,𝑣)=({𝑢_𝑛}_{𝑛∈ℤ𝛼},{𝑣_𝑛}_{𝑛∈ℤ𝛼}),(𝑤,𝑧)=({𝑥_𝑛}_{𝑛∈ℤ𝛼},{𝑦_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), which are bounded positive solutions of System (1.14) in 𝐴(𝑁,𝑀), respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀), we prove only that (𝑢,𝑣)≠(𝑤,𝑧). In light of (2.14) and (2.68)–(2.71), we have ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁=max{‖𝑢−𝑤‖,‖𝑣−𝑧‖}≥maxsup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑢_𝑛−𝑤_𝑛,sup_{𝑇𝑛≥max𝐿1,𝑇𝐿2}𝑣_𝑛−𝑧_𝑛≥maxsup_{𝑛≥max{𝑇𝐿1,𝑇𝐿2}}𝐿₁−𝐿₂𝑝_1𝑛+𝜏1−𝑢_𝑛+𝜏1−𝑤_𝑛+𝜏1𝑝_1𝑛+𝜏1−1𝑝_1𝑛+𝜏1^∞_{𝑚=𝑛+𝜏1}𝑓₁𝑚,𝑢_𝑎1𝑚,…,𝑢_𝑎ℎ𝑚,𝑣_𝑏1𝑚,…,𝑣_𝑏𝑘𝑚−𝑓₁𝑚,𝑤_𝑎1𝑚,…,𝑤_𝑎ℎ𝑚,𝑧_𝑏1𝑚,…,𝑧_𝑏𝑘𝑚,sup_{𝑛≥max{𝑇𝐿1,𝑇𝐿2}}𝐿₁−𝐿₂𝑝_2𝑛+𝜏2−𝑢_𝑛+𝜏2−𝑤_𝑛+𝜏2𝑝_2𝑛+𝜏2−1𝑝_2𝑛+𝜏2×^∞_{𝑚=𝑛+𝜏2}𝑓₂𝑚,𝑢_𝑐1𝑚,…,𝑢_𝑐ℎ𝑚,𝑣_𝑑1𝑚,…,𝑣_𝑑𝑘𝑚−𝑓₂𝑚,𝑤_𝑐1𝑚,…,𝑤_𝑐ℎ𝑚,𝑧_𝑑1𝑚,…,𝑧_𝑑𝑘𝑚𝐿≥max¹−𝐿₂𝑝−‖𝑢−𝑤‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_1𝑚max𝑢_𝑎𝑙𝑚−𝑤_𝑎𝑙𝑚,𝑣_𝑏𝑗𝑚−𝑧_𝑏𝑗𝑚∶𝑙∈Λ_ℎ,𝑗∈Λ_𝐾,𝐿₁−𝐿₂𝑝−‖𝑣−𝑧‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_2𝑚max𝑢_𝑐𝑙𝑚−𝑤_𝑐𝑙𝑚,𝑣_𝑑𝑗𝑚−𝑧_𝑑𝑗𝑚∶𝑙∈Λ_ℎ,𝑗∈Λ_𝐾𝐿≥max¹−𝐿₂𝑝−‖𝑢−𝑤‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_1𝑚‖,‖,𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂𝑝−‖𝑣−𝑧‖𝑝−1𝑝^∞_{𝑇𝑚=max𝐿1,𝑇𝐿2}𝑡_2𝑚‖,‖≥𝐿max{‖𝑢−𝑤𝑣−𝑧‖}₁−𝐿₂𝑝−1𝑝1+max𝜃_𝐿1,𝜃_𝐿2‖(𝑢,𝑣)−(𝑤,𝑧)‖₁, (2.76) which yields that ‖(𝑢,𝑣)−(𝑤,𝑧)‖₁≥𝑝𝐿₁−𝐿₂𝑝𝑝+max𝜃_𝐿1,𝜃_𝐿2>0, (2.77) that is, (𝑢,𝑣)≠(𝑤,𝑧). This completes the proof.

Remark 2.6. Let (𝑎₁,𝑏₁) and (𝑎₂,𝑏₂) be two arbitrary intervals in ℝ. It is easy to see that 𝑎₁,𝑏₁∩𝑎₂,𝑏₂≠∅⟺max𝑎₁,𝑎₂<min𝑏₁,𝑏₂. (2.78)

From Remark 2.6 and Theorems 2.1–2.5, we can obtain the following.

Theorem 2.7. Assume that there exist constants 𝑀,𝑁∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), 𝑛max𝑟^∞_{𝑛=𝑛01𝑛},𝑡_1𝑛,𝑞_1𝑛<+∞,_{∞𝑛=𝑛0}max𝑟_2𝑛,𝑡_2𝑛,𝑞_2𝑛<+∞,𝑁<𝑀,𝑝_1𝑛=−1,∀𝑛≥𝑛₁,𝑝_2𝑛=1,∀𝑛≥𝑛₁. (2.79)(2.80)(2.81)(2.82) Then,(a)for each 𝐿∈(𝑁,𝑀), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.5) and (2.27) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1] and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Theorem 2.8. Assume that there exist constants 𝑀,𝑁∈ℝ⁺⧵{0},𝑝,𝑝∈ℝ⁺, 𝑛₁∈ℕ_𝑛0, and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.79)–(2.81), 𝑁<1−𝑝−𝑝𝑀,𝑝+𝑝<1,−𝑝≤𝑝_2𝑛≤𝑝,∀𝑛≥𝑛₁. (2.83) Then,(a)for each 𝐿∈(𝑁,𝑀)∩(𝑁+𝑝𝑀,𝑀(1−𝑝)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.5) and (2.41) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1] and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Theorem 2.9. Assume that there exist constants 𝑀,𝑁,𝑝,𝑝∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0} and {𝑡_2𝑛}_n∈ℕ𝑛0, satisfying (2.1), (2.2), (2.79)–(2.81), 0<𝑁𝑝−1<𝑀𝑝−1,−𝑝≤𝑝_2𝑛≤−𝑝,∀𝑛≥𝑛₁𝑝,max𝑁,𝑁−1<min𝑀,𝑀𝑝−1. (2.84)(2.85) Then,(a)for each 𝐿∈(𝑁,𝑀)∩(𝑁(𝑝−1),𝑀(𝑝−1)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.5) and (2.54) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1] and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Theorem 2.10. Assume that there exist constants 𝑀,𝑁,𝑀₀,𝑁₀,𝑝₀,𝑝₀∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2) with 𝑖=1, (2.79)–(2.81), 𝑓₂𝑛,𝑢₁,…,𝑢_ℎ,𝑣₁,…,𝑣_𝑘≤𝑟_2𝑛,∀𝑛,𝑢_𝑙,𝑣_𝑗∈ℕ_𝑛1×𝑁₀,𝑀₀²,𝑙∈Λ_ℎ,𝑗∈Λ_𝑘𝑓₂,𝑛,𝑢₁,…,𝑢_ℎ,𝑣₁,…,𝑣_𝑘−𝑓₂𝑛,𝑤₁,…,𝑤_ℎ,𝑧₁,…,𝑧_𝑘≤𝑡_2𝑛max𝑢_𝑙−𝑤_𝑙,𝑣_𝑗−𝑧_𝑗∶𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,∀𝑛,𝑢_𝑙,𝑤_𝑙,𝑣_𝑗,𝑧_𝑗∈ℕ_𝑛1×𝑁₀,𝑀₀⁴,𝑙∈Λ_ℎ,𝑗∈Λ_𝑘,𝑁₀𝑝²₀−𝑝₀𝑝₀<𝑀₀𝑝²₀−𝑝₀𝑝₀,1<𝑝₀≤𝑝_2𝑛≤𝑝₀<𝑝²₀,∀𝑛≥𝑛₁,max𝑁,𝑝₀𝑀₀𝑝₀+𝑁₀<min𝑀,𝑝₀𝑁₀𝑝₀+𝑀₀. (2.86)(2.87)(2.88)(2.89) Then,(a)for each 𝐿∈(𝑁,𝑀)∩(𝑝₀(𝑀₀/𝑝₀+𝑁₀),𝑝₀(𝑁₀/𝑝₀+𝑀₀)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.5) and (2.67) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀,𝑁₀,𝑀₀) are arbitrary sequences with (2.8) and (2.9),(b)the system (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Theorem 2.11. Assume that there exist constants 𝑀,𝑁∈ℝ⁺⧵{0},𝑝,𝑝∈ℝ⁺, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.24), (2.83) and 𝑁<𝑀,𝑝_1𝑛=1,∀𝑛≥𝑛₁. (2.90) Then,(a)for each 𝐿∈(𝑁,𝑀)∩(𝑁+𝑝𝑀,𝑀(1−𝑝)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.26) and (2.41) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Theorem 2.12. Assume that there exist constants 𝑀,𝑁,𝑝,𝑝∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.24), (2.84), (2.85), and (2.90). Then,(a)for each 𝐿∈(𝑁,𝑀)∩(𝑁(𝑝−1),𝑀(𝑝−1)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.26) and (2.54) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Theorem 2.13. Assume that there exist 𝑀,𝑁,𝑀₀,𝑁₀,𝑝₀,𝑝₀∈ℝ⁺⧵{0}, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0}, {𝑟_2𝑛}_{𝑛∈ℕ𝑛0}, {𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2) with 𝑖=1, (2.24), and (2.86)–(2.90). Then,(a)for each 𝐿∈(𝑁,𝑀)∩(𝑝₀(𝑀₀/𝑝₀+𝑁₀),𝑝₀(𝑁₀/𝑝₀+𝑀₀)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.26) and (2.67) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀,𝑁₀,𝑀₀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Theorem 2.14. Assume that there exist constants 𝑀,𝑁,𝑝,𝑝∈ℝ⁺⧵{0},𝑝₁,𝑝₁∈ℝ⁺, 𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0},{𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2), (2.24), (2.84), 𝑁<1−𝑝₁−𝑝₁𝑀,𝑝₁+𝑝₁<1,−𝑝₁≤𝑝_1𝑛≤𝑝₁,∀𝑛≥𝑛₁,max𝑁+𝑀𝑝₁,𝑁𝑝−1<min𝑀1−𝑝₁,𝑀𝑝−1. (2.91)(2.92) Then,(a)for each 𝐿∈(𝑁+𝑀𝑝₁,𝑀(1−𝑝₁))∩(𝑁(𝑝−1),𝑀(𝑝−1)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.41) and (2.54) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,M) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Theorem 2.15. Assume that there exist constants 𝑀,𝑁,𝑀₀,𝑁₀,𝑝₀,𝑝₀,𝑝₁,𝑝₁∈ℝ⁺⧵{0},𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_{𝑛∈ℕ𝑛0},{𝑟_2𝑛}_{𝑛∈ℕ𝑛0}, {𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2) with 𝑖=1, (2.24), (2.86)–(2.88), (2.91), and max𝑁+𝑀𝑝₁,𝑝₀𝑀₀𝑝₀+𝑁₀<min𝑀1−𝑝₁,𝑝₀𝑁₀𝑝₀+𝑀₀. (2.93) Then,(a)for each 𝐿∈(𝑁+𝑀𝑝₁,𝑀(1−𝑝₁))∩(𝑝₀(𝑀₀/𝑝₀+𝑁₀),𝑝₀(𝑁₀/𝑝₀+𝑀₀)), there exist 𝜃_𝐿∈(0,1) and T_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.41) and (2.67) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀,𝑁₀,𝑀₀) are arbitrary sequences with (2.8) and (2.9)(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Theorem 2.16. Assume that there exist constants 𝑀,𝑁,𝑀₀,𝑁₀,𝑝₀,𝑝₀,𝑝₁,𝑝₁∈ℝ⁺⧵{0},𝑛₁∈ℕ_𝑛0 and four nonnegative sequences {𝑟_1𝑛}_𝑛∈ℕn0,{𝑟_2𝑛}_{𝑛∈ℕ𝑛0}, {𝑡_1𝑛}_{𝑛∈ℕ𝑛0}, and {𝑡_2𝑛}_{𝑛∈ℕ𝑛0} satisfying (2.1), (2.2) with 𝑖=1, (2.24), (2.86)–(2.88), 0<𝑁𝑝₁−1<𝑀𝑝₁−1,−𝑝₁≤𝑝_1𝑛≤−𝑝₁,∀𝑛≥𝑛₁𝑁𝑝,max₁−1,𝑝₀𝑀₀𝑝₀+𝑁₀<min𝑀𝑝₁−1,𝑝₀𝑁₀𝑝₀+𝑀₀. (2.94)(2.95) Then,(a)for each 𝐿∈(𝑁(𝑝₁−1),𝑀(𝑝₁−1))∩(𝑝₀(𝑀₀/𝑝₀+𝑁₀),𝑝₀(𝑁₀/𝑝₀+𝑀₀)), there exist 𝜃_𝐿∈(0,1) and 𝑇_𝐿≥max{𝜏₁,𝜏₂}+𝑛₁+|𝛼| such that for any (𝑥⁰,𝑦⁰)=({𝑥⁰_𝑛}_{𝑛∈ℤ𝛼},{𝑦⁰_𝑛}_{𝑛∈ℤ𝛼})∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀), the Mann iterative sequence with errors {(𝑥^𝜇,𝑦^𝜇)}_𝜇∈ℕ0={({𝑥^𝜇_𝑛}_{𝑛∈ℤ𝛼},{𝑦^𝜇_𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0 generated by Schemes (2.54) and (2.67) converges to a bounded positive solution (𝑥,𝑦)∈𝐴(𝑁,𝑀,𝑁₀,𝑀₀) of System (1.14) and has the error estimate (2.7), where {𝛽_𝜇}_𝜇∈ℕ0,{𝛾_𝜇}_𝜇∈ℕ0⊂[0,1], and {(𝛿^𝜇₁,𝛿^𝜇₂)}_𝜇∈ℕ0={({𝛿^𝜇_1𝑛}_{𝑛∈ℤ𝛼},{𝛿^𝜇_2𝑛}_{𝑛∈ℤ𝛼})}_𝜇∈ℕ0⊂𝐴(𝑁,𝑀,𝑁₀,𝑀₀) are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Remark 2.17. Theorems 2.1–2.5 extend and improve [14, Theorem 1] and [27, Theorems 2.1–2.7].

3. Examples

In this section we construct fifteen examples to explain the results presented in Section 2.

Example 3.1. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛−𝑥_𝑛−𝜏1+𝑥³_𝑛−3𝑦_𝑛−6−√𝑛𝑥_𝑛−4𝑦²_𝑛−5𝑛³+𝑛𝑥²_𝑛−3+𝑛²+1𝑥⁴_𝑛−4𝑦²_𝑛−6=(−1)^𝑛−1𝑛⁴+√,Δ𝑦𝑛+1_𝑛−𝑦_𝑛−𝜏2+(𝑛+1)𝑥²_𝑛−2𝑦_𝑛−5+(−1)^𝑛𝑦_𝑛−2𝑛⁴+(𝑛+2)𝑥²_𝑛−1=√𝑛+1𝑛³+(𝑛+2)ln𝑛,𝑛≥1, (3.1) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=𝑛₁=1,ℎ=𝑘=2,𝑀>𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,−5,𝑎_1𝑛𝑎=𝑛−3,_2𝑛=𝑛−4,𝑏_1𝑛=𝑛−6,𝑏_2𝑛=𝑛−5,𝑐_1𝑛=𝑛−2,𝑐_2𝑛=𝑛−1,𝑑_1𝑛𝑑=𝑛−5,_2𝑛=𝑛−2,𝑝_1𝑛=𝑝_2𝑛=−1,𝑞_1𝑛=(−1)^𝑛−1𝑛⁴+√𝑛+1,𝑞_2𝑛=√𝑛+1𝑛³+(),𝑓𝑛+2ln𝑛₁𝑛,𝑢₁,𝑢₂,𝑣₁,𝑣₂=𝑢³₁𝑣₁−√𝑛𝑢₂𝑣²₂𝑛³+𝑛𝑢²₁+𝑛²+1𝑢⁴₂𝑣²₁,𝑓₂𝑛,𝑢₁,𝑢₂,𝑣₁,𝑣₂=(𝑛+1)𝑢²₁𝑣₂+(−1)^𝑛𝑣₁𝑛⁴+(𝑛+2)𝑢²₂,𝑟_1𝑛=𝑀³𝑀+√𝑛𝑛³+𝑛𝑁²+𝑛²+1𝑁⁶,𝑟_2𝑛=𝑀1+(𝑛+1)𝑀²𝑛⁴+(𝑛+2)𝑁²,𝑡_1𝑛=𝑛𝑀²4𝑛²𝑀+3𝑛^5/2+2𝑀³+5√𝑛𝑀²+16𝑛𝑀⁷+14√𝑛𝑀⁶𝑛³+𝑛𝑁²+𝑛²+1𝑁⁶²,𝑡_2𝑛=1𝑛⁷6𝑛⁴𝑀²+𝑛³+30𝑛𝑀⁴+9𝑀²,∀𝑛,𝑢₁,𝑢₂,𝑣₁,𝑣₂∈ℕ_𝑛0×ℝ⁴. (3.2) It is easy to see that (2.1)–(2.4) are satisfied. Thus Theorem 2.1 implies that System (3.1) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀). But [14, Theorem 1] and [27, Theorem 2.1] are not valid for System (3.1).

Example 3.2. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+𝑥_𝑛−𝜏1+𝑛𝑥³_𝑛−1−(𝑛−1)𝑦_𝑛−4𝑛³+𝑥²_𝑛−1𝑦²_𝑛−4=𝑛²−1𝑛⁴,Δ𝑦+2𝑛+1_𝑛+𝑦_𝑛−𝜏2+𝑛⁹𝑥_𝑛−3𝑦²_𝑛−2𝑛¹²+𝑛⁶−1𝑥²_𝑛−3=(−1)^𝑛𝑛²ln³𝑛𝑛²+1²,𝑛≥1, (3.3) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=𝑛₁=1,ℎ=𝑘=1,𝑀>𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,−3,𝑎_1𝑛𝑏=𝑛−1,_1𝑛=𝑛−4,𝑐_1𝑛=𝑛−3,𝑑_1𝑛=𝑛−2,𝑝_1𝑛=𝑝_2𝑛=1,𝑞_1𝑛=𝑛²−1𝑛⁴,𝑞+2𝑛+1_2𝑛=(−1)^𝑛𝑛²ln³𝑛𝑛²+1²,𝑓₁(𝑛,𝑢,𝑣)=𝑛𝑢³−(𝑛−1)𝑣𝑛³+𝑢²𝑣²,𝑓₂()=𝑛𝑛,𝑢,𝑣⁹𝑢𝑣²𝑛¹²+𝑛⁶−1𝑢²,𝑟_1𝑛=𝑛𝑀(𝑀+1)𝑛³+𝑁⁴,𝑟_2𝑛=𝑛⁹𝑀³𝑛¹²+𝑛⁶−1𝑁²,𝑡_1𝑛=𝑛𝑛³(2𝑀+1)+3𝑀⁴+2𝑀⁵𝑛³+𝑁⁴²,𝑡_2𝑛=3𝑛¹⁵𝑛⁶+1𝑀²𝑛¹²+𝑛⁶−1𝑁²²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.4) It is easy to verify that (2.1), (2.2), (2.24), and (2.25) are fulfilled. Thus Theorem 2.2 implies that System (3.3) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀). But in [14, Theorem 1] and [27, Theorem 2.2] are unapplicable for System (3.3).

Example 3.3. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+𝑛²2𝑛²𝑥+3^𝑛−𝜏1+𝑥_𝑛−2−2𝑦_𝑛−6𝑛²+sin𝑥_𝑛−1𝑦_𝑛−6=√𝑛−2cos𝑛³−3𝑛²+6𝑛³,Δ𝑦+3𝑛+1_𝑛+(−1)^𝑛𝑦3𝑛+1^𝑛−𝜏2+𝑛𝑥_𝑛−1+(2𝑛−1)𝑦²_𝑛−3𝑛³+𝑥²_𝑛−1=𝑛²−3sin𝑛⁵−2𝑛+1𝑛⁴+ln³𝑛,𝑛≥2, (3.5) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=𝑛₁=2,ℎ=𝑘=1,𝑀>6𝑁>0,𝛼=min2−𝜏₁,2−𝜏₂,−4,𝑎_1𝑛𝑏=𝑛−2,_1𝑛=𝑛−6,𝑐_1𝑛=𝑛−1,𝑑_1𝑛=𝑛−3,𝑝_1𝑛=𝑛²2𝑛²,𝑝+3_2𝑛=(−1)^𝑛,3𝑛+11𝑝=2,𝑝=13,𝑞_1𝑛=√𝑛−2cos𝑛³−3𝑛²+6𝑛³+3𝑛+1,𝑞_2𝑛=𝑛²−3sin𝑛⁵−2𝑛+1𝑛⁴+ln³𝑛,𝑓₁(𝑛,𝑢,𝑣)=𝑢−2𝑣𝑛²(),𝑓+sin𝑢𝑣₂()=()𝑣𝑛,𝑢,𝑣𝑛𝑢+2𝑛−1²𝑛³+𝑢²,𝑟_1𝑛=3𝑀𝑛²−1,𝑟_2𝑛=𝑛𝑀(1+2𝑀)𝑛³+𝑁²,𝑡_1𝑛=6𝑛²+𝑀²𝑛²−1²,𝑡_2𝑛=𝑛(1+4𝑀)𝑛³+𝑀²+8𝑀³𝑛³+𝑁²²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.6) It is easy to see that (2.1), (2.2), (2.24), and (2.39) hold. Thus, Theorem 2.3 ensures that System (3.5) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀). But [14, Theorem 1] and [27, Theorems 2.3, 2.5, and 2.6] are not valid for System (3.5).

Example 3.4. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+−4+sin2𝑛²𝑥_𝑛−𝜏1+𝑛⁵𝑥²_𝑛−1𝑦⁵_𝑛−2𝑛¹⁰+5𝑛⁸=()+1−1^𝑛𝑛³ln(1+𝑛)𝑛⁹+𝑛⁵,Δ𝑦+1_𝑛+−4+cos5𝑛³𝑦_𝑛−𝜏2+𝑛²𝑥_𝑛−2−(𝑛+2)𝑦²_𝑛−3𝑛⁸+1+𝑛𝑥²_𝑛−2+𝑦²_𝑛−3=𝑛⁵−6(𝑛+1)⁷,𝑛≥1, (3.7) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=𝑛₁=ℎ=𝑘=1,𝑀>2𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,𝑎,−2_1𝑛=𝑛−1,𝑏_1𝑛𝑐=𝑛−2,_1𝑛=𝑛−2,𝑑_1𝑛=𝑛−3,𝑝_1𝑛=−4+sin2𝑛²,𝑝_2𝑛=−4+cos5𝑛³,𝑝=3,𝑝𝑞=5,_1𝑛=(−1)^𝑛𝑛³ln(1+𝑛)𝑛⁹+𝑛⁵+1,𝑞_2𝑛=𝑛⁵−6(𝑛+1)⁷,𝑓₁()=𝑛𝑛,𝑢,𝑣⁵𝑢²𝑣⁵𝑛¹⁰+5𝑛⁸,𝑓+1₂()=𝑛𝑛,𝑢,𝑣²𝑢−(𝑛+2)𝑣²𝑛⁸+1+𝑛𝑢²+𝑣²,𝑟_1𝑛=𝑀⁷𝑛⁵,𝑟_2𝑛=𝑛²𝑀+(𝑛+2)𝑀²𝑛⁸+1+𝑛𝑁²+𝑁²,𝑡_1𝑛=7𝑀⁶𝑛⁵,𝑡_2𝑛=𝑛²2𝑛⁸+8𝑛⁷𝑀+𝑛𝑀²+8𝑀³+3𝑀²𝑛⁸+1+𝑛𝑁²+𝑁²²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.8) Clearly (2.1), (2.2), (2.24), and (2.52) hold. Thus, Theorem 2.4 means that System (3.7) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀). But [14, Theorem 1] and [27, Theorem 2.4] are inapplicable for System (3.7).

Example 3.5. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+8𝑛²+1𝑛²𝑥+1^𝑛−𝜏1+𝑛−(𝑛−1)𝑦³_𝑏1𝑛𝑛³+𝑥²_𝑎1𝑛=√𝑛−(−1)^{𝑛(𝑛+1)/2}√3𝑛−2𝑛²+√,Δ𝑦2𝑛+1_𝑛+1+24ln³𝑛2+3ln³𝑛𝑦_𝑛−𝜏2+2𝑛𝑥_𝑐1𝑛−3√𝑛+1𝑦³_𝑑1𝑛𝑛⁴+𝑥²_𝑐1𝑛𝑦²_𝑑1𝑛=1+𝑛ln1+𝑛²𝑛³ln³1+𝑛²,𝑛≥1, (3.9) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑛2𝑏if𝑛iseven,𝑛−2if𝑛isodd,^1𝑛=𝑛−1if𝑛iseven,𝑛−12𝑐if𝑛isodd,_1𝑛=𝑑𝑛−2if𝑛iseven,𝑛−1if𝑛isodd,^1𝑛=𝑛2−1if𝑛iseven,𝑛+12if𝑛isodd. (3.10) Let 𝑛₀=1,𝑛₁=10,ℎ=𝑘=1,𝑀>399328𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,−1,𝑝_1𝑛=8𝑛²+1𝑛²,𝑝+1_2𝑛=1+24ln³𝑛2+3ln³𝑛,𝑝=8,𝑝𝑞=7,_1𝑛=√𝑛−(−1)^{𝑛(𝑛+1)/2}√3𝑛−2𝑛²+√2𝑛+1,𝑞_2𝑛=1+𝑛ln1+𝑛²𝑛³ln³1+𝑛²,𝑓₁()=()𝑣𝑛,𝑢,𝑣𝑛−𝑛−1³𝑛³+𝑢²,𝑓₂()=√𝑛,𝑢,𝑣2𝑛𝑢−3𝑛+1𝑣³𝑛⁴+𝑢²𝑣²,𝑟_1𝑛=𝑛1+𝑀³𝑛³+𝑁²,𝑟_2𝑛=𝑛𝑀2+3𝑀²𝑛⁴+𝑁⁴,𝑡_1𝑛=𝑛𝑀2+3𝑛³𝑀+5𝑀³𝑛³+𝑁²²,𝑡_2𝑛=𝑛2𝑛⁴+6𝑀²+9𝑛⁴𝑀²+9𝑀⁶𝑛⁴+𝑁⁴²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.11) Obviously (2.1), (2.2), (2.24), and (2.65) hold. Thus, Theorem 2.5 implies that System (3.9) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀). But [14, Theorem 1] and [27, Theorem 2.7] are useless for System (3.9).

Example 3.6. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛−𝑥_𝑛−𝜏1+2𝑛²𝑥³_𝑎1𝑛+𝑛−1𝑛⁵+𝑛𝑛𝑥_𝑎1𝑛−(3𝑛+2)𝑦_𝑏1𝑛²=(−1)^{(𝑛−1)𝑛(𝑛+1)/3}√2𝑛³+4𝑛⁴+sin²2𝑛⁴,Δ𝑦+1_𝑛+𝑦_𝑛−𝜏2+(−1)^𝑛𝑥²_𝑐1𝑛−𝑛𝑦³_𝑑1𝑛(𝑛+1)²√=()𝑛+1−1^𝑛−1ln𝑛³+2𝑛+1𝑛^3/2+ln³1+𝑛²,𝑛≥1, (3.12) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑏𝑛−1if𝑛≥100,𝑛−2if1≤𝑛<100,^1𝑛=𝑛2if𝑛iseven,𝑛−12𝑐if𝑛isodd,_1𝑛=𝑛2−2if𝑛iseven,𝑛−12𝑑if𝑛isodd,^1𝑛=𝑛−3if𝑛≥100,𝑛if1≤𝑛<100. (3.13) Let 𝑛₀=𝑛₁=ℎ=𝑘=1,𝑀>𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,−1,𝑝_1𝑛𝑝=−1,_2𝑛=1,𝑞_1𝑛=(−1)^{(𝑛−1)𝑛(𝑛+1)/3}√2𝑛³+4𝑛⁴+sin²2𝑛⁴+1,𝑞_2𝑛=(−1)^𝑛−1ln𝑛³+2𝑛+1𝑛^3/2+ln³1+𝑛²,𝑓₁(𝑛,𝑢,𝑣)=2𝑛²𝑢³+𝑛−1𝑛⁵+𝑛(𝑛𝑢−(3𝑛+2)𝑣)²,𝑓₂()=()𝑛,𝑢,𝑣−1^𝑛𝑢²−𝑛𝑣³(𝑛+1)²√,𝑟𝑛+1_1𝑛=1+2𝑛𝑀³𝑛⁴,𝑟_2𝑛=𝑀²+𝑛𝑀³𝑛^5/2,𝑡_1𝑛=6𝑀16+𝑛³𝑀+8(2𝑛+𝑀)𝑀²+3𝑛𝑀³𝑛⁶,𝑡_2𝑛=2𝑀+3𝑛𝑀²𝑛^5/2,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.14) It is easy to see that (2.1), (2.2), and (2.79)–(2.82) are satisfied. Thus, Theorem 2.7 implies that System (3.12) possesses uncountably many bounded positive solutions in 𝐴(N,𝑀).

Example 3.7. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛−𝑥_𝑛−𝜏1+𝑛𝑥³_𝑎1𝑛−𝑦_𝑏1𝑛(𝑛+1)⁴+√𝑛+1𝑦²_𝑏1𝑛=(−1)^{𝑛(𝑛+1)/2}1−√3𝑛+1𝑛³+√,Δ𝑦3𝑛+2_𝑛+𝑛²−13𝑛²𝑦+2^𝑛−𝜏2+√𝑛(𝑛+1)𝑦⁴_𝑑1𝑛(𝑛+2)³+𝑛𝑥²_𝑐1𝑛=√2𝑛+1(𝑛+3)²ln1+𝑛⁴(𝑛+1)⁴,𝑛≥1, (3.15) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑏𝑛−4if𝑛iseven,𝑛−1if𝑛isodd,^1𝑛=𝑛−2if𝑛iseven,𝑛−12𝑐if𝑛isodd,_1𝑛=𝑛−1if𝑛iseven,𝑛+12𝑑if𝑛isodd,^1𝑛=𝑛2if𝑛iseven,𝑛−2if𝑛isodd. (3.16) Let 𝑛₀=𝑛₁=ℎ=𝑘=1,𝑀>6𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,𝑝,−2_1𝑛=−1,𝑝_2𝑛=𝑛²−13𝑛²,+21𝑝=3,𝑝=12,𝑞_1𝑛=(−1)^{𝑛(𝑛+1)/2}1−√3𝑛+1𝑛³+√,𝑞3𝑛+2_2𝑛=√2𝑛+1(𝑛+3)²ln1+𝑛⁴(𝑛+1)⁴,𝑓₁(𝑛,𝑢,𝑣)=𝑛𝑢³−𝑣(𝑛+1)⁴+√𝑛+1𝑣²,𝑓₂()=√𝑛,𝑢,𝑣𝑛(𝑛+1)𝑣⁴(𝑛+2)³+𝑛𝑢²,𝑟_1𝑛=𝑛𝑀²+1𝑀(𝑛+1)⁴,𝑟_2𝑛=√𝑛(𝑛+1)𝑀⁴(𝑛+2)³+𝑛𝑁²,𝑡_1𝑛=(𝑛+1)⁴3𝑛𝑀²+1+5𝑛𝑀²+1√𝑛+1𝑀²(𝑛+1)⁴+√𝑛+1𝑁²²,𝑡_2𝑛=2√𝑛(𝑛+1)𝑀³2(𝑛+2)³+3𝑛𝑀²(𝑛+2)³+𝑛𝑁²²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.17) Clearly (2.1), (2.2), (2.79)–(2.81), and (2.83) are satisfied. Thus, Theorem 2.8 guarantees that System (3.15) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Example 3.8. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛−𝑥_𝑛−𝜏1+𝑛⁷−𝑥²_𝑎1𝑛𝑛¹⁰+2𝑦²_𝑏1𝑛=𝑛²⁰−(−1)^{(𝑛+1)(𝑛+2)/2}3𝑛⁵+1²𝑛²³+(𝑛+5)⁵ln⁸𝑛²,Δ𝑦+1_𝑛−6𝑛⁵2𝑛⁵𝑦+1^𝑛−𝜏2+𝑛³𝑥_𝑐1𝑛𝑦²_𝑑1𝑛𝑛⁶+(𝑛+1)𝑥⁴_𝑐1𝑛=𝑛⁸−(−1)^𝑛−1𝑛³+1²(𝑛+1)¹¹,𝑛≥2, (3.18) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑛2−1if𝑛iseven,𝑛−32𝑏if𝑛isodd,^1𝑛=𝑐𝑛−1if𝑛iseven,𝑛−2if𝑛isodd,_1𝑛=𝑛2+1if𝑛iseven,𝑛+12𝑑if𝑛isodd,^1𝑛=𝑛−5if𝑛iseven,𝑛−3if𝑛isodd. (3.19) Let 𝑛₀=𝑛₁=2,ℎ=𝑘=1,𝑀>6𝑁>0,𝛼=min2−𝜏₁,2−𝜏₂,−3,𝑝_1𝑛𝑝=−1,_2𝑛=−6𝑛⁵2𝑛⁵,+1𝑝=2,𝑝=3,𝑞_1𝑛=𝑛²⁰−(−1)^{(𝑛+1)(𝑛+2)/2}3𝑛⁵+1²𝑛²³+(𝑛+5)⁵ln⁸𝑛²,𝑞+1_2𝑛=𝑛⁸−(−1)^𝑛−1𝑛³+1²(𝑛+1)¹¹,𝑓₁()=𝑛𝑛,𝑢,𝑣⁷−𝑢²𝑛¹⁰+2𝑣²,𝑓₂()=𝑛𝑛,𝑢,𝑣³𝑢𝑣²𝑛⁶+(𝑛+1)𝑢⁴,𝑟_1𝑛=𝑛⁷+𝑀²𝑛¹⁰+2𝑁²,𝑟_2𝑛=𝑛³𝑀³𝑛⁶+(𝑛+1)𝑁⁴,𝑡_1𝑛=2𝑀2𝑛⁷+𝑛¹⁰+4𝑀²𝑛¹⁰+2𝑁²²,𝑡_2𝑛=3𝑛³𝑀²𝑛⁶+(𝑛+1)𝑀⁴𝑛⁶+(𝑛+1)𝑁⁴²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.20) Obviously (2.1), (2.2), (2.79)–(2.81), (2.84), and (2.85) hold. Thus, Theorem 2.9 implies that System (3.18) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Example 3.9. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛−𝑥_𝑛−𝜏1+𝑛𝑥²_𝑎1𝑛−𝑦³_𝑏1𝑛𝑛³ln²=𝑛𝑛+1²−(−1)^𝑛−1(3𝑛+1)𝑛⁵+8(𝑛+1)⁴,Δ𝑦+1_𝑛+8𝑛²2𝑛²𝑦+3^𝑛−𝜏2+𝑛²𝑛⁴+𝑥_𝑐1𝑛𝑦_𝑑1𝑛=(−1)^𝑛−1sin𝑛⁵+1𝑛²+1,𝑛≥1, (3.21) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑛−2if𝑛iseven,𝑛−32𝑏if𝑛isodd,^1𝑛=𝑛2if𝑛iseven,𝑛+12𝑐if𝑛isodd,_1𝑛=𝑛−1if𝑛iseven,𝑛−12𝑑if𝑛isodd,^1𝑛=𝑛−4if𝑛iseven,𝑛−2if𝑛isodd. (3.22) Let 𝑛₀=1,𝑛₁=3,ℎ=𝑘=1,𝑀=3000,𝑁=2000,𝑀₀=1200,𝑁₀=40,𝛼=min1−𝜏₁,1−𝜏₂,−2,𝑝_1𝑛=−1,𝑝_2𝑛=8𝑛²2𝑛²,+3𝑝₀=4,𝑝₀𝑞=3,_1𝑛=𝑛²−(−1)^𝑛−1(3𝑛+1)𝑛⁵+8(𝑛+1)⁴+1,𝑞_2𝑛=(−1)^𝑛−1sin𝑛⁵+1𝑛²+1,𝑓₁(𝑛,𝑢,𝑣)=𝑛𝑢²−𝑣³𝑛³ln²,𝑓𝑛+1₂()=𝑛𝑛,𝑢,𝑣²𝑛⁴+|𝑢𝑣|,𝑟_1𝑛=𝑀²(𝑛+𝑀)𝑛³ln²𝑛+1,𝑟_2𝑛=𝑛²𝑛⁴+𝑁²₀,𝑡_1𝑛=𝑀(2𝑛+3𝑀)𝑛³ln²𝑛+1,𝑡_2𝑛=2𝑛²𝑀₀𝑛⁴+𝑁²₀²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.23) It is easy to verify that (2.1), (2.2) with 𝑖=1, (2.79)–(2.81), and (2.86)–(2.89) are satisfied. Thus, Theorem 2.10 reveals that System (3.21) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Example 3.10. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+𝑥_𝑛−𝜏1+1−𝑛⁵𝑥²_𝑎1𝑛𝑛⁷+(𝑛+3)ln1+𝑦²_𝑏1𝑛=𝑛⁴−4𝑛³+𝑛−1𝑛⁶+(𝑛−1)⁴,Δ𝑦_𝑛−2𝑛²5𝑛²𝑦+6^𝑛−𝜏2+𝑛²𝑦_𝑑1𝑛𝑛⁴+sin²𝑥²_𝑐1𝑛+𝑛𝑦²_𝑑1𝑛=(−1)^𝑛√𝑛−1𝑛³+2𝑛²+1,𝑛≥2, (3.24) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑛−3if𝑛iseven,𝑛−12𝑏if𝑛isodd,^1𝑛=𝑛2+1if𝑛iseven,𝑛−12𝑐if𝑛isodd,_1𝑛=𝑛−1if𝑛iseven,𝑛+12𝑑if𝑛isodd,^1𝑛=𝑛−2if𝑛iseven,𝑛+12if𝑛isodd. (3.25) Let 𝑛₀=𝑛₁=2,ℎ=𝑘=1,𝑀>5𝑁3>0,𝛼=min1−𝜏₁,1−𝜏₂,−1,𝑝_1𝑛𝑝=1,_2𝑛=−2𝑛²5𝑛²,+6𝑝=0,𝑝=25,𝑞_1𝑛=𝑛⁴−4𝑛³+𝑛−1𝑛⁶+(𝑛−1)⁴,𝑞_2𝑛=(−1)^𝑛√𝑛−1𝑛³+2𝑛²,𝑓+1₁(𝑛,𝑢,𝑣)=1−𝑛⁵𝑢²𝑛⁷+(𝑛+3)ln1+𝑣²,𝑓₂()=𝑛𝑛,𝑢,𝑣²𝑣𝑛⁴+sin²𝑢²+𝑛𝑣²,𝑟_1𝑛=1+𝑛⁵𝑀²𝑛⁷+(𝑛+3)ln1+𝑁²,𝑟_2𝑛=𝑀𝑛²,𝑡_1𝑛=1𝑛⁷+(𝑛+3)ln1+𝑁²²2M(𝑛+3)1+𝑁²+𝑛¹²+𝑛⁵(𝑛+3)2𝑀²1+𝑁²+ln1+𝑀²,𝑡_2𝑛=𝑛⁴+4𝑛𝑀²+4𝑀²+1𝑛⁶,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.26) Obviously (2.1), (2.2), (2.24), (2.83), and (2.90) hold. Thus, Theorem 2.11 gives that System (3.24) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Example 3.11. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+𝑥_𝑛−𝜏1+𝑛²−𝑥²_𝑎1𝑛𝑦_𝑏1𝑛𝑛⁴+(𝑛+1)𝑦⁴_𝑏1𝑛=𝑛⁶−9𝑛⁵+(−1)^𝑛𝑛⁴−𝑛³+𝑛−1𝑛⁸+3𝑛⁷+𝑛⁵+4𝑛⁴+𝑛²,Δ𝑦+1_𝑛−3𝑛³𝑛³𝑦+1^𝑛−𝜏2+𝑛𝑦²_𝑑1𝑛𝑛³+𝑥²_𝑐1𝑛=(−1)^{𝑛(𝑛+1)/2}(2𝑛−5)ln²1+𝑛²𝑛³+3𝑛²+5𝑛+1,𝑛≥1, (3.27) where 𝜏₁,𝜏₂∈ℕ are fixed and 𝑎_1𝑛=𝑛−2if𝑛iseven,𝑛+12𝑏if𝑛isodd,^1𝑛=𝑛2−1if𝑛iseven,𝑛+12𝑐if𝑛isodd,_1𝑛=𝑛−3if𝑛iseven,𝑛−32𝑑if𝑛isodd,^1𝑛=𝑛−4if𝑛iseven,𝑛−12if𝑛isodd. (3.28) Let 𝑛₀=𝑛₁=ℎ=𝑘=1,𝑀>4𝑁>0,𝛼=min1−𝜏₁,1−𝜏₂,−2,𝑝_1𝑛𝑝=1,_2𝑛=−3𝑛³𝑛³,+13𝑝=2,𝑝=3,𝑞_1𝑛=𝑛⁶−9𝑛⁵+(−1)^𝑛𝑛⁴−𝑛³+𝑛−1𝑛⁸+3𝑛⁷+𝑛⁵+4𝑛⁴+𝑛²,𝑞+1_2𝑛=(−1)^{𝑛(𝑛+1)/2}(2𝑛−5)ln²1+𝑛²𝑛³+3𝑛²+5𝑛+1,𝑓₁()=𝑛𝑛,𝑢,𝑣²−𝑢²𝑣𝑛⁴+(𝑛+1)𝑣⁴,𝑓₂(𝑛,𝑢,𝑣)=𝑛𝑣²𝑛³+𝑢²,𝑟_1𝑛=𝑛²+𝑀³𝑛⁴+(𝑛+1)𝑁⁴,𝑟_2𝑛=𝑛𝑀²𝑛³+𝑁²,𝑡_1𝑛=𝑛𝑀²8𝑛²𝑀+3𝑛³+10𝑀⁴𝑛⁴+(𝑛+1)𝑁⁴²,𝑡_2𝑛=2𝑀𝑛³+2𝑀²𝑛³+𝑁²²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.29) Clearly (2.1), (2.2), (2.24), (2.84), (2.85), and (2.90) hold. Thus, Theorem 2.12 ensures that System (3.27) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Example 3.12. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+𝑥_𝑛−𝜏1+𝑛²⁵−𝑛¹⁵+1𝑦²_𝑛−1𝑛³⁰+𝑛¹⁸+1𝑥²_𝑛−2=𝑛³⁸−19𝑛²⁵+30𝑛¹⁸−3𝑛⁹+2𝑛−1𝑛⁴⁰+8𝑛³⁰+𝑛¹⁵+6𝑛²,Δ𝑦+1_𝑛+16𝑛³+14𝑛³𝑦+131^𝑛−𝜏2+𝑥³_𝑛−4𝑦²_𝑛−3𝑛ln¹⁰=𝑛𝑛+1³⁴−6𝑛³²−7𝑛¹⁵+(−1)^𝑛𝑛⁷−1𝑛³⁶+6𝑛³⁰+5𝑛¹²+6𝑛⁶+𝑛³+1,𝑛≥2, (3.30) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=2,𝑛₁=4,ℎ=𝑘=1,𝑀=800𝑁₀,𝑁=500𝑁₀,𝑀₀=300𝑁₀>0,𝛼=min2−𝜏₁,2−𝜏₂,−2,𝑎_1𝑛=𝑛−2,𝑏_1𝑛𝑐=𝑛−1,_1𝑛=𝑛−4,𝑑_1𝑛𝑝=𝑛−3,_1𝑛=1,𝑝_2𝑛=16𝑛³+14𝑛³,+131𝑝₀=4,𝑝₀𝑞=3,_1𝑛=𝑛³⁸−19𝑛²⁵+30𝑛¹⁸−3𝑛⁹+2𝑛−1𝑛⁴⁰+8𝑛³⁰+𝑛¹⁵+6𝑛²,𝑞+1_2𝑛=𝑛³⁴−6𝑛³²−7𝑛¹⁵+(−1)^𝑛𝑛⁷−1𝑛³⁶+6𝑛³⁰+5𝑛¹²+6𝑛⁶+𝑛³+1,𝑓₁()=𝑛𝑛,𝑢,𝑣²⁵−𝑛¹⁵+1𝑣²𝑛³⁰+𝑛¹⁸+1𝑢²,𝑓₂()=𝑢𝑛,𝑢,𝑣³𝑣²𝑛ln¹⁰𝑛+1,𝑟_1𝑛=𝑛²⁵+𝑛¹⁵+1𝑀²𝑛³⁰+𝑛¹⁸+1𝑁²,𝑟_2𝑛=𝑀⁵₀𝑛ln¹⁰,𝑡𝑛+1_1𝑛=4𝑛³³𝑀𝑛¹⁰+𝑛¹²+4𝑀²𝑛³⁰+𝑛¹⁸+1𝑁²²,𝑡_2𝑛=5𝑀⁴₀𝑛ln¹⁰𝑛+1²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.31) It is easy to verify that (2.1), (2.2) with 𝑖=1, (2.24), and (2.86)–(2.90) hold. Thus, Theorem 2.13 guarantees that System (3.30) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Example 3.13. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+3(−1)^𝑛𝑛¹⁰+2𝑛+19𝑛¹⁰+4𝑛⁸𝑥+1^𝑛−𝜏1+𝑥_𝑛−3−𝑛𝑦_𝑛−1𝑛⁴+𝑥²_𝑛−3=𝑛⁴−√2𝑛+5ln⁵𝑛𝑛⁶+6𝑛²,Δ𝑦+1_𝑛−4+(−1)^𝑛𝑛⁴+5𝑛⁴𝑦+1^𝑛−𝜏2+𝑛³𝑥_𝑛−2𝑦_𝑛−4𝑛⁵+(𝑛+1)𝑦²_𝑛−4=𝑛⁷−(−1)^𝑛𝑛⁴−1𝑛⁹+3𝑛⁶+5,𝑛≥2, (3.32) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=2,𝑛₁=4,ℎ=𝑘=1,𝑀>10𝑁>0,𝛼=min2−𝜏₁,2−𝜏₂,𝑎,−2_1𝑛=𝑛−3,𝑏_1𝑛=𝑛−1,𝑐_1𝑛=𝑛−2,𝑑_1𝑛=𝑛−4,𝑝_1𝑛=3(−1)^𝑛𝑛¹⁰+2𝑛+19𝑛¹⁰+4𝑛⁸,𝑝+1_2𝑛()=−4+−1^𝑛𝑛⁴+5𝑛⁴,+1𝑝₁=𝑝₁=13,𝑝=3,𝑝=5,𝑞_1𝑛=𝑛⁴−√2𝑛+5ln⁵𝑛𝑛⁶+6𝑛²,𝑞+1_2𝑛=𝑛⁷−(−1)^𝑛𝑛⁴−1𝑛⁹+3𝑛⁶+5,𝑓₁(𝑛,𝑢,𝑣)=𝑢−𝑛𝑣𝑛⁴+𝑢²,𝑓₂()=𝑛𝑛,𝑢,𝑣³𝑢𝑣𝑛⁵+(𝑛+1)𝑣²,𝑟_1𝑛=𝑀(𝑛+1)𝑛⁴+𝑁²,𝑟_2𝑛=𝑛³𝑀²𝑛⁵+(𝑛+1)𝑁²,𝑡_1𝑛=𝑛⁴+𝑀²+𝑛⁵+3𝑛𝑀²𝑛⁴+𝑁²²,𝑡_2𝑛=2𝑛³𝑀𝑛⁵+(𝑛+1)𝑀²𝑛⁵+(𝑛+1)𝑁²²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.33) Obviously (2.1), (2.2), (2.24), (2.84), (2.91), and (2.92) hold. Thus, Theorem 2.14 means that System (3.32) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Example 3.14. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+(−1)^𝑛𝑛+1𝑥4𝑛+100^𝑛−𝜏1+𝑥_𝑛−2−𝑛𝑦_𝑛−1𝑛³+𝑥²_𝑛−2𝑦²_𝑛−1=𝑛⁵ln⁹𝑛−(−1)^𝑛𝑛⁸+5𝑛⁷,Δ𝑦+1_𝑛+64𝑛⁸+18𝑛⁸+3𝑛⁴𝑦+7^𝑛−𝜏2+𝑛𝑥²_𝑛−5−𝑦_𝑛−3sin𝑛𝑛²ln²=𝑛𝑛+1⁶−(−1)^𝑛𝑛²+1𝑛⁸+7𝑛⁷+2,𝑛≥2, (3.34) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=𝑛₁=2,ℎ=𝑘=1,𝑀=40000,𝑁=5000,𝑀₀=4480,𝑁₀=80,𝛼=min2−𝜏₁,2−𝜏₂,−3,𝑎_1𝑛=𝑛−2,𝑏_1𝑛=𝑛−1,𝑐_1𝑛=𝑛−5,𝑑_1𝑛𝑝=𝑛−3,_1𝑛=(−1)^𝑛𝑛+14𝑛+100,𝑝_2𝑛=64𝑛⁸+18𝑛⁸+3𝑛⁴,+7𝑝₁=𝑝₁=14,𝑝₀=8,𝑝₀𝑞=7,_1𝑛=𝑛⁵ln⁹𝑛−(−1)^𝑛𝑛⁸+5𝑛⁷+1,𝑞_2𝑛=𝑛⁶−(−1)^𝑛𝑛²+1𝑛⁸+7𝑛⁷+2,𝑓₁(𝑛,𝑢,𝑣)=𝑢−𝑛𝑣𝑛³+𝑢²𝑣²,𝑓₂(𝑛,𝑢,𝑣)=𝑛𝑢²−𝑣sin𝑛𝑛²ln²𝑛+1,𝑟_1𝑛=𝑀(𝑛+1)𝑛³+𝑁⁴,𝑟_2𝑛=𝑀₀𝑀₀+1𝑛²ln²,𝑡𝑛+1_1𝑛=𝑛³+3𝑀⁴+𝑛⁴+3𝑛𝑀²𝑛³+𝑁⁴²,𝑡_2𝑛=2𝑀₀+1𝑛²ln²𝑛+1²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.35) It is easy to see that (2.1), (2.2) with 𝑖=1, (2.24), (2.86)–(2.88), (2.91), and (2.93) hold. Thus, Theorem 2.15 implies that System (3.34) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Example 3.15. Consider the nonlinear difference system with multiple delays: Δ𝑥_𝑛+−20𝑛+1𝑥4𝑛+9^𝑛−𝜏1+𝑛𝑥³_𝑛−1𝑛³+𝑦²_𝑛−2=𝑛³+3𝑛²√𝑛−1+1𝑛⁵+3𝑛⁴,Δ𝑦+1_𝑛+6𝑛²+12𝑛²𝑦+5𝑛+3^𝑛−𝜏2+𝑥_𝑛−2𝑦_𝑛−4𝑛²+cos𝑛²=𝑛+1⁴+5𝑛²−7𝑛⁶+4𝑛³+1,𝑛≥2, (3.36) where 𝜏₁,𝜏₂∈ℕ are fixed. Let 𝑛₀=2,𝑛₁=10,ℎ=𝑘=1,𝑀=2𝑁₀,𝑁=𝑁₀>0,𝑀₀=30𝑁₀,𝛼=min2−𝜏₁,2−𝜏₂,−2,𝑎_1𝑛=𝑛−1,𝑏_1𝑛=𝑛−2,𝑐_1𝑛=𝑛−2,𝑑_1𝑛𝑝=𝑛−4,_1𝑛=−20𝑛+14𝑛+9,𝑝_2𝑛=6𝑛²+12𝑛²,+5𝑛+3𝑝₁=4,𝑝₁=5,𝑝₀=3,𝑝₀𝑞=2,_1𝑛=𝑛³+3𝑛²√𝑛−1+1𝑛⁵+3𝑛⁴+1,𝑞_2𝑛=𝑛⁴+5𝑛²−7𝑛⁶+4𝑛³+1,𝑓₁(𝑛,𝑢,𝑣)=𝑛𝑢³𝑛³+𝑣²,𝑓₂(𝑛,𝑢,𝑣)=𝑢𝑣𝑛²+cos𝑛²+1,𝑟_1𝑛=𝑛𝑀³𝑛³+𝑁²,𝑟_2𝑛=𝑀²₀𝑛²,𝑡−1_1𝑛=𝑛𝑀²3𝑛³+5𝑀²𝑛³+𝑁²²,𝑡_2𝑛=2𝑀₀𝑛²−1²,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (3.37) Clearly (2.1), (2.2) with 𝑖=1, (2.24), (2.86)–(2.88), (2.94), and (2.95) hold. Thus, Theorem 2.16 implies that System (3.36) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀,𝑁₀,𝑀₀).

Acknowledgments

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