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

Authors: Zeqing Liu; Shin Min Kang; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper studies the nonlinear two-dimensional difference system with multiple delaysΔ(xn+p1nxn-τ1)+f1(n,xa1n,…,xahn,yb1n,…,ybkn)=q1n,Δ(yn+p2nyn-τ2)+f2(n,xc1n,…,xchn,yd1n,…,ydkn)=q2n,n≥n0. Using the Banach fixed point theorem and a few new analysis techniques, we show the existence of uncountably many bounded positive solutions for the system, suggest Mann iterative algorithms with errors, and discuss
the error estimates between the positive solutions and iterative sequences generated by the Mann iterative
algorithms. Examples to illustrates the results are included.


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## Body

## 1. Introduction and Preliminaries

In recent years, there has been an increasing interest in the study of oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, positive solutions, nonoscillatory solutions, and periodic solutions, respectively, for various difference equations and systems, see for example, [1–34] and the references cited therein.Jiang and Tang [18] and Graef and Thandapani [14] studied the oscillation of the linear two-dimensional difference system
(1.1)Δxn=pnyn,Δyn-1=-qnxn,n≥n0,
and nonlinear two-dimensional difference system
(1.2)Δxn=bng(yn),Δyn-1=-anf(xn),n≥n0,
where n0∈ℕ, {pn}n≥n0,{qn}n≥n0,{an}n≥n0, and {bn}n≥n0 are nonnegative sequences, f,g∈C(ℝ,ℝ) with uf(u)>0 and ug(u)>0 for all u≠0. Jiang and Tang [19] also gave some necessary and sufficient conditions for all solutions of System (1.2) to be oscillatory.  Agarwal et al. [2] discussed the two-dimensional nonlinear difference system of the form
(1.3)Δxn=anf(yn),Δyn-1=bng(xn),n≥n0,
where n0∈ℕ, {an}n≥n0 and {bn}n≥n0 are positive sequences, f,g∈C(ℝ,ℝ) are increasing with uf(u)>0 and ug(u)>0 for all u≠0, and they provided a classification scheme of positive solutions for System (1.3) and established conditions for the existence of solutions with designated asymptotic behavior. Li [21] introduced the two-dimensional nonlinear difference system:
(1.4)Δxn=anf(yn),Δyn=-bng(xn),n≥n0,
where n0∈ℕ, {an}n≥n0 and {bn}n≥n0 are nonnegative sequences, f,g∈C(ℝ,ℝ) with uf(u)>0, and ug(u)>0 for all u≠0, he showed both classification schemes for nonoscillatory solutions of System (1.4) and gave necessary and sufficient conditions for the existence of these solutions. Huo and Li [15, 16] considered the nonlinear two-dimensional difference system
(1.5)Δxn=bng(yn),Δyn=-anf(xn)+rn,n≥n0,
and the Emden-Fowler difference system
(1.6)Δxn=bng(yn),Δyn-1=-anf(xn)+rn,n≥1,
where n0∈ℕ, {an}n≥n0 is a real sequence, {bn}n≥n0 is a nonnegative sequence, ∑i=1∞‍|ri|<+∞, f,g∈C(ℝ,ℝ) with uf(u)>0, and ug(u)>0 for all u≠0, they proved some oscillation results for Systems (1.5) and (1.6). Jiang and Li [17] investigated the nonlinear two-dimensional difference system:
(1.7)Δxn=ang(yn),Δyn-1=f(n,xn),n≥n0,
where n0∈ℕ, {an}n≥n0 is a nonnegative sequence, f∈C(ℕn0×ℝ,ℝ),g∈C(ℝ,ℝ) with uf(n,u)>0 for all n∈ℕn0, and u≠0 and ug(u)>0 for all u≠0, they obtained some necessary and sufficient conditions for all solutions of System (1.7) to be oscillatory. Thandapani and Kumar [31] studied the oscillation for the nonlinear two-dimensional difference system of the neutral type
(1.8)Δ(xn-anxσn)=png(yn),Δyn=δqnf(xτn),n≥n0,
where δ=±1,n0∈ℕ, {an}n≥n0 is a positive sequence, {pn}n≥n0  and  {qn}n≥n0 are nonnegative sequences with ∑i=n0∞‍pi=+∞, limn→∞σn=limn→∞τn=+∞, f,g∈C(ℝ,ℝ) are nondecreasing with uf(u)>0 and ug(u)>0 for all u≠0. Wu and Liu [33] established the existence and multiplicity of periodic solutions for the first-order neutral difference system
(1.9)Δ(xn-cxn-τ)=a1ng1(xn)xn-λb1nf1(xn-τ1n,yn-ρ1n),Δ(yn-cyn-τ)=a2ng2(yn)yn-μb2nf2(xn-τ2n,yn-ρ2n),
where τ∈ℕ,λ,μ∈ℝ+∖{0}, c∈ℝ with |c|≠1, {a1n},{a2n},{b1n}, and {b2n} are positive T-periodic sequences, {τ1n},{τ2n},{ρ1n}, and {ρ2n} are positive T-periodic integer sequences. Tang [30] proved the existence results of a bounded nonoscillatory solution for the second-order linear delay difference equation
(1.10)Δ2xn=pnxn-k,n≥0,
where k∈ℕ and {pn}n≥0 is a nonnegative sequence. Cheng [20] utilized the Banach fixed point theorem to discuss the existence of a nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients:
(1.11)Δ2(xn+pxn-m)+pnxn-k-qnxn-l=0,n≥n0,
where m,k,l∈ℕ, p∈ℝ∖{-1}, {pn}n≥0 and {qn}n≥0 are nonnegative sequences with ∑n=n0∞‍nmax{pn,qn}<+∞. M. Migda and J. Migda [29] obtained the asymptotic behavior of the second-order neutral difference equation:
(1.12)Δ2(xn+pxn-k)+f(n,xn)=0,n≥1,
where p∈ℝ, k∈ℕ0, and f∈C(ℕ×ℝ,ℝ). Liu et al. [27] studied the global existence of uncountably many bounded nonoscillatory solutions for the second-order nonlinear neutral delay difference equation:
(1.13)Δ(anΔ(xn+bxn-τ))+f(n,xn-d1n,xn-d2n,…,xn-dkn)=cn,n≥n0,
where b∈ℝ,τ,k∈ℕ,n0∈ℕ0,{an}n∈ℕn0 is a positive sequence, {cn}n∈ℕn0 is a real sequence, ⋃l=1k‍{dln}n∈ℕn0⊆ℤ with limn→∞(n-dln)=+∞ for l∈{1,2,…,k}, and f:ℕn0×ℝk→ℝ is a mapping.The purpose of this paper is to study the below nonlinear two-dimensional difference system with multiple delays(1.14)Δ(xn+p1nxn-τ1)+f1(n,xa1n,…,xahn,yb1n,…,ybkn)=q1n,Δ(yn+p2nyn-τ2)+f2(n,xc1n,…,xchn,yd1n,…,ydkn)=q2n,n≥n0,
where h,k,τi∈ℕ, n0∈ℕ0,{pin}n∈ℕn0,{qin}n∈ℕn0⊂ℝ, and fi∈C(ℕn0×ℝh+k,ℝ) for i∈Λ2, {aln}n∈ℕn0,{cln}n∈ℕn0⊂ℤ with limn→∞aln=limn→∞cln=+∞ for l∈Λh and {bjn}n∈ℕn0,{djn}n∈ℕn0⊂ℤ with limn→∞bjn=limn→∞djn=+∞ for j∈Λk. It is easy to see that the system (1.14) includes Systems (1.1)–(1.9), (1.10)–(1.13), and a lot of the first- and second-order nonlinear, half-linear, and quasilinear difference equations as special cases. Using the Banach fixed point theorem and some analysis techniques, we prove the existence of uncountably many bounded positive solutions for  System (1.14), establish their iterative approximations, and discuss the error estimates between the positive solutions and iterative approximations. Our results sharp, and improve [14, Theorem 1] and [27, Theorems 2.1–2.7]. To illustrate our results, fifteen examples are also included.Throughout this paper, we assume thatΔ is the forward difference operator defined by Δxn=xn+1-xn, ℝ=(-∞,+∞),ℝ+=[0,+∞),ℤ,ℕ, and ℕ0 stand for the sets of all integers, positive integers, and nonnegative integers, respectively,
(1.15)Nn0={n:n∈N0withn≥n0},Λn={1,…,n},∀n∈N,α=inf⁡{n0-τ1,n0-τ2,aln,cln,bjn,djn:n∈Nn0,l∈Λh,j∈Λk},Zα={n:n∈Zwithn≥α},lα∞ denotes the Banach space of all bounded sequences on ℤα with norm
(1.16)‖x‖=sup⁡n∈Zα|xn|forx={xn}n∈Zα∈lα∞.
Let
(1.17)A(N,M)={(x,y)=({xn}n∈Zα,{yn}n∈Zα)∈lα∞×lα∞:N≤xn≤M,N≤yn≤M,n∈Zα},A(N,M,N0,M0)={(x,y)=({xn}n∈Zα,{yn}n∈Zα)∈lα∞×lα∞:N≤xn≤M,N0≤yn≤M0,n∈Zα},
for any M>N>0 and M0>N0>0. It is easy to see that A(N,M) and A(N,M,N0,M0) are bounded closed and convex subsets of the Banach space lα∞×lα∞ with norm
(1.18)‖(x,y)‖1=max⁡{‖x‖,‖y‖}for(x,y)∈lα∞×lα∞.By a solution of System (1.14), we mean a sequence ({xn}n∈ℤα,{yn}n∈ℤα) with a positive integer T≥n0+τ1+τ2+|α| such that System (1.14) is satisfied for all n≥T. A solution ({xn}n∈ℤα,{yn}n∈ℤα) of System (1.14) is said to be positive if both components are positive.Lemma 1.1 (see [35]).
Let{An}n≥0,{Bn}n≥0,{Cn}n≥0, and {Dn}n≥0 be four nonnegative real sequences satisfying the inequality
(1.19)An+1≤(1-Dn)An+DnBn+Cn,∀n≥0,
where {Dn}n≥0⊂[0,1],∑n=0∞‍Dn=+∞,limn→∞Bn=0, and ∑n=0∞‍Cn<+∞. Then limn→∞An=0.Lemma 1.2 (see [36]).
Letτ∈ℕ,n0∈ℕ0, and {Bn}n∈ℕn0 be a nonnegative sequence. Then
(1.20)∑i=0∞‍∑n=n0+iτ∞‍Bn<+∞⟺∑n=n0∞‍nBn<+∞.


## 2. Main Results

In this section, we investigate the existence of uncountably many bounded positive solutions for System (1.14) and their iterative approximations and the error estimates between the positive solutions and iterative approximations.Theorem 2.1.
Assume that there exist constantsM,N∈ℝ+∖{0}, n1∈ℕn0, and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying
(2.1)|fi(n,u1,…,uh,v1,…,vk)|≤rin,∀(n,ul,vj)∈Nn1×[N,M]2,l∈Λh,j∈Λk,i∈Λ2,(2.2)|fi(n,u1,…,uh,v1,…,vk)-fi(n,w1,…,wh,z1,…,zk)|≤tinmax⁡{|ul-wl|,|vj-zj|:l∈Λh,j∈Λk},∀(n,ul,wl,vj,zj)∈Nn1×[N,M]4,l∈Λh,j∈Λk,i∈Λ2,(2.3)∑n=n0∞‍nmax⁡{rin,tin,|qin|:i∈Λ2}<+∞,(2.4)N<M,pin=-1,∀n≥n1,i∈Λ2.
Then,(a)
for eachL∈(N,M), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by the schemes
(2.5)xnμ+1={(1-βμ-γμ)xnμ+βμ{L-∑l=0∞‍∑m=n+(l+1)τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1nμ,μ≥0,n≥TL,(1-βμ-γμ)xTLμ+βμ{L-∑l=0∞‍∑m=TL+(l+1)τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1TLμ,μ≥0,α≤n<TL,(2.6)ynμ+1={(1-βμ-γμ)ynμ+βμ{L-∑l=0∞‍∑m=n+(l+1)τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2nμ,μ≥0,n≥TL,(1-βμ-γμ)yTLμ+βμ{L-∑l=0∞‍∑m=TL+(l+1)τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2TLμ,μ≥0,α≤n<TLconverges to a bounded positive solution(x,y)∈A(N,M) of System (1.14) and has the error estimate:
(2.7)‖(xμ+1,yμ+1)-(x,y)‖1≤(1-βμ(1-θL))‖(xμ,yμ)-(x,y)‖1+2Mγμ,∀μ≥0,where{βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1] and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with
(2.8)∑μ=0∞‍βμ=+∞,(2.9)∑μ=0∞‍γμ<+∞orthereexistsasequence{ημ}μ∈N0⊂[0,+∞)satisfyingγμ=βμημ,∀μ∈N0,lim⁡μ→∞ημ=0,(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Proof.
(a) LetL∈(N,M). Now we construct a contraction mapping SL:A(N,M)→A(N,M) and prove that its fixed point is a bounded positive solution of System (1.14). It follows from (2.3) and (2.4) that there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| satisfying
(2.10)θL=∑m=TL∞‍(1+m)max⁡{tim:i∈Λ2},(2.11)∑m=TL∞‍(1+m)max⁡{rim+|qim|:i∈Λ2}<min⁡{M-L,L-N}.
Define three mappings SL:A(N,M)→lα∞×lα∞,S1L,S2L:A(N,M)→lα∞ by
(2.12)S1L(xn,yn)={L-∑l=0∞‍∑m=n+(l+1)τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,S1L(xTL,yTL),α≤n<TL,(2.13)S2L(xn,yn)={L-∑l=0∞‍∑m=n+(l+1)τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,S2L(xTL,yTL),α≤n<TL,(2.14)SL(xn,yn)=(S1L(xn,yn),S2L(xn,yn)),n∈Zα,
for all (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M). It follows from (2.1), (2.3), (2.12), (2.13), and Lemma 1.2 that S1L and S2L are well defined.
In view of (2.1), (2.2), and (2.10)–(2.14), we know that for any (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M) and (u,v)=({un}n∈ℤα,{vn}n∈ℤα)∈A(N,M),
(2.15)‖SL(x,y)-SL(u,v)‖1=max⁡{‖SiL(x,y)-SiL(u,v)‖:i∈Λ2}=max⁡{sup⁡n≥TL|SiL(xn,yn)-SiL(un,vn)|:i∈Λ2}≤max⁡{sup⁡n≥TL{∑l=0∞‍∑m=n+(l+1)τ1∞‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-f1(m,ua1m,…,uahm,vb1m,…,vbkm)yb1m|∑l=0∞‍},sup⁡n≥TL{∑l=0∞‍∑m=n+(l+1)τ2∞‍|f2(m,xc1m,…,xchm,ydb1m,…,ydkm)-f2(m,uc1m,…,uchm,vd1m,…,vdkm)|∑m=n+(l+1)τ1∞‍}}≤max⁡{∑l=0∞‍∑m=TL+(l+1)τ1∞‍t1mmax⁡{|xaim-uaim|,|ybjm-vbjm|:i∈Λh,j∈Λk},∑l=0∞‍∑m=TL+(l+1)τ2∞‍t2mmax⁡{|xcim-ucim|,|ydjm-vdjm|:i∈Λh,j∈Λk}∑m=TL+(l+1)τ1∞‍}≤max⁡{∑l=0∞‍∑m=TL+(l+1)τ1∞‍t1m,∑l=0∞‍∑m=TL+(l+1)τ2∞‍t2m}max⁡{‖x-u‖,‖y-v‖}≤max⁡{∑m=TL∞‍(1+mτ1)t1m,∑m=TL∞‍(1+mτ2)t2m}‖(x,y)-(u,v)‖1≤(∑m=TL∞‍(1+m)max⁡{t1m,t2m})‖(x,y)-(u,v)‖1=θL‖(x,y)-(u,v)‖1,max⁡{|S1L(xn,yn)-L|,|S2L(xn,yn)-L|}≤max⁡{∑l=0∞‍∑m=n+(l+1)τ1∞‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m|,∑l=0∞‍∑m=n+(l+1)τ2∞‍|f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m|}≤max⁡{∑l=0∞‍∑m=n+(l+1)τ1∞‍(r1m+|q1m|),∑l=0∞‍∑m=n+(l+1)τ2∞‍(r2m+|q2m|)}≤max⁡{∑m=TL∞‍(1+mτ1)(r1m+|q1m|),∑m=TL∞‍(1+mτ2)(r2m+|q2m|)}≤∑m=TL∞‍(1+m)max⁡{rim+|qim|:i∈Λ2}<min⁡{M-L,L-N},∀n≥TL,
which give that
(2.16)‖SL(x,y)-SL(u,v)‖1≤θL‖(x,y)-(u,v)‖1,∀(x,y),(u,v)∈A(N,M),(2.17)N≤S1L(xn,yn)≤M,N≤S2L(xn,yn)≤M,∀({xn}n∈Zα,{yn}n∈Zα)∈A(N,M),n≥TL.
Observe that θL∈(0,1). It is easy to see that (2.12)–(2.17) give that SL:A(N,M)→A(N,M) is a contraction. Consequently, SL possesses a unique fixed point (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M), which implies that
(2.18)xn=L-∑l=0∞‍∑m=n+(l+1)τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,yn=L-∑l=0∞‍∑m=n+(l+1)τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,
which yield that
(2.19)xn-xn-τ1=L-∑l=0∞‍∑m=n+(l+1)τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]-L+∑l=0∞‍∑m=n+lτ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]=∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],∀n≥TL+τ1,yn-yn-τ2=L-∑l=0∞‍∑m=n+(l+1)τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m]-L+∑l=0∞‍∑m=n+lτ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m]=∑m=n∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],∀n≥TL+τ2,
which give that
(2.20)Δ(xn-xn-τ1)=-f1(n,xa1n,…,xahn,yb1n,…,ybkn)+q1n,∀n≥TL+τ1,Δ(yn-yn-τ2)=-f2(n,xc1n,…,xchn,yd1n,…,ydkn)+q2n,∀n≥TL+τ2,
that is, (x,y) is a bounded positive solution of System (1.14) in A(N,M).
In light of (2.5), (2.6), and (2.11)–(2.16), we arrive at
(2.21)‖(xμ+1,yμ+1)-(x,y)‖1=max⁡{sup⁡n≥TL|xnμ+1-xn|,sup⁡n≥TL|ynμ+1-yn|}≤max⁡{sup⁡n≥TL[∑l=0∞‍(1-βμ-γμ)|xnμ-xn|+βμ|L-∑l=0∞‍∑m=n+(l+1)τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]-xn|+γμ|δ1nμ-xn|∑l=0∞‍],sup⁡n≥TL[∑l=0∞‍(1-βμ-γμ)|ynμ-yn|+βμ|L-∑l=0∞‍∑m=n+(l+1)τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2mxc1mμ]-yn∑m=n+(l+1)τ2∞‍|+γμ|δ2nμ-yn|∑l=0∞‍]}≤max⁡{sup⁡n≥TL[(1-βμ-γμ)|xnμ-xn|+βμ|S1L(xnμ,ynμ)-S1L(xn,yn)|+2Mγμ],sup⁡n≥TL[(1-βμ-γμ)|ynμ-yn|+βμ|S2L(xnμ,ynμ)-S2L(xn,yn)|+2Mγμ]}≤max⁡{(1-βμ-γμ)‖xμ-x‖+θLβμ‖(xμ,yμ)-(x,y)‖1+2Mγμ,(1-βμ-γμ)‖yμ-y‖+θLβμ‖(xμ,yμ)-(x,y)‖1+2Mγμv}≤(1-βμ(1-θL))‖(xμ,yμ)-(x,y)‖1+2Mγμ,∀μ≥0,
that is, (2.7) holds. It follows from (2.7)–(2.9) and Lemma 1.1 that limμ→∞∥(xμ,yμ)-(x,y)∥1=0.
(b) LetL1,L2∈(N,M), and L1≠L2. As in the proof of  (a), we similarly infer that for each i∈Λ2, there exist a constant θLi∈(0,1), a positive integer TLi≥max{τ1,τ2}+n1+|α|, and mappings SLi,S1Li,S2Li satisfying (2.10)–(2.14), where L,θL and TL are replaced by Li,θLi and TLi, respectively, and the contraction mappings SL1 and SL2 have the unique fixed points (u,v)=({un}n∈ℤα,{vn}n∈ℤα),(w,z)=({wn}n∈ℤα,{zn}n∈ℤα)∈A(N,M), respectively, (u,v) and (w,z) are bounded positive solutions of System (1.14) in A(N,M). In order to show that System (1.14) possesses uncountably many bounded positive solutions in A(N,M), we prove only that (u,v)≠(w,z). By means of (2.10) and (2.12)–(2.16), we conclude that
(2.22)‖(u,v)-(w,z)‖1=max⁡{‖u-w‖,‖v-z‖}≥max⁡{sup⁡n≥max⁡⁡{TL1,TL2}|un-wn|,sup⁡n≥max⁡⁡{TL1,TL2}|vn-zn|}≥max⁡{|L1-L2|-∑l=0∞‍∑m=max⁡⁡{TL1,TL2}+(l+1)τ1∞‍|f1(m,ua1m,…,uahm,vb1m,…,vbkm)-f1(m,wa1m,…,wahm,zb1m,…,zbkm)f1|,|L1-L2|-∑l=0∞‍∑m=max⁡{TL1,TL2}+(l+1)τ2∞‍|f2(m,uc1m,…,uchm,vd1m,…,vdkm)-f2(m,wc1m,…,wchm,zd1m,…,zdkm)f1|∑m=max⁡⁡{TL1,TL2}+(l+1)τ1∞‍}≥max⁡{|L1-L2|-∑l=0∞‍∑m=max⁡⁡{TL1,TL2}+(l+1)τ1∞‍t1mmax⁡{|uaim-waim|,|vbjm-zbjm|:i∈Λh,j∈ΛK},|L1-L2|-∑l=0∞∑m=max⁡⁡{TL1,TL2}+(l+1)τ2∞‍t2m‍×max⁡{|ucim-wcim|,|vdjm-zdjm|:i∈Λh,j∈ΛK}∑m=max⁡⁡{TL1,TL2}+(l+1)τ2∞‍}≥max⁡{|L1-L2|-∑l=0∞∑m=max⁡⁡{TL1,TL2}+(l+1)τ1∞‍‍t1mmax⁡{‖u-w‖,‖v-z‖},|L1-L2|-∑l=0∞‍∑m=max⁡⁡{TL1,TL2}+(l+1)τ2∞‍t2mmax⁡{‖u-w‖,‖v-z‖}}≥|L1-L2|-(∑m=max⁡⁡{TL1,TL2}∞‍(1+m)max⁡{tim:i∈Λ2})‖(u,v)-(w,z)‖1≥|L1-L2|-max⁡{θL1,θL2}‖(u,v)-(w,z)‖1,
which implies that
(2.23)‖(u,v)-(w,z)‖1≥|L1-L2|1+max⁡{θL1,θL2}>0,
that is, (u,v)≠(w,z). This completes the proof.Theorem 2.2.
Assume that there exist constantsM,N∈ℝ+∖{0}, n1∈ℕn0, and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2),
(2.24)∑n=n0∞‍max⁡{rin,tin,|qin|:i∈Λ2}<+∞,(2.25)N<M,pin=1,∀n≥n1,i∈Λ2.
Then,(a)
for eachL∈(N,M), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by the schemes
(2.26)xnμ+1={(1-βμ-γμ)xnμ+βμ{L+∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1nμ,μ≥0,n≥TL,(1-βμ-γμ)xTLμ+βμ{L+∑l=1∞‍∑m=TL+(2l-1)τ1T+2lτ1-1‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1TLμ,μ≥0,α≤n<TL,(2.27)ynμ+1={(1-βμ-γμ)ynμ+βμ{L+∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2nμ,μ≥0,n≥TL,(1-βμ-γμ)yTLμ+βμ{L+∑l=1∞‍∑m=TL+(2l-1)τ2TL+2lτ2-1‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2TLμ,μ≥0,α≤n<TLconverges to a bounded positive solution(x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Proof.
(a) LetL∈(N,M). Now we construct a contraction mapping SL:A(N,M)→A(N,M) and prove that its fixed point is a bounded positive solution of System (1.14). Note that (2.24) and (2.25) guarantee that there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| satisfying
(2.28)θL=∑m=TL∞‍max⁡{tim:i∈Λ2},(2.29)∑m=TL∞‍max⁡{rim+|qim|:i∈Λ2}<min⁡{M-L,L-N}.
Define three mappings SL:A(N,M)→lα∞×lα∞,S1L,S2L:A(N,M)→lα∞ by (2.14),
(2.30)S1L(xn,yn)={L+∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,S1L(xTL,yTL),α≤n<TL,(2.31)S2L(xn,yn)={L+∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,S2L(xTL,yTL),α≤n<TL,
for all (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M).
Using (2.1), (2.2), (2.14), and (2.28)–(2.31), we deduce that for any (x,y)=({xn}n∈ℤα,{yn}n∈ℤα) and (u,v)=({un}n∈ℤα,{vn}n∈ℤα)∈A(N,M),
(2.32)‖SL(x,y)-SL(u,v)‖1=max⁡{‖SiL(x,y)-SiL(u,v)‖:i∈Λ2}=max⁡{sup⁡n≥TL|SiL(xn,yn)-SiL(un,vn)|:i∈Λ2}≤max⁡{sup⁡n≥TL{∑l=1∞‍∑m=n+(2l-1)τ1m=n+2lτ1-1‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-f1(m,ua1m,…,uahm,vb1m,…,vbkm)f1|∑m=n+(2l-1)τ1m=n+2lτ1-1‍},sup⁡n≥TL{∑l=1∞‍∑m=n+(2l-1)τ2m=n+2lτ2-1‍|f2(m,xc1m,…,xchm,yd1m,…,ydkm)-f2(m,uc1m,…,uchm,vd1m,…,vdkm)f1|∑m=n+(2l-1)τ2m=n+2lτ2-1‍}}≤max⁡{∑m=TL+τ1∞‍t1mmax⁡{|xaim-uaim|,|ybjm-vbjm|:i∈Λh,j∈Λk},∑m=TL+τ2∞‍t2mmax⁡{|xcim-ucim|,|ydjm-vdjm|:i∈Λh,j∈Λk}∑m=TL+τ2∞‍}≤max⁡{∑m=TL+τ1∞‍t1m,∑m=TL+τ2∞‍t2m}max⁡{‖x-u‖,‖y-v‖}≤(∑m=TL∞‍max⁡{tim:i∈Λ2})‖(x,y)-(u,v)‖1=θL‖(x,y)-(u,v)‖1,max⁡{|S1L(xn,yn)-L|,|S2L(xn,yn)-L|}≤max⁡{∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m|,∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍|f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m|}≤max⁡{∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍(r1m+|q1m|),∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍(r2m+|q2m|)}≤max⁡{∑m=TL+τ1∞‍(r1m+|q1m|),∑m=TL+τ2∞‍(r2m+|q2m|)}≤∑m=TL∞‍max⁡{rim+|qim|:i∈Λ2}<min⁡{M-L,L-N},∀n≥TL,
which imply (2.16) and (2.17) and which ensure that SL is a self-mapping from A(N,M) into itself and is a contraction by θL∈(0,1). The Banach fixed point theorem means that SL has a unique fixed point (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M), that is,
(2.33)xn=L+∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,yn=L+∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,
which reveal that
(2.34)xn+xn-τ1=L+∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]+L+∑l=1∞‍∑m=n+2(l-1)τ1n+(2l-1)τ1-1‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]=2L+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],∀n≥TL+τ1,yn+yn-τ2=L+∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m]+L+∑l=1∞‍∑m=n+2(l-1)τ2n+(2l-1)τ2-1‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m]=2L+∑m=n∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],∀n≥TL+τ2,
which yield that
(2.35)Δ(xn-xn-τ1)=-f1(n,xa1n,…,xahn,yb1n,…,ybkn)+q1n,∀n≥TL+τ1,Δ(yn-yn-τ2)=-f2(n,xc1n,…,xchn,yd1n,…,ydkn)+q2n,∀n≥TL+τ2,
that is, (x,y) is a bounded positive solution of System (1.14) in A(N,M).
It follows from (2.14), (2.16), (2.26)–(2.28), (2.30), and (2.31) that
(2.36)‖(xμ+1,yμ+1)-(x,y)‖1=max⁡{sup⁡n≥TL|xnμ+1-xn|,sup⁡n≥TL|ynμ+1-yn|}≤max⁡{sup⁡n≥TL[∑m=n+(2l-1)τ2n+2lτ2-1‍(1-βμ-γμ)|xnμ-xn|+βμ|L+∑l=1∞‍∑m=n+(2l-1)τ1n+2lτ1-1‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]-xn|+γμ|δ1nμ-xn|∑m=n+(2l-1)τ2n+2lτ2-1‍],sup⁡n≥TL[∑m=n+(2l-1)τ2n+2lτ2-1‍(1-βμ-γμ)|ynμ-yn|+βμ|L+∑l=1∞‍∑m=n+(2l-1)τ2n+2lτ2-1‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]-yn|+γμ|δ2nμ-yn|∑m=n+(2l-1)τ2n+2lτ2-1‍ydkmμ]∑m=n+(2l-1)τ2n+2lτ2-1‍}≤max⁡{sup⁡n≥TL[(1-βμ-γμ)|xnμ-xn|+βμ|S1L(xnμ,ynμ)-S1L(xn,yn)|+2Mγμ],sup⁡n≥TL[(1-βμ-γμ)|ynμ-yn|+βμ|S2L(xnμ,ynμ)-S2L(xn,yn)|+2Mγμ]sup⁡n≥TL}≤(1-βμ(1-θL))‖(xμ,yμ)-(x,y)‖1+2Mγμ,∀μ≥0,
which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) mean that limμ→∞∥(xμ,yμ)-(x,y)∥1=0.
(b) LetL1,L2∈(N,M) and L1≠L2. As in the proof of  (a), we conclude that for each i∈Λ2, there exist a constant θLi∈(0,1), a positive integer TLi≥max{τ1,τ2}+n1+|α|, and mappings SLi,S1Li,S2Li satisfying (2.14) and (2.28)–(2.31), where L,θL, and TL are replaced by Li,θLi, and TLi, respectively, and the contraction mappings SL1 and SL2 have the unique fixed points (u,v)=({un}n∈ℤα,{vn}n∈ℤα),(w,z)=({wn}n∈ℤα,{zn}n∈ℤα)∈A(N,M), respectively, (u,v) and (w,z) are bounded positive solutions of the system (1.14) in A(N,M). In order to show that System (1.14) possesses uncountably many bounded positive solutions in A(N,M), we prove only that (u,v)≠(w,z). By virtue of (2.14) and (2.28)–(2.31), we infer that
(2.37)‖(u,v)-(w,z)‖1=max⁡{‖u-w‖,‖v-z‖}≥max⁡{sup⁡n≥max⁡⁡{TL1,TL2}|un-wn|,sup⁡n≥max⁡⁡{TL1,TL2}|vn-zn|}≥max⁡{|L1-L2|-∑l=1∞‍∑m=max⁡{TL1,TL2}+(2l-1)τ1max⁡{TL1,TL2}+2lτ1-1‍|f1(m,ua1m,…,uahm,vb1m,…,vbkm)-f1(m,wa1m,…,wahm,zb1m,…,zbkm)f1|,|L1-L2|-∑l=1∞‍∑m=max⁡{TL1,TL2}+(2l-1)τ2max⁡{TL1,TL2}+2lτ2-1‍|f2(m,uc1m,…,uchm,vd1m,…,vdkm)-f2(m,wc1m,…,wchm,zd1m,…,zdkm)f1|∑m=max⁡⁡{TL1,TL2}+(2l-1)τ1max⁡⁡{TL1,TL2}+2lτ1-1‍}≥max⁡{|L1-L2|-∑m=max⁡⁡{TL1,TL2}+τ1∞‍t1mmax⁡{|uaim-waim|,|vbjm-zbjm|:i∈Λh,j∈ΛK},|L1-L2|-∑m=max⁡⁡{TL1,TL2}+τ2∞‍t2mmax⁡{|ucim-wcim|,|vdjm-zdjm|:i∈Λh,j∈ΛK}}≥max⁡{|L1-L2|-∑m=max⁡⁡{TL1,TL2}+τ1∞‍t1mmax⁡{‖u-w‖,‖v-z‖},|L1-L2|-∑m=max⁡⁡{TL1,TL2}+τ2∞‍t2mmax⁡{‖u-w‖,‖v-z‖}}≥|L1-L2|-(∑m=max⁡⁡{TL1,TL2}∞‍max⁡{tim:i∈Λ2})‖(u,v)-(w,z)‖1≥|L1-L2|-max⁡{θL1,θL2}‖(u,v)-(w,z)‖1,
which yields that
(2.38)‖(u,v)-(w,z)‖1≥|L1-L2|1+max⁡{θL1,θL2}>0,
that is, (u,v)≠(w,z). This completes the proof.Theorem 2.3.
Assume that there exist constantsM,N∈ℝ+∖{0},p¯,p_∈ℝ+, n1∈ℕn0, and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.24), and
(2.39)N<(1-p_-p¯)M,p_+p¯<1,-p_≤pin≤p¯,∀n≥n1,i∈Λ2.
Then,(a)
for eachL∈(N+p¯M,M(1-p_)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by the schemes
(2.40)xnμ+1={(1-βμ-γμ)xnμ+βμ{∑m=n∞‍L-p1μxn-τ1μ+∑m=n∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1nμ,μ≥0,n≥TL,(1-βμ-γμ)xTLμ+βμ{∑m=TL∞‍L-p1μxTL-τ1μ+∑m=TL∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1TLμ,μ≥0,α≤n<TL,(2.41)ynμ+1={(1-βμ-γμ)ynμ+βμ{∑m=n∞‍L-p2μyn-τ2μ+∑m=n∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2nμ,μ≥0,n≥TL,(1-βμ-γμ)yTLμ+βμ{∑m=TL∞‍L-p2μyTL-τ2μ+∑m=TL∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2TLμ,μ≥0,α≤n<TLconverges to a bounded positive solution(x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Proof.
(a) LetL∈(N+p¯M,M(1-p_)). Now we construct a contraction mapping SL:A(N,M)→A(N,M) and prove that its fixed point is a bounded positive solution of the system (1.14). Notice that (2.24) and (2.39) mean that there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| satisfying
(2.42)θL=p_+p¯+∑m=TL∞‍max⁡{tim:i∈Λ2},(2.43)∑m=TL∞‍max⁡{rim+|qim|:i∈Λ2}<min⁡{M(1-p_)-L,L-N-p¯M}.
Define three mappings SL:A(N,M)→lα∞×lα∞,S1L,S2L:A(N,M)→lα∞ by (2.14),
(2.44)S1L(xn,yn)={L-p1nxn-τ1+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,S1L(xTL,yTL),α≤n<TL,(2.45)S2L(xn,yn)={L-p2nyn-τ2+∑m=n∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,S2L(xTL,yTL),α≤n<TL,
for all (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M).
It follows from (2.1), (2.2), (2.14), and (2.42)–(2.45) that for any (x,y)=({xn}n∈ℤα,{yn}n∈ℤα) and (u,v)=({un}n∈ℤα,{vn}n∈ℤα)∈A(N,M),
(2.46)‖SL(x,y)-SL(u,v)‖1=max⁡{‖SiL(x,y)-SiL(u,v)‖:i∈Λ2}=max⁡{sup⁡n≥TL|SiL(xn,yn)-SiL(un,vn)|:i∈Λ2}≤max⁡{sup⁡n≥TL{|p1n(xn-τ1-un-τ1)|+∑m=n∞‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-f1(m,ua1m,…,uahm,vb1m,…,vbkm)f1|∑m=n∞‍},sup⁡n≥TL{∑m=n∞‍|p2n(yn-τ2-vn-τ2)|+∑m=n∞‍|f2(m,xc1m,…,xchm,yd1m,…,ydkm)-f2(m,uc1m,…,uchm,vd1m,…,vdkm)f1|∑m=n∞‍}}≤max⁡{(p_+p¯)‖x-u‖+∑m=TL∞‍t1mmax⁡{|xalm-ualm|,|ybjm-vbjm|:l∈Λh,j∈Λk},(p_+p¯)‖y-v‖+∑m=TL∞‍t2mmax⁡{|xclm-uclm|,|ydjm-vdjm|:l∈Λh,j∈Λk}}≤max⁡{(p_+p¯)‖x-u‖+∑m=TL∞‍t1mmax⁡{‖x-u‖,‖y-v‖},(p_+p¯)‖y-v‖+∑m=TL∞‍t2mmax⁡{‖x-u‖,‖y-v‖}}≤(p_+p¯+∑m=TL∞‍max⁡{tim:i∈Λ2})‖(x,y)-(u,v)‖1=θL‖(x,y)-(u,v)‖1,S1L(xn,yn)=L-p1nxn-τ1+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]≤L+p_M+∑m=TL∞‍(r1m+|q1m|)≤L+p_M+min⁡{M(1-p_)-L,L-N-p¯M}≤M,∀n≥TL,S1L(xn,yn)=L-p1nxn-τ1+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]≥L-p¯M-∑m=TL∞‍(r1m+|q1m|)≥L-p¯M-min⁡{M(1-p_)-L,L-N-p¯M}≥N,∀n≥TL,N≤S2L(xn,yn)≤M,∀n≥TL,
which give (2.16) and (2.17), which together with θL∈(0,1) guarantee that SL:A(N,M)→A(N,M) is a contraction. Thus the Banach fixed point theorem ensures that SL has a unique fixed point (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M), that is,
(2.47)xn=L-p1nxn-τ1+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],yn=L-p2nyn-τ2+∑m=n∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,
which yield that
(2.48)Δ(xn+p1nxn-τ1)=-f1(n,xa1n,…,xahn,yb1n,…,ybkn)+q1n,Δ(yn+p2nyn-τ2)=f2(n,xc1n,…,xchn,yd1n,…,ydkn)+q2n,∀n≥TL,
that is, (x,y) is a bounded positive solution of System (1.14) in A(N,M).
In light of (2.14), (2.16), (2.40), (2.41), (2.44), and (2.45), we deduce that
(2.49)‖(xμ+1,yμ+1)-(x,y)‖1=max⁡{sup⁡n≥TL|xnμ+1-xn|,sup⁡n≥TL|ynμ+1-yn|}≤max⁡{sup⁡n≥TL[∑m=n∞‍(1-βμ-γμ)|xnμ-xn|+βμ|L-p1nxn-τ1+∑m=n∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]-xn|+γμ|δ1nμ-xn|∑m=n∞‍],sup⁡n≥TL[∑m=n∞‍(1-βμ-γμ)|ynμ-yn|+βμ|L-p2nyn-τ2+∑m=n∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]-yn|+γμ|δ2nμ-yn|∑m=n∞‍]}≤max⁡{sup⁡n≥TL[(1-βμ-γμ)‖xμ-x‖+βμ|S1L(xnμ,ynμ)-S1L(xn,yn)|+2Mγμ],sup⁡n≥TL[(1-βμ-γμ)‖yμ-y‖+βμ|S2L(xnμ,ynμ)-S2L(xn,yn)|+2Mγμ]sup⁡n≥TL}≤(1-βμ(1-θL))‖(xμ,yμ)-(x,y)‖1+2Mγμ,∀μ≥0,
which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) imply that limμ→∞∥(xμ,yμ)-(x,y)∥1=0.
(b) LetL1,L2∈(N,M) and L1≠L2. Similar to the proof of  (a), we know that for each i∈Λ2, there exist a constant θLi∈(0,1), a positive integer TLi≥max{τ1,τ2}+n1+|α| and mappings SLi,S1Li,S2Li satisfying (2.14) and (2.42)–(2.45), where L,θL, and TL are replaced by Li,θLi and TLi, respectively, and the contraction mappings SL1 and SL2 have the unique fixed points (u,v)=({un}n∈ℤα,{vn}n∈ℤα),(w,z)=({wn}n∈ℤα,{zn}n∈ℤα)∈A(N,M), which are bounded positive solutions of System (1.14) in A(N,M), respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in A(N,M), we prove only that (u,v)≠(w,z). In terms of (2.14) and (2.42)–(2.45), we get that
(2.50)‖(u,v)-(w,z)‖1=max⁡{‖u-w‖,‖v-z‖}≥max⁡{sup⁡n≥max⁡⁡{TL1,TL2}|un-wn|,sup⁡n≥max⁡⁡{TL1,TL2}|vn-zn|}≥max⁡{sup⁡n≥max⁡{TL1,TL2}{∑m=n∞‍|L1-L2|-|p1n||un-τ1-wn-τ1|-∑m=n∞‍|f1(m,ua1m,…,uahm,vb1m,…,vbkm)-f1(m,wa1m,…,wahm,zb1m,…,zbkm)|∑m=n∞‍},sup⁡n≥max⁡⁡{TL1,TL2}{∑m=n∞‍|L1-L2|-|p2n||vn-τ2-zn-τ2|-∑m=n∞‍|f2(m,uc1m,…,uchm,vd1m,…,vdkm)-f2(m,wc1m,…,wchm,zd1m,…,zdkm)f2|∑m=n∞‍}}≥max⁡{∑m=max⁡⁡{TL1,TL2}∞‍|L1-L2|-(p_+p¯)‖u-w‖-∑m=max⁡⁡{TL1,TL2}∞‍t1mmax⁡{|uaim-waim|,|vbjm-zbjm|:i∈Λh,j∈ΛK},|L1-L2|-(p_+p¯)‖v-z‖-∑m=max⁡⁡{TL1,TL2}∞‍t2mmax⁡{|ucim-wcim|,|vdjm-zdjm|:i∈Λh,j∈ΛK}}≥max⁡{|L1-L2|-(p_+p¯)‖u-w‖-∑m=max⁡⁡{TL1,TL2}∞‍t1mmax⁡{‖u-w‖,‖v-z‖},|L1-L2|-(p_+p¯)‖v-z‖-∑m=max⁡⁡{TL1,TL2}∞‍t2mmax⁡{‖u-w‖,‖v-z‖}}≥|L1-L2|-(p_+p¯+∑m=max⁡⁡{TL1,TL2}∞‍max⁡{tim:i∈Λ2})‖(u,v)-(w,z)‖1≥|L1-L2|-max⁡{θL1,θL2}‖(u,v)-(w,z)‖1,
which yields that
(2.51)‖(u,v)-(w,z)‖1≥|L1-L2|1+max⁡{θL1,θL2}>0,
that is, (u,v)≠(w,z). This completes the proof.Theorem 2.4.
Assume that there exist constantsM,N,p¯,p_∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.24), and
(2.52)0<N(p_-1)<M(p¯-1),-p_≤pin≤-p¯,∀n≥n1,i∈Λ2.
Then,(a)
for eachL∈(N(p_-1),M(p¯-1)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by the schemes
(2.53)xnμ+1={(1-βμ-γμ)xnμ+βμ{∑m=TL+τ1∞‍-Lp1n+τ1-xn+τ1μp1n+τ1+1p1n+τ1×∑m=n+τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1nμ,μ≥0,n≥TL,(1-βμ-γμ)xTLμ+βμ{∑m=TL+τ1∞‍-Lp1TL+τ1-xTL+τ1μp1TL+τ1+1p1TL+τ1×∑m=TL+τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1TLμ,μ≥0,α≤n<TL,(2.54)ynμ+1={(1-βμ-γμ)ynμ+βμ{∑m=n+τ2∞‍-Lp2n+τ2-yn+τ2μp2n+τ2+1p2n+τ2×∑m=n+τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2nμ,μ≥0,n≥TL,(1-βμ-γμ)yTLμ+βμ{∑m=n+τ2∞‍-Lp2TL+τ2-yTL+τ2μp2TL+τ2+1p2TL+τ2×∑m=TL+τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2TLμ,μ≥0,α≤n<TLconverges to a bounded positive solution(x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1] and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Proof.
(a) LetL∈(N(p_-1),M(p¯-1)). Now we construct a contraction mapping SL:A(N,M)→A(N,M) and prove that its fixed point is a bounded positive solution of the system (1.14). Observe that (2.24) and (2.52) guarantee that there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| satisfying
(2.55)θL=1p¯(1+∑m=TL∞‍max⁡{tim:i∈Λ2}),(2.56)∑m=TL∞‍max⁡{rim+|qim|:i∈Λ2}<min⁡{M(p¯-1)-L,p¯(L+Np_-N)}.
Define three mappings SL:A(N,M)→lα∞×lα∞,S1L,S2L:A(N,M)→lα∞ by (2.14),
(2.57)S1L(xn,yn)={-Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,S1L(xTL,yTL),α≤n<TL,(2.58)S2L(xn,yn)={-Lp2n+τ2-yn+τ2p2n+τ2+1p2n+τ2∑m=n+τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,S2L(xTL,yTL),α≤n<TL,
for all (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M).
It follows from (2.1), (2.2), (2.14), and (2.55)–(2.58) that for any (x,y)=({xn}n∈ℤα,{yn}n∈ℤα) and (u,v)=({un}n∈ℤα,{vn}n∈ℤα)∈A(N,M)(2.59)‖SL(x,y)-SL(u,v)‖1=max⁡{‖SiL(x,y)-SiL(u,v)‖:i∈Λ2}=max⁡{sup⁡n≥TL|SiL(xn,yn)-SiL(un,vn)|:i∈Λ2}≤max⁡{sup⁡n≥TL{1|p1n+τ1|(∑m=n+τ1∞‍|xn+τ1-un+τ1|+∑m=n+τ1∞‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-f1(m,ua1m,…,uahm,vb1m,…,vbkm)f1|∑m=n+τ1∞‍)},sup⁡n≥TL{∑m=n+τ2∞‍1|p2n+τ2|(∑m=n+τ2∞‍|yn+τ2-vn+τ2|+∑m=n+τ2∞‍|f2(m,xc1m,…,xchm,yd1m,…,ydkm)-f2(m,uc1m,…,uchm,vd1m,…,vdkm)f1|∑m=n+τ2∞‍)}}≤1p¯max⁡{‖x-u‖+∑m=TL∞‍t1mmax⁡{|xaim-uaim|,|ybjm-vbjm|:i∈Λh,j∈Λk},‖y-v‖+∑m=TL∞‍t2mmax⁡{|xcim-ucim|,|ydjm-vdjm|:i∈Λh,j∈Λk}}≤1p¯max⁡{‖x-u‖+∑m=TL∞‍t1mmax⁡{‖x-u‖,‖y-v‖},‖y-v‖+∑m=TL∞‍t2mmax⁡{‖x-u‖,‖y-v‖}}≤1p¯(1+∑m=TL∞‍max⁡{tim:i∈Λ2})‖(x,y)-(u,v)‖1=θL‖(x,y)-(u,v)‖1,S1L(xn,yn)=-Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]≤Lp¯+Mp¯+1p¯∑m=TL+τ1∞‍(r1m+|q1m|)≤Lp¯+Mp¯+1p¯min⁡{M(p¯-1)-L,p¯(L+Np_-N)}≤M,∀n≥TL,S1L(xn,yn)=-Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]≥Lp_+Np_-1p¯∑m=TL+τ1∞‍(r1m+|q1m|)≥Lp_+Np_-1p¯min⁡{M(p¯-1)-L,p¯(L+Np_-N)}≥N,∀n≥TL,N≤S2L(xn,yn)≤M,∀n≥TL,
which yield (2.16) and (2.17), which together with θL∈(0,1) ensure that SL is a contraction in A(N,M). By the Banach fixed point theorem, we deduce that SL has a unique fixed point (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M), that is,
(2.60)xn=-Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,yn=-Lp2n+τ2-yn+τ2p2n+τ2+1p2n+τ2∑m=n+τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,
which yield that
(2.61)xn+p1nxn-τ1=-L+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL+τ1,yn+p2nyn-τ2=-L+∑m=n∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL+τ2,Δ(xn+p1nxn-τ1)=-f1(n,xa1n,…,xahn,yb1n,…,ybkn)+q1n,∀n≥TL+τ1,Δ(yn+p2nyn-τ2)=-f2(n,xc1n,…,xchn,yd1n,…,ydkn)+q2n,∀n≥TL+τ2,
that is, (x,y) is a bounded positive solution of the system (1.14) in A(N,M).
In light of (2.14), (2.16), (2.53)–(2.55), (2.57), and (2.58), we get that
(2.62)‖(xμ+1,yμ+1)-(x,y)‖1=max⁡{sup⁡n≥TL|xnμ+1-xn|,sup⁡n≥TL|ynμ+1-yn|}≤max⁡{sup⁡n≥TL[∑m=n∞‍(1-βμ-γμ)|xnμ-xn|+βμ×|-Lp1n+τ1-xn+τ1μp1n+τ1+1p1n+τ1∑m=n∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]-xn|+γμ|δ1nμ-xn|∑m=n∞‍],sup⁡n≥TL[∑m=n∞‍(1-βμ-γμ)|ynμ-yn|+βμ×|-Lp2n+τ2-yn+τ2μp2n+τ2+1p2n+τ2∑m=n∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]-yn|+γμ|δ2nμ-yn|∑m=n∞‍]}≤max⁡{sup⁡n≥TL[(1-βμ-γμ)‖xμ-x‖+βμ|S1L(xnμ,ynμ)-S1L(xn,yn)|+2Mγμ],sup⁡n≥TL[(1-βμ-γμ)‖yμ-y‖+βμ|S2L(xnμ,ynμ)-S2L(xn,yn)|+2Mγμ]}≤(1-βμ(1-θL))‖(xμ,yμ)-(x,y)‖1+2Mγμ,∀μ≥0,
which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) imply that limμ→∞∥(xμ,yμ)-(x,y)∥1=0.
(b) LetL1,L2∈(N,M) and L1≠L2. Similar to the proof of  (a), we conclude that for each i∈Λ2, there exist a constant θLi∈(0,1), a positive integer TLi≥max{τ1,τ2}+n1+|α|, and mappings SLi,S1Li,S2Li satisfying (2.14) and (2.55)–(2.58), where L,θL, and TL are replaced by Li,θLi, and TLi, respectively, and the contraction mappings SL1 and SL2 have the unique fixed points (u,v)=({un}n∈ℤα,{vn}n∈ℤα),(w,z)=({wn}n∈ℤα,{zn}n∈ℤα)∈A(N,M), which are bounded positive solutions of System (1.14) in A(N,M), respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in A(N,M), we prove only that (u,v)≠(w,z). By virtue of (2.14) and (2.55)–(2.58), we get that
(2.63)‖(u,v)-(w,z)‖1=max⁡{‖u-w‖,‖v-z‖}≥max⁡{sup⁡n≥max⁡⁡{TL1,TL2}|un-wn|,sup⁡n≥max⁡⁡{TL1,TL2}|vn-zn|}≥max⁡{∑m=n+τ2∞‍sup⁡n≥max⁡{TL1,TL2}{|L1-L2||p1n+τ1|-|un+τ1-wn+τ1||p1n+τ1|-1|p1n+τ1|∑m=n+τ1∞‍|f1(m,ua1m,…,uahm,vb1m,…,vbkm)-f1(m,wa1m,…,wahm,zb1m,…,zbkm)f1||L1-L2||p1n+τ1|},sup⁡n≥max⁡{TL1,TL2}{∑m=n+τ2∞‍|L1-L2||p2n+τ2|-|un+τ2-wn+τ2||p2n+τ2|-1|p2n+τ2|×∑m=n+τ2∞‍|f2(m,uc1m,…,uchm,vd1m,…,vdkm)-f2(m,wc1m,…,wchm,zd1m,…,zdkm)f1|∑m=n+τ2∞‍}}≥max⁡{∑m=max⁡⁡{TL1,TL2}∞‍|L1-L2|p_-‖u-w‖p¯-1p¯∑m=max⁡⁡{TL1,TL2}∞‍t1mmax⁡{|uaim-waim|,|vbjm-zbjm|:i∈Λh,j∈ΛK},|L1-L2|p_-‖v-z‖p¯-1p¯∑m=max⁡⁡{TL1,TL2}∞‍t2mmax⁡{|ucim-wcim|,|vdjm-zdjm|:i∈Λh,j∈ΛK}}≥max⁡{|L1-L2|p_-‖u-w‖p¯-1p¯∑m=max⁡⁡{TL1,TL2}∞‍t1mmax⁡{‖u-w‖,‖v-z‖},|L1-L2|p_-‖v-z‖p¯-1p¯∑m=max⁡⁡{TL1,TL2}∞‍t2mmax⁡{‖u-w‖,‖v-z‖}}≥|L1-L2|p_-1p¯(1+max⁡{θL1,θL2})‖(u,v)-(w,z)‖1,
which yields that
(2.64)‖(u,v)-(w,z)‖1≥p¯|L1-L2|p_(p¯+max⁡{θL1,θL2})>0,
that is, (u,v)≠(w,z). This completes the proof.Theorem 2.5.
Assume that there exist constantsM,N,p¯,p_∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.24), and
(2.65)N(p¯2-p_)p¯<M(p_2-p¯)p_,1<p_≤pin≤p¯<p_2,∀n≥n1,i∈Λ2.
Then,(a)
for eachL∈(p¯(M/p_+N),p_(N/p¯+M)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by the schemes
(2.66)xnμ+1={(1-βμ-γμ)xnμ+βμ{Lp1n+τ1-xn+τ1μp1n+τ1+1p1n+τ1×∑m=n+τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1nμ,μ≥0,n≥TL,(1-βμ-γμ)xTLμ+βμ{Lp1TL+τ1-xTL+τ1μp1TL+τ1+1p1TL+τ1×∑m=TL+τ1∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]}+γμδ1TLμ,μ≥0,α≤n<TL,(2.67)ynμ+1={(1-βμ-γμ)ynμ+βμ{Lp2n+τ2-yn+τ2μp2n+τ2+1p2n+τ2×∑m=n+τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2nμ,μ≥0,n≥TL,(1-βμ-γμ)yTLμ+βμ{Lp2TL+τ2-yTL+τ2μp2TL+τ2+1p2TL+τ2×∑m=TL+τ2∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]}+γμδ2TLμ,μ≥0,α≤n<TLconverges to a bounded positive solution(x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Proof.
(a) LetL∈(p¯(M/p_+N),p_(N/p¯+M)). Now we construct a contraction mapping SL:A(N,M)→A(N,M) and prove that its fixed point is a bounded positive solution of System (1.14). Observe that (2.24) and (2.65) imply that there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| satisfying
(2.68)θL=1p_(1+∑m=TL∞‍max⁡{tim:i∈Λ2}),(2.69)∑m=TL∞‍max⁡{rim+|qim|:i∈Λ2}<min⁡{Np_p¯+Mp_-L,Lp_p¯-M-Np_}.
Define three mappings SL:A(N,M)→lα∞×lα∞,S1L,S2L:A(N,M)→lα∞ by (2.14),
(2.70)S1L(xn,yn)={Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,S1L(xTL,yTL),α≤n<TL,(2.71)S2L(xn,yn)={Lp2n+τ2-yn+τ2p2n+τ2+1p2n+τ2∑m=n+τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,S2L(xTL,yTL),α≤n<TL,
for all (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M).
Using (2.1), (2.2), (2.14), (2.65), and (2.68)–(2.71), we deduce that for any (x,y)=({xn}n∈ℤα,{yn}n∈ℤα) and (u,v)=({un}n∈ℤα,{vn}n∈ℤα)∈A(N,M),
(2.72)‖SL(x,y)-SL(u,v)‖1=max⁡{‖SiL(x,y)-SiL(u,v)‖:i∈Λ2}=max⁡{sup⁡n≥TL|SiL(xn,yn)-SiL(un,vn)|:i∈Λ2}≤max⁡{sup⁡n≥TL{1|p1n+τ1|(|xn+τ1-un+τ1|+∑m=n+τ1∞‍|f1(m,xa1m,…,xahm,yb1m,…,ybkm)-f1(m,ua1m,…,uahm,vb1m,…,vbkm)f1|∑m=n+τ1∞‍)},sup⁡n≥TL{1|p2n+τ2|(∑m=n+τ1∞‍|yn+τ2-vn+τ2|+∑m=n+τ2∞‍|f2(m,xc1m,…,xchm,yd1m,…,ydkm)-f2(m,uc1m,…,uchm,vd1m,…,vdkm)f1|∑m=n+τ1∞‍)}}≤1p_max⁡{‖x-u‖+∑m=TL∞‍t1mmax⁡{|xaim-uaim|,|ybjm-vbjm|:i∈Λh,j∈Λk},‖y-v‖+∑m=TL∞‍t2mmax⁡{|xcim-ucim|,|ydjm-vdjm|:i∈Λh,j∈Λk}}≤1p_max⁡{‖x-u‖+∑m=TL∞‍t1mmax⁡{‖x-u‖,‖y-v‖},‖y-v‖+∑m=TL∞‍t2mmax⁡{‖x-u‖,‖y-v‖}}≤1p_(1+∑m=TL∞‍max⁡{t1m,t2m})‖(x,y)-(u,v)‖1=θL‖(x,y)-(u,v)‖1,S1L(xn,yn)=Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]≤Lp_-Np¯+1p_∑m=TL+τ1∞‍(r1m+|q1m|)≤Lp_-Np¯+1p_min⁡{Np_p¯+Mp_-L,Lp_p¯-M-Np_}≤M,∀n≥TL,S1L(xn,yn)=Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m]≥Lp¯-Mp_-1p_∑m=TL+τ1∞‍(r1m+|q1m|)≥Lp¯-Mp_-1p_min⁡{Np_p¯+Mp_-L,Lp_p¯-M-Np_}≥N,∀n≥TL,N≤S2L(xn,yn)≤M,∀n≥TL,
which yield (2.16) and (2.17), and which guarantee that SL is a self-mapping from A(N,M) into itself and is a contraction by θL∈(0,1). Thus the Banach fixed point theorem means that SL has a unique fixed point (x,y)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M), that is,
(2.73)xn=Lp1n+τ1-xn+τ1p1n+τ1+1p1n+τ1∑m=n+τ1∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL,yn=Lp2n+τ2-yn+τ2p2n+τ2+1p2n+τ2∑m=n+τ2∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL,
which yield that
(2.74)xn+p1nxn-τ1=L+∑m=n∞‍[f1(m,xa1m,…,xahm,yb1m,…,ybkm)-q1m],n≥TL+τ1,yn+p2nyn-τ2=L+∑m=n∞‍[f2(m,xc1m,…,xchm,yd1m,…,ydkm)-q2m],n≥TL+τ2,Δ(xn+p1nxn-τ1)=-f1(n,xa1n,…,xahn,yb1n,…,ybkn)+q1n,∀n≥TL+τ1,Δ(yn+p2nyn-τ2)=-f2(n,xc1n,…,xchn,yd1n,…,ydkn)+q2n,∀n≥TL+τ2,
that is, (x,y) is a bounded positive solution of the system (1.14) in A(N,M).
In view of (2.14), (2.16), (2.65)–(2.67), (2.70), and (2.71), we gain that
(2.75)‖(xμ+1,yμ+1)-(x,y)‖1=max⁡{sup⁡n≥TL|xnμ+1-xn|,sup⁡n≥TL|ynμ+1-yn|}≤max⁡{sup⁡n≥TL[∑m=n∞‍(1-βμ-γμ)|xnμ-xn|+βμ×|Lp1n+τ1-xn+τ1μp1n+τ1+1p1n+τ1∑m=n∞‍[f1(m,xa1mμ,…,xahmμ,yb1mμ,…,ybkmμ)-q1m]-xn|+γμ|δ1nμ-xn|∑m=n∞‍],sup⁡n≥TL[∑m=n∞‍(1-βμ-γμ)|ynμ-yn|+βμ×|Lp2n+τ2-yn+τ2μp2n+τ2+1p2n+τ2∑m=n∞‍[f2(m,xc1mμ,…,xchmμ,yd1mμ,…,ydkmμ)-q2m]-yn|+γμ|δ2nμ-yn|∑m=n∞‍]}≤max⁡{sup⁡n≥TL[(1-βμ-γμ)‖xμ-x‖+βμ|S1L(xnμ,ynμ)-S1L(xn,yn)|+2Mγμ],sup⁡n≥TL[(1-βμ-γμ)‖yμ-y‖+βμ|S2L(xnμ,ynμ)-S2L(xn,yn)|+2Mγμ]}≤(1-βμ(1-θL))‖(xμ,yμ)-(x,y)‖1+2Mγμ,∀μ≥0,
which implies (2.7). Thus, Lemma 1.1 and (2.7)–(2.9) imply that limμ→∞∥(xμ,yμ)-(x,y)∥1=0.
(b) LetL1,L2∈(N,M) and L1≠L2. Similar to the proof of  (a), we conclude that for each i∈Λ2, there exist a constant θLi∈(0,1), a positive integer TLi≥max{τ1,τ2}+n1+|α|, and mappings SLi,S1Li,S2Li satisfying (2.14) and (2.68)–(2.71), where L,θL, and TL are replaced by Li,θLi, and TLi, respectively, and the contraction mappings SL1 and SL2 have the unique fixed points (u,v)=({un}n∈ℤα,{vn}n∈ℤα),(w,z)=({xn}n∈ℤα,{yn}n∈ℤα)∈A(N,M), which are bounded positive solutions of System (1.14) in A(N,M), respectively. In order to show that  System (1.14) possesses uncountably many bounded positive solutions in A(N,M), we prove only that (u,v)≠(w,z). In light of (2.14) and (2.68)–(2.71), we have
(2.76)‖(u,v)-(w,z)‖1=max⁡{‖u-w‖,‖v-z‖}≥max⁡{sup⁡n≥max⁡⁡{TL1,TL2}|un-wn|,sup⁡n≥max⁡⁡{TL1,TL2}|vn-zn|}≥max⁡{sup⁡n≥max⁡{TL1,TL2}{|L1-L2||p1n+τ1|-|un+τ1-wn+τ1||p1n+τ1|-1|p1n+τ1|∑m=n+τ1∞‍|f1(m,ua1m,…,uahm,vb1m,…,vbkm)-f1(m,wa1m,…,wahm,zb1m,…,zbkm)f1|∑m=n+τ1∞‍},sup⁡n≥max⁡{TL1,TL2}{|L1-L2||p2n+τ2|-|un+τ2-wn+τ2||p2n+τ2|-1|p2n+τ2|×∑m=n+τ2∞‍|f2(m,uc1m,…,uchm,vd1m,…,vdkm)-f2(m,wc1m,…,wchm,zd1m,…,zdkm)f1|∑m=n+τ2∞‍}}≥max⁡{|L1-L2|p¯-‖u-w‖p_-1p_∑m=max⁡⁡{TL1,TL2}∞‍t1mmax⁡{|ualm-walm|,|vbjm-zbjm|:l∈Λh,j∈ΛK},|L1-L2|p¯-‖v-z‖p_-1p_∑m=max⁡⁡{TL1,TL2}∞‍t2mmax⁡{|uclm-wclm|,|vdjm-zdjm|:l∈Λh,j∈ΛK}}≥max⁡{|L1-L2|p¯-‖u-w‖p_-1p_∑m=max⁡⁡{TL1,TL2}∞‍t1mmax⁡{‖u-w‖,‖v-z‖},|L1-L2|p¯-‖v-z‖p_-1p_∑m=max⁡⁡{TL1,TL2}∞‍t2mmax⁡{‖u-w‖,‖v-z‖}}≥|L1-L2|p¯-1p_(1+max⁡{θL1,θL2})‖(u,v)-(w,z)‖1,
which yields that
(2.77)‖(u,v)-(w,z)‖1≥p_|L1-L2|p¯(p_+max⁡{θL1,θL2})>0,
that is, (u,v)≠(w,z). This completes the proof.Remark 2.6.
Let(a1,b1) and (a2,b2) be two arbitrary intervals in ℝ. It is easy to see that
(2.78)(a1,b1)∩(a2,b2)≠∅⟺max⁡{a1,a2}<min⁡{b1,b2}.From Remark2.6 and Theorems 2.1–2.5, we can obtain the following.Theorem 2.7.
Assume that there exist constantsM,N∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2),
(2.79)∑n=n0∞‍nmax⁡{r1n,t1n,|q1n|}<+∞,(2.80)∑n=n0∞‍max⁡{r2n,t2n,|q2n|}<+∞,(2.81)N<M,p1n=-1,∀n≥n1,(2.82)p2n=1,∀n≥n1.
Then,(a)
for eachL∈(N,M), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.5) and (2.27) converges to a bounded positive solution (x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1] and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Theorem 2.8.
Assume that there exist constantsM,N∈ℝ+∖{0},p¯,p_∈ℝ+, n1∈ℕn0, and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.79)–(2.81),
(2.83)N<(1-p_-p¯)M,p_+p¯<1,-p_≤p2n≤p¯,∀n≥n1.
Then,(a)
for eachL∈(N,M)∩(N+p¯M,M(1-p_)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.5) and (2.41) converges to a bounded positive solution (x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1] and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Theorem 2.9.
Assume that there exist constantsM,N,p¯,p_∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0 and {t2n}n∈ℕn0, satisfying (2.1), (2.2), (2.79)–(2.81),
(2.84)0<N(p_-1)<M(p¯-1),-p_≤p2n≤-p¯,∀n≥n1,(2.85)max⁡⁡{N,N(p_-1)}<min⁡⁡{M,M(p¯-1)}.
Then,(a)
for eachL∈(N,M)∩(N(p_-1),M(p¯-1)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.5) and (2.54) converges to a bounded positive solution (x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1] and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Theorem 2.10.
Assume that there exist constantsM,N,M0,N0,p¯0,p_0∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2) with i=1, (2.79)–(2.81),
(2.86)|f2(n,u1,…,uh,v1,…,vk)|≤r2n,∀(n,ul,vj)∈Nn1×[N0,M0]2,l∈Λh,j∈Λk,(2.87)|f2(n,u1,…,uh,v1,…,vk)-f2(n,w1,…,wh,z1,…,zk)|≤t2nmax⁡{|ul-wl|,|vj-zj|:l∈Λh,j∈Λk},∀(n,ul,wl,vj,zj)∈Nn1×[N0,M0]4,l∈Λh,j∈Λk,(2.88)N0(p¯02-p_0)p¯0<M0(p_02-p¯0)p_0,1<p_0≤p2n≤p¯0<p_02,∀n≥n1,(2.89)max⁡{N,p¯0(M0p_0+N0)}<min⁡{M,p_0(N0p¯0+M0)}.
Then,(a)
for eachL∈(N,M)∩(p¯0(M0/p_0+N0),p_0(N0/p¯0+M0)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M,N0,M0), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.5) and (2.67) converges to a bounded positive solution (x,y)∈A(N,M,N0,M0) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M,N0,M0) are arbitrary sequences with (2.8) and (2.9),(b)
the system (1.14) has uncountably many bounded positive solutions in A(N,M,N0,M0).Theorem 2.11.
Assume that there exist constantsM,N∈ℝ+∖{0},p¯,p_∈ℝ+, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.24), (2.83) and
(2.90)N<M,p1n=1,∀n≥n1.
Then,(a)
for eachL∈(N,M)∩(N+p¯M,M(1-p_)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.26) and (2.41) converges to a bounded positive solution (x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Theorem 2.12.
Assume that there exist constantsM,N,p¯,p_∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.24), (2.84), (2.85), and (2.90). Then,(a)
for eachL∈(N,M)∩(N(p_-1),M(p¯-1)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.26) and (2.54) converges to a bounded positive solution (x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Theorem 2.13.
Assume that there existM,N,M0,N0,p¯0,p_0∈ℝ+∖{0}, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0, {r2n}n∈ℕn0, {t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2) with i=1, (2.24), and (2.86)–(2.90). Then,(a)
for eachL∈(N,M)∩(p¯0(M0/p_0+N0),p_0(N0/p¯0+M0)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M,N0,M0), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.26) and (2.67) converges to a bounded positive solution (x,y)∈A(N,M,N0,M0) of  System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M,N0,M0) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M,N0,M0).Theorem 2.14.
Assume that there exist constantsM,N,p¯,p_∈ℝ+∖{0},p¯1,p_1∈ℝ+, n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0,{t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2), (2.24), (2.84),
(2.91)N<(1-p_1-p¯1)M,p_1+p¯1<1,-p_1≤p1n≤p¯1,∀n≥n1,(2.92)max⁡{N+Mp¯1,N(p_-1)}<min⁡{M(1-p_1),M(p¯-1)}.
Then,(a)
for eachL∈(N+Mp¯1,M(1-p_1))∩(N(p_-1),M(p¯-1)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.41) and (2.54) converges to a bounded positive solution (x,y)∈A(N,M) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M).Theorem 2.15.
Assume that there exist constantsM,N,M0,N0,p¯0,p_0,p¯1,p_1∈ℝ+∖{0},n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0, {t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2) with i=1, (2.24), (2.86)–(2.88), (2.91), and
(2.93)max⁡{N+Mp¯1,p¯0(M0p_0+N0)}<min⁡{M(1-p_1),p_0(N0p¯0+M0)}.
Then,(a)
for eachL∈(N+Mp¯1,M(1-p_1))∩(p¯0(M0/p_0+N0),p_0(N0/p¯0+M0)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M,N0,M0), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.41) and (2.67) converges to a bounded positive solution (x,y)∈A(N,M,N0,M0) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M,N0,M0) are arbitrary sequences with (2.8) and (2.9)(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M,N0,M0).Theorem 2.16.
Assume that there exist constantsM,N,M0,N0,p¯0,p_0,p¯1,p_1∈ℝ+∖{0},n1∈ℕn0 and four nonnegative sequences {r1n}n∈ℕn0,{r2n}n∈ℕn0, {t1n}n∈ℕn0, and {t2n}n∈ℕn0 satisfying (2.1), (2.2) with i=1, (2.24), (2.86)–(2.88),
(2.94)0<N(p_1-1)<M(p¯1-1),-p_1≤p1n≤-p¯1,∀n≥n1,(2.95)max⁡{N(p_1-1),p¯0(M0p_0+N0)}<min⁡{M(p¯1-1),p_0(N0p¯0+M0)}.
Then,(a)
for eachL∈(N(p_1-1),M(p¯1-1))∩(p¯0(M0/p_0+N0),p_0(N0/p¯0+M0)), there exist θL∈(0,1) and TL≥max{τ1,τ2}+n1+|α| such that for any (x0,y0)=({xn0}n∈ℤα,{yn0}n∈ℤα)∈A(N,M,N0,M0), the Mann iterative sequence with errors {(xμ,yμ)}μ∈ℕ0={({xnμ}n∈ℤα,{ynμ}n∈ℤα)}μ∈ℕ0 generated by Schemes (2.54) and (2.67) converges to a bounded positive solution (x,y)∈A(N,M,N0,M0) of System (1.14) and has the error estimate (2.7), where {βμ}μ∈ℕ0,{γμ}μ∈ℕ0⊂[0,1], and {(δ1μ,δ2μ)}μ∈ℕ0={({δ1nμ}n∈ℤα,{δ2nμ}n∈ℤα)}μ∈ℕ0⊂A(N,M,N0,M0) are arbitrary sequences with (2.8) and (2.9),(b)
System (1.14) has uncountably many bounded positive solutions in A(N,M,N0,M0).Remark 2.17.
Theorems2.1–2.5 extend and improve [14, Theorem 1] and [27, Theorems 2.1–2.7].


## 3. Examples

In this section we construct fifteen examples to explain the results presented in Section2.Example 3.1.
Consider the nonlinear difference system with multiple delays:(3.1)Δ(xn-xn-τ1)+xn-33yn-6-nxn-4yn-52n3+nxn-32+(n2+1)xn-44yn-62=(-1)n-1n4+n+1,Δ(yn-yn-τ2)+(n+1)xn-22yn-5+(-1)nyn-2n4+(n+2)xn-12=n+1n3+(n+2)ln⁡n,n≥1,
where τ1,τ2∈ℕ are fixed. Let
(3.2)n0=n1=1,h=k=2,M>N>0,α=min⁡{1-τ1,1-τ2,-5},a1n=n-3,a2n=n-4,b1n=n-6,b2n=n-5,c1n=n-2,c2n=n-1,d1n=n-5,d2n=n-2,p1n=p2n=-1,q1n=(-1)n-1n4+n+1,q2n=n+1n3+(n+2)ln⁡n,f1(n,u1,u2,v1,v2)=u13v1-nu2v22n3+nu12+(n2+1)u24v12,f2(n,u1,u2,v1,v2)=(n+1)u12v2+(-1)nv1n4+(n+2)u22,r1n=M3(M+n)n3+nN2+(n2+1)N6,r2n=M(1+(n+1)M2)n4+(n+2)N2,t1n=nM2(4n2M+3n5/2+2M3+5nM2+16nM7+14nM6)(n3+nN2+(n2+1)N6)2,t2n=1n7(6n4M2+n3+30nM4+9M2),∀(n,u1,u2,v1,v2)∈Nn0×R4.
It is easy to see that (2.1)–(2.4) are satisfied. Thus Theorem 2.1 implies that System (3.1) possesses uncountably many bounded positive solutions in A(N,M). But [14, Theorem 1] and [27, Theorem 2.1] are not valid for System (3.1).Example 3.2.
Consider the nonlinear difference system with multiple delays:(3.3)Δ(xn+xn-τ1)+nxn-13-(n-1)yn-4n3+xn-12yn-42=n2-1n4+2n+1,Δ(yn+yn-τ2)+n9xn-3yn-22n12+(n6-1)xn-32=(-1)nn2ln⁡3n(n2+1)2,n≥1,
where τ1,τ2∈ℕ are fixed. Let
(3.4)n0=n1=1,h=k=1,M>N>0,α=min⁡{1-τ1,1-τ2,-3},a1n=n-1,b1n=n-4,c1n=n-3,d1n=n-2,p1n=p2n=1,q1n=n2-1n4+2n+1,q2n=(-1)nn2ln⁡3n(n2+1)2,f1(n,u,v)=nu3-(n-1)vn3+u2v2,f2(n,u,v)=n9uv2n12+(n6-1)u2,r1n=nM(M+1)n3+N4,r2n=n9M3n12+(n6-1)N2,t1n=n(n3(2M+1)+3M4+2M5)(n3+N4)2,t2n=3n15(n6+1)M2(n12+(n6-1)N2)2,∀(n,u,v)∈Nn0×R2.
It is easy to verify that (2.1), (2.2), (2.24), and (2.25) are fulfilled. Thus Theorem 2.2 implies that System (3.3) possesses uncountably many bounded positive solutions in A(N,M). But in [14, Theorem 1] and [27, Theorem 2.2] are unapplicable for System (3.3).Example 3.3.
Consider the nonlinear difference system with multiple delays:(3.5)Δ(xn+n22n2+3xn-τ1)+xn-2-2yn-6n2+sin⁡(xn-1yn-6)=n-2cos⁡(n3-3n2+6)n3+3n+1,Δ(yn+(-1)n3n+1yn-τ2)+nxn-1+(2n-1)yn-32n3+xn-12=(n2-3)sin⁡(n5-2n+1)n4+ln⁡3n,n≥2,
where τ1,τ2∈ℕ are fixed. Let
(3.6)n0=n1=2,h=k=1,M>6N>0,α=min⁡{2-τ1,2-τ2,-4},a1n=n-2,b1n=n-6,c1n=n-1,d1n=n-3,p1n=n22n2+3,p2n=(-1)n3n+1,p¯=12,p_=13,q1n=n-2cos⁡(n3-3n2+6)n3+3n+1,q2n=(n2-3)sin⁡(n5-2n+1)n4+ln⁡3n,f1(n,u,v)=u-2vn2+sin⁡(uv),f2(n,u,v)=nu+(2n-1)v2n3+u2,r1n=3Mn2-1,r2n=nM(1+2M)n3+N2,t1n=6(n2+M2)(n2-1)2,t2n=n((1+4M)n3+M2+8M3)(n3+N2)2,∀(n,u,v)∈Nn0×R2.
It is easy to see that (2.1), (2.2), (2.24), and (2.39) hold. Thus, Theorem 2.3 ensures that System (3.5) possesses uncountably many bounded positive solutions in A(N,M). But  [14, Theorem 1] and [27, Theorems 2.3, 2.5, and 2.6] are not valid for System (3.5).Example 3.4.
Consider the nonlinear difference system with multiple delays:(3.7)Δ(xn+(-4+sin⁡(2n2))xn-τ1)+n5xn-12yn-25n10+5n8+1=(-1)nn3ln⁡(1+n)n9+n5+1,Δ(yn+(-4+cos⁡(5n3))yn-τ2)+n2xn-2-(n+2)yn-32n8+1+nxn-22+yn-32=n5-6(n+1)7,n≥1,
where τ1,τ2∈ℕ are fixed. Let
(3.8)n0=n1=h=k=1,M>2N>0,α=min⁡{1-τ1,1-τ2,-2},a1n=n-1,b1n=n-2,c1n=n-2,d1n=n-3,p1n=-4+sin⁡(2n2),p2n=-4+cos⁡(5n3),p¯=3,p_=5,q1n=(-1)nn3ln⁡(1+n)n9+n5+1,q2n=n5-6(n+1)7,f1(n,u,v)=n5u2v5n10+5n8+1,f2(n,u,v)=n2u-(n+2)v2n8+1+nu2+v2,r1n=M7n5,r2n=n2M+(n+2)M2n8+1+nN2+N2,t1n=7M6n5,t2n=n2(2n8+8n7M+nM2+8M3+3M2)(n8+1+nN2+N2)2,∀(n,u,v)∈Nn0×R2.
Clearly (2.1), (2.2), (2.24), and (2.52) hold. Thus, Theorem 2.4 means that System (3.7) possesses uncountably many bounded positive solutions in A(N,M). But [14, Theorem 1] and [27, Theorem 2.4] are inapplicable for System (3.7).Example 3.5.
Consider the nonlinear difference system with multiple delays:(3.9)Δ(xn+8n2+1n2+1xn-τ1)+n-(n-1)yb1n3n3+xa1n2=n-(-1)n(n+1)/23n-2n2+2n+1,Δ(yn+1+24ln⁡3n2+3ln⁡3nyn-τ2)+2nxc1n-3n+1yd1n3n4+xc1n2yd1n2=1+nln⁡(1+n2)n3ln⁡3(1+n2),n≥1,
where τ1,τ2∈ℕ are fixed and
(3.10)a1n={n2ifniseven,n-2ifnisodd,b1n={n-1ifniseven,n-12ifnisodd,c1n={n-2ifniseven,n-1ifnisodd,d1n={n2-1ifniseven,n+12ifnisodd.
Let
(3.11)n0=1,n1=10,h=k=1,M>399328N>0,α=min⁡{1-τ1,1-τ2,-1},p1n=8n2+1n2+1,p2n=1+24ln⁡3n2+3ln⁡3n,p¯=8,p_=7,q1n=n-(-1)n(n+1)/23n-2n2+2n+1,q2n=1+nln⁡(1+n2)n3ln⁡3(1+n2),f1(n,u,v)=n-(n-1)v3n3+u2,f2(n,u,v)=2nu-3n+1v3n4+u2v2,r1n=n(1+M3)n3+N2,r2n=nM(2+3M2)n4+N4,t1n=nM(2+3n3M+5M3)(n3+N2)2,t2n=n(2n4+6M2+9n4M2+9M6)(n4+N4)2,∀(n,u,v)∈Nn0×R2.
Obviously (2.1), (2.2), (2.24), and (2.65) hold. Thus, Theorem 2.5 implies that System (3.9) possesses uncountably many bounded positive solutions in A(N,M). But [14, Theorem 1] and [27, Theorem 2.7] are useless for System (3.9).Example 3.6.
Consider the nonlinear difference system with multiple delays:(3.12)Δ(xn-xn-τ1)+2n2xa1n3+n-1n5+n(nxa1n-(3n+2)yb1n)2=(-1)(n-1)n(n+1)/32n3+4n4+sin⁡2(2n4+1),Δ(yn+yn-τ2)+(-1)nxc1n2-nyd1n3(n+1)2(n+1)=(-1)n-1ln⁡(n3+2n+1)n3/2+ln⁡3(1+n2),n≥1,
where τ1,τ2∈ℕ are fixed and
(3.13)a1n={n-1ifn≥100,n-2if1≤n<100,b1n={n2ifniseven,n-12ifnisodd,c1n={n2-2ifniseven,n-12ifnisodd,d1n={n-3ifn≥100,nif1≤n<100.
Let
(3.14)n0=n1=h=k=1,M>N>0,α=min⁡{1-τ1,1-τ2,-1},p1n=-1,p2n=1,q1n=(-1)(n-1)n(n+1)/32n3+4n4+sin⁡2(2n4+1),q2n=(-1)n-1ln⁡(n3+2n+1)n3/2+ln⁡3(1+n2),f1(n,u,v)=2n2u3+n-1n5+n(nu-(3n+2)v)2,f2(n,u,v)=(-1)nu2-nv3(n+1)2(n+1),r1n=1+2nM3n4,r2n=M2+nM3n5/2,t1n=6M(16+n3M+8(2n+M)M2+3nM3)n6,t2n=2M+3nM2n5/2,∀(n,u,v)∈Nn0×R2.
It is easy to see that (2.1), (2.2), and (2.79)–(2.82) are satisfied. Thus, Theorem 2.7 implies that System (3.12) possesses uncountably many bounded positive solutions in A(N,M).Example 3.7.
Consider the nonlinear difference system with multiple delays:(3.15)Δ(xn-xn-τ1)+nxa1n3-yb1n(n+1)4+(n+1)yb1n2=(-1)n(n+1)/2(1-3n+1)n3+3n+2,Δ(yn+n2-13n2+2yn-τ2)+n(n+1)yd1n4(n+2)3+nxc1n2=2n+1(n+3)2ln⁡(1+n4)(n+1)4,n≥1,
where τ1,τ2∈ℕ are fixed and
(3.16)a1n={n-4ifniseven,n-1ifnisodd,b1n={n-2ifniseven,n-12ifnisodd,c1n={n-1ifniseven,n+12ifnisodd,d1n={n2ifniseven,n-2ifnisodd.
Let
(3.17)n0=n1=h=k=1,M>6N>0,α=min⁡{1-τ1,1-τ2,-2},p1n=-1,p2n=n2-13n2+2,p¯=13,p_=12,q1n=(-1)n(n+1)/2(1-3n+1)n3+3n+2,q2n=2n+1(n+3)2ln⁡(1+n4)(n+1)4,f1(n,u,v)=nu3-v(n+1)4+(n+1)v2,f2(n,u,v)=n(n+1)v4(n+2)3+nu2,r1n=(nM2+1)M(n+1)4,r2n=n(n+1)M4(n+2)3+nN2,t1n=(n+1)4(3nM2+1)+(5nM2+1)(n+1)M2((n+1)4+(n+1)N2)2,t2n=2n(n+1)M3(2(n+2)3+3nM2)((n+2)3+nN2)2,∀(n,u,v)∈Nn0×R2.
Clearly (2.1), (2.2), (2.79)–(2.81), and (2.83) are satisfied. Thus, Theorem 2.8 guarantees that System (3.15) possesses uncountably many bounded positive solutions in A(N,M).Example 3.8.
Consider the nonlinear difference system with multiple delays:(3.18)Δ(xn-xn-τ1)+n7-xa1n2n10+2yb1n2=n20-(-1)(n+1)(n+2)/2(3n5+1)2n23+(n+5)5ln⁡8(n2+1),Δ(yn-6n52n5+1yn-τ2)+n3xc1nyd1n2n6+(n+1)xc1n4=n8-(-1)n-1(n3+1)2(n+1)11,n≥2,
where τ1,τ2∈ℕ are fixed and
(3.19)a1n={n2-1ifniseven,n-32ifnisodd,b1n={n-1ifniseven,n-2ifnisodd,c1n={n2+1ifniseven,n+12ifnisodd,d1n={n-5ifniseven,n-3ifnisodd.
Let
(3.20)n0=n1=2,h=k=1,M>6N>0,α=min⁡{2-τ1,2-τ2,-3},p1n=-1,p2n=-6n52n5+1,p¯=2,p_=3,q1n=n20-(-1)(n+1)(n+2)/2(3n5+1)2n23+(n+5)5ln⁡8(n2+1),q2n=n8-(-1)n-1(n3+1)2(n+1)11,f1(n,u,v)=n7-u2n10+2v2,f2(n,u,v)=n3uv2n6+(n+1)u4,r1n=n7+M2n10+2N2,r2n=n3M3n6+(n+1)N4,t1n=2M(2n7+n10+4M2)(n10+2N2)2,t2n=3n3M2(n6+(n+1)M4)(n6+(n+1)N4)2,∀(n,u,v)∈Nn0×R2.
Obviously (2.1), (2.2), (2.79)–(2.81), (2.84), and (2.85) hold. Thus, Theorem 2.9 implies that System (3.18) possesses uncountably many bounded positive solutions in A(N,M).Example 3.9.
Consider the nonlinear difference system with multiple delays:(3.21)Δ(xn-xn-τ1)+nxa1n2-yb1n3n3ln⁡2n+1=n2-(-1)n-1(3n+1)n5+8(n+1)4+1,Δ(yn+8n22n2+3yn-τ2)+n2n4+|xc1nyd1n|=(-1)n-1sin⁡(n5+1)n2+1,n≥1,
where τ1,τ2∈ℕ are fixed and
(3.22)a1n={n-2ifniseven,n-32ifnisodd,b1n={n2ifniseven,n+12ifnisodd,c1n={n-1ifniseven,n-12ifnisodd,d1n={n-4ifniseven,n-2ifnisodd.
Let
(3.23)n0=1,n1=3,h=k=1,M=3000,N=2000,M0=1200,N0=40,α=min⁡{1-τ1,1-τ2,-2},p1n=-1,p2n=8n22n2+3,p¯0=4,p_0=3,q1n=n2-(-1)n-1(3n+1)n5+8(n+1)4+1,q2n=(-1)n-1sin⁡(n5+1)n2+1,f1(n,u,v)=nu2-v3n3ln⁡2n+1,f2(n,u,v)=n2n4+|uv|,r1n=M2(n+M)n3ln⁡2n+1,r2n=n2n4+N02,t1n=M(2n+3M)n3ln⁡2n+1,t2n=2n2M0(n4+N02)2,∀(n,u,v)∈Nn0×R2.
It is easy to verify that (2.1), (2.2) with i=1, (2.79)–(2.81), and (2.86)–(2.89) are satisfied. Thus, Theorem 2.10 reveals that System (3.21) possesses uncountably many bounded positive solutions in A(N,M,N0,M0).Example 3.10.
Consider the nonlinear difference system with multiple delays:(3.24)Δ(xn+xn-τ1)+1-n5xa1n2n7+(n+3)ln⁡(1+yb1n2)=n4-4n3+n-1n6+(n-1)4,Δ(yn-2n25n2+6yn-τ2)+n2yd1nn4+sin⁡2(xc1n2+nyd1n2)=(-1)nn-1n3+2n2+1,n≥2,
where τ1,τ2∈ℕ are fixed and
(3.25)a1n={n-3ifniseven,n-12ifnisodd,b1n={n2+1ifniseven,n-12ifnisodd,c1n={n-1ifniseven,n+12ifnisodd,d1n={n-2ifniseven,n+12ifnisodd.
Let
(3.26)n0=n1=2,h=k=1,M>5N3>0,α=min⁡{1-τ1,1-τ2,-1},p1n=1,p2n=-2n25n2+6,p¯=0,p_=25,q1n=n4-4n3+n-1n6+(n-1)4,q2n=(-1)nn-1n3+2n2+1,f1(n,u,v)=1-n5u2n7+(n+3)ln⁡(1+v2),f2(n,u,v)=n2vn4+sin⁡2(u2+nv2),r1n=1+n5M2n7+(n+3)ln⁡(1+N2),r2n=Mn2,t1n=1(n7+(n+3)ln⁡(1+N2))2[2M(n+3)1+N2+n12+n5(n+3)(2M21+N2+ln⁡(1+M2))],t2n=n4+4nM2+4M2+1n6,∀(n,u,v)∈Nn0×R2.
Obviously (2.1), (2.2), (2.24), (2.83), and (2.90) hold. Thus, Theorem 2.11 gives that System (3.24) possesses uncountably many bounded positive solutions in A(N,M).Example 3.11.
Consider the nonlinear difference system with multiple delays:(3.27)Δ(xn+xn-τ1)+n2-xa1n2yb1nn4+(n+1)yb1n4=n6-9n5+(-1)nn4-n3+n-1n8+3n7+n5+4n4+n2+1,Δ(yn-3n3n3+1yn-τ2)+nyd1n2n3+xc1n2=(-1)n(n+1)/2(2n-5)ln⁡2(1+n2)n3+3n2+5n+1,n≥1,
where τ1,τ2∈ℕ are fixed and
(3.28)a1n={n-2ifniseven,n+12ifnisodd,b1n={n2-1ifniseven,n+12ifnisodd,c1n={n-3ifniseven,n-32ifnisodd,d1n={n-4ifniseven,n-12ifnisodd.
Let
(3.29)n0=n1=h=k=1,M>4N>0,α=min⁡{1-τ1,1-τ2,-2},p1n=1,p2n=-3n3n3+1,p¯=32,p_=3,q1n=n6-9n5+(-1)nn4-n3+n-1n8+3n7+n5+4n4+n2+1,q2n=(-1)n(n+1)/2(2n-5)ln⁡2(1+n2)n3+3n2+5n+1,f1(n,u,v)=n2-u2vn4+(n+1)v4,f2(n,u,v)=nv2n3+u2,r1n=n2+M3n4+(n+1)N4,r2n=nM2n3+N2,t1n=nM2(8n2M+3n3+10M4)(n4+(n+1)N4)2,t2n=2M(n3+2M2)(n3+N2)2,∀(n,u,v)∈Nn0×R2.
Clearly (2.1), (2.2), (2.24), (2.84), (2.85), and (2.90) hold. Thus, Theorem 2.12 ensures that System (3.27) possesses uncountably many bounded positive solutions in A(N,M).Example 3.12.
Consider the nonlinear difference system with multiple delays:(3.30)Δ(xn+xn-τ1)+n25-(n15+1)yn-12n30+(n18+1)xn-22=n38-19n25+30n18-3n9+2n-1n40+8n30+n15+6n2+1,Δ(yn+16n3+14n3+131yn-τ2)+xn-43yn-32nln⁡10n+1=n34-6n32-7n15+(-1)nn7-1n36+6n30+5n12+6n6+n3+1,n≥2,
where τ1,τ2∈ℕ are fixed. Let
(3.31)n0=2,n1=4,h=k=1,M=800N0,N=500N0,M0=300N0>0,α=min⁡{2-τ1,2-τ2,-2},a1n=n-2,b1n=n-1,c1n=n-4,d1n=n-3,p1n=1,p2n=16n3+14n3+131,p¯0=4,p_0=3,q1n=n38-19n25+30n18-3n9+2n-1n40+8n30+n15+6n2+1,q2n=n34-6n32-7n15+(-1)nn7-1n36+6n30+5n12+6n6+n3+1,f1(n,u,v)=n25-(n15+1)v2n30+(n18+1)u2,f2(n,u,v)=u3v2nln⁡10n+1,r1n=n25+(n15+1)M2n30+(n18+1)N2,r2n=M05nln⁡10n+1,t1n=4n33M(n10+n12+4M2)(n30+(n18+1)N2)2,t2n=5M04(nln⁡10n+1)2,∀(n,u,v)∈Nn0×R2.
It is easy to verify that (2.1), (2.2) with i=1, (2.24), and (2.86)–(2.90) hold. Thus, Theorem 2.13 guarantees that System (3.30) possesses uncountably many bounded positive solutions in A(N,M,N0,M0).Example 3.13.
Consider the nonlinear difference system with multiple delays:(3.32)Δ(xn+3(-1)nn10+2n+19n10+4n8+1xn-τ1)+xn-3-nyn-1n4+xn-32=n4-2n+5ln⁡5nn6+6n2+1,Δ(yn-(4+(-1)n)n4+5n4+1yn-τ2)+n3xn-2yn-4n5+(n+1)yn-42=n7-(-1)nn4-1n9+3n6+5,n≥2,
where τ1,τ2∈ℕ are fixed. Let
(3.33)n0=2,n1=4,h=k=1,M>10N>0,α=min⁡{2-τ1,2-τ2,-2},a1n=n-3,b1n=n-1,c1n=n-2,d1n=n-4,p1n=3(-1)nn10+2n+19n10+4n8+1,p2n=-(4+(-1)n)n4+5n4+1,p¯1=p_1=13,p¯=3,p_=5,q1n=n4-2n+5ln⁡5nn6+6n2+1,q2n=n7-(-1)nn4-1n9+3n6+5,f1(n,u,v)=u-nvn4+u2,f2(n,u,v)=n3uvn5+(n+1)v2,r1n=M(n+1)n4+N2,r2n=n3M2n5+(n+1)N2,t1n=n4+M2+n5+3nM2(n4+N2)2,t2n=2n3M(n5+(n+1)M2)(n5+(n+1)N2)2,∀(n,u,v)∈Nn0×R2.
Obviously (2.1), (2.2), (2.24), (2.84), (2.91), and (2.92) hold. Thus, Theorem 2.14 means that System (3.32) possesses uncountably many bounded positive solutions in A(N,M).Example 3.14.
Consider the nonlinear difference system with multiple delays:(3.34)Δ(xn+(-1)nn+14n+100xn-τ1)+xn-2-nyn-1n3+xn-22yn-12=n5ln⁡9n-(-1)nn8+5n7+1,Δ(yn+64n8+18n8+3n4+7yn-τ2)+nxn-52-yn-3sin⁡nn2ln⁡2n+1=n6-(-1)nn2+1n8+7n7+2,n≥2,
where τ1,τ2∈ℕ are fixed. Let
(3.35)n0=n1=2,h=k=1,M=40000,N=5000,M0=4480,N0=80,α=min⁡{2-τ1,2-τ2,-3},a1n=n-2,b1n=n-1,c1n=n-5,d1n=n-3,p1n=(-1)nn+14n+100,p2n=64n8+18n8+3n4+7,p¯1=p_1=14,p¯0=8,p_0=7,q1n=n5ln⁡9n-(-1)nn8+5n7+1,q2n=n6-(-1)nn2+1n8+7n7+2,f1(n,u,v)=u-nvn3+u2v2,f2(n,u,v)=nu2-vsin⁡nn2ln⁡2n+1,r1n=M(n+1)n3+N4,r2n=M0(M0+1)n2ln⁡2n+1,t1n=n3+3M4+n4+3nM2(n3+N4)2,t2n=2M0+1(n2ln⁡2n+1)2,∀(n,u,v)∈Nn0×R2.
It is easy to see that (2.1), (2.2) with i=1, (2.24), (2.86)–(2.88), (2.91), and (2.93) hold. Thus, Theorem 2.15 implies that System (3.34) possesses uncountably many bounded positive solutions in A(N,M,N0,M0).Example 3.15.
Consider the nonlinear difference system with multiple delays:(3.36)Δ(xn+-20n+14n+9xn-τ1)+nxn-13n3+yn-22=n3+3n2n-1+1n5+3n4+1,Δ(yn+6n2+12n2+5n+3yn-τ2)+xn-2yn-4n2+cos⁡(n2+1)=n4+5n2-7n6+4n3+1,n≥2,
where τ1,τ2∈ℕ are fixed. Let
(3.37)n0=2,n1=10,h=k=1,M=2N0,N=N0>0,M0=30N0,α=min⁡{2-τ1,2-τ2,-2},a1n=n-1,b1n=n-2,c1n=n-2,d1n=n-4,p1n=-20n+14n+9,p2n=6n2+12n2+5n+3,p¯1=4,p_1=5,p¯0=3,p_0=2,q1n=n3+3n2n-1+1n5+3n4+1,q2n=n4+5n2-7n6+4n3+1,f1(n,u,v)=nu3n3+v2,f2(n,u,v)=uvn2+cos⁡(n2+1),r1n=nM3n3+N2,r2n=M02n2-1,t1n=nM2(3n3+5M2)(n3+N2)2,t2n=2M0(n2-1)2,∀(n,u,v)∈Nn0×R2.
Clearly (2.1), (2.2) with i=1, (2.24), (2.86)–(2.88), (2.94), and (2.95) hold. Thus, Theorem 2.16 implies that System (3.36) possesses uncountably many bounded positive solutions in A(N,M,N0,M0).



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*Source: 240378-2012-08-09.xml*