Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

Abstract

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation Δ(𝑎(𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛)Δ(𝑦_𝑛+𝑏_𝑛𝑦_𝑛−𝜏))+𝑓(𝑛,𝑦_𝑓1𝑛,…,𝑦_𝑓𝑘𝑛)=𝑐_𝑛,∀𝑛≥𝑛₀. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.

1. Introduction

It is well known that the oscillation, nonoscillation, asymptotic behavior, and existence of solutions for second-order difference equations with delays have been widely studied in many papers over the last 20 years, see, for example, [1–9] and the references cited therein.

Recently, Cheng [5] considered the second-order neutral delay linear difference equation with positive and negative coefficients Δ²𝑦_𝑛+𝑝𝑦_𝑛−𝑚+𝑝_𝑛𝑦_𝑛−𝑘−𝑞_𝑛𝑦_𝑛−𝑙=0,∀𝑛≥𝑛₀ (1.1) and investigated the existence of a nonoscillatory solution of (1.1) under the condition 𝑝≠−1 by using the Banach fixed point theorem. M. Migda and J. Migda [9] and Luo and Bainov [8] discussed the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima Δ²𝑦_𝑛+𝑝_𝑛𝑦_𝑛−𝑘+𝑞_𝑛max𝑦_𝑠∶𝑛−𝑙≤𝑠≤𝑛=0,∀𝑛≥1 (1.2) and the second-order neutral difference equation Δ²𝑦_𝑛+𝑝𝑦_𝑛−𝑘+𝑓𝑛,𝑦_𝑛=0,∀𝑛≥1. (1.3) Cheng and Chu [2] got sufficient and necessary conditions of the oscillatory solutions for the second-order difference equation Δ𝑟_𝑛−1Δ𝑦_𝑛−1+𝑝_𝑛𝑦^𝛾_𝑛=0,∀𝑛≥1. (1.4) Li and Yeh [6] established some oscillation criteria of the second-order delay difference equation Δ𝑎_𝑛−1Δ𝑦_𝑛−1+𝑝_𝑛−1𝑦_{𝑛−1−𝜎}+𝑞_𝑛𝑓𝑦_𝑛−𝜏=0,∀𝑛≥1. (1.5) Using the Leray-Schauder nonlinear alternative theorem, Agarwal et al. [1] studied the existence of nonoscillatory solutions for the discrete equation Δ𝑎_𝑛Δ𝑦_𝑛+𝑝𝑦_𝑛−𝜏+𝐹𝑛+1,𝑦_{𝑛+1−𝜎}=0,∀𝑛≥1 (1.6) under the condition |𝑝|≠1. Very recently, Liu et al. [7] utilized the Banach contraction principle to establish the global existence and multiplicity of bounded nonoscillatory solutions for the second-order nonlinear neutral delay difference equation Δ𝑎_𝑛Δ𝑦_𝑛+𝑏𝑦_𝑛−𝜏+𝑓𝑛,𝑦_{𝑛−𝑑1𝑛},𝑦_{𝑛−𝑑2𝑛},…,𝑦_{𝑛−𝑑𝑘𝑛}=𝑐_𝑛,∀𝑛≥𝑛₀. (1.7)

Motivated by the results in [1–9], in this paper, we discuss the solvability of the second-order nonlinear neutral delay difference equation Δ𝑎𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛Δ𝑦_𝑛+𝑏_𝑛𝑦_𝑛−𝜏+𝑓𝑛,𝑦_𝑓1𝑛,…,𝑦_𝑓𝑘𝑛=𝑐_𝑛,∀𝑛≥𝑛₀, (1.8) where 𝜏,𝑟,𝑘∈ℕ, 𝑛₀∈ℕ₀,  {𝑏_𝑛}_{𝑛∈ℕ𝑛0}∪{𝑐_𝑛}_{𝑛∈ℕ𝑛0}⊂ℝ, 𝑎∈𝐶(ℕ_𝑛0×ℝ^𝑟,ℝ⧵{0}),𝑓∈𝐶(ℕ_𝑛0×ℝ^𝑘,ℝ), ∪^𝑟_𝑑=1{𝑎_𝑑𝑛}_{𝑛∈ℕ𝑛0}⊆ℤ,  ∪^𝑘_𝑗=1{𝑓_𝑗𝑛}_{𝑛∈ℕ𝑛0}⊆ℤ and lim_𝑛→∞𝑎_𝑑𝑛=lim_𝑛→∞𝑓_𝑗𝑛=+∞,𝑑,𝑗∈𝐽_𝑟×𝐽_𝑘. (1.9) It is clear that (1.1)–(1.7) are special cases of (1.8). By utilizing the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and a few new techniques, we prove the existence of uncountably many bounded positive solutions of (1.8). Five examples are constructed to illuminate our results which extend essentially the corresponding results in [1, 7].

2. Preliminaries

Throughout this paper, we assume that Δ is the forward difference operator defined by Δ𝑦_𝑛=𝑦_𝑛+1−𝑦_𝑛,  ℝ=(−∞,+∞),  ℝ⁺=[0,+∞),  ℤ,  ℕ and ℕ₀ stand for the sets of all integers, positive integers, and nonnegative integers, respectively, 𝛽=min𝑛₀−𝜏,inf𝑎_𝑑𝑛∶𝑑∈𝐽_𝑟,𝑛∈ℕ_𝑛0,inf𝑓_𝑗𝑛∶𝑗∈𝐽_𝑘,𝑛∈ℕ_𝑛0,ℤ_𝛽=𝑛∶𝑛∈ℤwith𝑛≥𝛽,ℕ_𝑛0=𝑛∶𝑛∈ℕ₀with𝑛≥𝑛₀,𝐽_𝑙={1,…,𝑙}for𝑙∈{𝑟,𝑘}. (2.1)𝑙^∞_𝛽 denotes the Banach space of all bounded sequences 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽} with the norm 𝑦=sup_{𝑛∈ℤ𝛽}𝑦_𝑛for𝑦=𝑦_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽. (2.2) For any 𝑀>𝑁>0, put 𝑉(𝑁)=𝑦=𝑦_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽∶𝑦_𝑛≥𝑁,∀𝑛∈ℤ_𝛽,𝑈(𝑀)=(𝑁)∶𝑦,𝐵()=𝑦𝑦∈𝑉<𝑀𝑀,𝑁𝑦=_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽∶,𝐴()=𝑦𝑦−𝑀<𝑁𝑁,𝑀𝑦=_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽∶𝑁≤𝑦_𝑛≤𝑀,∀𝑛∈ℤ_𝛽. (2.3) It is easy to see that 𝑉(𝑁) is a closed convex subset of 𝑙^∞_𝛽,  𝑈(𝑀) is a bounded open subset of 𝑉(𝑁) and 𝐵(𝑀,𝑁) is a bounded open convex subset of 𝑙^∞_𝛽 and 𝐴(𝑁,𝑀) is a bounded closed and convex subset of 𝑙^∞_𝛽.

By a solution of (1.8), we mean a sequence {𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑙^∞_𝛽 with a positive integer 𝑇≥𝑛₀+𝜏+|𝛽| such that (1.8) is satisfied for all 𝑛≥𝑇.

The following Lemmas play important roles in this paper.

Lemma 2.1 (Discrete Arzela-Ascoli’s Theorem [3]). A bounded, uniformly Cauchy subset 𝑌 of 𝑙^∞_𝛽 is relatively compact.

Lemma 2.2 (Rothe Fixed Point Theorem [10]). Let 𝐷 be a bounded convex open subset of a Banach space 𝐸 and 𝐴∶𝐷→𝐸 be a continuous, condensing mapping, and 𝐴(𝜕𝐷)⊆𝐷. Then 𝐴 has a fixed point in 𝐷.

Lemma 2.3 (Leray-Schauder Nonlinear Alternative Theorem [1]). Let 𝑈 be an open subset of a closed convex set 𝐾 in a Banach space 𝐸 with 𝑝^∗∈𝑈. Let 𝑓∶𝑈→𝐾 be a continuous, condensing mapping with 𝑓(𝑈) bounded. Then either(a)𝑓has a fixed point in 𝑈; or(b)there exist an 𝑥∈𝜕𝑈 and a 𝜆∈(0,1) such that 𝑥=(1−𝜆)𝑝^∗+𝜆𝑓𝑥.

Lemma 2.4 (Krasnoselskill Fixed Point Theorem [5]). Let 𝑌 be a nonempty bounded closed convex subset of a Banach space 𝑋 and 𝑓,𝑔 be mappings from 𝑌 into 𝑋 such that 𝑓𝑥+𝑔𝑦∈𝑌 for every pair 𝑥,𝑦∈𝑌. If 𝑓 is a contraction mapping and 𝑔 is completely continuous, then the equation 𝑓𝑥+𝑔𝑥=𝑥 has at least one solution in 𝑌.

3. Main Results

Now we use the Rothe fixed point theorem to show the existence and multiplicity of bounded positive solutions of (1.8).

Theorem 3.1. Assume that there exist two constants 𝑁 and 𝑀 with 𝑀>𝑁>0 and two positive sequences {𝑎_𝑛}_{𝑛∈ℕ𝑛0} and {𝑝_𝑛}_{𝑛∈ℕ𝑛0} satisfying 𝑎𝑛,𝑢₁,𝑢₂,…,𝑢_𝑟≥𝑎_𝑛,∀𝑛,𝑢_𝑑∈ℕ_𝑛0×[𝑁,𝑀],𝑑∈𝐽_𝑟;𝑓𝑛,𝑢₁,𝑢₂,…,𝑢_𝑘≤𝑝_𝑛,∀𝑛,𝑢_𝑗∈ℕ_𝑛0×[𝑁,𝑀],𝑗∈𝐽_𝑘;_{∞𝑠=𝑛0}1𝑎_𝑠^∞_𝑖=𝑠max𝑝_𝑖,𝑐_𝑖𝑏<+∞;_𝑛=1eventually. (3.1)(3.2)(3.3) Then (1.8) has uncountably many bounded positive solutions in 𝐵(𝑀,𝑁).

Proof. Let 𝐿∈(𝑀−𝑁,𝑀+𝑁). First of all, we show that there exists a mapping 𝑆_𝐿∶𝐵(𝑀,𝑁)→𝑙^∞_𝛽 with 𝑆_𝐿(𝜕𝐵(𝑀,𝑁))⊆𝐵(𝑀,𝑁) such that 𝑆_𝐿 has a fixed point 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁), which is also a bounded positive solution of (1.8).
It follows from (3.2) and (3.3) that there exists 𝑇≥max{1,𝑛₀+𝜏+|𝛽|} satisfying 𝑏_𝑛=1,∀𝑛≥𝑇;_∞𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<12min{𝑀+𝑁−𝐿,𝑁−𝑀+𝐿}. (3.4)(3.5) Define a mapping 𝑆_𝐿∶𝐵(𝑀,𝑁)→𝑙^∞_𝛽 as follows: 𝑆_𝐿𝑦_𝑛=𝐿−∑^∞∑_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖∑_=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖𝑆,𝑛≥𝑇_𝐿𝑦_𝑇,𝛽≤𝑛<𝑇 (3.6) for each 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁). On account of (3.1), (3.5), and (3.6), we conclude that for every 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝜕𝐵(𝑀,𝑁)⊆𝐵(𝑀,𝑁) and 𝑛≥𝑇𝑆_𝐿𝑦_𝑛−𝑀=𝐿−𝑀−^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=s𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤|𝐿−𝑀|+^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤|𝐿−𝑀|+^∞_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<||+1𝐿−𝑀2{}≤𝑁min𝑀+𝑁−𝐿,𝑁−𝑀+𝐿2, (3.7) which means that 𝑆_𝐿≤𝑁𝑦−𝑀2<𝑁, (3.8) that is, 𝑆_𝐿(𝜕𝐵(𝑀,𝑁))⊆𝐵(𝑀,𝑁).
Now we assert that 𝑆_𝐿 is a continuous and condensing mapping in 𝐵(𝑀,𝑁). Let 𝑦^𝜔={𝑦^𝜔_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁) for each 𝜔∈ℕ and 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁) with lim_𝜔→∞𝑦^𝜔=𝑦. Let 𝜀>0. It follows from (3.2) and the continuity of 𝑎 and 𝑓 that there exist 𝑇₁,𝑇₂,𝑇₃∈ℕ with 𝑇₁>𝑇 and 𝑇₂>𝑇₁+𝜏−1 satisfying max^∞_{𝑠=𝑇1+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖<𝜀16;max^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎_𝑠^𝑇2−1_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖,_{𝑇1+𝜏−1𝑠=𝑇+𝜏}𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠𝑎²_𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜀16,∀𝜔≥𝑇₃. (3.9)(3.10) In view of (3.1) and (3.6)–(3.10), we deduce that for any 𝜔≥𝑇₃𝑆_𝐿𝑦^𝜔−𝑆_𝐿𝑦=sup_𝑛≥𝑇^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑐_𝑖−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖=sup_𝑛≥𝑇^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑐_𝑖−1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖+1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖=sup_𝑛≥𝑇^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1s,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤maxsup_𝑛≥𝑇1^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖+𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+sup_𝑛≥𝑇1^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠+1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠f𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖,sup_{𝑇≤𝑛≤𝑇1}^∞_−1𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+sup_{𝑇≤𝑛≤𝑇1}^∞_−1𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤max2^∞_{𝑠=𝑇1+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+2^∞_{𝑠=𝑇1+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,_{∞𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞_{𝑠=𝑇+𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,^∞_{𝑠=𝑇1+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎_𝑠^𝑇2−1_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞_{𝑠=𝑇1+𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖+^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,2^∞_{𝑠=𝑇1+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+2^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝜀16+2^∞_{𝑠=𝑇1+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+2^{𝑇1+𝜏−1}_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖+𝜀𝜀16≤max4,5𝜀8<𝜀, (3.11) which gives that lim_𝜔→∞𝑆_𝐿𝑦^𝜔=𝑆_𝐿𝑦, that is, 𝑆_𝐿 is continuous in 𝐵(𝑀,𝑁).
In light of (3.1), (3.5), and (3.6), we get that for any 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁)𝑆_𝐿𝑦=sup_𝑛≥𝑇𝐿−^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤𝐿+^∞_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖1<𝐿+2{}≤1min𝑀+𝑁−𝐿,𝑁−𝑀+𝐿2(𝑀+𝑁+𝐿), (3.12) which implies that 𝑆_𝐿(𝐵(𝑀,𝑁)) is uniformly bounded.
Given 𝜀>0. Clearly (3.2) ensures that there exists 𝑇^∗>𝑇 satisfying ^∞_𝑠=𝑇∗1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜖2, (3.13) which together with (3.1) and (3.6) implies that for all 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁) and 𝑡₂>𝑡₁≥𝑇^∗𝑆_𝐿𝑦_𝑡2−𝑆_𝐿𝑦_𝑡1=^∞𝑡_𝑙=1^{2+2𝑙𝜏−1}_{𝑠=𝑡2+()𝜏2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖−^∞𝑡_𝑙=1^{1+2𝑙𝜏−1}_{𝑠=𝑡1+()𝜏2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤^∞𝑡_𝑙=1^{2+2𝑙𝜏−1}_{𝑠=𝑡2+()𝜏2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖+^∞𝑡_𝑙=1^{1+2𝑙𝜏−1}_{𝑠=𝑡1+()𝜏2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤2^∞_{𝑠=𝑇∗+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜖, (3.14) which yields that 𝑆_𝐿(𝐵(𝑀,𝑁)) is uniformly Cauchy. Thus Lemma 2.1 means that 𝑆_𝐿(𝐵(𝑀,𝑁)) is relatively compact. Consequently 𝑆_𝐿 is condensing in 𝐵(𝑀,𝑁).
It follows from Lemma 2.2 that 𝑆_𝐿 has a fixed point 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁), that is, 𝑦_𝑛=𝐿−^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇, (3.15) which yields that 𝑦_𝑛+𝑦_𝑛−𝜏=2𝐿−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖𝑎,∀𝑛≥𝑇+𝜏,𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛Δ𝑦_𝑛+𝑦_𝑛−𝜏=^∞_𝑖=𝑛𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇+𝜏, (3.16) which together with (3.3) implies that Δ𝑎𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛Δ𝑦_𝑛+𝑏_𝑛𝑦_𝑛−𝜏=−𝑓𝑛,𝑦_𝑓1𝑛,…,𝑦_𝑓𝑘𝑛+𝑐_𝑛,∀𝑛≥𝑇+𝜏, (3.17) that is, (1.8) has a bounded positive solution 𝑦∈𝐵(𝑀,𝑁).
Next we show that (1.8) has uncountably many bounded positive solutions in 𝐵(𝑀,𝑁). Let 𝐿₁,𝐿₂∈(𝑀−𝑁,𝑀+𝑁) and 𝐿₁≠𝐿₂. For every 𝜃∈{1,2}, we infer similarly that there exist a constant 𝑇_𝐿𝜃 and a mapping 𝑆_𝐿𝜃 satisfying (3.4)–(3.6), where 𝐿,  𝑇, and 𝑆_𝐿 are replaced by 𝐿_𝜃,𝑇_𝐿𝜃, and 𝑆_𝐿𝜃, respectively, and the mapping 𝑆_𝐿𝜃 has a fixed point 𝑦^𝜃={𝑦^𝜃_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁), which is a bounded positive solution of (1.8) in 𝐵(𝑀,𝑁), that is, 𝑦^𝜃_𝑛=𝐿_𝜃−^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{𝑠=𝑛+(2𝑙−1)𝜏}1𝑎𝑠,𝑦^𝜃_𝑎1𝑠,…,𝑦^𝜃_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜃_𝑓1𝑖,…,𝑦^𝜃_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇_𝐿𝜃. (3.18)Equation (3.2) ensures that there exists 𝑇_∗>max{𝑇_𝐿1,𝑇_𝐿2} satisfying ^∞_𝑠=𝑇∗1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝐿₁−𝐿₂4. (3.19) In order to show that the set of bounded positive solutions of (1.8) is uncountable, it is sufficient to prove that 𝑦¹≠𝑦². It follows from (3.1), (3.18), and (3.19) that for all 𝑛≥𝑇_∗𝑦¹_𝑛−𝑦²_𝑛=𝐿₁−^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦¹_𝑎1𝑠,…,𝑦¹_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦¹_𝑓1𝑖,…,𝑦¹_𝑓𝑘𝑖−𝑐_𝑖−𝐿₂+^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎𝑠,𝑦²_𝑎1𝑠,…,𝑦²_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦²_𝑓1𝑖,…,𝑦²_𝑓𝑘𝑖−𝑐_𝑖≥𝐿₁−𝐿₂−^∞_𝑙=1^{𝑛+2𝑙𝜏−1}_{()𝜏𝑠=𝑛+2𝑙−1}1𝑎_𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦¹_𝑓1𝑖,…,𝑦¹_𝑓𝑘𝑖+𝑐_𝑖+𝑓𝑖,𝑦²_𝑓1𝑖,…,𝑦²_𝑓𝑘𝑖+𝑐_𝑖≥𝐿₁−𝐿₂−2^∞_{𝑠=𝑇∗+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖>12𝐿₁−𝐿₂, (3.20) that is, 𝑦¹≠𝑦². This completes the proof.

Theorem 3.2. Assume that there exist two constants 𝑁 and 𝑀 with 𝑀>𝑁>0 and two positive sequences {𝑎_𝑛}_{𝑛∈ℕ𝑛0} and {𝑝_𝑛}_{𝑛∈ℕ𝑛0} satisfying (3.1) and ^∞∞_{𝑙=1𝑠=𝑛0+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠max𝑝_𝑖,𝑐_𝑖𝑏<+∞;_𝑛=−1eventually. (3.21)(3.22) Then (1.8) has uncountably many bounded positive solutions.

Proof. Let 𝐿∈(𝑀−𝑁,𝑀+𝑁). Firstly, we show that there exists a mapping 𝑆_𝐿∶𝐵(𝑀,𝑁)→𝑙^∞_𝛽 with 𝑆_𝐿(𝜕𝐵(𝑀,𝑁))⊆𝐵(𝑀,𝑁) such that 𝑆_𝐿 has a fixed point 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁), which is also a bounded positive solution of (1.8). In view of (3.21) and (3.22), we choose a sufficiently large integer 𝑇≥max{1,𝑛₀+𝜏+|𝛽|} such that 𝑏_𝑛=−1,∀𝑛≥𝑇;_{∞∞𝑙=1𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<12min{𝑀+𝑁−𝐿,𝑁−𝑀+𝐿}. (3.23)(3.24) Define a mapping 𝑆_𝐿∶𝐵(𝑀,𝑁)→𝑙^∞_𝛽 as follows: 𝑆_𝐿𝑦_𝑛=𝐿+∑^∞∞_𝑙=1∑_{𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖∑_=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖𝑆,𝑛≥𝑇_𝐿𝑦_𝑇,𝛽≤𝑛<𝑇 (3.25) for each 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁). It follows from (3.1), (3.24), and (3.25) that for every 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝜕𝐵(𝑀,𝑁)⊆𝐵(𝑀,𝑁) and 𝑛≥𝑇𝑆_𝐿𝑦_𝑛−𝑀=𝐿−𝑀+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤|𝐿−𝑀|+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤|𝐿−𝑀|+^∞∞_{𝑙=1𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<||+1𝐿−𝑀2{}≤𝑁min𝑀+𝑁−𝐿,𝑁−𝑀+𝐿2, (3.26) which means that 𝑆_𝐿≤𝑁𝑦−𝑀2<𝑁, (3.27) that is, 𝑆_𝐿(𝜕𝐵(𝑀,𝑁)⊆𝐵(𝑀,𝑁).
Now we prove that 𝑆_L is a continuous and condensing mapping in 𝐵(𝑀,𝑁). Put 𝑦^𝜔={𝑦^𝜔_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁) for each 𝜔∈ℕ and 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁) with lim_𝜔→∞𝑦^𝜔=𝑦. Let 𝜀>0. Using (3.21) and the continuity of 𝑎 and 𝑓, we conclude that there exist four positive integers 𝑇₁,  𝑇₂,  𝑇₃, and 𝑇₄ with 𝑇₃>𝑇,   𝑇₂>𝑇₃+𝑇₁𝜏 satisfying max^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,^∞_𝑙=𝑇1^{𝑇3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,_{𝑇1−1𝑇𝑙=13+𝑙𝜏−1s=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖<𝜀;16max^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^𝑇2−1_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖,_{𝑇1−1𝑇𝑙=13+𝑙𝜏−1𝑠=𝑇+𝑙𝜏}𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠𝑎²_𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜀16,∀𝜔≥𝑇₄. (3.28)(3.29) By virtue of (3.1) and (3.25)–(3.29), we infer that for each 𝜔≥𝑇₄𝑆_𝐿𝑦^𝜔−𝑆_𝐿𝑦=sup_𝑛≥𝑇^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑐_𝑖−^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖=sup_𝑛≥𝑇^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤maxsup_𝑛≥𝑇3^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖+𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+sup_𝑛≥𝑇3^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠+1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖,sup_{𝑇≤𝑛≤𝑇3}^∞₋₁^∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+sup_{𝑇≤𝑛≤𝑇3}^∞₋₁^∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟s×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤max2^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+2^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,_{∞∞𝑙=1𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞∞_{𝑙=1𝑠=𝑇+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^𝑇2−1_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞_𝑙=𝑇1^{𝑇3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖+^∞_𝑙=𝑇1^{𝑇3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,2^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝜀16+2^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+2^∞_𝑙=𝑇1^{𝑇3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+2^∞∞_{𝑙=1𝑠=𝑇3+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+𝜀16+2^{𝑇1−1𝑇}_𝑙=1^{3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖+2^∞_𝑙=𝑇1^{𝑇3+𝑙𝜏−1}_{𝑠=𝑇+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀<max4,7𝜀8<𝜀, (3.30) which implies that lim_𝜔→∞𝑆_𝐿𝑦^𝜔=𝑆_𝐿𝑦, that is, 𝑆_𝐿 is continuous in 𝐵(𝑀,𝑁).
From (3.1), (3.24), and (3.25), we infer that for any 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁)𝑆_𝐿𝑦=sup_𝑛≥𝑇𝐿+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤𝐿+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖1<𝐿+2{}≤1min𝑀+𝑁−𝐿,𝑁−𝑀+𝐿2(𝑀+𝑁+𝐿), (3.31) which implies that 𝑆_𝐿(𝐵(𝑀,𝑁)) is uniformly bounded.
Let 𝜀>0. It follows from (3.21) that there exists 𝑇^∗>𝑇 satisfying ^∞∞_{𝑙=1𝑠=𝑇∗+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜀2, (3.32) which together with (3.1) and (3.25) yields that for all 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁) and 𝑡₂>𝑡₁≥𝑇^∗𝑆_𝐿𝑦_𝑡2−𝑆_𝐿𝑦_𝑡1=^∞∞_{𝑙=1𝑠=𝑡2+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖−^∞∞_{𝑙=1𝑠=𝑡1+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤^∞∞_{𝑙=1𝑠=𝑡2+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖+^∞∞_{𝑙=1𝑠=𝑡1+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟s^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤2^∞∞_{𝑙=1𝑠=𝑇∗+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜀, (3.33) which gives that 𝑆_𝐿(𝐵(𝑀,𝑁)) is uniformly Cauchy. Hence Lemma 2.1implies that 𝑆_𝐿(𝐵(𝑀,𝑁)) is relatively compact, that is, 𝑆_𝐿 is condensing in 𝐵(𝑀,𝑁).
It is clear that Lemma 2.2 means that 𝑆_𝐿 possesses a fixed point 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁), that is, 𝑦_𝑛=𝐿+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖𝑦,∀𝑛≥𝑇,_𝑛−𝜏=𝐿+^∞∞_{𝑙=1𝑠=𝑛+(𝑙−1)𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇+𝜏 (3.34) which lead to 𝑦_𝑛−𝑦_𝑛−𝜏=−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖Δ𝑦,∀𝑛≥𝑇+𝜏,_𝑛−𝑦_𝑛−𝜏=1𝑎𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛^∞_𝑖=𝑛𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇+𝜏, (3.35) which together with (3.23) yields that Δ𝑎𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛Δ𝑦_𝑛+𝑏_𝑛𝑦_𝑛−𝜏=−𝑓𝑛,𝑦_𝑓1𝑛,…,𝑦_𝑓𝑘𝑛+𝑐_𝑛,∀𝑛≥𝑇+𝜏, (3.36) that is, (1.8) has a bounded positive solution in 𝐵(𝑀,𝑁).
Next we show that (1.8) has uncountably many bounded positive solutions in 𝐵(𝑀,𝑁). Let 𝐿₁,𝐿₂∈(𝑀−𝑁,𝑀+𝑁) and 𝐿₁≠𝐿₂. Similarly we infer that for each 𝜃∈{1,2}, there exist a constant 𝑇_𝐿𝜃 and a mapping 𝑆_𝐿𝜃 satisfying (3.23)–(3.25), where 𝐿,  𝑇 and 𝑆_𝐿 are replaced by 𝐿_𝜃,𝑇_𝐿𝜃, and 𝑆_𝐿𝜃, respectively, and the mapping 𝑆_𝐿𝜃 has a fixed point 𝑦^𝜃={𝑦^𝜃_𝑛}_{𝑛∈ℤ𝛽}∈𝐵(𝑀,𝑁), which is a bounded positive solution of (1.8) in 𝐵(𝑀,𝑁), that is, 𝑦^𝜃_𝑛=𝐿_𝜃+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦^𝜃_𝑎1𝑠,…,𝑦^𝜃_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜃_𝑓1𝑖,…,𝑦^𝜃_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇_𝐿𝜃. (3.37) It follows from (3.21) that there exists 𝑇_∗>max{𝑇_𝐿1,𝑇_𝐿2} such that ^∞∞_{𝑙=1𝑠=𝑇∗+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝐿₁−𝐿₂4. (3.38) In order to show that the set of bounded positive solutions of (1.8) is uncountable, it is sufficient to prove that 𝑦¹≠𝑦². By means of (3.1), (3.37) and (3.38), we infer that for each 𝑛≥𝑇_∗𝑦¹_𝑛−𝑦²_𝑛=𝐿₁+^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦¹_𝑎1𝑠,…,𝑦¹_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦¹_𝑓1𝑖,…,𝑦¹_𝑓𝑘𝑖−𝑐_𝑖−𝐿₂−^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎𝑠,𝑦²_𝑎1𝑠,…,𝑦²_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦²_𝑓1𝑖,…,𝑦²_𝑓𝑘𝑖−𝑐_𝑖≥𝐿₁−𝐿₂−^∞∞_{𝑙=1𝑠=𝑛+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦¹_𝑓1𝑖,…,𝑦¹_𝑓𝑘𝑖+𝑐_𝑖+𝑓𝑖,𝑦²_𝑓1𝑖,…,𝑦²_𝑓𝑘𝑖+𝑐_𝑖≥𝐿₁−𝐿₂−2^∞∞_{𝑙=1𝑠=𝑇∗+𝑙𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖>12𝐿₁−𝐿₂, (3.39) that is, 𝑦¹≠𝑦². This completes the proof.

Next we use the Leray-Schauder nonlinear alternative theorem to show the existence and multiplicity of bounded positive solutions of (1.8).

Theorem 3.3. Assume that there exist four constants 𝑁,  𝑀,  𝑏_∗, and 𝑏^∗ and two positive sequences {𝑎_𝑛}_{𝑛∈ℕ𝑛0} and {𝑝_𝑛}_{𝑛∈ℕ𝑛0} satisfying (3.1), (3.2) and 0<𝑁<1−𝑏_∗−𝑏^∗𝑀,𝑏_∗≥0,𝑏^∗≥0,𝑏_∗+𝑏^∗<1,−𝑏_∗≤𝑏_𝑛≤𝑏^∗eventually. (3.40) Then (1.8) has uncountably many bounded positive solutions in 𝑈(𝑀).

Proof. Let 𝐿∈(𝑏^∗𝑀+𝑁,(1−𝑏_∗)𝑀). Now we prove that there exists a mapping 𝑆_𝐿∶𝑈(𝑀)→𝑉(𝑁) such that it has a fixed point 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀), which is also a bounded positive solution of (1.8). It follows from (3.2), (3.40) and that there exists a sufficiently large number 𝑇≥max{1,𝑛₀+𝜏+|𝛽|} satisfying −𝑏_∗≤𝑏_𝑛≤𝑏^∗,∀𝑛≥𝑇;_∞𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<min𝐿−𝑏^∗𝑀−𝑁,1−𝑏_∗𝑀−𝐿. (3.41)(3.42) Put 𝑝^∗=𝑀−𝜀^∗, where 𝜀^∗∈(0,min{𝐿−𝑏^∗𝑀−𝑁,(1−𝑏_∗)𝑀−𝐿,(𝑀−𝑁)/2}) is enough small and ^∞_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<min𝐿−𝑏^∗𝑀−𝑁,1−𝑏_∗𝑀−𝐿−𝜀^∗. (3.43) Obviously, 𝑝^∗∈𝑈(𝑀). Define a mapping 𝑆_𝐿∶𝑈(𝑀)→𝑙^∞_𝛽 by 𝑆_𝐿𝑦_𝑛=𝑆_1𝐿𝑦_𝑛+𝑆_2𝐿𝑦_𝑛,𝑛≥𝛽 (3.44) for each 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀), where the mappings 𝑆_1𝐿,𝑆_2𝐿∶𝑈(𝑀)→𝑙^∞_𝛽 are defined by 𝑆_1𝐿𝑦_𝑛=𝐿−𝑏_𝑛𝑦_𝑛−𝜏𝑆,𝑛≥𝑇_1𝐿𝑦_𝑇𝑆,𝛽≤𝑛<𝑇,_2𝐿𝑦_𝑛=−∑^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖∑_=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖𝑆,𝑛≥𝑇_2𝐿𝑦_𝑇,𝛽≤𝑛<𝑇. (3.45)(3.46) It follows from (3.1), (3.41), and (3.43)–(3.46) that for any 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) and 𝑛≥𝑇𝑆_𝐿𝑦_𝑛=𝑆_1𝐿𝑦_𝑛+𝑆_2𝐿𝑦_𝑛=𝐿−𝑏_𝑛𝑦_𝑛−𝜏−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≥𝐿−𝑏^∗𝑀−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖≥𝐿−𝑏^∗𝑀−^∞_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖>𝐿−𝑏^∗𝑀−min𝐿−𝑏^∗𝑀−𝑁,1−𝑏_∗𝑀−𝐿+𝜀^∗≥𝑁+𝜀^∗>𝑁, (3.47) which yields that 𝑆_𝐿(𝑈(𝑀))⊆𝑉(𝑁).
Next we show that 𝑆_2𝐿∶𝑈(𝑀)→𝑙^∞_𝛽 is a continuous and relatively compact mapping. Let 𝑦^𝜔={𝑦^𝜔_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) and 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) with lim_𝜔→∞𝑦^𝜔=𝑦. By virtue of (3.2) and the continuity of 𝑎 and 𝑓, we infer that there exist 𝑇₁,𝑇₂,𝑇₃∈ℕ with 𝑇₂>𝑇₁−1>𝑇 satisfying max^∞_𝑠=𝑇11𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,^𝑇1−1_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖<𝜀16;max^𝑇1−1_𝑠=𝑇1𝑎_𝑠^𝑇2−1_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖,_{𝑇1−1𝑠=𝑇}𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠𝑎²_𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜀16,∀𝜔≥𝑇₃. (3.48)(3.49) It follows from (3.1) and (3.46)–(3.49) that for each 𝜔≥𝑇₃𝑆_2𝐿𝑦^𝜔−𝑆_2𝐿𝑦=sup_𝑛≥𝑇^∞_𝑠=𝑛1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑐_𝑖−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖=sup_𝑛≥𝑇^∞_𝑠=𝑛1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞_𝑠=𝑛1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤maxsup_𝑛≥𝑇1^∞_𝑠=𝑛1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖+𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+sup_𝑛≥𝑇1^∞_𝑠=𝑛1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠+1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖,sup_{𝑇≤𝑛≤𝑇1}^∞_{−1𝑠=𝑛}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+sup_{𝑇≤𝑛≤𝑇1}^∞_{−1𝑠=𝑛}1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠×^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤max2^∞_𝑠=𝑇11𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+2^∞_𝑠=𝑇11𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖,_∞𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,𝑦^𝜔_𝑓2𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,𝑦_𝑓2𝑖,…,𝑦_𝑓𝑘𝑖+^∞_𝑠=𝑇1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,^𝑇1−1_𝑠=𝑇1𝑎_𝑠^𝑇2−1_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^𝑇1−1_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑇2𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^∞_𝑠=𝑇11𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜔_𝑓1𝑖,…,𝑦^𝜔_𝑓𝑘𝑖−𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+^𝑇1−1_𝑠=𝑇1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+^𝑇1−1_𝑠=𝑇1𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖+^∞_𝑠=𝑇11𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,𝜀16+2^𝑇1−1_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+2^∞_𝑠=𝑇11𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+^𝑇1−1_𝑠=𝑇𝑎𝑠,𝑦^𝜔_𝑎1𝑠,…,𝑦^𝜔_𝑎𝑟𝑠−𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠𝑎²_𝑠^𝑇2−1_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+2^𝑇1−1_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑇2𝑝_𝑖+𝑐_𝑖+2^∞_𝑠=𝑇11𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖𝜀≤max4,5𝜀8<𝜀, (3.50) which yields that lim_𝜔→∞‖𝑆_2𝐿𝑦^𝜔−𝑆_2𝐿𝑦‖=0, that is, 𝑆_2𝐿 is continuous in 𝑈(𝑀).
In light of (3.1) and (3.43)–(3.46), we deduce that for all 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀)𝑆_𝐿𝑦=sup_𝑛≥𝑇𝑆_1𝐿𝑦_𝑛+𝑆_2𝐿𝑦_𝑛≤sup_𝑛≥𝑇𝑆_1𝐿𝑦_𝑛+sup_𝑛≥𝑇𝑆_2𝐿𝑦_𝑛≤sup_𝑛≥𝑇𝐿−𝑏_𝑛𝑦_𝑛−𝜏+sup_𝑛≥𝑇^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤sup_𝑛≥𝑇𝐿+𝑏_𝑛𝑦_𝑛−𝜏+sup^∞_{𝑛≥𝑇𝑠=𝑛}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤𝐿+𝑏_∗+𝑏^∗𝑀+^∞_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖≤𝐿+𝑀+min𝐿−𝑏^∗𝑀−𝑁,1−𝑏_∗𝑀−𝐿−𝜀^∗<2𝐿+𝑀, (3.51) which means that 𝑆_𝐿(𝑈(𝑀)) and 𝑆_2𝐿(𝑈(𝑀)) are bounded.
Let 𝜀>0. Notice that (3.2) ensures that there exists 𝑇^∗>𝑇 satisfying ^∞_𝑠=𝑇∗1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜖2, (3.52) which together with (3.1) and (3.46) implies that for all 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) and 𝑡₂>𝑡₁≥𝑇^∗𝑆_2𝐿𝑦_𝑡2−𝑆_2𝐿𝑦_𝑡1=^∞_𝑠=𝑡21𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖−^∞_𝑠=𝑡11𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤^∞_𝑠=𝑡21𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖+^∞_𝑠=𝑡11𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖≤2^∞_𝑠=𝑇∗1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝜖, (3.53) which means that 𝑆_2𝐿(𝑈(𝑀)) is uniformly Cauchy. Thus 𝑆_2𝐿(𝑈(𝑀)) is relatively compact.
By virtue of (3.41) and (3.45), we infer that for all 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽},𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) and 𝑛≥𝑇𝑆_1𝐿𝑥_𝑛−𝑆_1𝐿𝑦_𝑛=𝑏_𝑛𝑥_𝑛−𝜏−𝑦_𝑛−𝜏≤𝑏_∗+𝑏^∗𝑥−𝑦, (3.54) which yields that 𝑆_1𝐿𝑥−𝑆_1𝐿𝑦≤𝑏_∗+𝑏^∗𝑥−𝑦, (3.55) that is, 𝑆_1𝐿 is a contraction mapping in 𝑈(𝑀). It follows that 𝑆_𝐿 is a continuous and condensing mapping.
Put 𝑃=𝑦=𝑦_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽∶𝑁≤𝑦_𝑛𝑦,𝑦≤𝑀,𝑛≥𝛽,=𝑀𝑄=𝑦=_{𝑛𝑛∈ℤ𝛽}∈𝑙^∞_𝛽∶𝑁≤𝑦_𝑛≤𝑀,𝑛≥𝛽andthereexists𝑛^∗≥𝛽satisfying𝑦_𝑛∗=𝑁. (3.56)(3.57) It is easy to see that 𝜕𝑈(𝑀)=𝑃∪𝑄. Suppose that there exist 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝜕𝑈(𝑀) and 𝜆∈(0,1) with 𝑦=(1−𝜆)𝑝^∗+𝜆𝑆_𝐿𝑦. (3.58) Now we consider two possible cases as follows.
Case  1. Let 𝑦∈𝑃. Obviously (3.41), (3.43)–(3.46), (3.56), and (3.58) guarantee that 𝑦_𝑛=(1−𝜆)𝑝^∗+𝜆𝑆_𝐿𝑦_𝑛=(1−𝜆)𝑝^∗+𝜆𝐿−𝑏_𝑛𝑦_𝑛−𝜏−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤(1−𝜆)𝑀−𝜀^∗+𝜆𝐿+𝑏_∗𝑀+^∞_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<(1−𝜆)𝑀−𝜀^∗+𝜆𝐿+𝑏_∗𝑀+min𝐿−𝑏^∗𝑀−𝑁,1−𝑏_∗𝑀−𝐿−𝜀^∗≤𝑀−𝜀^∗,∀𝑛≥𝑇, (3.59) which implies that 𝑀=𝑦=sup_𝑛≥𝛽𝑦_𝑛≤𝑀−𝜀^∗<𝑀, (3.60) which is a contradiction.
Case  2. Let 𝑦∈𝑄. It follows from (3.41), (3.43)–(3.46), (3.57), and (3.58) that 𝑁=𝑦_𝑛∗=(1−𝜆)𝑝^∗+𝜆𝑆_𝐿𝑦_𝑛∗=(1−𝜆)𝑝^∗+𝜆𝐿−𝑏_𝑛∗𝑦_{𝑛∗−𝜏}−^∞_{{𝑛𝑠=max∗},𝑇}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≥(1−𝜆)𝑀−𝜀^∗+𝜆𝐿−𝑏^∗𝑀−^∞_{{𝑛𝑠=max∗},𝑇}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖>(1−𝜆)𝑀−𝜀^∗+𝜆𝐿−𝑏^∗𝑀−min𝐿−𝑏^∗𝑀−𝑁,1−𝑏_∗𝑀−𝐿+𝜀^∗≥(1−𝜆)𝑀−𝜀^∗+𝜆𝑁+𝜀^∗≥min𝑀−𝜀^∗,𝑁+𝜀^∗=𝑁+𝜀^∗, (3.61) which is absurd. Thus Lemma 2.3 ensures that there exists 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) satisfying 𝑆_𝐿𝑦=𝑆_1𝐿𝑦+𝑆_2𝐿𝑦=𝑦, that is, 𝑦_𝑛=𝐿−𝑏_𝑛𝑦_𝑛−𝜏−^∞_𝑠=𝑛1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑥_ℎ𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇, (3.62) which means that 𝑎𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛Δ𝑦_𝑛+𝑏_𝑛𝑦_𝑛−𝜏=^∞_𝑖=𝑛𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇, (3.63) which yields that Δ𝑎𝑛,𝑦_𝑎1𝑛,…,𝑦_𝑎𝑟𝑛Δ𝑦_𝑛+𝑏_𝑛𝑦_𝑛−𝜏=−𝑓𝑛,𝑦_𝑓1𝑛,…,𝑦_𝑓𝑘𝑛+𝑐_𝑛,∀𝑛≥𝑇, (3.64) that is, 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) is a bounded positive solution of (1.8).
Finally we show that (1.8) has uncountably many bounded positive solutions in 𝑈(𝑀). Let 𝐿₁,𝐿₂∈(𝑏^∗𝑀+𝑁,(1−𝑏_∗)𝑀) and 𝐿₁≠𝐿₂. Similarly we infer that for each 𝜃∈{1,2}, there exists a mapping 𝑆_𝐿𝜃∶𝑈(𝑀)→𝑉(𝑁) satisfying (3.41)–(3.46), where 𝐿,  𝛽,  𝑇,  𝑆_1𝐿,  𝑆_2𝐿, and 𝑆_𝐿 are replaced by 𝐿_𝜃,  𝛽_𝜃,  𝑇_𝐿𝜃,  𝑆_1𝐿𝜃,  𝑆_2𝐿𝜃 and 𝑆_𝐿𝜃, respectively, and the mapping 𝑆_𝐿𝜃 has a fixed point 𝑦^𝜃={𝑦^𝜃_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀), which is a bounded positive solution of (1.8) in 𝑈(𝑀), that is, 𝑦^𝜃_𝑛=𝐿_𝜃−𝑏_𝑛𝑦^𝜃_𝑛−𝜏−^∞_𝑠=𝑛1𝑎𝑠,𝑦^𝜃_𝑎1𝑠,…,𝑦^𝜃_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦^𝜃_𝑓1𝑖,…,𝑦^𝜃_𝑓𝑘𝑖−𝑐_𝑖,∀𝑛≥𝑇_𝐿𝜃. (3.65) It follows from (3.2) that there exists 𝑇_∗>max{𝑇_𝐿1,𝑇_𝐿2} satisfying ^∞_𝑠=𝑇∗1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝐿₁−𝐿₂4. (3.66) In order to prove that the set of bounded positive solutions of (1.8) is uncountable, it is sufficient to verify that 𝑦¹≠𝑦². In terms of (3.1), (3.65), and (3.66), we deduce that for 𝑛≥𝑇_∗𝑦¹_𝑛−𝑦²_𝑛=𝐿₁−𝐿₂−𝑏_𝑛𝑦¹_𝑛−𝜏+𝑏_𝑛𝑦²_𝑛−𝜏−^∞_𝑠=𝑛1𝑎𝑠,𝑦¹_𝑎1𝑠,…,𝑦¹_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦¹_𝑓1𝑖,…,𝑦¹_𝑓𝑘𝑖−𝑐_𝑖+^∞_𝑠=𝑛1𝑎𝑠,𝑦²_𝑎1𝑠,…,𝑦²_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦²_𝑓1𝑖,…,𝑦²_𝑓𝑘𝑖−𝑐_𝑖≥𝐿₁−𝐿₂−𝑏_𝑛𝑦¹_𝑛−𝜏−𝑦²_𝑛−𝜏−^∞_𝑠=𝑛1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦¹_𝑓1𝑖,…,𝑦¹_𝑓𝑘𝑖+𝑐_𝑖+𝑓𝑖,𝑦²_𝑓1𝑖,𝑦²_𝑓2𝑖,…,𝑦²_𝑓𝑘𝑖+𝑐_𝑖≥𝐿₁−𝐿₂−𝑏_∗+𝑏^∗𝑦¹−𝑦²−2^∞_𝑠=𝑇∗1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖>12𝐿₁−𝐿₂−𝑏_∗+𝑏^∗𝑦¹−𝑦², (3.67) which means that 𝑦¹−𝑦²>𝐿₁−𝐿₂21+𝑏_∗+𝑏^∗>0, (3.68) that is, 𝑦¹≠𝑦². This completes the proof.

Theorem 3.4. Assume that there exist four constants 𝑁,  𝑀,  𝑏_∗, and 𝑏^∗ and two positive sequences {𝑎_𝑛}_{𝑛∈ℕ𝑛0} and {𝑝_𝑛}_{𝑛∈ℕ𝑛0} satisfying (3.1), (3.2) and 1+𝑏^∗𝑀<1+𝑏_∗𝑁<0,𝑏_∗≤𝑏_𝑛≤𝑏^∗<−1eventually. (3.69) Then (1.8) has uncountably many bounded positive solutions in 𝑈(𝑀).

Proof. Let 𝐿∈((1+𝑏^∗)𝑀,(1+𝑏_∗)𝑁). Now we show that there exists a mapping 𝑆_𝐿∶𝑈(𝑀)→𝑉(𝑁) such that it has a fixed point 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀), which is also a bounded positive solution of (1.8). It follows from (3.2) and (3.69) that there exists 𝑇≥max{1,𝑛₀+𝜏+|𝛽|} satisfying 𝑏_∗≤𝑏_𝑛≤𝑏^∗<−1,∀𝑛≥𝑇;_∞𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<min𝐿−1+𝑏^∗𝑏𝑀,^∗𝑏_∗𝑁1+𝑏_∗−𝐿. (3.70)(3.71) Let 𝑝^∗=𝑀−𝜀^∗, where 𝜀^∗∈(0,min{𝐿−(1+𝑏^∗)𝑀,(𝑏^∗/𝑏_∗)[𝑁(1+𝑏_∗)−𝐿,𝑏^∗(𝑀−𝑁)/(𝑏^∗−1)]}) is enough small and ^∞_𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<min𝐿−1+𝑏^∗𝑏𝑀,^∗𝑏_∗𝑁1+𝑏_∗−𝐿−𝜀^∗. (3.72) Obviously, 𝑝^∗∈𝑈(𝑀). Define a mapping 𝑆_𝐿∶𝑈(𝑀)→𝑙^∞_𝛽 by (3.44), where the mappings 𝑆_1𝐿,𝑆_2𝐿∶𝑈(𝑀)→𝑙^∞_𝛽 are defined by 𝑆_1𝐿𝑦_𝑛=𝐿𝑏_𝑛+𝜏−𝑦_𝑛+𝜏𝑏_𝑛+𝜏𝑆,𝑛≥𝑇_1𝐿𝑦_𝑇𝑆,𝛽≤𝑛<𝑇,_2𝐿𝑦_𝑛=−1𝑏^∞_{𝑛+𝜏𝑠=}∑_𝑛+𝜏1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖∑_=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖𝑆,𝑛≥𝑇_2𝐿𝑦_𝑇,𝛽≤𝑛<𝑇 (3.73)(3.74) for each 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀). By virtue of (3.1), (3.70), and (3.72)–(3.74), we get that for any 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝑈(𝑀) and 𝑛≥𝑇𝑆_𝐿𝑦_𝑛=𝑆_1𝐿𝑦_𝑛+𝑆_2𝐿𝑦_𝑛=𝐿𝑏_𝑛+𝜏−𝑦_𝑛+𝜏𝑏_𝑛+𝜏−1𝑏^∞_{𝑛+𝜏𝑠=𝑛+𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≥𝐿𝑏_∗−𝑁𝑏_∗+1𝑏^∗∞_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖≥𝐿𝑏_∗−𝑁𝑏_∗+1𝑏^∗min𝐿−1+𝑏^∗𝑏𝑀,^∗𝑏_∗𝑁1+𝑏_∗−1−𝐿𝑏^∗𝜀^∗1≥𝑁−𝑏^∗𝜀^∗>𝑁, (3.75) which gives that 𝑆_𝐿(𝑈(𝑀))⊆𝑉(𝑁). The rest of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof.

Now we employ the Krasnoselskii fixed point theorem to prove the existence and multiplicity of bounded positive solutions of (1.8).

Theorem 3.5. Assume that there exist four constants 𝑁,  𝑀,  𝑏_∗ and 𝑏^∗ and two positive sequences {𝑎_𝑛}_{𝑛∈ℕ𝑛0},  {𝑝_𝑛}_{𝑛∈ℕ𝑛0} satisfying (3.1), (3.2) and 0<𝑁𝑏^∗𝑏_∗<𝑀𝑏²_∗−𝑏^∗,1<𝑏_∗≤𝑏_𝑛≤𝑏^∗<𝑏²_∗eventually. (3.76) Then (1.8) has uncountably many bounded positive solutions in 𝐴(𝑁,𝑀).

Proof. Let 𝐿∈((𝑏^∗/𝑏_∗)𝑀+𝑏^∗𝑁,𝑏_∗𝑀). Now we show that there exist two mappings 𝑆_1𝐿,𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛽 such that the equation 𝑆_1𝐿𝑦+𝑆_2𝐿𝑦=𝑦 has a solution 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐴(𝑁,𝑀), which is also a bounded positive solution of (1.8). It follows from (3.2) and (3.76) that there exists 𝑇≥max{1,𝑛₀+𝜏+|𝛽|} satisfying 1<𝑏_∗≤𝑏_𝑛≤𝑏^∗<𝑏²_∗,∀𝑛≥𝑇;_∞𝑠=𝑇1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<min𝑏_∗𝑏𝑀−𝐿,^∗𝐿𝑏^∗−𝑀−𝑏_∗𝑁. (3.77)(3.78) Define two mappings 𝑆_1𝐿 and 𝑆_2𝐿∶𝐴(𝑁,𝑀)→𝑙^∞_𝛽 by (3.73) and (3.74), respectively. It follows from (3.1), (3.73), (3.74), (3.77), and (3.78) that for any 𝑥={𝑥_𝑛}_{𝑛∈ℤ𝛽}, 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐴(𝑁,𝑀) and 𝑛≥𝑇𝑆_1𝐿𝑥_𝑛−𝑆_1𝐿𝑦_𝑛=1𝑏_𝑛+𝜏𝑥_𝑛+𝜏−𝑦_𝑛+𝜏≤1𝑏_∗,𝑆𝑥−𝑦_1𝐿𝑥_𝑛+𝑆_2𝐿𝑦_𝑛=𝐿𝑏_𝑛+𝜏−𝑥_𝑛+𝜏𝑏_𝑛+𝜏−1𝑏^∞_{𝑛+𝜏𝑠=𝑛+𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≥𝐿𝑏^∗−𝑀𝑏_∗−1𝑏_∗^∞_{𝑠=𝑛+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≥𝐿𝑏^∗−𝑀𝑏_∗−1𝑏_∗^∞_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖>𝐿𝑏^∗−𝑀𝑏_∗−1𝑏_∗min𝑏_∗𝑏𝑀−𝐿,^∗𝐿𝑏^∗−𝑀−𝑏_∗𝑁𝑆≥𝑁,_1𝐿𝑥_𝑛+𝑆_2𝐿𝑦_𝑛=𝐿𝑏_𝑛+𝜏−𝑥_𝑛+𝜏𝑏_𝑛+𝜏−1𝑏^∞_{𝑛+𝜏𝑠=𝑛+𝜏}1𝑎𝑠,𝑦_𝑎1𝑠,…,𝑦_𝑎𝑟𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖−𝑐_𝑖≤𝐿𝑏_∗+1𝑏_∗^∞_{𝑠=𝑛+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑓𝑖,𝑦_𝑓1𝑖,…,𝑦_𝑓𝑘𝑖+𝑐_𝑖≤𝐿𝑏_∗+1𝑏_∗^∞_{𝑠=𝑇+𝜏}1𝑎_𝑠^∞_𝑖=𝑠𝑝_𝑖+𝑐_𝑖<𝐿𝑏_∗+1𝑏_∗min𝑏_∗𝑏𝑀−𝐿,^∗𝐿𝑏^∗−𝑀−𝑏_∗𝑁≤𝑀, (3.79) which yield that 𝑆_1𝐿𝑥−𝑆_1𝐿𝑦≤1𝑏_∗𝑥−𝑦,𝑆_1𝐿𝑥+𝑆_2𝐿𝑦∈𝐴(𝑁,𝑀),∀𝑥,𝑦∈𝐴(𝑁,𝑀). (3.80) As in the proof of Theorem 3.3, we infer similarly that 𝑆_2𝐿 is continuous in 𝐴(𝑁,𝑀) and 𝑆_2𝐿(𝐴(𝑁,𝑀)) is relatively compact. Thus 𝑆_2𝐿 is completely continuous, which together with (3.77), (3.80), and Lemma 2.4, ensures that the equation 𝑆_1𝐿𝑦+𝑆_2𝐿𝑦=𝑦 has a solution 𝑦={𝑦_𝑛}_{𝑛∈ℤ𝛽}∈𝐴(𝑁,𝑀), which is also a bounded positive solution of (1.8) in 𝐴(𝑁,𝑀). The rest of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof.

Remark 3.6. Theorems 3.1–3.5 extend and improve Theorem  2.1 in [1] and Theorems 2.1–2.7 in [7], respectively. The examples in Section 4 show that our results are indeed generalizations of the corresponding results in [1, 7].

4. Examples

Now we construct five examples to show the applications of the results presented in Section 3. Note that none of the known results can be applied to the five examples.

Example 4.1. Consider the second-order nonlinear neutral delay difference equation Δ(−1)^𝑛𝑛⁶−𝑛⁵+1𝑦²_𝑛+1Δ𝑦_𝑛+𝑦_𝑛−𝜏+−3𝑛³𝑦⁴_𝑛+1+√𝑛𝑦_𝑛+1+𝑦_𝑛+1−1^4/5𝑛⁵ln²(𝑛+2)+𝑦_𝑛2−𝑛²³=()+1−1^𝑛𝑛²−5𝑛+1𝑛⁴+𝑛²+sin√𝑛³+1,∀𝑛≥1, (4.1) where 𝜏∈ℕ is fixed. Let 𝑛₀=1,  𝑟=1,  𝑘=2,  𝑁=2,  𝑀=3,  𝛽=1−𝜏 and 𝑎_1𝑛=𝑛,𝑓_1𝑛=𝑛+1,𝑓_2𝑛=𝑛²,𝑏_𝑛=1,𝑐_𝑛=(−1)^𝑛𝑛²−5𝑛+1𝑛⁴+𝑛²+sin√𝑛³,𝑎()=()+1𝑛,𝑢−1^𝑛𝑛⁶−𝑛⁵+1𝑢²+1,𝑎_𝑛=3𝑛⁶−𝑛⁵,𝑓()=+1𝑛,𝑢,𝑣−3𝑛³𝑢⁴+√𝑛𝑢+(𝑢−1)^4/5𝑛⁵ln²(𝑛+2)+𝑣−𝑛²³+1,𝑝_𝑛=243𝑛³+3√𝑛+2𝑛⁵,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (4.2) It is easy to verify that (3.1)–(3.3) hold. Thus Theorem 3.1 guarantees that (4.1) has uncountably many bounded positive solutions in 𝐵(𝑀,𝑁). But the results in [1, 7] are not applicable for (4.1).

Example 4.2. Consider the second-order nonlinear neutral delay difference equation Δ𝑛⁵√𝑛+2+𝑦²_𝑛2−60𝑦⁴_2𝑛+(−1)^𝑛Δ𝑦_𝑛−𝑦_𝑛−𝜏+𝑛³𝑦_𝑛+4−2⁹−𝑛³−2𝑛+1𝑦³_2𝑛−1𝑛⁵+𝑦⁴_𝑛2−3−3𝑛²+sin𝑛𝑦_𝑛+4=𝑛+2₅−6𝑛³√2𝑛−4−5𝑛¹¹+4,∀𝑛≥5, (4.3) where 𝜏∈ℕ is fixed. Let 𝑛₀=5,  𝑟=2,  𝑘=3,  𝑁=1,  𝑀=3,  𝛽=min{5−𝜏,−35} and 𝑎_1𝑛=𝑛²−60,𝑎_2𝑛=2𝑛+(−1)^𝑛,𝑓_1𝑛=𝑛+4,𝑓_2𝑛𝑓=2𝑛−1,_3𝑛=𝑛²−3,𝑏_𝑛=−1,𝑐_𝑛=𝑛⁵−6𝑛³√2𝑛−4−5𝑛¹¹()+4,𝑎𝑛,𝑢,𝑣=𝑛⁵√𝑛+2+𝑢²𝑣⁴,𝑎_𝑛=𝑛⁵√()=𝑛𝑛+2+1,𝑓𝑛,𝑢,𝑣,𝑤³(𝑢−2)⁹−𝑛³−2𝑛+1𝑣³𝑛⁵+𝑤⁴−3𝑛²(),𝑝+sin𝑛𝑢+2_𝑛=28𝑛³+54𝑛+27𝑛⁵,∀(𝑛,𝑢,𝑣,𝑤)∈ℕ_𝑛0×ℝ³. (4.4) It is clear that (3.1), (3.21), and (3.22) hold. Hence Theorem 3.2 ensures that (4.3) has uncountably many bounded positive solutions in 𝐵(𝑀,𝑁). But the results in [1, 7] are not valid for (4.3).

Example 4.3. Consider the second-order nonlinear neutral delay difference equation Δ(−1)^{𝑛(𝑛−1)/2}𝑛³+2𝑦³_3𝑛+𝑛²𝑦²_𝑛−8+1Δ𝑥_𝑛+(−1)^𝑛𝑛⁷−13𝑛⁷𝑥+1^𝑛−𝜏+𝑥_𝑛−5−1^1/3+𝑥²_{𝑛()𝑛+1/2}−2𝑛ln²=𝑛𝑛+1₆−30𝑛⁴+8cos⁵4𝑛⁸−1𝑛⁹+𝑛⁵+2ln²𝑛,∀𝑛≥3, (4.5) where 𝜏∈ℕ is fixed. Let 𝑛₀=3,  𝑟=𝑘=2,𝑏_∗=𝑏^∗=1/3,𝑁=2,𝑀=7,𝛽=min{3−𝜏,−5} and 𝑎_1𝑛=3𝑛,𝑎_2𝑛=𝑛−8,𝑓_1𝑛=𝑛−5,𝑓_2𝑛=𝑛(𝑛+1)2,𝑏_𝑛=(−1)^𝑛𝑛⁷−13𝑛⁷,𝑐+1_𝑛=𝑛⁶−30𝑛⁴+8cos⁵4𝑛⁸−1𝑛⁹+𝑛⁵+2ln²𝑛,𝑎(𝑛,𝑢,𝑣)=(−1)^{𝑛(𝑛−1)/2}𝑛³+2𝑢³+𝑛²𝑣²,𝑎+1_𝑛=8𝑛³+4𝑛²()=()+1,𝑓𝑛,𝑢,𝑣𝑢−1^1/3+𝑣²−2𝑛ln²,𝑝𝑛+1_𝑛=6^1/3+√47𝑛ln²𝑛,∀(𝑛,𝑢,𝑣)∈ℕ_𝑛0×ℝ². (4.6) It is clear that (3.1), (3.2) and (3.40) are satisfied. Hence Theorem 3.3 implies that (4.5) has uncountably many bounded positive solutions in 𝑈(𝑀). But the results in [1, 7] are unapplicable for (4.5).

Example 4.4. Consider the second-order nonlinear neutral delay difference equation Δ(−1)^𝑛−1𝑛²𝑦⁴_𝑛3−1+1Δ𝑥_𝑛−3𝑛²−3𝑛−3cos(𝑛/2−1)+1𝑛²()𝑥−𝑛−cos𝑛/2−1^𝑛−𝜏+3−𝑥_𝑛2−9^1/3−√𝑛−1𝑥³_2𝑛−3𝑛⁴+𝑛𝑥⁴_3𝑛−7=𝑛+1₃+(−1)^𝑛𝑛²+ln³1+𝑛²𝑛⁶+4𝑛⁵−3𝑛⁴+sin³4𝑛²−5+1,𝑛≥2, (4.7) where 𝜏∈ℕ is fixed. Let 𝑛₀=2,𝑟=1,𝑘=3,𝑏_∗=−4,𝑏^∗=−3,𝑁=1,𝑀=3,𝛽=min{2−𝜏,−5} and 𝑎_1𝑛=𝑛³−1,𝑓_1𝑛=𝑛²−9,𝑓_2𝑛=3𝑛−7,𝑓_3𝑛𝑏=2𝑛−3,_𝑛=−3𝑛²−3𝑛−3cos(𝑛/2−1)+1𝑛²−𝑛−cos(𝑛/2−1),𝑐_𝑛=𝑛³+(−1)^𝑛𝑛²+ln³1+𝑛²𝑛⁶+4𝑛⁵−3𝑛⁴+sin³4𝑛²,𝑎()=()−5+1𝑛,𝑢−1^𝑛−1𝑛²𝑢⁴+1,𝑎_𝑛=𝑛²ln³()=()𝑛,𝑓𝑛,𝑢,𝑣,𝑤3−𝑢^1/3−√2𝑛−3𝑣³𝑛⁴+𝑛𝑤⁴,𝑝+1_𝑛=2+27√2𝑛−3𝑛⁴,∀(𝑛,𝑢,𝑣,𝑤)∈ℕ_𝑛0×ℝ³. (4.8) It is easy to verify that (3.1), (3.2), and (3.69) hold. Hence Theorem 3.4 implies that (4.7) has uncountably many bounded positive solutions in 𝑈(𝑀). But the results in [1, 7] are not valid for (4.7).

Example 4.5. Consider the second-order nonlinear neutral delay difference equation Δ(−1)^{𝑛(𝑛−1)/2}(𝑛+1)⁵+𝑦⁸_2𝑛−3+2𝑛³𝑦²_𝑛2−1+ln1+𝑛𝑦_{𝑛(𝑛−1)}×Δ𝑥_𝑛+11ln1+𝑛²+11sin𝑛+10ln1+𝑛²𝑥+sin𝑛+1^𝑛−𝜏+𝑥_𝑛−5−2^3/4+𝑥_{𝑛(𝑛−2)}−2²𝑛²+3³+𝑥_𝑛−5−𝑛²𝑥_{𝑛(𝑛−2)}=𝑛+1₇−5𝑛⁴−9𝑛¹¹+7𝑛⁸−6𝑛⁷+5𝑛³+1,𝑛≥0, (4.9) where 𝜏∈ℕ is fixed. Let 𝑛₀=0,𝑟=3,𝑘=2,𝑏_∗=10,𝑏^∗=11,𝑁=2,𝑀=3,𝛽=min{−𝜏,−5} and 𝑎_1𝑛=2𝑛−3,𝑎_2𝑛=𝑛²−1,𝑎_3𝑛=𝑛(𝑛−1),𝑓_1𝑛=𝑛−5,𝑓_2𝑛(),𝑏=𝑛𝑛−2_𝑛=11ln1+𝑛²+11sin𝑛+10ln1+𝑛²+sin𝑛+1,𝑐_𝑛=𝑛⁷−5𝑛⁴−9𝑛¹¹+7𝑛⁸−6𝑛⁷+5𝑛³,𝑎()=()+1𝑛,𝑢,𝑣,𝑤−1^{𝑛(𝑛−1)/2}(𝑛+1)⁵+𝑢⁸+2𝑛³𝑣²+ln(1+𝑛|𝑤|),𝑎_𝑛=𝑛⁵,𝑓()=||𝑛,𝑢,𝑣𝑢−2^3/4+(𝑣−2)²𝑛²+3³+𝑢−𝑛²𝑣+1,𝑝_𝑛=2𝑛⁶()+1,∀𝑛,𝑢,𝑣,𝑤∈ℕ_𝑛0×ℝ³. (4.10) It is easy to see that (3.1), (3.2), and (3.76) hold. Hence Theorem 3.5 guarantees that (4.9) possesses uncountably many bounded positive solutions in 𝐴(𝑁,𝑀). But the results in [1, 7] are inapplicable for (4.9).

Acknowledgments

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