# On Positive Solutions of a Fourth Order Nonlinear Neutral Delay Difference Equation

Authors: Zeqing Liu; Xiaoping Zhang; Shin Min Kang; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.


---

## Body

## 1. Introduction

In the past few decades, the researchers [1–31] and others studied oscillation, asymptotic behavior, and solvability for a lot of second and third order nonlinear difference equations, some of which are as follows:
(1)Δ(anΔ(xn+pxn-τ))+F(n+1,xn+1-σ)=0,n≥1,Δ(anΔ(xn+bxn-τ))+f(n,xn-d1n,…,xn-dkn)=cn,n≥n0,Δ3xn+f(n,xn,xn-r)=0,n≥n0,Δ(anΔ2(xn+Pnxn-τ))+f(n,xn-d1n,…,xn-dkn)=gn,n≥n0,Δ2(anΔ(xn+γnxn-τ))+Δ2f(n,xn-d1n,…,xn-dkn)+Δg(n,xn-d1n,…,xn-dkn)=h(n,xn-d1n,…,xn-dkn),n≥n0.By employing a few famous tools in nonlinear analysis including the nonlinear alternative of Leray-Schauder type, Banach's fixed point theorem, Schauder's fixed point theorem, Krasnoselskii's fixed point theorem, coincidence degree theory and critical point theory, the authors [3, 4, 9, 13–16, 18, 25, 27, 28] and others proved the existence of nonoscillatory solutions, uncountably many bounded nonoscillatory solutions and periodic solutions for the difference equations above, where Lipschitz conditions were used in [14, 16]. Recently, the authors [32] used the Krasnoselskii's fixed point theorem to obtain h-asymptotic stability results about the zero solution for a very general first order nonlinear neutral differential equation with functional delay.However, to our knowledge, no one studied the following fourth order nonlinear neutral delay difference equation:(2)Δ3(anΔ(xn+γnxn-τ))+Δ3f(n,xb1n,…,xbkn)+Δ2g(n,xc1n,…,xckn)+Δh(n,xd1n,…,xdkn)+p(n,xo1n,…,xokn)=rn,n≥n0,
where τ,k∈ℕ,n0∈ℕ0,f,g,h,p∈C(ℕn0×ℝk,ℝ), {an}n∈ℕn0, and {γn}n∈ℕn0 are real sequences with an≠0 for n∈ℕn0 and {bln,cln,dln,oln:n∈ℕn0,l∈{1,2,…,k}}⊂ℤ with
(3)lim⁡n→∞bln=lim⁡n→∞cln=lim⁡n→∞dln=lim⁡n→∞oln=+∞,lim⁡n→∞bln=lim⁡n→∞cln=lim⁡n→∞dln=l∈{1,2,…,k}.
The purpose of this paper is to fill this gap in the literature and to study solvability of (2). Under certain conditions, we prove the existence of uncountably many bounded positive solutions of (2) by means of the Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, respectively. Nine examples are included.This paper is organized as follows. In Section2 we present some notations, definitions, and lemmas. In Section 3 we establish nine sufficient conditions which guarantee the existence of uncountably many bounded positive solutions of (2) by using fixed point theorems and new techniques. In Section 4 we give nine examples to illustrate the effectiveness and applications of the results presented in Section 3.


## 2. Preliminaries

Throughout this paper, we assume thatℝ=(-∞,+∞),ℤ,ℕ and ℕ0 stand for the sets of all integers, positive integers, and nonnegative integers, respectively:
(4)ℕn0={n:n∈ℕwithn≥n0},n0∈ℕ0,α=inf⁡⁡{bln,dln,cln,oln:1≤l≤k,n∈ℕn0},β=min⁡⁡{n0-τ,α},ℤβ={n:n∈ℤwithn≥β};Δ denotes the forward different operator defined by Δxn=xn+1-xn and Δixn=Δ(Δi-1xn) for i∈{2,3,4}. Let lβ∞ denote the Banach space of all bounded sequences on ℤβ with norm:
(5)∥x∥=sup⁡n∈ℤβ|xn|forx={xn}n∈ℤβ∈lβ∞,Ω1(N,M)={x={xn}n∈ℤβ∈lβ∞:N≤xn≤M,n∈ℤβ};Ω2T(N,M)={x={xn}n∈ℤβ∈lβ∞:Nγn≤xn≤Mγn,n≥T;NγT≤xn≤MγT,β≤n<T};Ω3T(N,M)={x={xn}n∈ℤβ∈lβ∞:-Nγn≤xn≤-Mγn,n≥T;-NγT≤xn≤-MγT,β≤n<T}.
Obviously, Ω1(N,M), Ω2T(N,M), and Ω3T(N,M), are closed bounded and convex subsets of lβ∞ for any M>N>0.By a solution of (2), we mean a real sequence {xn}n∈ℤβ with a positive integer T≥n0+τ+|β| such that (2) is satisfied for all n≥T.Definition 1 (see [5]).
A subsetD of lβ∞ is said to be uniformly Cauchy (or equi-Cauchy) if for every ε>0 there exists a positive integer K>β such that
(6)|xi-xj|<ε,
whenever i,j>K for any x={xn}n∈ℤβ∈lβ∞.Lemma 2 (see [5]).
Each bounded and uniformly Cauchy subset oflβ∞ is relatively compact.Lemma 3 (Krasnoselskii's fixed point theorem).
LetX be a Banach space, let D be a bounded closed convex subset of X, and let S,G be mappings from D into X such that Sx+Gy∈D for every pair x,y∈D. If S is a contraction and G is completely continuous, then the equation
(7)Sx+Gx=x
has a solution in D.Lemma 4 (Schauder's fixed point theorem).
LetD be a nonempty closed convex subset of a Banach space X, T:D→D continuous, and f(D) relatively compact. Then f has at least one fixed point in D.Lemma 5.
Letτ∈ℕ, n0∈ℕ0, {an}n∈ℕn0, {bn}n∈ℕn0, and {cn}n∈ℕn0 be nonnegative sequences. If
(8)∑j=n0∞‍∑s=j∞‍∑t=s∞‍jcjasbt<+∞,
then
(9)∑i=1∞‍∑j=n0+iτ∞‍∑s=j∞‍∑t=s∞‍cjasbt=∑j=n0+τ∞‍∑s=j∞‍∑t=s∞‍[j-n0τ]cjasbt≤1τ∑j=n0+τ∞‍∑s=j∞‍∑t=s∞‍jcjasbt,
where [(j-n0)/τ] denotes the integer part of number (j-n0)/τ.Proof.
Notice that(10)∑i=1∞‍∑j=n0+iτ∞‍∑s=j∞‍∑t=s∞‍cjasbt=∑j=n0+τ∞‍∑s=j∞‍∑t=s∞‍cjasbt+∑j=n0+2τ∞∑s=j∞‍∑t=s∞‍cjasbt+∑j=n0+3τ∞‍∑s=j∞‍∑t=s∞‍cjasbt+⋯=∑j=n0+τn0+2τ-1‍∑s=j∞‍∑t=s∞‍[j-n0τ]cjasbt+∑j=n0+2τn0+3τ-1‍∑s=j∞‍∑t=s∞‍[j-n0τ]cjasbt+∑j=n0+3τn0+4τ-1‍∑s=j∞‍∑t=s∞‍[j-n0τ]cjasbt+⋯=∑j=n0+τ∞‍∑s=j∞‍∑t=s∞‍[j-n0τ]cjasbt≤1τ∑j=n0+τ∞‍∑s=j∞‍∑t=s∞‍jcjasbt.
That is, (9) holds. This completes the proof.


## 3. Existence of Uncountably Many Bounded Positive Solutions

Now we study the existence of uncountably many bounded positive solutions for (2) by using the Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, respectively.Theorem 6.
Assume that there exist constantsn1∈ℕn0, M, N, and c with M>N>0 and c∈[0,(M-N)/2M) and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying
(11)|γn|≤c,∀n≥n1,(12)|f(n,u1,u2,…,uk)|≤Wn,|g(n,u1,u2,…,uk)|≤Pn,|h(n,u1,u2,…,uk)|≤Qn,|p(n,u1,u2,…,uk)|≤Rn,|h(n,u1,u2,…,uk)|e∀(n,u1,u2,…,uk)∈ℕn0×[N,M]k,(13)max⁡{∑t=n0∞‍Wt|at|,∑s=n0∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt],maxww∑j=n0∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)}<+∞.
Then (2) possesses uncountably many bounded positive solutions in Ω1(N,M).Proof.
SetL∈(N+cM,M(1-c)). It follows from (13) that there exists T≥n0+n1+τ+|β| sufficiently large such that
(14)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M(1-c)-L,L-cM-N}.
Define two mappings UL and SL: Ω1(N,M)→lβ∞ by
(15)(ULx)n={12L-γnxn-τ,n≥T,(ULx)T,β≤n<T,(16)(SLx)n={12L+∑t=n∞‍1atf(t,xb1t,…,xbkt)12L+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑s=n∞‍∑t=s∞‍www1as-g(t,xc1t,…,xckt)]12L-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt],-∑j=n∞‍∑s=j∞‍∑t=s∞‍∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwt-s+1ajn≥T,(SLx)T,-∑j=n∞‍∑s=j∞‍∑t=s∞‍∑j=n∞‍∑s=j∞‍∑t=s∞‍∑s=j∞‍et-s+1ajβ≤n<T
for each x={xn}n∈ℤβ∈Ω1(N,M).
Now we prove that(17)ULx+SLy∈Ω1(N,M),∀x,y∈Ω1(N,M),∥ULx-ULy∥≤c∥x-y∥,∀x,y∈Ω1(N,M),∥SLx∥<M,∀x,y∈Ω1(N,M).
In view of (11), (12), and (14)–(16), we conclude that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω1(N,M), and n≥T,
(18)|(ULx)n+(SLy)n-L|=|-γnxn-τ+∑t=n∞‍1atf(t,yb1t,…,ybkt)=e+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n∞‍∑t=s∞‍1aswwww-g(t,yc1t,…,yckt)]=e-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑j=n∞‍∑s=j∞‍∑t=s∞‍wew×[p(t,yo1t,…,yokt)-rt]∑s=j∞‍∑t=s∞‍t-s+1aj|≤cM+∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|≤cM+∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍×[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<cM+min⁡{M(1-c)-L,L-cM-N},∥ULx-ULy∥=sup⁡n∈ℤβ|(ULx)n-(ULy)n|∥ULx-ULy∥=max⁡{sup⁡T>n≥β|(ULx)n-(ULy)n|,∥ULx-ULy∥sup⁡n≥T|(ULx)n-(ULy)n|}∥ULx-ULy∥=sup⁡n≥T(|γn||xn-τ-yn-τ|)∥ULx-ULy∥≤c∥x-y∥,∥SLx∥=sup⁡n∈ℤβ|(SLx)n|=max⁡{sup⁡T>n≥β|(SLx)n|,sup⁡n≥T|(SLx)n|}=sup⁡n≥T|12L+∑t=n∞‍1atf(t,xb1t,…,xbkt)=sup⁡n≥Tw+∑s=n∞‍∑t=s∞‍1as[(t,xc1t,…,xckt)(t-s+1)+∑s=n∞‍∑t=s∞‍1aswwwww×h(t,xd1t,…,xdkt)+∑s=n∞‍∑t=s∞‍1aswwwww-g(t,xc1t,…,xckt)]=sup⁡n≥Tw-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwww-rt(t,xc1t,…,xckt)]∑t=s∞‍|≤sup⁡n≥T(+∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)t-s+1|aj|12L+∑t=n∞‍Wt|at|+∑s=n∞‍∑t=s∞‍1|as|[(t-s+1)Qt12L+∑t=n∞‍Wt|at|+∑s=n∞‍∑t=s∞‍1|as|wwwww+Pt]≤sup⁡n≥Tww+∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|))≤12L+∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<12L+min⁡{M(1-c)-L,L-cM-N}<M,
which yield that (17) hold
In order to prove thatSL is completely continuous in Ω1(N,M), we have to show that SL is continuous in Ω1(N,M) and SL(Ω1(N,M)) is relatively compact. Suppose that {xm}m∈ℕ is an arbitrary sequence in Ω1(N,M) and x∈Ω1(N,M) with lim⁡m→∞xm=x, where xm={xnm}n∈ℤβ for each m∈ℕ and x={xn}n∈ℤβ. With the help of (12), (13), lim⁡m→∞xm=x, and the continuity of f,g,h, and p, we know that for given ε>0, there exist T1,T2,T3, and T4∈ℕ with T4>T3>T2>T1>T satisfying
(19)∑t=T1∞‍Wt|at|+∑s=T1∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T1∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<ε18;max⁡{∑t=T3∞‍t-s+1|aj|Rt:T≤j≤T1,j≤s≤T2}<ε18T1T2;max⁡{∑s=T2∞‍∑t=s∞‍t-s+1|aj|Rt:T≤j≤T1}<ε18T1;max⁡⁡{∑t=T2∞‍1|as|[(t-s+1)Qt+Pt]:T≤s≤T1}<ε18T1;max⁡⁡{|q(t,xb1tm,…,xbktm)maxw-q(t,xb1t,…,xbkt)|:q∈{f,g,h,p}}<ε18T1T2T3(A+B+E),∀m≥T4,T≤t≤T3,
where
(20)A=max⁡⁡{1|as|:T≤s≤T1},B=max⁡⁡{t-s+1|as|:T≤s≤T1,s≤t≤T2},E=max⁡{t-s+1|aj|:T≤j≤T1,j≤s≤T2,s≤t≤T3}.
By virtue of (16), (19), and (20), we get that
(21)∥SLxm-SLx∥=sup⁡n∈Zβ|(SLxm)n-(SLx)n|=max⁡{sup⁡T>n≥β|(SLxm)n-(SLx)n|,=maxwwwsup⁡n≥T|(SLxm)n-(SLx)n|}=sup⁡n≥T|∑t=n∞‍1atf(t,xb1tm,…,xbktm)=sup⁡n≥Twwr+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1tm,…,xdktm)=maxwwwwwwwww-g(t,xc1tm,…,xcktm)]=sup⁡n≥Twwr-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1tm,…,xoktm)-rt]=sup⁡n≥Twwr-(∑t=n∞‍1atf(t,xb1t,…,xbkt)+∑s=n∞‍∑t=s∞‍1asww+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)=sup⁡n≥Twwwwwwwwww-g(t,xc1t,…,xckt)]=sup⁡n≥Twwww-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwwwww-rt(t,xo1t,…,xokt)]∑t=n∞‍)|≤∑t=T∞‍1|at||f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=T∞‍∑t=s∞‍1|as|wwwwwwwwww-h(t,xd1t,…,xdkt)|+∑s=T∞‍∑t=s∞‍1|as|wwww+|g(t,xc1tm,…,xcktm)+∑s=T∞‍∑t=s∞‍1|as|wwwwww-g(t,xc1t,…,xckt)|]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwwew-p(t,xo1t,…,xokt)|=∑t=TT1‍1|at||f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑t=T1+1∞‍1|at||f(t,xb1tm,…,xbktm)+∑t=T1+1∞‍1|at|wew-f(t,xb1t,…,xbkt)|+∑s=TT1‍∑t=sT2‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=TT1‍∑t=sT2‍1|as|wwwwwwwww-h(t,xd1t,…,xdkt)|+∑s=TT1‍∑t=sT2‍1|as|wew+|g(t,xc1tm,…,xcktm)+∑s=TT1‍∑t=T2+1∞‍1|as|www-g(t,xc1t,…,xckt)|]+∑s=TT1‍∑t=T2+1∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=TT1‍∑t=sT2‍1|as|wwwwwwwwwww-h(t,xd1t,…,xdkt)|+∑s=TT1‍∑t=sT2‍1|as|wwww+|g(t,xc1tm,…,xcktm)+∑s=TT1‍∑t=T2+1∞‍1|as|wwww-g(t,xc1t,…,xckt)|]+∑s=T1+1∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=TT1‍∑t=sT2‍1|as|wwwwwwwwwww-h(t,xd1t,…,xdkt)|+∑s=TT1‍∑t=sT2‍1|as|wwww+|g(t,xc1tm,…,xcktm)+∑s=TT1‍∑t=T2+1∞‍1|as|wwwww-g(t,xc1t,…,xckt)|]+∑j=TT1‍∑s=jT2‍∑t=sT3‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=TT1‍∑s=jT2‍∑t=sT3‍t-s+1|aj|www-p(t,xo1t,…,xokt)|+∑j=TT1‍∑s=jT2‍∑t=T3+1∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=TT1‍∑s=jT2‍∑t=sT3‍t-s+1|aj|wwwW-p(t,xo1t,…,xokt)|+∑j=TT1‍∑s=T2+1∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=TT1‍∑s=jT2‍∑t=sT3‍t-s+1|aj|wwwW-p(t,xo1t,…,xokt)|+∑j=T1+1∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=TT1‍∑s=jT2‍∑t=sT3‍t-s+1|aj|wwwW-p(t,xo1t,…,xokt)|≤∑t=TT1‍A|f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑s=TT1‍∑t=sT2‍[B|h(t,xd1tm,…,xdktm)-h(t,xd1t,…,xdkt)|+∑s=TT1‍∑t=sT2‍wewi+A|g(t,xc1tm,…,xcktm)-g(t,xc1t,…,xckt)|]+∑j=TT1‍∑s=jT2‍∑t=sT3‍E|p(t,xo1tm,…,xoktm)-p(t,xo1t,…,xokt)|+2∑t=T1+1∞‍Wt|at|+2∑s=TT1‍∑t=T2+1∞‍1|as|[(t-s+1)Qt+Pt]+2∑s=T1+1∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+2∑j=TT1‍∑s=jT2‍∑t=T3+1∞‍t-s+1|aj|Rt+2∑j=TT1‍∑s=T2+1∞‍∑t=s∞‍t-s+1|aj|Rt+2∑j=T1+1∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<εA(T1-T+1)18T1T2T3(A+B+E)+ε(B+A)(T1-T+1)(T2-T+1)18T1T2T3(A+B+E)+εE(T1-T+1)(T2-T+1)(T3-T+1)18T1T2T3(A+B+E)+2ε(T1-T+1)18T1+2ε(T1-T+1)(T2-T+1)18T1T2+2ε(T1-T+1)18T1+ε18<ε,+2ε(T1-T+1)18T1ww+∀m≥T4,
which means that SL is continuous in Ω1(N,M).
Next we prove thatSL(Ω1(N,M)) is uniformly Cauchy. It follows from (13) that for given ε>0, there exists V>T satisfying
(22)∑t=V∞‍Wt|at|+∑s=V∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=V∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<ε4.
Using (16) and (22), we know that for any x={xn}n∈ℤβ∈Ω1(N,M) and m,n≥V,
(23)|(SLx)m-(SLx)n|=|∑t=m∞‍1atf(t,xb1t,…,xbkt)-∑t=n∞‍1atf(t,xb1t,…,xbkt)+∑s=m∞‍+∑s=m∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑s=m∞‍∑t=s∞‍1aswwww-g(t,xc1t,…,xckt)]+∑s=m∞‍-∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑s=m∞‍∑t=s∞‍1aswwww-g(t,xc1t,…,xckt)]+∑s=m∞‍+∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]+∑s=m∞‍-∑j=m∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]|≤∑t=m∞‍Wt|at|+∑t=n∞‍Wt|at|+∑s=m∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑s=n∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)+∑j=m∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)(24)≤2∑t=V∞‍Wt|at|+2∑s=V∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+2∑j=V∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<ε,
which yields that SL(Ω1(N,M)) is uniformly Cauchy. It follows from Lemma 2 that SL(Ω1(N,M)) is relatively compact. Consequently Lemma 3 guarantees that there exists x={xn}n∈ℤβ∈Ω1(N,M) satisfying ULx+SLx=x, which together with (15) and (16) gives that
(25)xn=L-γnxn-τ+∑t=n∞‍1atf(t,xb1t,…,xbkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)L-γnxn-τ-g(t,xc1t,…,xckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt],-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwwwwwwww∀n≥T,
which yields that
(26)Δ(xn+γnxn-τ)+1anf(n,xb1n,…,xbkn)=-∑t=n∞‍1an[(t-n+1)h(t,xd1t,…,xdkt)=-∑t=n∞‍1anwww-g(t,xc1t,…,xckt)]+∑s=n∞‍∑t=s∞‍t-s+1an[p(t,xo1t,…,xokt)-rt],+∑s=n∞‍∑t=s∞‍t-s+1an[p(t,xo1t,…,xokt)ee-]∀n≥T,anΔ(xn+γnxn-τ)+f(n,xb1n,…,xbkn)=-∑t=n∞‍(t-n+1)h(t,xd1t,…,xdkt)+∑t=n∞‍g(t,xc1t,…,xckt)+∑s=n∞‍∑t=s∞‍(t-s+1)[p(t,xo1t,…,xokt)-rt],+∑s=n∞‍∑t=s∞‍(t-s+1)[p(t,xo1t,…,xokt)we-]∀n≥T,
which means that
(27)Δ3(anΔ(xn+γnxn-τ))+Δ3f(n,xb1n,…,xbkn)+Δ2g(n,xc1n,…,xckn)+Δh(n,xd1n,…,xdkn)+p(n,xo1n,…,xokn)=rt,∀n≥T;
that is, x={xn}n∈ℤβ is a bounded positive solution of (2) in Ω1(N,M).
Finally we prove that (2) has uncountably many bounded positive solutions in Ω1(N,M). Let L1,L2∈(cM+N,M(1-c)) and L1≠L2. For each j∈{1,2}, we conclude similarly that there exist a positive integer Tj≥n0+n1+τ+|β| and two mappings ULj and SLj satisfying (14)–(16), where L and T are replaced by Lj and Tj, respectively, and ULj+SLj has a fixed point zj={zjn}n∈ℤβ, which is a bounded positive solution (2) in Ω1(N,M); that is,
(28)zjn=Lj-γnzjn-τ+∑t=n∞‍1atf(t,zjb1t,…,zjbkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,zjd1t,…,zjdkt)+∑s=n∞‍∑t=s∞‍1asw-g(t,zjc1t,…,zjckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,zjo1t,…,zjokt)-rt],-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[pww]∀n≥Tj,j∈{1,2}.
Equation (13) ensures that there exists some T3>max⁡{T1,T2} satisfying
(29)∑t=T3∞‍Wt|at|+∑s=T3∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T3∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<|L1-L2|4.
Combining (11), (12), (28), and (29), we deduce that for any n≥T3,
(30)|z1n-z2n|=|∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,z2o1t,…,z2okt)-rt]L1-L2-γnz1n-τ+γnz2n-τ=ww+∑t=n∞‍1atf(t,z1b1t,…,z1bkt)=ww-∑t=n∞‍1atf(t,z2b1t,…,z2bkt)=ww+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,z1d1t,…,z1dkt)+∑s=n∞‍∑t=s∞‍1aswwwww-g(t,z1c1t,…,z1ckt)]=ww-∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,z2d1t,…,z2dkt)+∑s=n∞‍∑t=s∞‍1aswwwww-g(t,z2c1t,…,z2ckt)]=ww-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,z1o1t,…,z1okt)-rt]=ww+∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,z2o1t,…,z2okt)-rt]|≥|L1-L2|-|γn||z1n-τ-z2n-τ|-∑t=n∞‍1at|f(t,z1b1t,…,z1bkt)-f(t,z2b1t,…,z2bkt)|-∑s=n∞‍∑t=s∞‍1as[(t-s+1)|h(t,z1d1t,…,z1dkt)-∑s=n∞‍∑t=s∞‍1aswwwwwwwwww-h(t,z2d1t,…,z2dkt)|-∑s=n∞‍∑t=s∞‍1aswwww+|g(t,z1c1t,…,z1ckt)-∑s=n∞‍∑t=s∞‍1aswwwwww-g(t,z2c1t,…,z2ckt)|]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj|p(t,z1o1t,…,z1okt)-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwww-p(t,z2o1t,…,z2okt)|≥|L1-L2|-c∥z1-z2∥-2(∑t=n∞‍Wt|at|+∑s=n∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]-2wwww+∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt)≥|L1-L2|-c∥z1-z2∥-2(∑t=T3∞‍Wt|at|+∑s=T3∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]-2wwww+∑j=T3∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt)>|L1-L2|2-c∥z1-z2∥,
which implies that
(31)∥z1-z2∥>|L1-L2|2(1+c)>0.
That is, z1≠z2. Thus (2) has uncountably many bounded positive solutions in Ω1(N,M). This completes the proof.Theorem 7.
Assume that there exist constantsn1∈ℕn0, M, N, and c with M>N>0 and c∈[0,(M-N)/M) and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (12), (13), and
(32)0≤γn≤c,∀n≥n1.
Then (2) possesses uncountably many bounded positive solutions in Ω1(N,M).Proof.
LetL∈(N+cM,M). It follows from (13) that there exists T≥n0+n1+τ+|β| sufficiently large such that
(33)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡⁡{M-L,L-cM-N}.
Let the mappings UL and SL:Ω1(N,M)→lβ∞ be defined by (15) and (16), respectively. By means of (12), (15), (16), (32), and (33), we infer that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω1(N,M), and n≥T,
(34)(ULx)n+(SLy)n=L-γnxn-τ+∑t=n∞‍1atf(t,yb1t,…,ybkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n∞‍∑t=s∞‍1aswww-g(t,yc1t,…,yckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt]≤L+∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<L+min⁡{M-L,L-cM-N}≤M,(ULx)n+(SLy)n=L-γnxn-τ+∑t=n∞‍1atf(t,yb1t,…,ybkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n∞‍∑t=s∞‍1aswwww-g(t,yc1t,…,yckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt]≥L-cM-∑t=T∞‍Wt|at|-∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]-∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)>L-cM-min⁡{M-L,L-cM-N}≥N;
that is, ULx+SLy∈Ω1(N,M) for any x,y∈Ω1(N,M). The rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof.Theorem 8.
Assume that there exist constantsn1∈ℕn0, M, N, and c with M>N>0 and c∈[0,(M-N)/M) and four nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (12), (13), and
(35)-c≤γn≤0,∀n≥n1
Then (2) possesses uncountably many bounded positive solutions in Ω1(N,M).Proof.
LetL∈(N,(1-c)M). It follows from (13) that there exists T≥n0+n1+τ+|β| sufficiently large such that
(36)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{(1-c)M-L,L-N}.
Let the mappings UL and SL:Ω1(N,M)→lβ∞ be defined by (15) and (16), respectively. Making use of (12), (15), (16), (35), and (36), we derive that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω1(N,M) and n≥T,
(37)(ULx)n+(SLy)n=L-γnxn-τ+∑t=n∞‍1atf(t,yb1t,…,ybkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n∞‍∑t=s∞‍1aswwww-g(t,yc1t,…,yckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt]≤L+cM+∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<L+cM+min⁡{(1-c)M-L,L-N}≤M,(ULx)n+(SLy)n=L-γnxn-τ+∑t=n∞‍1atf(t,yb1t,…,ybkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n∞‍∑t=s∞‍1aswwww-g(t,yc1t,…,yckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt]≥L-∑t=T∞‍Wt|at|-∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]-∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)>L-min⁡{(1-c)M-L,L-N}≥N;
that is, ULx+SLy∈Ω1(N,M) for any x,y∈Ω1(N,M). The rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof.Theorem 9.
Assume that there exist constantsn1∈ℕn0, M, N, and c with M>N>0 and c>M/(M-N) and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (13):
(38)γn≥c,∀n≥n1,(39)|f(n,u1,u2,…,uk)|≤Wn,|g(n,u1,u2,…,uk)|≤Pn,|h(n,u1,u2,…,uk)|≤Qn,|p(n,u1,u2,…,uk)|≤Rn,|h(,u1,u2,…,uk)|≤∀(n,u1,u2,…,uk)∈ℕn0×[0,Mc]k.
Then (2) possesses uncountably many bounded positive solutions in lβ∞.Proof.
LetL∈(N+M/c,M). It follows from (13) that there exists an integer T≥n0+n1+τ+|β| satisfying
(40)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M-L,L-N-Mc}.
Define two mappings UL and SL:Ω2T(N,M)→lβ∞ by
(41)(ULx)n={1γn+τ(L-xn+τ),n≥T,(ULx)T,β≤n<T(42)(SLx)n={1γn+τ{∑t=n+τ∞‍1atf(t,xb1t,…,xbkt)1γn+τw+∑s=n+τ∞‍∑t=s∞‍1as[(t,xc1t,…,xckt)(t-s+1)+∑s=n+τ∞‍∑t=s∞‍1aswwwww×h(t,xd1t,…,xdkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwww-g(t,xc1t,…,xckt)]1γn+τw-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍wwww×[p(t,xo1t,…,xokt)-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍eet-s+1aj-rt(t,xc1t,…,xckt)]∑t=s∞‍},-∑j=n+τ∞‍∑s=j∞‍∑s=j∞‍∑t=s∞‍∑s=j∞‍∑t=s∞‍ewet-s+1aj∑t=s∞‍n≥T,(SLx)T,-∑j=n+τ∞‍∑s=j∞‍eet-s+1aj∑t=s∞‍β≤n<T,
for each x={xn}n∈ℤβ∈Ω2T(N,M). By means of (13) and (39)–(42), we get that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω2T(N,M), and n≥T,
(43)(ULx)n+(SLy)n=1γn+τ(L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τwwe+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwwww-g(t,yc1t,…,yckt)]=1γn+τwwe-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwwwwwwwww-rt(t,xc1t,…,xckt)]∑t=n+τ∞‍)≤1γn+τ(L-Nγn+τ+∑t=T+τ∞‍Wt|at|≤1γn+τwew+∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≤1γn+τwew+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))<1γn+τ(L+min⁡{M-L,L-N-Mc})≤Mγn+τ,(ULx)n+(SLy)n=1γn+τ(L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τwww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwwwww-g(t,yc1t,…,yckt)]=1γn+τwww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt])≥1γn+τ(L-Mγn+τ-∑t=T+τ∞‍Wt|at|≥1γn+τwww-∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≥1γn+τwww-∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))>1γn+τ(L-Mc-min⁡{M-L,L-N-Mc})≥Nγn+τ,
which yield that ULx+SLy∈Ω2T(N,M) for any x,y∈Ω2T(N,M). By virtue of (38) and (41), we infer that
(44)∥ULx-ULy∥=sup⁡n∈ℤβ|(ULx)n-(ULy)n|=max⁡{sup⁡T>n≥β|(ULx)n-(ULy)n|,=maxwsup⁡n≥T|(ULx)n-(ULy)n|}=sup⁡n≥T(1|γn+τ||xn+τ-yn+τ|)≤1c∥x-y∥;
that is, UL is a contraction in Ω2T(N,M) because c>M/(M-N)>1.
In order to prove thatSL is completely continuous in Ω2T(N,M), we have to show that SL is continuous in Ω2T(N,M) and SL(Ω2T(N,M) is relatively compact. Suppose that {xm}m∈ℕ is an arbitrary sequence in Ω2T(N,M) and x∈Ω2T(N,M) with lim⁡m→∞xm=x, where xm={xnm}n∈ℤβ for each m∈ℕ and x={xn}n∈ℤβ. On account of (13), (39), lim⁡m→∞xm=x, and the continuity of f,g,h, and p, we obtain that for given ε>0, there exist T1,T2,T3, and T4∈ℕ with T4>T3>T2>T1>T satisfying
(45)∑t=T1+τ∞‍Wt|at|+∑s=T1+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T1+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<εc18;max⁡{∑t=T3+τ∞‍t-s+1|aj|Rt:T≤j≤T1+τ,j≤s≤T2+τ}<εc18T1T2;max⁡{∑s=T2+τ∞‍∑t=s∞‍t-s+1|aj|Rt:T≤j≤T1+τ}<εc18T1;max⁡{∑t=T2+τ∞‍1|as|[(t-s+1)Qt+Pt]+www:T≤s≤T1+τ∑t=T2+τ∞‍}<εc18T1;max⁡{|q(t,xb1tm,…,xbktm)maxw-q(t,xb1t,…,xbkt)|:q∈{f,g,h,p}}<εc18T1T2T3(A+B+E),∀m≥T4,T≤t≤T3,
where
(46)A=max⁡{1|as|:T+τ≤s≤T1+τ},B=max⁡{t-s+1|as|:T+τ≤s≤T1+τ,s≤t≤T2+τ},E=max⁡{t-s+1|aj|:T+τ≤j≤T1+τ,=maxwewj≤s≤T2+τ,s≤t≤T3+τt-s+1|aj|}.
It follows from (42)–(46) that
(47)∥SLxm-SLx∥=sup⁡n∈ℤβ|(SLxm)n-(SLx)n|=max⁡{sup⁡T>n≥β|(SLxm)n-(SLx)n|,=maxwwsup⁡n≥T|(SLxm)n-(SLx)n|}=sup⁡n≥T|1γn+τ(-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍ts+1aj[p(t,xo1tm,…,xoktm)-rt]∑t=n+τ∞‍1atf(t,xb1tm,…,xbktm)=sup⁡n≥Twwwww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1tm,…,xdktm)=sup⁡n≥Twwwwwwwwwwwwi-g(t,xc1tm,…,xcktm)]=sup⁡n≥Twwwww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1tm,…,xoktm)-rt])=sup⁡n≥Tww-1γn+τ(∑t=n+τ∞‍1atf(t,xb1t,…,xbkt)=sup⁡n≥Twwwwwww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwwwwwww-g(t,xc1t,…,xckt)]+∑s=n+τ∞‍∑t=s∞‍1aswww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑j=n+τ∞‍∑s=j∞‍×[p(t,xo1t,…,xokt)-rt]∑t=n+τ∞‍1at)∑s=j∞‍∑t=s∞‍t-s+1aj|≤1c(∑t=T+τ∞‍1|at||f(t,xb1tm,…,xbktm)∑t=T+τ∞‍1|at|wweww-f(t,xb1t,…,xbkt)|≤1cww+∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=T+τ∞‍∑t=s∞‍1|as|wwwwwwwwwwwe-h(t,xd1t,…,xdkt)|+∑s=T+τ∞‍∑t=s∞‍1|as|wwwwww+|g(t,xc1tm,…,xcktm)+∑s=T+τ∞‍∑t=s∞‍1|as|wwwwwwwe-g(t,xc1t,…,xckt)|]≤1cww+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwww-p(t,xo1t,…,xokt)|∑t=T+τ∞‍)≤1c(∑t=T+τT1+τ-1‍1|at||f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|∑t=T+τT1+τ-1‍e+∑t=T1+τ∞‍1|at||f(t,xb1tm,…,xbktm)+∑s=T+τ∞‍∑t=s∞‍1|as|www-f(t,xb1t,…,xbkt)|∑t=T+τT1+τ-1‍e+∑s=T+τT1+τ-1‍∑t=sT2+τ-1‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwwwwwwwe-h(t,xd1t,…,xdkt)|+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|www+|g(t,xc1tm,…,xcktm)+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwwww-g(t,xc1t,…,xckt)|]∑s=T+τT1+τ-1‍e+∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|wwwwwwwwwww-h(t,xd1t,…,xdkt)|+∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|wwwwww+|g(t,xc1tm,…,xcktm)+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwwwww-g(t,xc1t,…,xckt)|]∑s=T+τT1+τ-1‍e+∑s=T1+τ∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|wwwwwwwwww-h(t,xd1t,…,xdkt)|hhhhhhhhhhhhhhhhhhhhhhi+|g(t,xc1tm,…,xcktm)+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wjjw-g(t,xc1t,…,xckt)|]∑s=T+τT1+τ-1‍e+∑j=T+τT1+τ-1‍∑s=jT2+τ-1‍∑t=sT3+τ-1‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|wwwwwwwwww-p(t,xo1t,…,xokt)|∑s=T+τT1+τ-1‍e+∑j=T+τT1+τ-1‍∑s=jT2+τ-1‍∑t=T3+τ∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|wwwwwwwwwew-p(t,xo1t,…,xokt)|∑s=T+τT1+τ-1‍e+∑j=T+τT1+τ-1‍∑s=T2+τ∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)∑j=T+τT1+τ-1‍∑s=T2+τ∞‍∑t=s∞‍wwwwwewwwww-p(t,xo1t,…,xokt)|∑s=j∞‍e+∑j=T1+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑j=T1+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwwr-p(t,xo1t,…,xokt)|∑t=T2+τ∞‍)≤1c(∑t=T+τT1+τ-1‍A|f(t,xb1tm,…,xbktm)∑t=T+τT1+τ-1≤1c‍Awe-f(t,xb1t,…,xbkt)|≤1cww+∑s=T+τT1+τ-1‍∑t=sT2+τ-1‍[B|h(t,xd1tm,…,xdktm)+∑s=T+τT1+τ-1‍∑t=sT2+τ-1‍wwwwwww-h(t,xd1t,…,xdkt)|+∑s=T+τT1+τ-1‍∑t=sT2+τ-1‍wwwwww+A|g(t,xc1tm,…,xcktm)+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwww-g(t,xc1t,…,xckt)|]∑s=T+τT1+τ-1‍e+∑j=T+τT1+τ-1‍∑s=jT2+τ-1‍∑t=sT3+τ-1‍E|p(t,xo1tm,…,xoktm)+∑j=T+τT1+τ-1‍∑s=jT2+τ-1‍∑t=sT3+τ-1‍Ewwewww-p(t,xo1t,…,xokt)|+∑s=T+τT1+τ-1‍+2∑t=T1+τ∞‍Wt|at|+∑t=T1+τ∞‍+2∑s=T+τT1+τ-1‍∑t=T2+τ∞‍1|as|[(t-s+1)Qt+Pt]+∑s=T+τT1+τ-1‍+2∑s=T1+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑s=T+τT1+τ-1‍+2∑j=T+τT1+τ-1‍∑s=jT2+τ-1‍∑t=T3+τ∞‍t-s+1|aj|Rt+∑s=T+τT1+τ-1‍+2∑j=T+τT1+τ-1‍∑s=T2+τ∞‍∑t=s∞‍t-s+1|aj|Rt+∑s=T+τT1+τ-1‍+2∑j=T1+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt)<εA(T1-T)18T1T2T3(A+B+E)+ε(B+A)(T1-T)(T2-T)18T1T2T3(A+B+E)+εE(T1-T)(T2-T)(T3-T)18T1T2T3(A+B+E)+2ε(T1-T)18T1+2ε(T1-T)(T2-T)18T1T2+2ε(T1-T)18T1+ε18<ε,∀m≥T4,
which means that SL is continuous in Ω2T(N,M). It follows from (13) that for given ε>0, there exists V>T satisfying(48)∑t=V+τ∞‍Wt|at|+∑s=V+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=V+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<ε4.
Next we prove thatSL(Ω2T(N,M)) is uniformly Cauchy. In view of (42) and (48), we infer that for any x={xn}n∈ℤβ∈Ω2T(N,M) and m,n≥V,
(49)|(SLx)m-(SLx)n|=|1γm+τ∑t=m+τ∞‍1atf(t,xb1t,…,xbkt)∑t=m+τ∞‍-1γn+τ∑t=n+τ∞‍1atf(t,xb1t,…,xbkt)∑t=m+τ∞‍+1γm+τ∑s=m+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+1γm+τ∑s=m+τ∞‍∑t=s∞‍1aswwww-g(t,xc1t,…,xckt)]∑t=m+τ∞‍-1γn+τ∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+1γm+τ∑s=m+τ∞‍∑t=s∞‍1aswwww-g(t,xc1t,…,xckt)]∑t=m+τ∞‍+1γn+τ∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]∑t=m+τ∞‍-1γm+τ∑j=m+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]|≤1c(∑t=m+τ∞‍Wt|at|+∑t=n+τ∞‍Wt|at|∑j=m+τ∞‍∑s=j∞‍+∑s=m+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]∑j=m+τ∞‍∑s=j∞‍+∑s=n+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]∑j=m+τ∞‍∑s=j∞‍+∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)∑j=m+τ∞‍∑s=j∞‍+∑j=m+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|))≤2c(∑t=V+τ∞‍Wt|at|∑j=m+τ∞‍∑s=j∞‍+∑s=V+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]∑j=m+τ∞‍∑s=j∞‍+∑j=V+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|))<ε.
Note that (40) and (42) yield that
(50)∥SLx∥=sup⁡n∈ℤβ|(SLx)n|=max⁡{sup⁡T>n≥β|(SLx)n|,sup⁡n≥T|(SLx)n|}=sup⁡n≥T|1γn+τ(∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]∑t=n+τ∞‍1atf(t,xb1t,…,xbkt)=maxwwwwe+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwwwww-g(t,xc1t,…,xckt)]=maxwwwwe-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt])|≤1c(∑t=T+τ∞‍Wt|at|≤1cww+∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≤1cww+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|))<1cmin⁡{M-L,L-N-Mc}<Mc,∀x={xn}n∈ℤβ∈Ω2T(N,M),
which gives that SL(Ω2T(N,M)) is bounded. Thus Lemma 3 means that there exists x={xn}n∈ℤβ∈Ω2T(N,M) such that ULx+SLx=x, which is a bounded positive solution of (2) in Ω2T(N,M)⊂lβ∞.
LetL1,L2∈(N+M/c,M) and L1≠L2. For any j∈{1,2}, we deduce similarly that there exist a positive integer Tj≥n0+n1+τ+|β|, a closed bounded and convex subset Ω2Tj(N,M) of lβ∞, and two mappings ULj and SLj satisfying (40)–(42), where L and T are replaced by Lj and Tj, respectively, and ULj+SLj possesses a fixed point zj={zjn}n∈ℤβ∈Ω2Tj(N,M), which is a bounded positive solution of (2); that is,
(51)zjn=1γn+τ(Lj-zjn+τ+∑t=n+τ∞‍1atf(t,zjb1t,…,zjbkt)=1γn+τe+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,zjd1t,…,zjdkt)=1γn+τwwwwwwww-g(t,zjc1t,…,zjckt)]=1γn+τwe-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,zjo1t,…,zjokt)-rt]),∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[pwww(t,zjc1t,)∀n≥Tj,j∈{1,2}.
Observe that (13) implies that there exists T3∈ℕ with T3>max⁡{T1,T2} satisfying
(52)∑t=T3+τ∞‍Wt|at|+∑s=T3+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T3+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<|L1-L2|4,
which together with (51) yields that for each n≥T3(53)|z1n-z2n+z1n+τ-z2n+τγn+τ|=|1γn+τ(L1-L2+∑t=n+τ∞‍1atf(t,z1b1t,…,z1bkt)1γn+τwww-∑t=n+τ∞‍1atf(t,z2b1t,…,z2bkt)1γn+τwww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,z1d1t,…,z1dkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwww-g(t,z1c1t,…,z1ckt)]1γn+τwww-∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,z2d1t,…,z2dkt)-∑s=n+τ∞‍∑t=s∞‍1aswwwwwww-g(t,z2c1t,…,z2ckt)]1γn+τwww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,z1o1t,…,z1okt)-rt]1γn+τwww+∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,z2o1t,…,z2okt)-rt])|≥1γn+τ(|L1-L2|-∑t=n+τ∞‍1at|f(t,z1b1t,…,z1bkt)|L1-L2|-∑t=n+τ∞‍1atwwwwwww-f(t,z2b1t,…,z2bkt)|1γn+τwwwe-∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)|h(t,z1d1t,…,z1dkt)-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwwwww-h(t,z2d1t,…,z2dkt)|-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajww+|g(t,z1c1t,…,z1ckt)-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwww-g(t,z2c1t,…,z2ckt)|]1γn+τwwe-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj|p(t,z1o1t,…,z1okt)-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwww-p(t,z2o1t,…,z2okt)||L1-L2|-∑t=n+τ∞‍1at|f(t,z1b1t,…,z1bkt))≥1γn+τ(|L1-L2|-2∑t=T3+τ∞‍Wt|at|≥1γn+τwww-2∑s=T3+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≥1γn+τwww-2∑j=T3+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt)≥1γn+τ(|L1-L2|-|L1-L2|2)>|L1-L2|2γn+τ,
which implies that
(54)|z1n-z2n+z1n+τ-z2n+τγn+τ|>0,∀n≥T3;
that is, z1≠z2. Consequently, (2) has uncountably many bounded positive solutions in lβ∞. This completes the proof.Theorem 10.
Assume that there exist constantsn1∈ℕn0, M, N, c, and d with d>c>1 and M(1-1/c)>N(1-1/d) and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (13):
(55)c≤γn≤d,∀n≥n1,(56)|f(n,u1,u2,…,uk)|≤Wn,|g(n,u1,u2,…,uk)|≤Pn,|h(n,u1,u2,…,uk)|≤Qn,|p(n,u1,u2,…,uk)|≤Rn,|h(n,u1,u2,…,uk)|∀(n,u1,u2,…,uk)∈ℕn0×[Nd,Mc]k.
Then (2) possesses uncountably many bounded positive solutions in lβ∞.Proof.
LetL∈(N+M/c,M+N/d). It follows from (13) that there exists an integer T≥n0+n1+τ+|β| satisfying
(57)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M+Nd-L,L-N-Mc}.
Let the mappings UL and SL:Ω2T(N,M)→lβ∞ be defined by (41) and (42), respectively. Using (13) and (41), (42) and (57), we obtain that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω2T(N,M) and n≥T,
(58)(ULx)n+(SLy)n=1γn+τ(L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwew-g(t,yc1t,…,yckt)]=1γn+τww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt])≤1γn+τ(L-Nγn+τ+∑t=T+τ∞‍Wt|at|+∑s=n+τ∞‍∑t=s∞‍w+∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑s=n+τ∞‍∑t=s∞‍w+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))<1γn+τ(L-Nd+min⁡{M+Nd-L,L-N-Mc})≤Mγn+τ,(ULx)n+(SLy)n=1γn+τ(L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τw+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwewww-g(t,yc1t,…,yckt)]=1γn+τw-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt])≥1γn+τ(L-Mγn+τ-∑t=T+τ∞‍Wt|at|∑s=n+τ∞‍∑t=s∞‍wr-∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]∑s=n+τ∞‍∑t=s∞‍wr-∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))>1γn+τ(L-Mc-min⁡{M+Nd-L,L-N-Mc})≥Nγn+τ,
which imply that ULx+SLy∈Ω2T(N,M) for any x,y∈Ω2T(N,M). The rest of the proof is similar to that of Theorem 9 and is omitted. This completes the proof.Theorem 11.
Assume that there exist constantsn1∈ℕn0, M, N, and c with M>N>0 and c>M/(M-N) and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (13),(39), and
(59)γn≤-c,∀n≥n1.
Then (2) possesses uncountably many bounded positive solutions in lβ∞.Proof.
LetL∈(N,M(1-1/c)). It follows from (13) that there exists an integer T≥n0+n1+τ+|β| satisfying
(60)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]∑t=T∞‍Wt|at|+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)∑t=T∞‍Wt|at|w<min⁡{L-N,M-Mc-L}.
Define two mappings UL and SL:Ω3T(N,M)→lβ∞ by (42) and
(61)(ULx)n={1γn+τ(-L-xn+τ),n≥T,(ULx)T,β≤n<T,
for each x={xn}n∈ℤβ∈Ω3T(N,M). It follows from (42) and (59)–(61) that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω3T(N,M), and n≥T,
(62)(ULx)n+(SLy)n=1γn+τ(-L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwww-g(t,yc1t,…,yckt)]=1γn+τww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwwww-rt(t,yc1t,…,yckt)]∑t=n+τ∞‍)≤1γn+τ(-L+Mγn+τ-∑t=T+τ∞‍Wt|at|≤1γn+τwww-∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≤1γn+τwww-∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))<1γn+τ(-L-Mc-min⁡{L-N,M-Mc-L})≤-Mγn+τ,(ULx)n+(SLy)n=1γn+τ(-L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwww-g(t,yc1t,…,yckt)]=1γn+τww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwwwww-rt(t,yc1t,…,yckt)]∑t=n+τ∞‍)≥1γn+τ(-L-xn+τ+∑t=T+τ∞‍Wt|at|≥1γn+τww+∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≥1γn+τww+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))>1γn+τ(-L+min⁡{L-N,M-Mc-L})≥-Nγn+τ,
which yield that ULx+SLy∈Ω3T(N,M). The rest of the proof is similar to that of Theorem 9 and hence is omitted. This completes the proof.Theorem 12.
Assume that there exist constantsn1∈ℕn0, M, N, c, and d with M>N>0 and M(1-1/c)>N(1+1/d), d>c>1, and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (13), (56), and
(63)-d≤γn≤-c,∀n≥n1.
Then (2) possesses uncountably many bounded positive solutions in lβ∞.Proof.
LetL∈(N(1+1/d),M(1-1/c)). It follows from (13) that there exists an integer T≥n0+n1+τ+|β| satisfying
(64)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{L-Nd-N,M-Mc-L}.
Let the mappings UL and SL:Ω3T(N,M)→lβ∞ be defined by (42) and (61), respectively. By means of (42), (56), (61), and (64), we deduce that for any x={xn}n∈ℤβ, y={yn}n∈ℤβ∈Ω3T(N,M), and n≥T,
(65)(ULx)n+(SLy)n=1γn+τ(∑t=n+τ∞‍-L-xn+τ=1γn+τww+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwww-g(t,yc1t,…,yckt)]=1γn+τww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt])≤1γn+τ(-L+Mγn+τ-∑t=T+τ∞‍Wt|at|-∑j=n+τ∞‍∑s=j∞‍w-∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]-∑j=n+τ∞‍∑s=j∞‍w-∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))<1γn+τ(-L-Mc-min⁡{L-Nd-N,M-Mc-L})≤-Mγn+τ,(ULx)n+(SLy)n=1γn+τ(-L-xn+τ+∑t=n+τ∞‍1atf(t,yb1t,…,ybkt)=1γn+τww+∑s=n+τ∞‍∑t=s∞‍1as[(t-s+1)h(t,yd1t,…,ydkt)+∑s=n+τ∞‍∑t=s∞‍1aswwwwwww-g(t,yc1t,…,yckt)]=1γn+τww-∑j=n+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,yo1t,…,yokt)-rt])≥1γn+τ(-L+Nd-xn+τ+∑t=T+τ∞‍Wt|at|≥1γn+τww+∑s=T+τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]≥1γn+τww+∑j=T+τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj(Rt+|rt|))>1γn+τ(-L+Nd+min⁡{L-Nd-N,M-Mc-L})≥-Nγn+τ,
which mean that ULx+SLy∈Ω3T(N,M). The rest of the proof is similar to that of Theorem 9 and hence is omitted. This completes the proof.Theorem 13.
Assume that there exist constantsn1∈ℕn0, M, and N with M>N>0 and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (12), (13), and
(66)γn=1,∀n≥n1.
Then (2) possesses uncountably many bounded positive solutions in Ω1(N,M)Proof.
LetL∈(N,M). Equation (13) ensures that there exists T≥n0+n1+τ+|β| sufficiently large such that
(67)∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M-L,L-N}.
Define a mapping SL:Ω1(N,M)→lβ∞ by
(68)(SLx)n={L+∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1atf(t,xb1t,…,xbkt)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1as[(t,yb1t,…,ybkt)(t-s+1)∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍wwew∑t=s∞‍1as×h(t,xd1t,…,xdkt)∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍wwew∑t=s∞‍1as-g(t,xc1t,…,xckt)]-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswww×[p(t,xo1t,…,xokt)∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍w∑t=s∞‍1aswwww-rt],-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍e∑j=n+(2i-1)τn+2iτ-1‍∑s=n+(2i-1)τn+2iτ-1‍w∑s=j∞‍n≥T,(SLx)T,-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=n+(2i-1)τn+2iτ-1‍∑s=j∞‍β≤n<T,
for each x={xn}n∈ℤβ∈Ω1(N,M). In view of (12), (67), and (68), we deduce that for every x={xn}n∈ℤβ∈Ω1(N,M) and n≥T,
(69)|(SLx)n-L|=|∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1atf(t,xb1t,…,xbkt)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswwewe-g(t,xc1t,…,xckt)]-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍wwww×[p(t,xo1t,…,xokt)-rt]∑t=s∞‍|≤∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍Wt|at|+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)≤∑t=T∞‍Wt|at|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M-L,L-N},
which means that SL(Ω1(N,M))⊆Ω1(N,M) and ∥SLx∥≤M for all x∈Ω1(N,M). It follows from (13) that for each ε>0, there exists V>T satisfying (22). Using (22) and (68), we obtain that for any x={xn}n∈ℤβ∈Ω1(N,M) and m,n>V,
(70)|(SLx)m-(SLx)n|=|∑i=1∞‍∑t=m+(2i-1)τm+2iτ-1‍1atf(t,xb1t,…,xbkt)1atfw-∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1atf(t,xb1t,…,xbkt)1atfw+∑i=1∞‍∑s=m+(2i-1)τm+2iτ-1‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑i=1∞‍∑s=m+(2i-1)τm+2iτ-1‍∑t=s∞‍1aswwwww-g(t,xc1t,…,xckt)]1atfw-∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)-∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswwwww-g(t,xc1t,…,xckt)]1atfw+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍www×[p(t,xo1t,…,xokt)-rt]1atfw-∑i=1∞‍∑j=m+(2i-1)τm+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=m+(2i-1)τm+2iτ-1‍∑s=j∞‍∑t=s∞‍www×[p(t,xo1t,…,xokt)-rt]∑t=s∞‍|≤∑i=1∞‍∑t=m+(2i-1)τm+2iτ-1‍Wt|at|+∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍Wt|at|+∑i=1∞‍∑s=m+(2i-1)τm+2iτ-1‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)+∑i=1∞‍∑j=m+(2i-1)τm+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)≤2∑t=V∞‍Wt|at|+2∑s=V∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+2∑j=V∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<ε,
which yields that SL(Ω1(N,M)) is uniformly Cauchy.
Now we prove thatSL is continuous in Ω1(N,M). Suppose that {xm}m∈N is an arbitrary sequence in Ω1(N,M) and x∈Ω1(N,M) with lim⁡m→∞xm=x, where xm={xnm}n∈Zβ for each m∈N and x={xn}n∈ℤβ. Using (12), (13), lim⁡m→∞xm=x, and the continuity of f,g,h, and p, we conclude that for given ε>0, there exist T1,T2,T3, and T4∈ℕ with T4>T3>T2>T1>T satisfying (19) and (20). It follows from (19), (20), and (68) that
(71)∥SLxm-SLx∥=sup⁡n∈ℤβ|(SLxm)n-(SLx)n|=max⁡{sup⁡T>n≥β|(SLxm)n-(SLx)n|,=maxwesup⁡n≥T|(SLxm)n-(SLx)n|}=sup⁡n≥T|L+∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1atf(t,xb1tm,…,xbktm)=sup⁡n≥Tw+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1as[(t-s+1)h(t,xd1tm,…,xdktm)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswwwww-g(t,xc1tm,…,xcktm)]=sup⁡n∈Zβw-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1tm,…,xoktm)-rt]=sup⁡n∈Zβw-(L+∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1atf(t,xb1t,…,xbkt)=sup⁡n∈Zβwwwe+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1as[(t,xd1tm,…,xdktm)(t-s+1)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswwwwwwwe×h(t,xd1t,…,xdkt)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswwwwwwww-g(t,xc1t,…,xckt)]=sup⁡n∈Zβwwwe-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1ajww×[p(t,xo1t,…,xokt)-rt]∑t=s∞‍)|≤∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1|at||f(t,xb1tm,…,xbktm)≤∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1|at|w-f(t,xb1t,…,xbkt)|+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1|as|wwwwwwwwe-h(t,xd1t,…,xdkt)|+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1|as|ww+|g(t,xc1tm,…,xcktx)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1|as|wwww-g(t,xc1t,…,xckt)|]+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1|aj|+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍w×|p(t,xo1tm,…,xoktm)+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍www-p(t,xo1t,…,xokt)|≤∑t=T∞‍1|at||f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑s=T∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍wwwww-h(t,xd1t,…,xdkt)|+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍+|g(t,xc1tm,…,xcktx)-g(t,xc1t,…,xckt)|]+∑j=T∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍weww-p(t,xo1t,…,xokt)|<ε,
which implies that SL is continuous in Ω1(N,M). Thus Lemma 4 means that SL possesses a fixed point x={xn}n∈ℤβ∈Ω1(N,M); that is,
(72)xn=L+∑i=1∞‍∑t=n+(2i-1)τn+2iτ-1‍1atf(t,xb1t,…,xbkt)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1as[(t,xd1t,…,xdkt)(t-s+1)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1asww×h(t,xd1t,…,xdkt)+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1asww-g(t,xc1t,…,xckt)]-∑i=1∞‍∑j=n+(2i-1)τn+2iτ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj+∑i=1∞‍∑s=n+(2i-1)τn+2iτ-1‍∑t=s∞‍1aswe×[p(t,xo1t,…,xokt)-rt],Δ(anΔ(xn+xn-τ))+wwwwwwwww∀n≥T+τ,xn-τ=L+∑i=1∞‍∑t=n+(2i-2)τn+(2i-1)τ-1‍1atf(t,xb1t,…,xbkt)+∑i=1∞‍∑s=n+(2i-2)τn+(2i-1)τ-1‍∑t=s∞‍1as[(t,xd1t,…,xdkt)(t-s+1)+∑i=1∞‍∑s=n+(2i-2)τn+(2i-1)τ-1‍∑t=s∞‍1asww×h(t,xd1t,…,xdkt)+∑i=1∞‍∑s=n+(2i-2)τn+(2i-1)τ-1‍∑t=s∞‍1asww-g(t,xc1t,…,xckt)]-∑i=1∞‍∑j=n+(2i-2)τn+(2i-1)τ-1‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=n+(2i-2)τn+(2i-1)τ-1‍∑s=j∞‍∑t=s∞‍w×[p(t,xo1t,…,xokt)-rt],Δ(anΔ(xn+xn-τ))+wwwwwwww∀n≥T+τ,
which imply that
(73)xn+xn-τ=2L+∑t=n∞‍1atf(t,xb1t,…,xbkt)+∑s=n∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑s=n∞‍∑t=s∞‍1aswwww-g(t,xc1t,…,xckt)]-∑j=n∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt],-∑j=n∞‍∑s=j∞‍w∑t=s∞‍t-s+1aj[pw(t,xo1t,…,)]∀n≥T+τ,
which means that
(74)Δ(anΔ(xn+xn-τ))+Δf(n,xb1n,…,xbkn)+g(n,xc1n,…,xckn)=∑t=n∞‍h(n,xd1n,…,xdkn)-∑t=n∞‍(t-n+1)[p(t,xo1t,…,xokt)-rt],Δ(anΔ(xn+xn-τ))+wwwwww∀n≥T+τ,Δ3(anΔ(xn+xn-τ))+Δ3f(n,xb1n,…,xbkn)+Δ2g(n,xc1n,…,xckn)+Δh(n,xd1n,…,xdkn)+p(n,xo1n,…,xokn)=rn,∀n≥T+τ,
which yields that x={xn}n∈ℤβ∈Ω1(N,M) is bounded positive solution of (2). The rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof.Theorem 14.
Assume that there exist constantsn1∈ℕn0, M, and N with M>N>0 and nonnegative sequences {Wn}n∈ℕn0, {Pn}n∈ℕn0, {Qn}n∈ℕn0, and {Rn}n∈ℕn0 satisfying (12) and
(75)γn=-1,∀n≥n1;(76)max⁡{∑i=1∞‍∑t=n0+iτ∞‍Wt|at|,maxw∑i=1∞‍∑s=n0+iτ∞‍∑t=s∞‍1|as|[Pt+(t-s+1)Qt],maxw∑i=1∞‍∑j=n0+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|∑i=1∞‍∑j=n0+iτ∞‍∑s=j∞‍∑t=s∞‍wwww×(Rt+|rt|)∑i=1∞‍}<+∞.
Then (2) possesses uncountably many bounded positive solution in Ω1(N,M).Proof.
LetL∈(N,M). It follows from (76) that there exists T≥n0+n1+τ+|β| sufficiently large such that
(77)∑i=1∞‍∑t=T+iτ∞‍Wt|at|+∑i=1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M-L,L-N}.
Define a mappingSL:Ω1(N,M)→lβ∞ by
(78)(SLx)n={L-∑i=1∞‍∑t=n+iτ∞‍1atf(t,xb1t,…,xbkt)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1as-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍w×[(t-s+1)h(t,xd1t,…,xdkt)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍www-g(t,xc1t,…,xckt)]-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍w×[p(t,xo1t,…,xokt)-rt],-∑i=1∞‍∑j=n+iτ∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍∑t=s∞‍w∑t=s∞‍∑s=j∞‍∑s=j∞‍∑t=s∞‍∑s=j∞‍∑t=s∞‍n≥T,(SLx)T,-∑i=1∞‍∑j=n+iτ∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍∑s=j∞‍∑s=j∞‍∑t=s∞‍β≤n<T,
for all x={xn}n∈ℤβ∈Ω1(N,M). By virtue of (12), (77), and (78), we know that for every x={xn}n∈ℤβ∈Ω1(N,M) and n≥T,
(79)|(SLx)n-L|=|-∑i=1∞‍∑t=n+iτ∞‍1atf(t,xb1t,…,xbkt)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswwwew-g(t,xc1t,…,xckt)]-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍wwew×[p(t,xo1t,…,xokt)-rt]∑t=n+iτ∞‍|≤∑i=1∞‍∑t=n+iτ∞‍Wt|at|+∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)≤∑i=1∞‍∑t=T+iτ∞‍Wt|at|+∑i=1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<min⁡{M-L,L-N},
which means that SL(Ω1(N,M))⊆Ω1(N,M) and ∥SLx∥≤M for each x∈Ω1(N,M).
Next we prove thatSL(Ω1(N,M)) is uniformly Cauchy. It follows from (76) that for any given ε>0 there exists T*≥T with
(80)∑i=1∞‍∑t=T*+iτ∞‍Wt|at|+∑i=1∞‍∑s=T*+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=T*+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<ε2.
It follows from (12), (78), and (80) that for any x={xn}n∈ℤβ∈Ω1(N,M) and m,n≥T*,
(81)|(SLx)m-(SLx)n|=|∑i=1∞‍∑t=n+iτ∞‍1atf(t,xb1t,…,xbkt)1atfw-∑i=1∞‍∑t=m+iτ∞‍1atf(t,xb1t,…,xbkt)1atfw+∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)+∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswweww-g(t,xc1t,…,xckt)]1atfw-∑i=1∞‍∑s=m+iτ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)-∑i=1∞‍∑s=m+iτ∞‍∑t=s∞‍1aswweww-g(t,yc1t,…,yckt)]1atfw+∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]1atfw-∑i=1∞‍∑j=m+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=m+iτ∞‍∑s=j∞‍∑t=s∞‍www×[p(t,xo1t,…,xokt)-rt]∑t=s∞‍|≤∑i=1∞‍∑t=n+iτ∞‍Wt|at|+∑i=1∞‍∑t=m+iτ∞‍Wt|at|+∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑s=m+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)+∑i=1∞‍∑j=m+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)≤2∑i=1∞‍∑t=T*+iτ∞‍Wt|at|+2∑i=1∞‍∑s=T*+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+2∑i=1∞‍∑j=T*+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|(Rt+|rt|)<ε,
which yields that SL(Ω1(N,M)) is uniformly Cauchy.
Now we prove thatSL is continuous in Ω1(N,M). Suppose that {xm}m∈ℕ is an arbitrary sequence in Ω1(N,M) and x∈Ω1(N,M) with lim⁡m→∞xm=x, where xm={xnm}n∈ℤβ for each m∈ℕ and x={xn}n∈ℤβ. By (12), (76), lim⁡m→∞xm=x, and the continuity of f,g,h, and p, we get that for given ε>0, there exist T1,T2,T3,T4, and T5∈ℕ with T5>T4>T3>T2>T1>T satisfying
(82)∑i=T1∞‍∑t=T+iτ∞‍Wt|at|+∑i=T1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑i=T1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<ε18;∑t=T+T2τ∞‍Wt|at|+∑s=T+T2τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+∑j=T+T2τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<ε18T1;max⁡{∑t=T+T3τ∞‍1|as|[(t-s+1)Qt+Pt]max⁡w+∑s=T+T3τ∞‍∑t=s∞‍t-s+1|aj|Rtmaxwe:T+τ≤s≤T+T2τ,T+τ≤j≤T+T2τ∑t=T+T3τ∞‍}<ε18T1T2τ;max⁡{∑t=T+T4τ∞‍t-s+1|aj|Rtmaxwe:T+τ≤j≤T+T2τ,j≤s≤T+T3τ∑t=T+T4τ∞‍}<ε18T1T2T3τ2;∑i=1T1-1‍∑t=T+iτT+T2τ-1‍A|f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍[B|h(t,xd1tm,…,xdktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍wewww-h(t,xd1t,…,xdkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍www+A|g(t,xc1tm,…,xcktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍wwwwww-g(t,xc1t,…,xckt)|]+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍E|p(t,xo1tm,…,xoktm)+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍www-p(t,xo1t,…,xokt)|<ε18,
where
(83)A=max⁡{1|as|:T≤s≤T+T2τ},B=max⁡{t-s+1|as|:T≤s≤T+T2τ,B=maxws≤t≤T+T3τt-s+1|as|},E=max⁡{t-s+1|aj|:T≤j≤T+T2τ,E=maxwj≤s≤T+T3τ,s≤t≤T+T4τt-s+1|as|}.
In terms of (12), (78), and (83), we have
(84)∥SLxm-SLx∥=sup⁡n∈ℤβ|(SLxm)n-(SLx)n|=max⁡{sup⁡β≥n>T|(SLxm)n-(SLx)n|,=maxwwwsup⁡n≥T|(SLxm)n-(SLx)n|}=sup⁡n≥T|L-∑i=1∞‍∑t=n+iτ∞‍1atf(t,xb1tm,…,xbktm)=sup⁡n≥Tww-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1tm,…,xdktm)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswwwwww-g(t,xc1tm,…,xcktm)]=sup⁡n≥Tww-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1tm,…,xoktm)-rt]=sup⁡n≥Tww-(∑s=j∞‍∑t=s∞‍t-s+1aj[p(t,xo1t,…,xokt)-rt]L-∑i=1∞‍∑t=n+iτ∞‍1atf(t,xb1t,…,xbkt)=sup⁡n≥Twwww-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1as[(t-s+1)h(t,xd1t,…,xdkt)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswwwwwwww-g(t,xc1t,…,xckt)]-∑i=1∞‍∑s=n+iτ∞‍w-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwe×[p(t,xo1t,…,xokt)-rt]-∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj)|≤∑i=1∞‍∑t=T+iτ∞‍1|at||f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑i=1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|wwwwwwwwww-h(t,xd1t,…,xdkt)|+∑i=1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|wwwe+|g(t,xc1tm,⋯,xcktm)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswwwwww-g(t,xc1t,…,xckt)|]+∑i=T1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj||p(t,xo1tm,…,xoktm)+∑i=T1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|wwww-p(t,xo1t,…,xokt)|≤∑i=1T1-1‍∑t=T+iτT+T2τ-1‍1|at||f(t,xb1tm,…,xbktm)≤∑i=1T1-1‍∑t=T+iτT+T2τ-1‍1|at|ww-f(t,xb1t,…,xbkt)|+∑i=1T1-1‍∑t=T+T2τ∞‍1|at||f(t,xb1tm,…,xbktm)+∑i=1T1-1‍∑t=T+T2τ∞‍1|at|wwww-f(t,xb1t,…,xbkt)|+∑i=T1∞‍∑t=T+iτ∞‍1|at||f(t,xb1tm,…,xbktm)+∑i=1T1-1‍∑t=T+T2τ∞‍1|at|www-f(t,xb1t,…,xbkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|w×[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|wwwewewww-h(t,xd1t,…,xdkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|wwee+|g(t,xc1tm,…,xcktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|wwwew-g(t,xc1t,…,xckt)|]+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=T+T3τ∞‍1|as|∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|w×[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=T+T3τ∞‍1|as|wwwwwweww-h(t,xd1t,…,xdkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|ww+|g(t,xc1tm,…,xcktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|wwww-g(t,xc1t,…,xckt)|]+∑i=1T1-1‍∑s=T+T2τ∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=1T1-1‍∑s=T+T2τ∞‍∑t=s∞‍1|as|wwwwwwwwww-h(t,xd1t,…,xdkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|wwe+|g(t,xc1tm,…,xcktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|wwww-g(t,xc1t,…,xckt)|]+∑i=T1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|[(t-s+1)|h(t,xd1tm,…,xdktm)+∑i=T1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|wwwwwwwwww-h(t,xd1t,…,xdkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|w+|g(t,xc1tm,…,xcktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍1|as|www-g(t,xc1t,…,xckt)|]+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍t-s+1|aj|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍w∑t=sT+T4τ-1‍ew×|p(t,xo1tm,…,xoktm)+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍wweww-p(t,xo1t,…,xokt)|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=T+T4τ∞‍t-s+1|aj|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍wew×|p(t,xo1tm,…,xoktm)+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍wweww-p(t,xo1t,…,xokt)|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=T+T3τ∞‍∑t=s∞‍t-s+1|aj|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍w×|p(t,xo1tm,…,xoktm)+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍www-p(t,xo1t,…,xokt)|+∑i=1T1-1‍∑j=T+T2τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍ww×|p(t,xo1tm,…,xoktm)+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑t=sT+T4τ-1‍wwww-p(t,xo1t,…,xokt)|+∑i=T1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑t=sT+T4τ-1‍ew×|p(t,xo1tm,…,xoktm)+∑i=1T1-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍weww-p(t,xo1t,…,xokt)|≤∑i=1T1-1‍∑t=T+iτT+T2τ-1‍A|f(t,xb1tm,…,xbktm)-f(t,xb1t,…,xbkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍[B|h(t,xd1tm,…,xdktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍wwwwwe-h(t,xd1t,…,xdkt)|+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍wwww+A|g(t,xc1tm,…,xcktm)+∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=sT+T3τ-1‍wwwwwww-g(t,xc1t,…,xckt)|]+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍E|p(t,xo1tm,⋯,xoktm)+∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=sT+T4τ-1‍Ewrww-p(t,xo1t,⋯,xokt)|+2∑i=1T1-1‍∑t=T+T2τ∞‍Wt|at|+2∑i=T1∞‍∑t=T+iτ∞‍Wt|at|+2∑i=1T1-1‍∑s=T+iτT+T2τ-1‍∑t=T+T3τ∞‍1|as|[(t-s+1)Qt+Pt]+2∑i=1T1-1‍∑s=T+T2τ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+2∑i=T1∞‍∑s=T+iτ∞‍∑t=s∞‍1|as|[(t-s+1)Qt+Pt]+2∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=jT+T3τ-1‍∑t=T+T4τ∞‍t-s+1|aj|Rt+2∑i=1T1-1‍∑j=T+iτT+T2τ-1‍∑s=T+T3τ∞‍∑t=s∞‍t-s+1|aj|Rt+2∑i=1T1-1‍∑j=T+T2τ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt+2∑i=T1∞‍∑j=T+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1|aj|Rt<ε18+2(T1-1)(T2-1)τε18T1T2τ+2(T1-1)ε18T1+2(T1-1)(T2-1)(T3-1)τ2ε18T1T2T3τ2+2ε18<ε,∀m≥T5,
which yields that SL is continuous in Ω1(N,M). It follows from Lemmas 2 and 4 that SL has a fixed point x={xn}n∈ℤβ∈Ω1(N,M); that is,
(85)xn=L-∑i=1∞‍∑t=n+iτ∞‍1atf(t,xb1t,…,xbkt)-∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1as[(t,xc1t,…,xckt)(t-s+1)∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswwww×h(t,xd1t,…,xdkt)∑i=1∞‍∑s=n+iτ∞‍∑t=s∞‍1aswwww-g(t,xc1t,…,xckt)]+∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj+∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍w×[p(t,xo1t,…,xokt)-rt],+∑i=1∞‍∑j=n+iτ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwwww∀n≥T+τ,xn-τ=L-∑i=1∞‍∑t=n+(i-1)τ∞‍1atf(t,xb1t,…,xbkt)-∑i=1∞‍∑s=n+(i-1)τ∞‍∑t=s∞‍1as-∑i=1∞‍∑s=n+(i-1)τ∞‍∑t=s∞‍rw×[(t-s+1)h(t,xd1t,…,xdkt)-∑i=1∞‍∑s=n+(i-1)τ∞‍∑t=s∞‍1asw-g(t,xc1t,…,xckt)]+∑i=1∞‍∑j=n+(i-1)τ∞‍∑s=j∞‍∑t=s∞‍t-s+1aj+∑i=1∞‍∑j=n+(i-1)τ∞‍∑s=j∞‍∑t=s∞‍ew×[p(t,xo1t,…,xokt)-rt],+∑i=1∞‍∑j=n+(i-1)τ∞‍∑s=j∞‍∑t=s∞‍t-s+1ajwwewww∀n≥T+τ,
which imply that
(86)Δ(xn-xn-τ)=-1anf(n,xb1n,…,xbkn)-∑t=n∞‍1an[(t-n+1)h(t,xd1t,…,xdkt)-∑t=n∞‍1anwwww-g(t,xc1t,…,xckt)]+∑s=n∞‍∑t=s∞‍t-s+1an[p(t,xo1t,…,xokt)-rt],+∑s=n∞‍∑t=s∞‍t-s+1an[p(t,ww,…,xokt)]∀n≥T+τ,
which yields that
(87)Δ3(anΔ(xn-xn-τ))+Δ3f(n,xb1n,…,xbkn)+Δ2g(n,xc1n,…,xckn)+Δh(n,xd1n,…,xdkn)+p(n,xo1n,…,xokn)=rn,∀n≥T+τ;
that is, x={xn}n∈ℤβ∈Ω1(N,M) is a bounded positive solution of (2). The rest of the proof is similar to that of Theorem 6 and is omitted. This is completes proof.


## 4. Applications

Now we display nine examples as applications of the results presented in Section3.Example 1.
Consider the fourth order nonlinear neutral delay difference equation:(88)Δ3((n5+2n)Δ(xn+(-1)nn24n2+n+3xn-τ))+Δ3(nxn2-n17+nxn2-n5-n-3)+Δ2((-1)nxn-321+5n2+1n7-17(xn-34+3n2+2))+Δ(12n-x2n-115cos⁡2(n3x2n-15)2+n5|x2n-1|)+(-1)n-1nsin11(nxn3-3n9)n2+2n+5(1+xn3-3n22)+n5=4n3+2n-1(n+1)123n2+7,=4n3+2n-1(n+1)123n2+7,wwwwwwe∀n≥7.
It follows from Theorem 6 that (88) possesses uncountably many bounded positive solutions in Ω1(N,M).Example 2.
Consider the fourth order nonlinear neutral delay difference equation:(89)Δ3((-1)n+1n5/2Δ(xn+6n-310n+2xn-τ))+Δ3(xn3-55+2n3xn3-58+(n+3)4)+Δ2((-1)n-1n2x2n-114+n5+|cos⁡(x2n-12n3)|)+Δ(xn-14sinn+xn-12-n4n7+xn-16cos⁡4|xn-13|+n)+x5n-98n6n9+|nx5n-9sin3(n2+1)|=cos⁡(n3-2n+3)n4+3n-1,=4n3+2n-1(n+1)123n2+7,wwwwwe∀n≥1.
It follows from Theorem 7 that (89) possesses uncountably many bounded positive solutions in Ω1(N,M).Example 3.
Consider the fourth order nonlinear neutral delay difference equation:(90)Δ3(4n2cos⁡4(3n2+2)5n2+2n(n7-n3+2)×Δ(xn-4n2cos⁡4(3n2+2)5n2+2nxn-τ))+Δ3(n-9xn3-53xn+115)+Δ2(n+3xn3-421-2n+5xn2-112n5+3nxn3-42xn2-12)+Δ((xn-7+xn4-1+n)4|(n+xn-7)5(n+xn4-1)3|+1)+n2xn2-33xn-45+ln⁡nn2+xn-44+(n+xn2-3)6=(-1)n+1(4n5-3)n9+4n2+5,=4n3+2n-1(n+1)123n2+7,wwwwwe∀n≥9.
Theorem 8 implies that (90) possesses uncountably many bounded positive solutions in Ω1(N,M).Example 4.
Consider the fourth order nonlinear neutral delay difference equation:(91)Δ3(n3Δ(xn+ln⁡(3n3-134n)xn-τ))+Δ3(2n+x2n-33cos⁡n(n+x2n-3)4(n+1)5+72)+Δ2(nxn57sin3(n2xn53)(n+2+3)6)+Δ(xn+62-(-1)n+1nn2+1+(xn+6+n)10)+n2+x2n-1827n13+x2n-182|cos⁡3(nx2n-184)|=n7-n11-2n15+36,=4n3+2n-1(n+1)123n2+7,wwwe∀n≥15.
Theorem 9 yields that (91) has uncountably many bounded positive solutions in lβ∞.Example 5.
Consider the fourth order nonlinear neutral delay difference equation:(92)Δ3((-1)nnln⁡2nΔ(xn+(1+1n)nxn-τ))+Δ3(nxn-43-xn-42+1n5+ln⁡(1+xn-42))+Δ2(n2xn-93-nxn-9-nn8+(xn-94-n)4)+Δ(xn-65-(-1)nnn6+ln⁡3(n+xn-68))+cos⁡⁡(n-ln⁡(1+xn-72))n5+cos⁡3(nxn-77)+1=n2-n+(-1)nn9+3n+1,=4n3+2n-1(n+1)123n2+7,wwwwwe∀n≥10.
Theorem 10 guarantees that (92) possesses uncountably many bounded positive solutions in lβ∞.Example 6.
Consider the fourth order nonlinear neutral delay difference equation:(93)Δ3(n4+2nln⁡(n+5)Δ(xn-(n-16)3xn-τ))Δ3w+Δ3(n2-2ncos⁡2(nxn-52))Δ3w+Δ2(xn2-2n+33ln⁡2(n-13)n11+|xn2-2n+3+1|)Δ3w+Δ(nxn-194sin(xn-192+n)n9+2n+xn-196)Δ3w+n2sin16(n+xn-134)(n+xn-132)8=(-1)n+1(n3+11)(n+2)5(n+5)4,∀n≥18.
Theorem 11 ensures that (93) possesses uncountably many bounded positive solutions in lβ∞.Example 7.
Consider the fourth order nonlinear neutral delay difference equation:(94)Δ3(-n8Δ(xn-(3-sin9n)xn-τ))+Δ3(n5ln⁡(n+xn-302)-xn-303n25+xn-308)+Δ2(n3xn2-13-nxn2-12-cos⁡⁡(n5)n81+(nxn2-13-1)2)+Δ(n6xn-23-n+3n16+(n3-xn-27)2)+n8+3n5xn-42-xn-43n58+n22-sin(n5xn-43)=n25+(-1)nn11-1n91+33n+5,∀n≥1.
Theorem 12 guarantees that (94) possesses uncountably many bounded positive solutions in lβ∞.Example 8.
Consider the fourth order nonlinear neutral delay difference equation:(95)Δ3((-1)n-1(n+2)3Δ(xn+xn-τ))+Δ3(n4xn2-95xn-42n3+|xn2-925-nxn-437|)+Δ2(n4+xn-12xn3-33n9+ln⁡3(n2+xn3-36))+Δ(sin(n9xn-28x5n-69)n5+(n+2xn-282x5n-619)2)+ln⁡(n2+xn2-176xn-614)n16+(2xn2-17+xn-6)2=(-1)n(n+1)/2cos⁡7(3n5+1)n5+n3+3,(-1)n(n+1)/2cos⁡7(3n5+1)n5+n3+3e∀n≥8.
It follows from Theorem 13 that (95) possesses uncountably many bounded positive solutions in Ω(N,M).Example 9.
Consider the fourth order nonlinear neutral delay difference equation:(96)Δ3(n17Δ(xn-xn-τ))+Δ3(nx9n-53x2n-12-sin3(arctan(n2+1)))+Δ2(n3+xn-74xn2-25(n+xn-76)13+|xn-7-nxn2-22|)+Δ(xn-103ln⁡4(n2+xn3-32)(n+xn-102xn3-32)9(n+2)31)+(3-n2)x3n-22-nxn-99n60+|x3n-2xn-9|=(-1)n(n2-1)cos⁡5(n-n)n19+15n8+1,=(-1)n(n+1)/2cos⁡7(3n5+1)n5+n3+3w∀n≥11.
Theorem 14 means that (96) possesses uncountably many bounded positive solutions in Ω(N,M).



---

*Source: 238036-2014-03-04.xml*