# Existence of Bounded Positive Solutions for Partial Difference Equations with Delays

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

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## Abstract

This paper deals with solvability of the third-order nonlinear partial difference equation with delaysΔn(am,nΔm2(xm,n+bm,nxm-τ0,n-σ0))+f(m,n,xm-τ1,m,n-σ1,n,…,xm-τk,m,n-σk,n)=cm,n,m≥m0,n≥n0. With the help of the Banach fixed-point theorem, the existence results of uncountably many bounded positive solutions for the partial difference equation are given; some Mann iterative schemes with errors are suggested, and the error estimates between the iterative schemes and the bounded positive solutions are discussed. Three nontrivial examples illustrating the results presented in this paper are also provided.


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## Body

## 1. Introduction and Preliminaries

In the past twenty years many authors studied the oscillation, nonoscillation, asymptotic behavior, and solvability for various neutral delay difference and partial difference equations; see, for example, [1–14] and the references cited therein.By using the Banach fixed-point theorem, Cheng [2] investigated the existence of a nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients(1.1)Δ2(xn+pxn-m)+pnxn-k-qnxn-l=0,n≥n0
under the condition p∈ℝ∖{-1}. Applying a nonlinear alternative of Leray-Schauder type for condensing operators, Agarwal et al. [1] discussed the existence of a bounded nonoscillatory solution for the discrete equation:(1.2)Δ(anΔ(xn+pxn-τ))+F(n+1,xn+1-σ)=0,n≥0.
Liu et al. [6] introduced the second-order nonlinear neutral delay difference equation(1.3)Δ(anΔ(xn+bxn-τ))+f(n,xn-d1n,xn-d2n,…,xn-dkn)=cn,n≥0
with respect to all b∈ℝ and gave the existence of uncountably many bounded nonoscillatory solutions for (1.3) by utilizing the Banach fixed-point theorem. Kong et al. [3] investigated a class of BVPs for the third-order functional difference equation(1.4)Δ3xn+anf(n,xw(n))=0,n≥0
and established the existence of positive solutions for (1.4) under certain conditions. Using the Schauder fixed-point theorem, Yan and Liu [12] studied the existence of a bounded nonoscillatory solution for third order nonlinear delay difference equation(1.5)Δ3xn+f(n,xn,xn-r)=0,n≥n0
and provided also a necessary and sufficient condition for the existence of a bounded nonoscillatory solution of (1.5).Karpuz and Öcalan [4] discussed the first-order linear partial difference equation:(1.6)xm+1,n+xm,n+1-xm,n+pm,nxm-k,n-l=0,(m,n)∈Z0,0,
where {pm,n}(m,n)∈ℤ0,0 is a nonnegative sequence and k,l∈ℕ1 and obtained sufficient conditions under which every solution of  (1.6) is oscillatory. Yang and Zhang [14] considered oscillations of the partial difference equation with several nonlinear terms of the form(1.7)xm+1,n+xm,n+1-xm,n+∑i=1hpi(m,n)|xm-ki,n-li|αisgn⁡xm-ki,n-li=0
and established some new oscillatory criteria by making use of frequency measures. Wong and Agarwal [10] considered the partial difference equations(1.8)xm+1,n+βm,nxm,n+1-δm,nxm,n+p(m,n,xm-k,n-l)=Q(m,n,xm-k,n-l),m≥m0,n≥n0,(1.9)xm+1,n+βm,nxm,n+1-δm,nxm,n+∑i=1τpi(m,n,xm-k,n-l)=∑i=1τQ(m,n,xm-k,n-l),m≥m0,n≥n0
and offered sufficient conditions for the oscillation of all solutions for (1.8) and (1.9), respectively. Wong [9] established the existence of eventually positive and monotone decreasing solutions for the partial difference inequalities(1.10)ΔmΔnxm,n+∑i=1rpi(m,n,xgi(m),hi(n))≥(≤)∑i=1rQi(m,n,xgi(m),hi(n)),m≥m0,n≥n0,
where gi(m) and hi(m) are some deviating arguments for 1≤i≤τ.However, to the best of our knowledge, there is no literature referred to the following third order nonlinear partial difference equation with delays:(1.11)Δn(am,nΔm2(xm,n+bm,nxm-τ0,n-σ0))+f(m,n,xm-τ1,m,n-σ1,n,…,xm-τk,m,n-σk,n)=cm,n,m≥m0,n≥n0,
where m0,n0∈ℕ0,k,τ0,σ0∈ℕ,{am,n}(m,n)∈ℕm0,n0,{bm,n}(m,n)∈ℕm0,n0,{cm,n}(m,n)∈ℕm0,n0 are real sequences with am,n≠0, bm,n≠±1 for (m,n)∈ℕm0,n0, f:ℕm0,n0×ℝk→ℝ and {τl,m,σl,n:(m,n)∈ℕm0,n0,l∈{1,2,…,k}}⊆ℤ with(1.12)lim⁡m→∞(m-τl,m)=lim⁡n→∞(n-σl,n)=+∞,l∈{1,2,…,k}.The aim of this paper is to establish three sufficient conditions of the existence of uncountably many bounded positive solutions for (1.11) by using the Banach fixed-point theorem, to suggest some Mann iterative methods with errors for these bounded positive solutions and to compute the error estimates between the bounded positive solutions and the sequences generated by the Mann iterative methods with errors. In order to explain the results presented in this paper, three nontrivial examples are constructed.Throughout this paper, the forward partial difference operatorsΔm and Δn are defined by Δmxm,n=xm+1,n-xm,n and Δnxm,n=xm,n+1-xm,n, respectively the second and third-order partial difference operators are defined by Δm2xm,n=Δm(Δmxm,n) and ΔnΔm2xm,n=Δn(Δm2xm,n), respectively. Let ℝ=(-∞,+∞), ℕ and ℤ denote the sets of all positive integers and integers, respectively,(1.13)N0={0}∪N,Ns={n:n∈N0withn≥s},s∈N0,Ns,t={(m,n):m,n∈N0withm≥s,n≥t},s,t∈N0,Zs,t={(m,n):m,n∈Zwithm≥s,n≥t},s,t∈Z,α=min⁡{m-τ0,m-τl,m:1≤l≤k,m∈Nm0},β=min⁡{n-σ0,n-σl,n:1≤l≤k,n∈Nn0}.lα,β∞ represents the Banach space of all bounded sequences on ℤα,β with the norm
(1.14)‖x‖=sup⁡m,n∈Zα,β|xm,n|forx={xm,n}(m,n)∈Zα,β∈lα,β∞,A(N,M)={x={xm,n}(m,n)∈Zα,β∈lα,β∞:N≤xm,n≤M,(m,n)∈Zα,β}forM>N>0.It is not difficult to see thatA(N,M) is a bounded closed and convex subset of the Banach space lα,β∞. By a solution of (1.11), we mean a sequence {xm,n}(m,n)∈ℤα,β with positive integers m1≥m0+τ0+|α| and n1≥n0+σ0+|β| such that (1.11) is satisfied for all m≥m1 and n≥n1.Lemma 1.1 (see [15]).
Let{α(n)}n∈ℕ0,{β(n)}n∈ℕ0,{γ(n)}n∈ℕ0,and{t(n)}n∈ℕ0 be nonnegative sequences satisfying the inequality
(1.15)α(n+1)≤(1-t(n))α(n)+t(n)β(n)+γ(n),n∈N0,
where {t(n)}n∈ℕ0⊂[0,1] with ∑n=0∞t(n)=+∞,limn→∞β(n)=0 and ∑n=0∞γ(n)<+∞. Then limn→∞α(n)=0.


## 2. Existence of Uncountably Many Bounded Positive Solutions and Mann Iterative Schemes with Errors

Utilizing the Banach fixed-point theorem, we now investigate the existence of uncountably many bounded positive solutions for (1.11), suggest the Mann type iterative schemes with errors and discuss the error estimates between the bounded positive solutions and the sequences generated by the Mann iterative schemes.Theorem 2.1.
Assume that there exists positive constantsM and N, nonnegative constants b1 and b2, and nonnegative sequences {Pm,n}(m,n)∈ℕm0,n0 and {Qm,n}(m,n)∈ℕm0,n0 satisfying
(2.1)b1+b2<1,N<[1-(b1+b2)]M,(2.2)-b2≤bm,n≤b1,eventually,(2.3)|f(m,n,u1,u2,…,uk)-f(m,n,u̅1,u̅2,…,u̅k)|≤Pm,nmax⁡{|ul-u̅l|:1≤l≤k},(m,n,ul,u̅l)∈Nm0,n0×[N,M]2,1≤l≤k,(2.4)|f(m,n,u1,u2,…,uk)|≤Qm,n,(m,n,ul)∈Nm0,n0×[N,M],1≤l≤k;(2.5)∑j=m0∞∑i=j∞sup⁡n∈Nn0{1|ai,n|∑t=n∞max⁡{Pi,t,Qi,t,|ci,t|}}<+∞.
Then
(a)
for anyL∈(N+b1M,(1-b2)M), there exist θ∈(0,1), m1≥m0+τ0+|α| and n1≥n0+σ0+|β| such that for any x(0)={xm,n(0)}(m,n)∈ℤα,β∈A(N,M), the Mann iterative sequence with errors {x(s)}s∈ℕ0={xm,n(s)}(m,n,s)∈ℤα,β×ℕ0 generated by the scheme:(2.6)xm,n(s+1)={[1-α(s)-β(s)]xm,n(s)+α(s)×{L-bm,nxm-τ0,n-σ0(s)+∑j=m∞∑i=j∞1ai,n×∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t(s),…,xi-τk,i,t-σk,t(s))-ci,t]∑j=m∞}+β(s)γm,n(s),(m,n)∈Zm1,n1,s∈N0,[1-α(s)-β(s)]xm1,n1(s)+α(s)×{L-bm1,n1xm1-τ0,n1-σ0(s)+∑j=m1∞∑i=j∞1ai,n1∑t=n1∞×[f(i,t,xi-τ1,i,t-σ1,t(s),…,xi-τk,i,t-σk,t(s))-ci,t]∑j=m1∞}+β(s)γm1,n1(s),(m,n)∈Zα,β∖Zm1,n1,s∈N0,
converges to a bounded positive solution x∈A(M,N) of (1.11) and has the following error estimate:
(2.7)‖x(s+1)-x‖≤[1-(1-θ)α(s)]‖x(s)-x‖+2Mβ(s),s∈N0,
where {γ(s)}s∈ℕ0 is an arbitrary sequence in A(M,N), {α(s)}s∈ℕ0 and {β(s)}s∈ℕ0 are any sequences in [0,1] such that
(2.8)∑s=0∞α(s)=+∞,(2.9)∑s=0∞β(s)<+∞orthereexistsasequence{ξ(s)}s∈N0⊆[0,+∞)satisfyingβ(s)=ξ(s)α(s),s∈N0,lim⁡s→∞ξ(s)=0;
(b)
(1.11) possesses uncountably many bounded positive solutions in A(M,N).Proof.
First of all we show that (a) holds. SetL∈(N+b1M,(1-b2)M). It follows from (2.1), (2.2), and (2.5) that there exist θ∈(0,1), m1≥m0+τ0+|α| and n1≥n0+σ0+|β| such that
(2.10)θ=b1+b2+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞Pi,t},(2.11)-b2≤bm,n≤b1,(m,n)∈Nm1,n1,(2.12)∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≤min⁡{(1-b2)M-L,L-b1M-N}.
Define a mapping TL:A(N,M)→lα,β∞ by
(2.13)TLxm,n={L-bm,nxm-τ0,n-σ0+∑j=m∞∑i=j∞1ai,n×∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],(m,n)∈Zm1,n1,TLxm1,n1,(m,n)∈Zα,β∖Zm1,n1
for each x={xm,n}(m,n)∈ℤα,β∈A(N,M). By employing (2.1)–(2.4) and (2.10)–(2.13), we infer that for x={xm,n}(m,n)∈ℤα,β,y={ym,n}(m,n)∈ℤα,β∈A(N,M) and (m,n)∈ℤm1,n1(2.14)|TLxm,n-TLym,n|=|∑i=j∞bm,n(xm-τ0,n-σ0-ym-τ0,n-σ0)-∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-f(i,t,yi-τ1,i,t-σ1,t,…,yi-τk,i,t-σk,t)]∑i=j∞|≤|bm,n||xm-τ0,n-σ0-ym-τ0,n-σ0|+∑j=m∞∑i=j∞1|ai,n|∑t=n∞|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-f(i,t,yi-τ1,i,t-σ1,t,…,yi-τk,i,t-σk,t)|≤(b1+b2)‖x-y‖+∑j=m∞∑i=j∞1|ai,n|×∑t=n∞Pi,tmax⁡{|xi-τl,i,t-σl,t-yi-τl,i,t-σl,t|:1≤l≤k}≤(b1+b2+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞Pi,t})‖x-y‖=θ‖x-y‖,TLxm,n=L-bm,nxm-τ0,n-σ0+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t]≤L+b2M+∑j=m∞∑i=j∞1|ai,n|∑t=n∞[|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)|+|ci,t|]≤L+b2M+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≤L+b2M+min⁡{(1-b2)M-L,L-b1M-N}≤M,TLxm,n=L-bm,nxm-τ0,n-σ0+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t]≥L-b1M-∑j=m∞∑i=j∞1|ai,n|∑t=n∞[|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)|+|ci,t|]≥L-b1M-∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≥L-b1M-min⁡{(1-b2)M-L,L-b1M-N}≥N,
which lead to
(2.15)TL(A(N,M))⊆A(N,M),‖TLx-TLy‖≤θ‖x-y‖,x,y∈A(N,M).
Consequently, (2.15) means that TL is a contraction mapping in A(N,M) and it has a unique fixed-point x={xm,n}(m,n)∈ℤα,β∈A(N,M), which together with (2.13) gives that for (m,n)∈ℤm1,n1(2.16)xm,n=L-bm,nxm-τ0,n-σ0+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],
which yields that for (m,n)∈ℤm1,n1(2.17)Δm(xm,n+bm,nxm-τ0,n-σ0)=-∑i=m∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],Δm2(xm,n+bm,nxm-τ0,n-σ0)=1am,n∑t=n∞[f(m,t,xm-τ1,m,t-σ1,t,…,xm-τk,m,t-σk,t)-cm,t],Δn(am,nΔm2(xm,n+bm,nxm-τ0,n-σ0))=-f(m,n,xm-τ1,m,n-σ1,t,…,xm-τk,m,t-σk,t)+cm,n,
that is, x={xm,n}(m,n)∈ℤα,β is a bounded positive solution of (1.11) in A(N,M).
Using (2.6), (2.13), and (2.15), we infer that for any s∈ℕ0 and (m,n)∈ℤm1,n1(2.18)|xm,n(s+1)-xm,n|=|∑j=m∞[1-α(s)-β(s)]xm,n(s)+α(s)×{∑i=j∞L-bm,nxm-τ0,n-σ0(s)+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t(s),…,xi-τk,i,t-σk,t(s))-ci,t]}+β(s)γm,n(s)-xm,n∑j=m∞|≤[1-α(s)-β(s)]|xm,n(s)-xm,n|+α(s)|TLxm,n(s)-TLxm,n|+β(s)|γm,n(s)-xm,n|≤[1-α(s)-β(s)]‖x(s)-x‖+α(s)θ‖x(s)-x‖+2Mβ(s)≤[1-(1-θ)α(s)]‖x(s)-x‖+2Mβ(s),
which yields that
(2.19)‖x(s+1)-x‖≤[1-(1-θ)α(s)]‖x(s)-x‖+2Mβ(s),s∈N0.
That is, (2.7) holds. Consequently, Lemma 1.1 and (2.7)–(2.9) imply that lims→∞x(s)=x.
Next we show that (b) holds. LetL1,L2∈(N+b1M,(1-b2)M) and let L1≠L2. As in the proof of (a), we infer that for each i∈{1,2}, there exist θi,mi+1,ni+1 and TLi satisfying (2.10)–(2.13), where θ, m1, n1, L and TL are replaced by θi,mi+1,ni+1,Li, and TLi, respectively, and the mapping TLi has a fixed-point xi={xm,ni}(m,n)∈ℤα,β∈A(N,M), which is a bounded positive solution of (1.11), that is,
(2.20)xm,n1=L1-bm,nxm-τ0,n-σ01+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-ci,t],(m,n)∈Zm2,n2,xm,n2=L2-bm,nxm-τ0,n-σ02+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)-ci,t],(m,n)∈Zm3,n3.
In order to show that the set of bounded positive solutions of (1.11) is uncountable, it is sufficient to prove that x1≠x2. It follows from (2.3), (2.10), (2.11), (2.20) that for (m,n)∈ℤmax{m2,m3},max{n2,n3}(2.21)|xm,n1-xm,n2|=|∑j=m∞L1-L2-bm,n(xm-τ0,n-σ01-xm-τ0,n-σ02)+∑j=m∞∑i=j∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)]∑j=m∞|≥|L1-L2|-|bm,n||xm-τ0,n-σ01-xm-τ0,n-σ02|-∑j=m∞∑i=j∞1|ai,n|∑t=n∞[|f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)|]≥|L1-L2|-(b1+b2)‖x1-x2‖-∑j=m∞∑i=j∞1|ai,n|∑t=n∞Pi,tmax⁡{|xi-τl,i,t-σl,t1-xi-τl,i,t-σl,t2|:1≤l≤k}≥|L1-L2|-(b1+b2+∑j=m∞∑i=j∞1|ai,n|∑t=n∞Pi,t)‖x1-x2‖≥|L1-L2|-(b1+b2+∑j=max⁡{m2,m3}∞∑i=j∞sup⁡n∈Nmax⁡{n2,n3}{1|ai,n|∑t=n∞Pi,t})‖x1-x2‖≥|L1-L2|-max⁡{θ1,θ2}‖x1-x2‖,
which implies that
(2.22)‖x1-x2‖≥|L1-L2|1+max⁡{θ1,θ2}>0,
that is, x1≠x2. This completes the proof.Theorem 2.2.
Assume that there exist positive constantsM and N, negative constants b1 and b2 and nonnegative sequences {Pm,n}(m,n)∈ℕm0,n0 and {Qm,n}(m,n)∈ℕm0,n0 satisfying (2.3)–(2.5) and
(2.23)b1<-1,N(1+b2)>M(1+b1);(2.24)b2≤bm,n≤b1,eventually.
Then
(a) for anyL∈(M(1+b1),N(1+b2)), there exist θ∈(0,1), m1≥m0+τ0+|α| and n1≥n0+σ0+|β| such that for each x(0)={xm,n(0)}(m,n)∈ℤα,β∈A(N,M), the Mann iterative sequence with errors {x(s)}s∈ℕ0={xm,n(s)}(m,n,s)∈ℤα,β×ℕ0 generated by the scheme:
(2.25)xm,n(s+1)={[1-α(s)-β(s)]xm,n(s)+α(s)×{Lbm+τ0,n+σ0-xm+τ0,n+σ0(s)bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t(s),…,xi-τk,i,t-σk,t(s))-ci,t]}+β(s)γm,n(s),(m,n)∈Zm1,n1,s∈N0,[1-α(s)-β(s)]xm1,n1(s)+α(s)×{Lbm1+τ0,n1+σ0-xm1+τ0,n1+σ0(s)bm1+τ0,n1+σ0+1bm1+τ0,n1+σ0∑j=m1+τ0∞∑i=j∞1ai,n1+σ0×∑t=n1+σ0∞[f(i,t,xi-τ1,i,t-σ1,t(s),…,xi-τk,i,t-σk,t(s))-ci,t]}+β(s)γm1,n1(s),(m,n)∈Zα,β∖Zm1,n1,s∈N0
converges to a bounded positive solution x∈A(N,M) of (1.11) and has the error estimate (2.7), where {γ(s)}s∈ℕ0 is an arbitrary sequence in A(N,M), {α(s)}s∈ℕ0 and {β(s)}s∈ℕ0 are any sequences in [0,1] satisfying (2.8) and (2.9);
(b) (1.11) possesses uncountably many bounded positive solutions in A(M,N).Proof.
First of all we show (a). TakingL∈(M(1+b1),N(1+b2)), from (2.5), (2.23), and (2.24) we infer that there exist θ∈(0,1), m1≥m0+τ0+|α| and n1≥n0+σ0+|β| such that(2.26)θ=-1b1(1+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞Pi,t}),(2.27)b2≤bm,n≤b1,(m,n)∈Nm1,n1,(2.28)∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≤min⁡{L-M(1+b1),b1N(1+1b2)-b1Lb2}.
Define a mapping TL:A(N,M)→lα,β∞ by(2.29)TLxm,n={Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],(m,n)∈Zm1,n1,TLxm1,n1,(m,n)∈Zα,β∖Zm1,n1
for each x={xm,n}(m,n)∈ℤα,β∈A(N,M). It follows from (2.3), (2.4), (2.23), (2.24), and (2.26)–(2.29) that for x={xm,n}(m,n)∈ℤα,β, y={ym,n}(m,n)∈ℤα,β∈A(N,M) and (m,n)∈ℤm1,n1:(2.30)|TLxm,n-TLym,n|=|xm+τ0,n+σ0-ym+τ0,n+σ0bm+τ0,n+σ0-1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-f(i,t,yi-τ1,i,t-σ1,t,…,yi-τk,i,t-σk,t)]∑t=n+σ0∞|≤-|xm+τ0,n+σ0-ym+τ0,n+σ0|bm+τ0,n+σ0-1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-f(i,t,yi-τ1,i,t-σ1,t,…,yi-τk,i,t-σk,t)|≤-‖x-y‖b1-1b1∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞Pi,tmax⁡{|xi-τl,i,t-σl,t-yi-τl,i,t-σl,t|:1≤l≤k}≤-1b1(1+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞Pi,t})‖x-y‖=θ‖x-y‖,TLxm,n=Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t]≤Lb1-Mb1-1b1∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞[|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)|+|ci,t|]≤Lb1-Mb1-1b1∑j=m+τ0∞∑i=j∞1|ai,n+σ0|∑t=n+σ0∞(Qi,t+|ci,t|)≤Lb1-Mb1-1b1∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≤Lb1-Mb1-1b1min⁡{L-M(1+b1),b1N(1+1b2)-b1Lb2}≤M,TLxm,n=Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t]≥Lb2-Nb2+1b1∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞[|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)|+|ci,t|]≥Lb2-Nb2+1b1∑j=m+τ0∞∑i=j∞1|ai,n+σ0|∑t=n+σ0∞(Qi,t+|ci,t|)≥Lb2-Nb2+1b1∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≥Lb2-Nb2+1b1min⁡{L-M(1+b1),b1N(1+1b2)-b1Lb2}≥N,
which imply that (2.15) holds. Consequently, (2.15) ensures that TL is a contraction mapping in A(N,M) and it has a unique fixed-point x={xm,n}(m,n)∈ℤα,β∈A(N,M), which together with (2.29) gives that
(2.31)xm,n=Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],(m,n)∈Zm1,n1,
which yields that for (m,n)∈ℤm1,n1(2.32)Δm(xm,n+bm,nxm-τ0,n-σ0)=-∑i=m∞1ai,n∑t=n∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],Δm2(xm,n+bm,nxm-τ0,n-σ0)=1am,n∑t=n∞[f(m,t,xm-τ1,m,t-σ1,t,…,xm-τk,m,t-σk,t)-cm,t],Δn(am,nΔm2(xm,n+bm,nxm-τ0,n-σ0))=-f(m,n,xm-τ1,m,n-σ1,t,…,xm-τk,m,t-σk,t)+cm,n,
which implies that x={xm,n}(m,n)∈ℤα,β is a bounded positive solution of (1.11) in A(N,M).
It follows from (2.15), (2.25) and (2.29) that for any s∈ℕ0 and (m,n)∈ℤm1,n1(2.33)|xm,n(s+1)-xm,n|=|∑i=j∞[1-α(s)-β(s)]xm,n(s)+α(s)×{∑i=j∞Lbm+τ0,n+σ0-xm+τ0,n+σ0(s)bm+τ0,n+σ0+1bm+τ0,n+σ0×∑j=m+τ0∞∑i=j∞1ai,n+σ0∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t(s),…,xi-τk,i,t-σk,t(s))-ci,t]}+β(s)γm,n(s)-xm,n∑i=j∞|≤[1-α(s)-β(s)]|xm,n(s)-xm,n|+α(s)|TLxm,n(s)-TLxm,n|+β(s)|γm,n(s)-xm,n|≤[1-α(s)-β(s)]‖x(s)-x‖+α(s)θ‖x(s)-x‖+2Mβ(s)≤[1-(1-θ)α(s)]‖x(s)-x‖+2Mβ(s),
which yields (2.7). Thus Lemma 1.1 and (2.7)–(2.9) ensure that lims→∞x(s)=x.
Next we show that (b) holds. LetL1,L2∈(M(1+b1),N(1+b2))  let and L1≠L2. As in the proof of (a), we infer that for each i∈{1,2}, there exist θi,mi+1,ni+1 and TLi satisfying (2.26)–(2.29), where θ,m1,n1,L and TL are replaced by θi,mi+1,ni+1,Li and TLi, respectively, and the mapping TLi has a fixed-point xi={xm,ni}(m,n)∈ℤα,β∈A(N,M), which is a bounded positive solution of  (1.11), that is: (2.34)xm,n1=L1bm+τ0,n+σ0-xm+τ0,n+σ01bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-ci,t],(m,n)∈Zm2,n2,(2.35)xm,n2=L2bm+τ0,n+σ0-xm+τ0,n+σ02bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)-ci,t],(m,n)∈Zm3,n3.
In order to show that the set of bounded positive solutions of (1.11) is uncountable, it is sufficient to prove that x1≠x2. It follows from (2.3), (2.26), (2.27), (2.34), and (2.35) that for (m,n)∈ℤmax{m2,m3},max{n2,n3}(2.36)|xm,n1-xm,n2|=|L1-L2bm+τ0,n+σ0-xm+τ0,n+σ01-xm+τ0,n+σ02bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)]|≥-|L1-L2|bm+τ0,n+σ0+|xm+τ0,n+σ01-xm+τ0,n+σ02|bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞|f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)|≥-|L1-L2|b2+‖x1-x2‖b1+1b1∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞Pi,tmax⁡{|xi-τ1,i,t-σ1,t1-xi-τ1,i,t-σ1,t2|:1≤l≤k}≥-|L1-L2|b2+1b1(1+∑j=m+τ0∞∑i=j∞1|ai,n+σ0|∑t=n+σ0∞Pi,t)‖x1-x2‖≥-|L1-L2|b2+1b1(1+∑j=max⁡{m2,m3}∞∑i=j∞sup⁡n∈Nmax⁡{n2,n3}{1|ai,n|∑t=n∞Pi,t})‖x1-x2‖≥-|L1-L2|b2-max⁡{θ1,θ2}‖x1-x2‖,
which implies that
(2.37)‖x1-x2‖≥-|L1-L2|b2(1+max⁡{θ1,θ2})>0,
that is, x1≠x2. This completes the proof.Theorem 2.3.
Assume that there exist positive constantsM and N, nonnegative constants b1 and b2, and nonnegative sequences {Pm,n}(m,n)∈ℕm0,n0 and {Qm,n}(m,n)∈ℕm0,n0 satisfying (2.3)–(2.5), (2.24) and
(2.38)1<b2,b1<b22,Mb1(b22-b1)>Nb2(b12-b2).
Then
(a) for anyL∈(b1N+b1M/b2,b2M+b2N/b1), there exist θ∈(0,1),  m1≥m0+τ0+|α| and n1≥n0+σ0+|β| such that for each x(0)={xm,n(0)}(m,n)∈ℤα,β∈A(N,M), the Mann iterative sequence with errors {x(s)}s∈ℕ0={xm,n(s)}(m,n,s)∈ℤα,β×ℕ0 generated by (2.25) converges to a bounded positive solution x∈A(N,M) of (1.11) and has the error estimate (2.7), where {γ(s)}s≥0 is an arbitrary sequence in A(N,M),{α(s)}s≥0 and {β(s)}s≥0 are any sequences in [0,1] satisfying (2.8) and (2.9);
(b) (1.11) possesses uncountably many bounded positive solutions in A(M,N).Proof.
SetL∈(b1N+b1M/b2),b2M+b2N/b1). It follows from (2.5), (2.24), and (2.38) that there exist θ∈(0,1), m1≥m0+τ0+|α| and n1≥n0+σ0+|β| satisfying (2.27):
(2.39)θ=1b2(1+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞Pi,t}),(2.40)∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≤min⁡{b2M-L+b2Nb1,b2Lb1-M-b2N}.
Let the mapping TL:A(N,M)→lα,β∞ be defined by (2.29). It follows from (2.3), (2.4), (2.24), (2.27), (2.29), and (2.38)–(2.40) that for x={xm,n}(m,n)∈ℤα,β, y={ym,n}(m,n)∈ℤα,β∈A(N,M) and (m,n)∈ℤm1,n1(2.41)|TLxm,n-TLym,n|=|xm+τ0,n+σ0-ym+τ0,n+σ0bm+τ0,n+σ0-1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-f(i,t,yi-τ1,i,t-σ1,t,…,yi-τk,i,t-σk,t)]∑t=n+σ0∞|≤|xm+τ0,n+σ0-ym+τ0,n+σ0|bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-f(i,t,yi-τ1,i,t-σ1,t,…,yi-τk,i,t-σk,t)|≤‖x-y‖b2+1b2∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞Pi,tmax⁡{|xi-τl,i,t-σl,t-yi-τl,i,t-σl,t|:1≤l≤k}≤1b2(1+∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞Pi,t})‖x-y‖=θ‖x-y‖,TLxm,n=Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0,m∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t]≤Lb2-Nb1+1b2∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞[|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)|+|ci,t|]≤Lb2-Nb1+1b2∑j=m1∞∑i=j∞sup⁡n∈Nn1{1|ai,n|∑t=n∞(Qi,t+|ci,t|)}≤Lb2-Nb1+1b2min⁡{b2M-L+b2Nb1,b2Lb1-M-b2N}≤M,TLxm,n=Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0,m∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t]≥Lb1-Mb2-1b2∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞[|f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)|+|ci,t|]≥Lb1-Mb2-1b2∑j=m1∞∑i=j∞sup⁡n∈Nn1{1ai,n∑t=n∞(Qi,t+|ci,t|)}≥Lb1-Mb2-1b2min⁡{b2M-L+b2Nb1,b2Lb1-M-b2N}≥N,
which imply that (2.15) holds. Consequently (2.15) ensures that TL is a contraction mapping, and hence it has a unique fixed-point x={xm,n}(m,n)∈ℤα,β∈A(N,M), which gives that
(2.42)xm,n=Lbm+τ0,n+σ0-xm+τ0,n+σ0bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t,…,xi-τk,i,t-σk,t)-ci,t],(m,n)∈Zm1,n1.
As in the proof of Theorem 2.2, it is easy to verify that x={xm,n}(m,n)∈ℤα,β is a bounded positive solution of (1.11) in A(N,M); (2.7) holds and lims→∞x(s)=x.
Next we show that (b) holds. LetL1,L2∈(b1N+b1M/b2,b2M+b2N/b1) and L1≠L2. As in the proof of (a), we infer that for each i∈{1,2}, there exist θi,mi+1,ni+1 and TLi satisfying (2.27), (2.29), (2.39), and (2.40), where θ,m1,n1,L and TL are replaced by θi,mi+1,ni+1,Li, and TLi, respectively, and the mapping TLi has a fixed-point xi={xm,ni}(m,n)∈ℤα,β∈A(N,M), which is a bounded positive solution of (1.11) and satisfies (2.34) and (2.35). In order to show that the set of bounded positive solutions of (1.11) is uncountable, it is sufficient to prove that x1≠x2. It follows from (2.3), (2.27), (2.34), (2.35), and (2.39) that for (m,n)∈ℤmax{m2,m3},max{n2,n3}(2.43)|xm,n1-xm,n2|=|∑j=m+τ0∞L1-L2bm+τ0,n+σ0-xm+τ0,n+σ01-xm+τ0,n+σ02bm+τ0,n+σ0+1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1ai,n+σ0×∑t=n+σ0∞[f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)]∑j=m+τ0∞|≥|L1-L2|bm+τ0,n+σ0-|xm+τ0,n+σ01-xm+τ0,n+σ02|bm+τ0,n+σ0-1bm+τ0,n+σ0∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞|f(i,t,xi-τ1,i,t-σ1,t1,…,xi-τk,i,t-σk,t1)-f(i,t,xi-τ1,i,t-σ1,t2,…,xi-τk,i,t-σk,t2)|≥|L1-L2|b1-‖x1-x2‖b2-1b2∑j=m+τ0∞∑i=j∞1|ai,n+σ0|×∑t=n+σ0∞Pi,tmax⁡{|xi-τ1,i,t-σ1,t1-xi-τ1,i,t-σ1,t2|:1≤l≤k}≥|L1-L2|b1-1b2(1+∑j=m+τ0∞∑i=j∞1|ai,n+σ0|∑t=n+σ0∞Pi,t)‖x1-x2‖≥|L1-L2|b1-1b2(1+∑j=max⁡{m2,m3}∞∑i=j∞sup⁡n∈Nmax⁡{n2,n3}{1|ai,n|∑t=n∞Pi,t})‖x1-x2‖≥|L1-L2|b1-max⁡{θ1,θ2}‖x1-x2‖,
which implies that
(2.44)‖x1-x2‖≥|L1-L2|b1(1+max⁡{θ1,θ2})>0,
that is, x1≠x2. This completes the proof.


## 3. Examples

Now we illustrate the results presented in Section2 with the following three examples. Note that none of the known results can be applied to the examples.Example 3.1.
Consider the third-order nonlinear partial difference equation with delays:(3.1)Δn((-1)mnm4n3Δm2(xm,n+(-1)m+n3xm-τ0,n-σ0))+nm2(n2+1)xm2,n(n+1)/23-cos⁡(m3n5-ln⁡m)(m+1)2n2xm3-2m,n2-n2=(-1)nsin⁡(m2-2n)m5n4+1,m≥1,n≥1,
where τ0,σ0∈ℕ are fixed. Let m0=n0=1, k=2, b1=b2=1/3, α=min{1-τ0,-1},β=1-σ0, M and let N be two positive constants with M>3N and
(3.2)am,n=(-1)mnm4n3,bm,n=(-1)m+n3,cm,n=(-1)m+nsin⁡(m2-2n)m5n4+1,f(m,n,u,v)=nm2(n2+1)u3-cos⁡(m3n5-ln⁡m)(m+1)2n2v2,τ1,m=m(1-m),τ2,m=m(3-m2),σ1,n=n(1-n)2,σ2,n=n(2-n),Pm,n=3M2nm2(n2+1)+2M(m+1)2n2,Qm,n=M3nm2(n2+1)+M2(m+1)2n2,(m,n,u,v)∈Nm0,n0×R2.It is easy to verify that (2.1)–(2.4) hold. Note that
(3.3)∑j=m0∞∑i=j∞sup⁡n≥n0{1|ai,n|∑t=n∞max⁡{Pi,t,Qi,t,|ci,t|}}=∑j=m0∞∑i=j∞sup⁡n≥n0{1i4n3∑t=n∞max⁡{3M2ti2(t2+1)+2M(i+1)2t2,M3ti2(t2+1)+M2(i+1)2t2,|sin⁡(i2-2t)|i5t4+1}}<(1+2M+4M2+M3)(∑t=n0∞1t3)∑j=m0∞∑i=j∞1i4<+∞.Hence the conditions of Theorem 2.1 are fulfilled. It follows from Theorem 2.1 that (3.1) possesses uncountably many bounded positive solutions in A(N,M). On the other hand, for any L∈(N+(1/3)M,(2/3)M), there exist θ∈(0,1) and m1≥m0+τ0+|α|,n1≥n0+σ0+|β| such that the Mann iterative sequence with errors {x(s)}s≥0 generated by (2.6) converges to a bounded positive solution x∈A(N,M) of (3.1) and has the error estimate (2.7), where {γ(s)}s≥0 is an arbitrary sequence in A(N,M),{α(s)}s≥0 and {β(s)}s≥0 are any sequences in [0,1] satisfying (2.8) and (2.9).Example 3.2.
Consider the third-order nonlinear partial difference equation with delays:(3.4)Δn((-1)nm3ln⁡2(m+n)Δm2(xm,n-4n+(-1)mnn+1xm-τ0,n-σ0))+xm-2,n-32x3m2-2,n5-13m3n2=cos⁡(nm3-m)n3+1,m≥1,n≥1,
where τ0,σ0∈ℕ are fixed. Let m0=n0=1,k=2,b1=-2,   b2=-5,  α=min{1-τ0,-1},β=min{1-σ0,-2}, M and let N be two positive constants with M>4N and
(3.5)am,n=(-1)nm3ln⁡2(m+n),bm,n=-4n+(-1)mnn+1,cm,n=cos⁡(nm3-m)n3+1,f(m,n,u,v)=u2v3m3n2,τ1,m=2,τ2,m=-3m2+m+2,σ1,n=3,σ2,n=-n5+n+1,Pm,n=5M4m3n2,Qm,n=M5m3n2,(m,n,u,v)∈Nm0,n0×R2.It is clear that (2.3), (2.4), (2.23), and (2.24) hold. Observe that
(3.6)∑j=m0∞∑i=j∞sup⁡n≥n0{1|ai,n|∑t=n∞max⁡{Pi,t,Qi,t,|ci,t|}}=∑j=m0∞∑i=j∞sup⁡n≥n0{1i3ln⁡2(i+n)∑t=n∞max⁡{5M4i3t2,M5i3t2,|cos⁡(ti3-i)|t3+1}}<1+5M4+M5ln⁡22(∑t=n0∞1t3)∑j=m0∞∑i=j∞1i3<+∞.That is, the conditions of Theorem 2.2 are fulfilled. Thus Theorem 2.2 ensures that (3.4) has uncountably many bounded positive solutions in A(N,M). On the other hand, for any L∈(-M,-4N), there exist θ∈(0,1) and m1≥m0+τ0+|α|,n1≥n0+σ0+|β| such that the Mann iterative sequence with errors {x(s)}s≥0 generated by (2.25) converges to a bounded positive solution x∈A(N,M) of (3.4) and has the error estimate (2.7), where {γ(s)}s≥0 is an arbitrary sequence in A(N,M),{α(s)}s≥0 and {β(s)}s≥0 are any sequences in [0,1] satisfying (2.8) and (2.9).Example 3.3.
Consider the third-order nonlinear partial difference equation with delays:(3.7)Δn((-1)n(m5n)Δm2(xm,n+2mn+3mn+1xm-τ0,n-σ0))+((-1)m-1/n)xm-4,n-34m3n2+xm2-2m,n3-n2=(-1)n+m(m2-3n3)m5(n7+1),m≥1,n≥1,
where τ0,σ0∈ℕ are fixed. Let m0=n0=1,k=2,  b1=3,b2=2,α=min{1-τ0,-3},β=min{1-σ0,-2}, M and N be two positive constants with M>(14/3)N and
(3.8)am,n=(-1)n(m5n),bm,n=2mn+3mn+1,cm,n=(-1)n+m(m2-3n3)m5(n7+1),f(m,n,u,v)=((-1)m-1n)u4m3n2+v2,τ1,m=4,τ2,m=m(3-m),σ1,n=3,σ2,n=-n3+2n,Pm,n=2M3(2m3n2+2M2+1)(m3n2+N2)2,Qm,n=2M4m3n2+N2,(m,n,u,v)∈Nm0,n0×R2.
Clearly (2.3), (2.4), (2.24), and (2.38) hold. Notice that
(3.9)∑j=m0∞∑i=j∞sup⁡n≥n0{1|ai,n|∑t=n∞max⁡{Pi,t,Qi,t,|ci,t|}}=∑j=m0∞∑i=j∞sup⁡n≥n0{1i5n∑t=n∞max⁡{2M3(2i3t2+2M2+1)(i3t2+N2)2,2M4i3t2+N2,|i2-3t3|i5(t7+1)}}<max⁡{4,2M4,4M3(1+M2)}(∑t=n0∞1t2)∑j=m0∞∑i=j∞1i5<+∞.Hence the conditions of Theorem 2.3 are fulfilled. Consequently Theorem 2.3 implies that (3.7) possesses uncountably many bounded positive solutions in A(N,M). On the other hand, for any L∈(3N+(3/2)M,2M+(2/3)N), there exist θ∈(0,1) and m1≥m0+τ0+|α|,n1≥n0+σ0+|β| such that the Mann iterative sequence with errors {x(s)}s≥0 generated by (2.25) converges to a bounded positive solution x∈A(N,M) of (3.7) and has the error estimate (2.7), where {γ(s)}s≥0 is an arbitrary sequence in A(N,M),{α(s)}s≥0 and {β(s)}s≥0 are any sequences in [0,1] satisfying (2.8) and (2.9).



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*Source: 191254-2012-05-22.xml*