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

Authors: Zeqing Liu; Wei Sun; Jeong Sheok Ume; Shin Min Kang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equationΔ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))+f(n,yf1n,…,yfkn)=cn,∀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.


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## Body

## 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
(1.1)Δ2(yn+pyn-m)+pnyn-k-qnyn-l=0,∀n≥n0
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
(1.2)Δ2(yn+pnyn-k)+qnmax{ys:n-l≤s≤n}=0,∀n≥1
and the second-order neutral difference equation
(1.3)Δ2(yn+pyn-k)+f(n,yn)=0,∀n≥1.
Cheng and Chu [2] got sufficient and necessary conditions of the oscillatory solutions for the second-order difference equation
(1.4)Δ(rn-1Δyn-1)+pnynγ=0,∀n≥1.
Li and Yeh [6] established some oscillation criteria of the second-order delay difference equation
(1.5)Δ(an-1Δ(yn-1+pn-1yn-1-σ))+qnf(yn-τ)=0,∀n≥1.
Using the Leray-Schauder nonlinear alternative theorem, Agarwal et al. [1] studied the existence of nonoscillatory solutions for the discrete equation
(1.6)Δ(anΔ(yn+pyn-τ))+F(n+1,yn+1-σ)=0,∀n≥1
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
(1.7)Δ(anΔ(yn+byn-τ))+f(n,yn-d1n,yn-d2n,…,yn-dkn)=cn,∀n≥n0.Motivated by the results in [1–9], in this paper, we discuss the solvability of the second-order nonlinear neutral delay difference equation
(1.8)Δ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))+f(n,yf1n,…,yfkn)=cn,∀n≥n0,
where τ,r,k∈ℕ, n0∈ℕ0,  {bn}n∈ℕn0∪{cn}n∈ℕn0⊂ℝ, a∈C(ℕn0×ℝr,ℝ∖{0}),f∈C(ℕn0×ℝk,ℝ), ∪d=1r{adn}n∈ℕn0⊆ℤ,  ∪j=1k{fjn}n∈ℕn0⊆ℤ and
(1.9)limn→∞adn=limn→∞fjn=+∞,(d,j)∈Jr×Jk.
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 thatΔ is the forward difference operator defined by Δyn=yn+1-yn,  ℝ=(-∞,+∞),  ℝ+=[0,+∞),  ℤ,  ℕ and ℕ0 stand for the sets of all integers, positive integers, and nonnegative integers, respectively,
(2.1)β=min{n0-τ,inf{adn:d∈Jr,n∈ℕn0},inf{fjn:j∈Jk,n∈ℕn0}},ℤβ={n:n∈ℤwithn≥β},ℕn0={n:n∈ℕ0withn≥n0},Jl={1,…,l}forl∈{r,k}.lβ∞ denotes the Banach space of all bounded sequences y={yn}n∈ℤβ with the norm
(2.2)∥y∥=supn∈ℤβ|yn|fory={yn}n∈ℤβ∈lβ∞.
For any M>N>0, put
(2.3)V(N)={y={yn}n∈ℤβ∈lβ∞:yn≥N,∀n∈ℤβ},U(M)={y∈V(N):∥y∥<M},B(M,N)={y={yn}n∈ℤβ∈lβ∞:∥y-M∥<N},A(N,M)={y={yn}n∈ℤβ∈lβ∞:N≤yn≤M,∀n∈ℤβ}.
It is easy to see that V(N) is a closed convex subset of lβ∞,  U(M) is a bounded open subset of V(N) and B(M,N) is a bounded open convex subset of lβ∞ and A(N,M) is a bounded closed and convex subset of lβ∞.By a solution of (1.8), we mean a sequence {yn}n∈ℤβ∈lβ∞ 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 subsetY of lβ∞ is relatively compact.Lemma 2.2 (Rothe Fixed Point Theorem [10]).
LetD 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]).
LetU 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 anx∈∂U and a λ∈(0,1) such that x=(1-λ)p*+λfx.Lemma 2.4 (Krasnoselskill Fixed Point Theorem [5]).
LetY 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 constantsN and M with M>N>0 and two positive sequences {an}n∈ℕn0 and {pn}n∈ℕn0 satisfying
(3.1)|a(n,u1,u2,…,ur)|≥an,∀(n,ud)∈ℕn0×[N,M],d∈Jr;|f(n,u1,u2,…,uk)|≤pn,∀(n,uj)∈ℕn0×[N,M],j∈Jk;(3.2)∑s=n0∞‍1as∑i=s∞‍max{pi,|ci|}<+∞;(3.3)bn=1eventually.
Then (1.8) has uncountably many bounded positive solutions in B(M,N)¯.Proof.
LetL∈(M-N,M+N). First of all, we show that there exists a mapping SL:B(M,N)¯→lβ∞ with SL(∂B(M,N))⊆B(M,N)¯ such that SL has a fixed point y={yn}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
(3.4)bn=1,∀n≥T;(3.5)∑s=T∞‍1as∑i=s∞‍(pi+|ci|)<12min{M+N-L,N-M+L}.
Define a mapping SL:B(M,N)¯→lβ∞ as follows:
(3.6)(SLy)n={L-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],n≥T(SLy)T,β≤n<T
for each y={yn}n∈ℤβ∈B(M,N)¯. On account of (3.1), (3.5), and (3.6), we conclude that for every y={yn}n∈ℤβ∈∂B(M,N)⊆B(M,N)¯ and n≥T(3.7)|(SLy)n-M|=|L-M-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤|L-M|+∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≤|L-M|+∑s=T+τ∞‍1as∑i=s∞‍(pi+|ci|)<|L-M|+12min{M+N-L,N-M+L}≤N2,
which means that
(3.8)∥SLy-M∥≤N2<N,
that is, SL(∂B(M,N))⊆B(M,N)¯.
Now we assert thatSL is a continuous and condensing mapping in B(M,N)¯. Let yω={ynω}n∈ℤβ∈B(M,N)¯ for each ω∈ℕ and y={yn}n∈ℤβ∈B(M,N)¯ with limω→∞yω=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
(3.9)max{∑s=T1+τ∞‍1as∑i=s∞‍(pi+|ci|),∑s=T+τT1+τ-1‍1as∑i=T2∞‍(pi+|ci|)}<ε16;(3.10)max{∑s=T+τT1+τ-1‍1as∑i=sT2-1‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|,∑s=T+τT1+τ-1‍|a(s,ya1sω,…,yarsω)-a(s,ya1s,…,yars)|as2∑i=sT2-1‍(pi+|ci|)}<ε16,∀ω≥T3.
In view of (3.1) and (3.6)–(3.10), we deduce that for any ω≥T3(3.11)∥SLyω-SLy∥=supn≥T|∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍[∑1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1iω,…,yfkiω)-ci]-1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]∑]|=supn≥T|∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍[1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1iω,…,yfkiω)-ci]-1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]+1a(s,ya1sω,…,yarsω)×∑i=s∞‍[f(i,yf1i,…,yfki)-ci]-1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]∑]|=supn≥T|∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍[1a(s,ya1sω,…,yarsω)×∑i=s∞‍[f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)]+(1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars))×∑i=s∞‍[f(i,yf1i,…,yfki)-ci]]|≤max{supn≥T1∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1|a(s,ya1sω,…,yarsω)|×∑i=s∞‍[|f(i,yf1iω,…,yfkiω)|+|f(i,yf1i,…,yfki)|]+supn≥T1∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍(1|a(s,ya1sω,…,yarsω)|+1|a(s,ya1s,…,yars)|)×∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|],supT≤n≤T1-1∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1|a(s,ya1sω,…,yarsω)|×∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+supT≤n≤T1-1∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|×∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]}≤max{2∑s=T1+τ∞‍1as∑i=s∞‍pi+2∑s=T1+τ∞‍1as∑i=s∞‍(pi+|ci|),∑s=T+τ∞‍1as∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=T+τ∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)}≤max{ε4,∑s=T1+τ∞‍1as∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=T+τT1+τ-1‍1as∑i=T2∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=T+τT1+τ-1‍1as∑i=sT2-1‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=T1+τ∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)+∑s=T+τT1+τ-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=T2∞‍(pi+|ci|)+∑s=T+τT1+τ-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=sT2-1‍(pi+|ci|)}≤max{ε4,2∑s=T1+τ∞‍1as∑i=s∞‍pi+2∑s=T+τT1+τ-1‍1as∑i=T2∞‍pi+ε16+2∑s=T1+τ∞‍1as∑i=s∞‍(pi+|ci|)+2∑s=T+τT1+τ-1‍1as∑i=T2∞‍(pi+|ci|)+ε16}≤max{ε4,5ε8}<ε,
which gives that limω→∞SLyω=SLy, that is, SL is continuous in B(M,N)¯.
In light of (3.1), (3.5), and (3.6), we get that for any y={yn}n∈ℤβ∈B(M,N)¯(3.12)∥SLy∥=supn≥T|L-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤L+∑s=T+τ∞‍1as∑i=s∞‍(pi+|ci|)<L+12min{M+N-L,N-M+L}≤12(M+N+L),
which implies that SL(B(M,N)¯) is uniformly bounded.
Givenε>0. Clearly (3.2) ensures that there exists T*>T satisfying
(3.13)∑s=T*∞‍1as∑i=s∞‍(pi+|ci|)<ϵ2,
which together with (3.1) and (3.6) implies that for all y={yn}n∈ℤβ∈B(M,N)¯ and t2>t1≥T*(3.14)|(SLy)t2-(SLy)t1|=|∑l=1∞‍∑s=t2+(2l-1)τt2+2lτ-1‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]-∑l=1∞‍∑s=t1+(2l-1)τt1+2lτ-1‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤∑l=1∞‍∑s=t2+(2l-1)τt2+2lτ-1‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]+∑l=1∞‍∑s=t1+(2l-1)τt1+2lτ-1‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≤2∑s=T*+τ∞‍1as∑i=s∞‍(pi+|ci|)<ϵ,
which yields that SL(B(M,N)¯) is uniformly Cauchy. Thus Lemma 2.1 means that SL(B(M,N)¯) is relatively compact. Consequently SL is condensing in B(M,N)¯.
It follows from Lemma2.2 that SL has a fixed point y={yn}n∈ℤβ∈B(M,N)¯, that is,
(3.15)yn=L-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T,
which yields that
(3.16)yn+yn-τ=2L-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T+τ,a(n,ya1n,…,yarn)Δ(yn+yn-τ)=∑i=n∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T+τ,
which together with (3.3) implies that
(3.17)Δ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))=-f(n,yf1n,…,yfkn)+cn,∀n≥T+τ,
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 TLθ and a mapping SLθ satisfying (3.4)–(3.6), where L,  T, and SL are replaced by Lθ,TLθ, and SLθ, respectively, and the mapping SLθ has a fixed point yθ={ynθ}n∈ℤβ∈B(M,N)¯, which is a bounded positive solution of (1.8) in B(M,N)¯, that is,
(3.18)ynθ=Lθ-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1sθ,…,yarsθ)∑i=s∞‍[f(i,yf1iθ,…,yfkiθ)-ci],∀n≥TLθ.Equation (3.2) ensures that there exists T*>max{TL1,TL2} satisfying
(3.19)∑s=T*∞‍1as∑i=s∞‍(pi+|ci|)<|L1-L2|4.
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*(3.20)|yn1-yn2|=|L1-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1s1,…,yars1)∑i=s∞‍[f(i,yf1i1,…,yfki1)-ci]-L2+∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1a(s,ya1s2,…,yars2)∑i=s∞‍[f(i,yf1i2,…,yfki2)-ci]|≥|L1-L2|-∑l=1∞‍∑s=n+(2l-1)τn+2lτ-1‍1as×∑i=s∞‍[|f(i,yf1i1,…,yfki1)|+|ci|+|f(i,yf1i2,…,yfki2)|+|ci|]≥|L1-L2|-2∑s=T*+τ∞‍1as∑i=s∞‍(pi+|ci|)>12|L1-L2|,
that is, y1≠y2. This completes the proof.Theorem 3.2.
Assume that there exist two constantsN and M with M>N>0 and two positive sequences {an}n∈ℕn0 and {pn}n∈ℕn0 satisfying (3.1) and
(3.21)∑l=1∞‍∑s=n0+lτ∞‍1as∑i=s∞‍max{pi,|ci|}<+∞;(3.22)bn=-1eventually.
Then (1.8) has uncountably many bounded positive solutions.Proof.
LetL∈(M-N,M+N). Firstly, we show that there exists a mapping SL:B(M,N)¯→lβ∞ with SL(∂B(M,N))⊆B(M,N)¯ such that SL has a fixed point y={yn}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
(3.23)bn=-1,∀n≥T;(3.24)∑l=1∞‍∑s=T+lτ∞‍1as∑i=s∞‍(pi+|ci|)<12min{M+N-L,N-M+L}.
Define a mapping SL:B(M,N)¯→lβ∞ as follows:
(3.25)(SLy)n={L+∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],n≥T(SLy)T,β≤n<T
for each y={yn}n∈ℤβ∈B(M,N)¯. It follows from (3.1), (3.24), and (3.25) that for every y={yn}n∈ℤβ∈∂B(M,N)⊆B(M,N)¯ and n≥T(3.26)|(SLy)n-M|=|L-M+∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤|L-M|+∑l=1∞‍∑s=n+lτ∞‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≤|L-M|+∑l=1∞‍∑s=T+lτ∞‍1as∑i=s∞‍(pi+|ci|)<|L-M|+12min{M+N-L,N-M+L}≤N2,
which means that
(3.27)∥SLy-M∥≤N2<N,
that is, SL(∂B(M,N)¯⊆B(M,N)¯.
Now we prove thatSL is a continuous and condensing mapping in B(M,N)¯. Put yω={ynω}n∈ℤβ∈B(M,N)¯ for each ω∈ℕ and y={yn}n∈ℤβ∈B(M,N)¯ with limω→∞yω=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
(3.28)max{∑l=1∞‍∑s=T3+lτ∞‍1as∑i=s∞‍(pi+|ci|),∑l=T1∞‍∑s=T+lτT3+lτ-1‍1as∑i=s∞‍(pi+|ci|),∑l=1T1-1‍∑s=T+lτT3+lτ-1‍1as∑i=T2∞‍(pi+|ci|)}<ε16;(3.29)max{∑l=1T1-1‍∑s=T+lτT3+lτ-1‍1as∑i=sT2-1‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|,∑l=1T1-1‍∑s=T+lτT3+lτ-1‍|a(s,ya1sω,…,yarsω)-a(s,ya1s,…,yars)|as2∑i=sT2-1‍(pi+|ci|)}<ε16,∀ω≥T4.
By virtue of (3.1) and (3.25)–(3.29), we infer that for each ω≥T4(3.30)∥SLyω-SLy∥=supn≥T|∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1iω,…,yfkiω)-ci]-∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|=supn≥T|∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)]+∑l=1∞‍∑s=n+lτ∞‍(1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars))×∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤max{supn≥T3∑l=1∞‍∑s=n+lτ∞‍1|a(s,ya1sω,…,yarsω)|×∑i=s∞‍[|f(i,yf1iω,…,yfkiω)|+|f(i,yf1i,…,yfki)|]+supn≥T3∑l=1∞‍∑s=n+lτ∞‍(1|a(s,ya1sω,…,yarsω)|+1|a(s,ya1s,…,yars)|)×∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|],supT≤n≤T3-1∑l=1∞‍∑s=n+lτ∞‍1|a(s,ya1sω,…,yarsω)|×∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+supT≤n≤T3-1∑l=1∞‍∑s=n+lτ∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|×∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]}≤max{2∑l=1∞‍∑s=T3+lτ∞‍1as∑i=s∞‍pi+2∑l=1∞‍∑s=T3+lτ∞‍1as∑i=s∞‍(pi+|ci|),∑l=1∞‍∑s=T+lτ∞‍1as∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑l=1∞‍∑s=T+lτ∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)}≤max{ε4,∑l=1∞‍∑s=T3+lτ∞‍1as∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑l=1T1-1‍∑s=T+lτT3+lτ-1‍1as∑i=sT2-1‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑l=1T1-1‍∑s=T+lτT3+lτ-1‍1as∑i=T2∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑l=T1∞‍∑s=T+lτT3+lτ-1‍1as∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑l=1∞‍∑s=T3+lτ∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)+∑l=1T1-1‍∑s=T+lτT3+lτ-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=sT2-1‍(pi+|ci|)+∑l=1T1-1‍∑s=T+lτT3+lτ-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=T2∞‍(pi+|ci|)+∑l=T1∞‍∑s=T+lτT3+lτ-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)}≤max{ε4,2∑l=1∞‍∑s=T3+lτ∞‍1as∑i=s∞‍pi+ε16+2∑l=1T1-1‍∑s=T+lτT3+lτ-1‍1as∑i=T2∞‍pi+2∑l=T1∞‍∑s=T+lτT3+lτ-1‍1as∑i=s∞‍pi+2∑l=1∞‍∑s=T3+lτ∞‍1as∑i=s∞‍(pi+|ci|)+ε16+2∑l=1T1-1‍∑s=T+lτT3+lτ-1‍1as∑i=T2∞‍(pi+|ci|)+2∑l=T1∞‍∑s=T+lτT3+lτ-1‍1as∑i=s∞‍(pi+|ci|)}<max{ε4,7ε8}<ε,
which implies that limω→∞SLyω=SLy, that is, SL is continuous in B(M,N)¯.
From (3.1), (3.24), and (3.25), we infer that for any y={yn}n∈ℤβ∈B(M,N)¯(3.31)∥SLy∥=supn≥T|L+∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤L+∑l=1∞‍∑s=n+lτ∞‍1as∑i=s∞‍(pi+|ci|)<L+12min{M+N-L,N-M+L}≤12(M+N+L),
which implies that SL(B(M,N)¯) is uniformly bounded.
Letε>0. It follows from (3.21) that there exists T*>T satisfying
(3.32)∑l=1∞‍∑s=T*+lτ∞‍1as∑i=s∞‍(pi+|ci|)<ε2,
which together with (3.1) and (3.25) yields that for all y={yn}n∈ℤβ∈B(M,N)¯ and t2>t1≥T*(3.33)|(SLy)t2-(SLy)t1|=|∑l=1∞‍∑s=t2+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]-∑l=1∞‍∑s=t1+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤∑l=1∞‍∑s=t2+lτ∞‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]+∑l=1∞‍∑s=t1+lτ∞‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≤2∑l=1∞‍∑s=T*+lτ∞‍1as∑i=s∞‍(pi+|ci|)<ε,
which gives that SL(B(M,N)¯) is uniformly Cauchy. Hence Lemma 2.1implies that SL(B(M,N)¯) is relatively compact, that is, SL is condensing in B(M,N)¯.
It is clear that Lemma2.2 means that SL possesses a fixed point y={yn}n∈ℤβ∈B(M,N)¯, that is,
(3.34)yn=L+∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T,yn-τ=L+∑l=1∞‍∑s=n+(l-1)τ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T+τ
which lead to
(3.35)yn-yn-τ=-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T+τ,Δ(yn-yn-τ)=1a(n,ya1n,…,yarn)∑i=n∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T+τ,
which together with (3.23) yields that
(3.36)Δ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))=-f(n,yf1n,…,yfkn)+cn,∀n≥T+τ,
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 TLθ and a mapping SLθ satisfying (3.23)–(3.25), where L,  T and SL are replaced by Lθ,TLθ, and SLθ, respectively, and the mapping SLθ has a fixed point yθ={ynθ}n∈ℤβ∈B(M,N)¯, which is a bounded positive solution of (1.8) in B(M,N)¯, that is,
(3.37)ynθ=Lθ+∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1sθ,…,yarsθ)∑i=s∞‍[f(i,yf1iθ,…,yfkiθ)-ci],∀n≥TLθ.
It follows from (3.21) that there exists T*>max{TL1,TL2} such that
(3.38)∑l=1∞‍∑s=T*+lτ∞‍1as∑i=s∞‍(pi+|ci|)<|L1-L2|4.
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*(3.39)|yn1-yn2|=|L1+∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s1,…,yars1)∑i=s∞‍[f(i,yf1i1,…,yfki1)-ci]-L2-∑l=1∞‍∑s=n+lτ∞‍1a(s,ya1s2,…,yars2)∑i=s∞‍[f(i,yf1i2,…,yfki2)-ci]|≥|L1-L2|-∑l=1∞‍∑s=n+lτ∞‍1as∑i=s∞‍[|f(i,yf1i1,…,yfki1)|+|ci|+|f(i,yf1i2,…,yfki2)|+|ci|]≥|L1-L2|-2∑l=1∞‍∑s=T*+lτ∞‍1as∑i=s∞‍(pi+|ci|)>12|L1-L2|,
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 constantsN,  M,  b*, and b* and two positive sequences {an}n∈ℕn0 and {pn}n∈ℕn0 satisfying (3.1), (3.2) and
(3.40)0<N<(1-b*-b*)M,b*≥0,b*≥0,b*+b*<1,-b*≤bn≤b*eventually.
Then (1.8) has uncountably many bounded positive solutions in U(M)¯.Proof.
LetL∈(b*M+N,(1-b*)M). Now we prove that there exists a mapping SL:U(M)¯→V(N) such that it has a fixed point y={yn}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
(3.41)-b*≤bn≤b*,∀n≥T;(3.42)∑s=T∞‍1as∑i=s∞‍(pi+|ci|)<min{L-b*M-N,(1-b*)M-L}.
Put p*=M-ε*, where ε*∈(0,min{L-b*M-N,(1-b*)M-L,(M-N)/2}) is enough small and
(3.43)∑s=T∞‍1as∑i=s∞‍(pi+|ci|)<min{L-b*M-N,(1-b*)M-L}-ε*.
Obviously, p*∈U(M). Define a mapping SL:U(M)¯→lβ∞ by
(3.44)(SLy)n=(S1Ly)n+(S2Ly)n,n≥β
for each y={yn}n∈ℤβ∈U(M)¯, where the mappings S1L,S2L:U(M)¯→lβ∞ are defined by
(3.45)(S1Ly)n={L-bnyn-τ,n≥T(S1Ly)T,β≤n<T,(3.46)(S2Ly)n={-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],n≥T(S2Ly)T,β≤n<T.
It follows from (3.1), (3.41), and (3.43)–(3.46) that for any y={yn}n∈ℤβ∈U(M)¯ and n≥T(3.47)(SLy)n=(S1Ly)n+(S2Ly)n=L-bnyn-τ-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]≥L-b*M-∑s=n∞‍1|a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)≥L-b*M-∑s=T∞‍1as∑i=s∞‍(pi+|ci|)>L-b*M-min{L-b*M-N,(1-b*)M-L}+ε*≥N+ε*>N,
which yields that SL(U(M)¯)⊆V(N).
Next we show thatS2L:U(M)¯→lβ∞ is a continuous and relatively compact mapping. Let yω={ynω}n∈ℤβ∈U(M)¯ and y={yn}n∈ℤβ∈U(M)¯ with limω→∞yω=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
(3.48)max{∑s=T1∞‍1as∑i=s∞‍(pi+|ci|),∑s=TT1-1‍1as∑i=T2∞‍(pi+|ci|)}<ε16;(3.49)max{∑s=TT1-1‍1|as|∑i=sT2-1‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|,∑s=TT1-1‍|a(s,ya1sω,…,yarsω)-a(s,ya1s,…,yars)|as2∑i=sT2-1‍(pi+|ci|)}<ε16,∀ω≥T3.
It follows from (3.1) and (3.46)–(3.49) that for each ω≥T3(3.50)∥S2Lyω-S2Ly∥=supn≥T|∑s=n∞‍1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1iω,…,yfkiω)-ci]-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|=supn≥T|∑s=n∞‍1a(s,ya1sω,…,yarsω)∑i=s∞‍[f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)]+∑s=n∞‍(1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars))×∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤max{supn≥T1∑s=n∞‍1|a(s,ya1sω,…,yarsω)|×∑i=s∞‍[|f(i,yf1iω,…,yfkiω)|+|f(i,yf1i,…,yfki)|]+supn≥T1∑s=n∞‍(1|a(s,ya1sω,…,yarsω)|+1|a(s,ya1s,…,yars)|)×∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|],supT≤n≤T1-1∑s=n∞‍1|a(s,ya1sω,…,yarsω)|×∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+supT≤n≤T1-1∑s=n∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|×∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]}≤max{2∑s=T1∞‍1as∑i=s∞‍pi+2∑s=T1∞‍1as∑i=s∞‍(pi+|ci|),∑s=T∞‍1as∑i=s∞‍|f(i,yf1iω,yf2iω,…,yfkiω)-f(i,yf1i,yf2i,…,yfki)|+∑s=T∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)}≤max{ε4,∑s=TT1-1‍1as∑i=sT2-1‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=TT1-1‍1as∑i=T2∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=T1∞‍1as∑i=s∞‍|f(i,yf1iω,…,yfkiω)-f(i,yf1i,…,yfki)|+∑s=TT1-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=sT2-1‍(pi+|ci|)+∑s=TT1-1‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=T2∞‍(pi+|ci|)+∑s=T1∞‍|1a(s,ya1sω,…,yarsω)-1a(s,ya1s,…,yars)|∑i=s∞‍(pi+|ci|)}≤max{ε4,ε16+2∑s=TT1-1‍1as∑i=T2∞‍pi+2∑s=T1∞‍1as∑i=s∞‍pi+∑s=TT1-1‍|a(s,ya1sω,…,yarsω)-a(s,ya1s,…,yars)|as2∑i=sT2-1‍(pi+|ci|)+2∑s=TT1-1‍1as∑i=T2∞‍(pi+|ci|)+2∑s=T1∞‍1as∑i=s∞‍(pi+|ci|)}≤max{ε4,5ε8}<ε,
which yields that limω→∞∥S2Lyω-S2Ly∥=0, that is, S2L is continuous in U(M)¯.
In light of (3.1) and (3.43)–(3.46), we deduce that for all y={yn}n∈ℤβ∈U(M)¯(3.51)∥SLy∥=supn≥T|(S1Ly)n+(S2Ly)n|≤supn≥T|(S1Ly)n|+supn≥T|(S2Ly)n|≤supn≥T|L-bnyn-τ|+supn≥T|∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤supn≥T(L+|bn||yn-τ|)+supn≥T∑s=n∞‍1|a(s,ya1s,…,yars)|∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≤L+(b*+b*)M+∑s=T∞‍1as∑i=s∞‍(pi+|ci|)≤L+M+min{L-b*M-N,(1-b*)M-L}-ε*<2L+M,
which means that SL(U(M)¯) and S2L(U(M)¯) are bounded.
Letε>0. Notice that (3.2) ensures that there exists T*>T satisfying
(3.52)∑s=T*∞‍1as∑i=s∞‍(pi+|ci|)<ϵ2,
which together with (3.1) and (3.46) implies that for all y={yn}n∈ℤβ∈U(M)¯ and t2>t1≥T*(3.53)|(S2Ly)t2-(S2Ly)t1|=|∑s=t2∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]-∑s=t1∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]|≤∑s=t2∞‍1as∑i=s∞‍(pi+|ci|)+∑s=t1∞‍1as∑i=s∞‍(pi+|ci|)≤2∑s=T*∞‍1as∑i=s∞‍(pi+|ci|)<ϵ,
which means that S2L(U(M)¯) is uniformly Cauchy. Thus S2L(U(M)¯) is relatively compact.
By virtue of (3.41) and (3.45), we infer that for all x={xn}n∈ℤβ,y={yn}n∈ℤβ∈U(M)¯ and n≥T(3.54)|(S1Lx)n-(S1Ly)n|=|bn||xn-τ-yn-τ|≤(b*+b*)∥x-y∥,
which yields that
(3.55)∥S1Lx-S1Ly∥≤(b*+b*)∥x-y∥,
that is, S1L is a contraction mapping in U(M)¯. It follows that SL is a continuous and condensing mapping.
Put(3.56)P={y={yn}n∈ℤβ∈lβ∞:N≤yn≤M,n≥β,∥y∥=M},(3.57)Q={y={yn}n∈ℤβ∈lβ∞:N≤yn≤M,n≥βandthereexistsn*≥βsatisfyingyn*=N}.
It is easy to see that ∂U(M)=P∪Q. Suppose that there exist y={yn}n∈ℤβ∈∂U(M) and λ∈(0,1) with
(3.58)y=(1-λ)p*+λSLy.
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
(3.59)yn=(1-λ)p*+λSLyn=(1-λ)p*+λ[L-bnyn-τ-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]]≤(1-λ)(M-ε*)+λ[L+b*M+∑s=T∞‍1as∑i=s∞‍(pi+|ci|)]<(1-λ)(M-ε*)+λ[L+b*M+min{L-b*M-N,(1-b*)M-L}-ε*]≤M-ε*,∀n≥T,
which implies that
(3.60)M=∥y∥=supn≥β|yn|≤M-ε*<M,
which is a contradiction.
Case  2. Let y∈Q. It follows from (3.41), (3.43)–(3.46), (3.57), and (3.58) that
(3.61)N=yn*=(1-λ)p*+λSLyn*=(1-λ)p*+λ[L-bn*yn*-τ-∑s=max{n*,T}∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]]≥(1-λ)(M-ε*)+λ[L-b*M-∑s=max{n*,T}∞‍1as∑i=s∞‍(pi+|ci|)]>(1-λ)(M-ε*)+λ[L-b*M-min{L-b*M-N,(1-b*)M-L}+ε*]≥(1-λ)(M-ε*)+λ(N+ε*)≥min{M-ε*,N+ε*}=N+ε*,
which is absurd. Thus Lemma 2.3 ensures that there exists y={yn}n∈ℤβ∈U(M)¯ satisfying SLy=S1Ly+S2Ly=y, that is,
(3.62)yn=L-bnyn-τ-∑s=n∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,xhki)-ci],∀n≥T,
which means that
(3.63)a(n,ya1n,…,yarn)Δ(yn+bnyn-τ)=∑i=n∞‍[f(i,yf1i,…,yfki)-ci],∀n≥T,
which yields that
(3.64)Δ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))=-f(n,yf1n,…,yfkn)+cn,∀n≥T,
that is, y={yn}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 SLθ:U(M)¯→V(N) satisfying (3.41)–(3.46), where L,  β,  T,  S1L,  S2L, and SL are replaced by Lθ,  βθ,  TLθ,  S1Lθ,  S2Lθ and SLθ, respectively, and the mapping SLθ has a fixed point yθ={ynθ}n∈ℤβ∈U(M)¯, which is a bounded positive solution of (1.8) in U(M)¯, that is,
(3.65)ynθ=Lθ-bnyn-τθ-∑s=n∞‍1a(s,ya1sθ,…,yarsθ)∑i=s∞‍[f(i,yf1iθ,…,yfkiθ)-ci],∀n≥TLθ.
It follows from (3.2) that there exists T*>max{TL1,TL2} satisfying
(3.66)∑s=T*∞‍1as∑i=s∞‍(pi+|ci|)<|L1-L2|4.
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*(3.67)|yn1-yn2|=|L1-L2-bnyn-τ1+bnyn-τ2-∑s=n∞‍1a(s,ya1s1,…,yars1)∑i=s∞‍[f(i,yf1i1,…,yfki1)-ci]+∑s=n∞‍1a(s,ya1s2,…,yars2)∑i=s∞‍[f(i,yf1i2,…,yfki2)-ci]|≥|L1-L2|-|bn||yn-τ1-yn-τ2|-∑s=n∞‍1as∑i=s∞‍[|f(i,yf1i1,…,yfki1)|+|ci|+|f(i,yf1i2,yf2i2,…,yfki2)|+|ci|]≥|L1-L2|-(b*+b*)∥y1-y2∥-2∑s=T*∞‍1as∑i=s∞‍(pi+|ci|)>12|L1-L2|-(b*+b*)∥y1-y2∥,
which means that
(3.68)∥y1-y2∥>|L1-L2|2(1+b*+b*)>0,
that is, y1≠y2. This completes the proof.Theorem 3.4.
Assume that there exist four constantsN,  M,  b*, and b* and two positive sequences {an}n∈ℕn0 and {pn}n∈ℕn0 satisfying (3.1), (3.2) and
(3.69)(1+b*)M<(1+b*)N<0,b*≤bn≤b*<-1eventually.
Then (1.8) has uncountably many bounded positive solutions in U(M)¯.Proof.
LetL∈((1+b*)M,(1+b*)N). Now we show that there exists a mapping SL:U(M)¯→V(N) such that it has a fixed point y={yn}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
(3.70)b*≤bn≤b*<-1,∀n≥T;(3.71)∑s=T∞‍1as∑i=s∞‍(pi+|ci|)<min{L-(1+b*)M,b*b*[N(1+b*)-L]}.
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
(3.72)∑s=T∞‍1as∑i=s∞‍(pi+|ci|)<min{L-(1+b*)M,b*b*[N(1+b*)-L]}-ε*.
Obviously, p*∈U(M). Define a mapping SL:U(M)¯→lβ∞ by (3.44), where the mappings S1L,S2L:U(M)¯→lβ∞ are defined by
(3.73)(S1Ly)n={Lbn+τ-yn+τbn+τ,n≥T(S1Ly)T,β≤n<T,(3.74)(S2Ly)n={-1bn+τ∑s=n+τ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci],n≥T(S2Ly)T,β≤n<T
for each y={yn}n∈ℤβ∈U(M)¯. By virtue of (3.1), (3.70), and (3.72)–(3.74), we get that for any y={yn}n∈ℤβ∈U(M)¯ and n≥T(3.75)(SLy)n=(S1Ly)n+(S2Ly)n=Lbn+τ-yn+τbn+τ-1bn+τ∑s=n+τ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]≥Lb*-Nb*+1b*∑s=T+τ∞‍1as∑i=s∞‍(pi+|ci|)≥Lb*-Nb*+1b*min{L-(1+b*)M,b*b*[N(1+b*)-L]}-1b*ε*≥N-1b*ε*>N,
which gives that SL(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 constantsN,  M,  b* and b* and two positive sequences {an}n∈ℕn0,  {pn}n∈ℕn0 satisfying (3.1), (3.2) and
(3.76)0<Nb*b*<M(b*2-b*),1<b*≤bn≤b*<b*2eventually.
Then (1.8) has uncountably many bounded positive solutions in A(N,M).Proof.
LetL∈((b*/b*)M+b*N,b*M). Now we show that there exist two mappings S1L,S2L:A(N,M)→lβ∞ such that the equation S1Ly+S2Ly=y has a solution y={yn}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
(3.77)1<b*≤bn≤b*<b*2,∀n≥T;(3.78)∑s=T∞‍1as∑i=s∞‍(pi+|ci|)<min{b*M-L,b*Lb*-M-b*N}.
Define two mappings S1L and S2L:A(N,M)→lβ∞ 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={xn}n∈ℤβ, y={yn}n∈ℤβ∈A(N,M) and n≥T(3.79)|(S1Lx)n-(S1Ly)n|=1bn+τ|xn+τ-yn+τ|≤1b*∥x-y∥,(S1Lx)n+(S2Ly)n=Lbn+τ-xn+τbn+τ-1bn+τ∑s=n+τ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]≥Lb*-Mb*-1b*∑s=n+τ∞‍1as∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≥Lb*-Mb*-1b*∑s=T+τ∞‍1as∑i=s∞‍(pi+|ci|)>Lb*-Mb*-1b*min{b*M-L,b*Lb*-M-b*N}≥N,(S1Lx)n+(S2Ly)n=Lbn+τ-xn+τbn+τ-1bn+τ∑s=n+τ∞‍1a(s,ya1s,…,yars)∑i=s∞‍[f(i,yf1i,…,yfki)-ci]≤Lb*+1b*∑s=n+τ∞‍1as∑i=s∞‍[|f(i,yf1i,…,yfki)|+|ci|]≤Lb*+1b*∑s=T+τ∞‍1as∑i=s∞‍(pi+|ci|)<Lb*+1b*min{b*M-L,b*Lb*-M-b*N}≤M,
which yield that
(3.80)∥S1Lx-S1Ly∥≤1b*∥x-y∥,S1Lx+S2Ly∈A(N,M),∀x,y∈A(N,M).
As in the proof of Theorem 3.3, we infer similarly that S2L is continuous in A(N,M) and S2L(A(N,M)) is relatively compact. Thus S2L is completely continuous, which together with (3.77), (3.80), and Lemma 2.4, ensures that the equation S1Ly+S2Ly=y has a solution y={yn}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.
Theorems3.1–3.5 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 Section3. 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(4.1)Δ((-1)n(n6-n5+1)(yn2+1)Δ(yn+yn-τ))+-3n3yn+14+nyn+1+(yn+1-1)4/5n5ln2(n+2)+|yn2-n2|3+1=(-1)nn2-5n+1n4+n2+sinn3+1,∀n≥1,
where τ∈ℕ is fixed. Let n0=1,  r=1,  k=2,  N=2,  M=3,  β=1-τ and
(4.2)a1n=n,f1n=n+1,f2n=n2,bn=1,cn=(-1)nn2-5n+1n4+n2+sinn3+1,a(n,u)=(-1)n(n6-n5+1)(u2+1),an=3(n6-n5+1),f(n,u,v)=-3n3u4+nu+(u-1)4/5n5ln2(n+2)+|v-n2|3+1,pn=243n3+3n+2n5,∀(n,u,v)∈ℕn0×ℝ2.
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(4.3)Δ((n5n+2+yn2-602y2n+(-1)n4)Δ(yn-yn-τ))+n3(yn+4-2)9-(n3-2n+1)y2n-13n5+(yn2-34-3n)2+sin(nyn+4)+2=n5-6n32n-4-5n11+4,∀n≥5,
where τ∈ℕ is fixed. Let n0=5,  r=2,  k=3,  N=1,  M=3,  β=min{5-τ,-35} and
(4.4)a1n=n2-60,a2n=2n+(-1)n,f1n=n+4,f2n=2n-1,f3n=n2-3,bn=-1,cn=n5-6n32n-4-5n11+4,a(n,u,v)=n5n+2+u2v4,an=n5n+2+1,f(n,u,v,w)=n3(u-2)9-(n3-2n+1)v3n5+(w4-3n)2+sin(nu)+2,pn=28n3+54n+27n5,∀(n,u,v,w)∈ℕn0×ℝ3.
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(4.5)Δ((-1)n(n-1)/2((n3+2)|y3n3|+n2yn-82+1)Δ(xn+(-1)n(n7-1)3n7+1xn-τ))+(xn-5-1)1/3+|xn(n+1)/22-2|nln2n+1=n6-30n4+8cos5(4n8-1)n9+n5+2ln2n,∀n≥3,
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
(4.6)a1n=3n,a2n=n-8,f1n=n-5,f2n=n(n+1)2,bn=(-1)n(n7-1)3n7+1,cn=n6-30n4+8cos5(4n8-1)n9+n5+2ln2n,a(n,u,v)=(-1)n(n-1)/2((n3+2)|u3|+n2v2+1),an=8n3+4n2+1,f(n,u,v)=(u-1)1/3+|v2-2|nln2n+1,pn=61/3+47nln2n,∀(n,u,v)∈ℕn0×ℝ2.
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(4.7)Δ((-1)n-1n2(yn3-14+1)Δ(xn-3n2-3n-3cos(n/2-1)+1n2-n-cos(n/2-1)xn-τ))+(3-xn2-9)1/3-n-1x2n-33n4+nx3n-74+1=n3+(-1)nn2+ln3(1+n2)n6+4n5-3n4+sin3(4n2-5)+1,n≥2,
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
(4.8)a1n=n3-1,f1n=n2-9,f2n=3n-7,f3n=2n-3,bn=-3n2-3n-3cos(n/2-1)+1n2-n-cos(n/2-1),cn=n3+(-1)nn2+ln3(1+n2)n6+4n5-3n4+sin3(4n2-5)+1,a(n,u)=(-1)n-1n2(u4+1),an=n2ln3n,f(n,u,v,w)=(3-u)1/3-2n-3v3n4+nw4+1,pn=2+272n-3n4,∀(n,u,v,w)∈ℕn0×ℝ3.
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(4.9)Δ(∑(-1)n(n-1)/2((n+1)5+y2n-38+2n3yn2-12+ln(1+n|yn(n-1)|))×Δ(xn+11ln(1+n2)+11sinn+10ln(1+n2)+sinn+1xn-τ)∑)+|xn-5-2|3/4+(xn(n-2)-2)2(n2+3)3+|xn-5-n2xn(n-2)|+1=n7-5n4-9n11+7n8-6n7+5n3+1,n≥0,
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
(4.10)a1n=2n-3,a2n=n2-1,a3n=n(n-1),f1n=n-5,f2n=n(n-2),bn=11ln(1+n2)+11sinn+10ln(1+n2)+sinn+1,cn=n7-5n4-9n11+7n8-6n7+5n3+1,a(n,u,v,w)=(-1)n(n-1)/2((n+1)5+u8+2n3v2+ln(1+n|w|)),an=n5,f(n,u,v)=|u-2|3/4+(v-2)2(n2+3)3+|u-n2v|+1,pn=2n6+1,∀(n,u,v,w)∈ℕn0×ℝ3.
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).



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*Source: 172939-2012-12-26.xml*