Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 245012, 12 pages
http://dx.doi.org/10.1155/2013/245012
Research Article
Unbounded Positive Solutions and Mann Iterative Schemes of
a Second-Order Nonlinear Neutral Delay Difference Equation
Zeqing Liu,1 Xiaochuan Hou,2 Jeong Sheok Ume,3 and Shin Min Kang4
1
Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
Fushun No. 10 High School, Fushun, Liaoning 113004, China
3
Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea
4
Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea
2
Correspondence should be addressed to Jeong Sheok Ume; jsume@changwon.ac.kr
Received 12 May 2012; Accepted 12 December 2012
Academic Editor: Alexander I. Domoshnitsky
Copyright © 2013 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation Δ2 (𝑥𝑛 + 𝑎𝑛 𝑥𝑛−𝜏 ) +
Δℎ(𝑛, 𝑥ℎ1𝑛 , 𝑥ℎ2𝑛 , . . . , 𝑥ℎ𝑘𝑛 ) + 𝑓(𝑛, 𝑥𝑓1𝑛 , 𝑥𝑓2𝑛 , . . . , 𝑥𝑓𝑘𝑛 ) = 𝑏𝑛 , ∀𝑛 ≥ 𝑛0 . Utilizing the Banach fixed point theorem and some new
techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several
Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the
sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this
paper.
1. Introduction and Preliminaries
and the second-order neutral difference equation:
Recently, the oscillation, nonoscillation, asymptotic behavior,
and existence of solutions of different classes of linear
and nonlinear second-order difference equations have been
studied by many authors; see, for example, [1–26] and
the references cited therein. Using the Banach fixed point
theorem, Jinfa [5] discussed the existence of a bounded
nonoscillatory solution for the second-order neutral delay
difference equation with positive and negative coefficients:
Δ2 (𝑥𝑛 + 𝑝𝑥𝑛−𝑚 ) + 𝑝𝑛 𝑥𝑛−𝑘 − 𝑞𝑛 𝑥𝑛−𝑙 = 0,
∀𝑛 ≥ 𝑛0
∀𝑛 ≥ 1,
∀𝑛 ≥ 1
(2)
(3)
respectively. Meng and Yan [15] studied the existence of
bounded nonoscillatory solutions for the second-order nonlinear nonautonomous neutral delay difference equation:
𝑚
Δ2 (𝑥𝑛 − 𝑝𝑥𝑛−𝜏 ) = ∑ 𝑞𝑖 𝑥𝑛−𝜎𝑖 + 𝑓 (𝑛, 𝑥𝑛−𝜂1𝑛 , . . . , 𝑥𝑛−𝜂𝑙𝑛 ) ,
𝑛=1
∀𝑛 ≥ 𝑛0 .
(4)
(1)
under the condition 𝑝 ≠ − 1. Luo and Bainov [13] and M.
Migda and J. Migda [16] considered the asymptotic behaviors
of nonoscillatory solutions for the second-order neutral
difference equation with maxima:
Δ2 (𝑥𝑛 + 𝑝𝑛 𝑥𝑛−𝑘 ) + 𝑞𝑛 max {𝑥𝑠 : 𝑛 − 𝑙 ≤ 𝑠 ≤ 𝑛} = 0,
Δ2 (𝑥𝑛 + 𝑝𝑥𝑛−𝑘 ) + 𝑓 (𝑛, 𝑥𝑛 ) = 0,
Applying the cone compression and expansion theorem in
Fréchet spaces, Tian and Ge [21] established the existence
of multiple positive solutions of the second-order discrete
equation on the half-line:
Δ2 𝑥𝑛−1 − 𝑝Δ𝑥𝑛−1 − 𝑞𝑥𝑛−1 + 𝑓 (𝑛, 𝑥𝑛 ) = 0,
∀𝑛 ≥ 1
(5)
with certain boundary value conditions. But to the best of our
knowledge, results on multiplicity of unbounded solutions
2
Abstract and Applied Analysis
for neutral delay difference equations are very scarce in
the literature. Nothing has been done with the existence of
uncountably many unbounded positive solutions for (1)∼
(5) and any other second-order neutral delay difference
equations:
Inspired and motivated by the results in [1–26], in this
paper we introduce and study the second-order nonlinear
neutral delay difference equation:
Δ2 (𝑥𝑛 + 𝑎𝑛 𝑥𝑛−𝜏 ) + Δℎ (𝑛, 𝑥ℎ1𝑛 , 𝑥ℎ2𝑛 , . . . , 𝑥ℎ𝑘𝑛 )
Lemma 1 (see [27]). Let {𝛼𝑛 }𝑛∈N0 , {𝛽𝑛 }𝑛∈N0 , {𝛾𝑛 }𝑛∈N0 , and
{𝑡𝑛 }𝑛∈N0 be four nonnegative sequences satisfying the inequality
𝛼𝑛+1 ≤ (1 − 𝑡𝑛 ) 𝛼𝑛 + 𝑡𝑛 𝛽𝑛 + 𝛾𝑛 ,
∀𝑛 ∈ N0 ,
(10)
(6)
where {𝑡𝑛 }𝑛∈N0 ⊂ [0, 1], ∑∞
𝑛=0 𝑡𝑛 = +∞, lim𝑛 → ∞ 𝛽𝑛 = 0, and
∞
∑𝑛=0 𝛾𝑛 < +∞. Then lim𝑛 → ∞ 𝛼𝑛 = 0.
where 𝜏, 𝑘, 𝑛0 ∈ N, {𝑎𝑛 }𝑛∈N𝑛 , {𝑏𝑛 }𝑛∈N𝑛 ⊂ R, ℎ, 𝑓 ∈ 𝐶(N𝑛0 ×
Lemma 2 (see [11]). Let 𝜏, 𝑛0 ∈ N and {𝑏𝑛 }𝑛∈N𝑛 be a
0
nonnegative sequence. Then
∞
∞
(a) ∑∞
𝑖=0 ∑𝑠=𝑛0 +𝑖𝜏 𝑏𝑠 < +∞ ⇔ ∑𝑠=𝑛0 𝑠𝑏𝑠 < +∞;
∞
∞
∞
∞
(b) ∑𝑖=0 ∑𝑠=𝑛0 +𝑖𝜏 ∑𝑡=𝑠 𝑏𝑡 < +∞ ⇔ ∑∞
𝑠=𝑛0 ∑𝑡=𝑠 𝑠𝑏𝑡 < +∞.
+ 𝑓 (𝑛, 𝑥𝑓1𝑛 , 𝑥𝑓2𝑛 , . . . , 𝑥𝑓𝑘𝑛 ) = 𝑏𝑛 ,
0
𝑘
R , R), {ℎ𝑙𝑛 }𝑛∈N𝑛 , {𝑓𝑙𝑛 }𝑛∈N𝑛 ⊆ N, and
0
lim ℎ
𝑛 → ∞ 𝑙𝑛
∀𝑛 ≥ 𝑛0 ,
0
0
= lim 𝑓𝑙𝑛 = +∞,
𝑛→∞
𝑙 ∈ {1, 2, . . . , 𝑘} .
(7)
By means of the Banach fixed point theorem and some
new techniques, we establish sufficient conditions for the
existence of uncountably many unbounded positive solutions
of (6), suggest a few Mann iterative schemes with errors
for approximating these unbounded positive solutions, and
prove their convergence and the error estimates. The results
obtained in this paper extend the result in [5]. Four nontrivial
examples are interested in the text to illustrate the importance
of our results.
Throughout this paper, we assume that Δ is the forward
difference operator defined by Δ𝑥𝑛 = 𝑥𝑛+1 − 𝑥𝑛 , R =
(−∞, +∞), R+ = [0, +∞), Z, N0 , and N denote the sets
of all integers, nonnegative integers, and positive integers,
respectively,
N𝑡 = {𝑛 : 𝑛 ∈ N with 𝑛 ≥ 𝑡} ,
∀𝑡 ∈ N,
2. Existence of Uncountably Many Unbounded
Positive Solutions
Using the Banach fixed point theorem and the Mann iterative schemes with errors, we next discuss the existence
of uncountably many unbounded positive solutions of (6),
prove that the Mann iterative schemes with errors converge
to these unbounded positive solutions, and compute the error
estimates between the Mann iterative schemes with errors
and the unbounded positive solutions.
Theorem 3. Assume that there exist two constants 𝑀 and
𝑁 with 𝑀 > 𝑁 > 0 and four nonnegative sequences
{𝑃𝑛 }𝑛∈N𝑛 , {𝑄𝑛 }𝑛∈N𝑛 , {𝑅𝑛 }𝑛∈N𝑛 , and {𝑊𝑛 }𝑛∈N𝑛 satisfying
0
0
0
0
𝛽 = min {𝑛0 − 𝜏, inf {ℎ𝑙𝑛 , 𝑓𝑙𝑛 : 1 ≤ 𝑙 ≤ 𝑘, 𝑛 ∈ N𝑛0 }} ∈ N,
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑛, 𝑢1 , 𝑢2 , . . . , 𝑢𝑘 ) − 𝑓 (𝑛, 𝑢1 , 𝑢2 , . . . , 𝑢𝑘 )󵄨󵄨󵄨
𝐻𝑛 = max {ℎ𝑙𝑛 : 𝑙 ∈ {1, 2, . . . , 𝑘}} ,
𝐹𝑛 = max {𝑓𝑙𝑛 : 𝑙 ∈ {1, 2, . . . , 𝑘}} ,
(8)
𝑙𝛽∞
represents the Banach space of all real sequences on N𝛽
with norm
󵄨󵄨 𝑥 󵄨󵄨
‖𝑥‖ = sup 󵄨󵄨󵄨󵄨 𝑛 󵄨󵄨󵄨󵄨 < +∞ for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈ 𝑙𝛽∞ ,
𝑛∈N 󵄨 𝑛 󵄨
𝛽
𝐴 (𝑁, 𝑀) = {𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈ 𝑙𝛽∞ : 𝑁 ≤
󵄨
󵄨
≤ 𝑃𝑛 max {󵄨󵄨󵄨𝑢𝑙 − 𝑢𝑙 󵄨󵄨󵄨 : 1 ≤ 𝑙 ≤ 𝑘} ,
∀𝑛 ∈ N𝑛0 ,
𝑥𝑛
≤ 𝑀, 𝑛 ∈ N𝛽 }
𝑛
for any 𝑀 > 𝑁 > 0.
(9)
It is clear that 𝐴(𝑁, 𝑀) is a closed and convex subset of 𝑙𝛽∞ .
By a solution of (6), we mean a sequence {𝑥𝑛 }𝑛∈N𝛽 with a
positive integer 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 such that (6) holds for all
𝑛 ≥ 𝑇.
The following lemmas play important roles in this paper.
󵄨󵄨
󵄨
󵄨󵄨ℎ (𝑛, 𝑢1 , 𝑢2 , . . . , 𝑢𝑘 ) − ℎ (𝑛, 𝑢1 , 𝑢2 , . . . , 𝑢𝑘 )󵄨󵄨󵄨
󵄨
󵄨
≤ 𝑅𝑛 max {󵄨󵄨󵄨𝑢𝑙 − 𝑢𝑙 󵄨󵄨󵄨 : 1 ≤ 𝑙 ≤ 𝑘} ,
2
∀ (𝑛, 𝑢𝑙 , 𝑢𝑙 ) ∈ N𝑛0 × (R+ \ {0}) ,
󵄨󵄨
󵄨
󵄨󵄨𝑓 (𝑛, 𝑢1 , 𝑢2 , . . . , 𝑢𝑘 )󵄨󵄨󵄨 ≤ 𝑄𝑛 ,
(11)
1 ≤ 𝑙 ≤ 𝑘,
󵄨󵄨
󵄨
󵄨󵄨ℎ (𝑛, 𝑢1 , 𝑢2 , . . . , 𝑢𝑘 )󵄨󵄨󵄨 ≤ 𝑊𝑛 ,
∀ (𝑛, 𝑢𝑙 ) ∈ N𝑛0 × (R+ \ {0}) ,
1 ≤ 𝑙 ≤ 𝑘,
(12)
1 ∞ ∞
∑ ∑ max {𝑅𝑠 𝐻𝑠 , 𝑊𝑠 } = 0,
𝑛→∞𝑛
𝑖=0 𝑠=𝑛+𝑖𝜏
(13)
1 ∞ ∞ ∞
󵄨 󵄨
∑ ∑ ∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨} = 0,
𝑛→∞𝑛
𝑖=0 𝑠=𝑛+𝑖𝜏 𝑡=𝑠
(14)
𝑎𝑛 = −1 𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.
(15)
lim
lim
Abstract and Applied Analysis
3
Then
(a) for any 𝐿 ∈ (𝑁, 𝑀), there exist 𝜃 ∈ (0, 1) and 𝑇 ≥ 𝑛0 +
𝜏 + 𝛽 such that for each 𝑥0 = {𝑥0𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀), the Mann
iterative sequence with errors {𝑥𝑚 }𝑚∈N0 = {𝑥𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽
generated by the scheme
Proof. First of all we show that (a) holds. Set 𝐿 ∈ (𝑁, 𝑀). It
follows from (13), (14), and (15) that there exist 𝜃 ∈ (0, 1) and
𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 satisfying
𝜃=
𝑥𝑚+1𝑛
(1 − 𝛼𝑚 − 𝛽𝑚 ) 𝑥𝑚𝑛
{
{
{
{
{
{
{ +𝛼𝑚 {𝑛𝐿
{
{
{
{
∞ ∞
{
{
{
{
−
∑
∑ [ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
{
{
{
{
𝑖=1 𝑠=𝑛+𝑖𝜏
{
{
∞
{
{
{
{
− ∑ (𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
{
{
{
𝑡=𝑠
{
{
{
{
{
{
𝑥𝑚𝑓𝑘𝑡 )−𝑏𝑡 ) ]}
{
{
{
{
{
{
{ +𝛽𝑚 𝛾𝑚𝑛 ,
𝑛 ≥ 𝑇, 𝑚 ≥ 0,
={
{(1 − 𝛼𝑚 − 𝛽𝑚 ) 𝑥𝑚𝑇
{
{
{
{
{
{
{
+𝛼𝑚 {𝑇𝐿
{
{
{
{
{
∞ ∞
{
{
{
{
− ∑ ∑ [ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
{
{
{
𝑖=1 𝑠=𝑇+𝑖𝜏
{
{
∞
{
{
{
{
− ∑ (𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
{
{
{
{
𝑡=𝑠
{
{
{
{
{
𝑥𝑚𝑓𝑘𝑡 ) − 𝑏𝑡 ) ]}
{
{
{
{
{
𝛽 ≤ 𝑛 < 𝑇, 𝑚 ≥ 0
{ +𝛽𝑚 𝛾𝑚𝑇 ,
∞
1∞ ∞
∑ ∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 ) ,
𝑇 𝑖=1 𝑠=𝑇+𝑖𝜏
𝑡=𝑠
(19)
∞
∞
1 ∞ ∞
󵄨 󵄨
∑ ∑ (𝑊𝑠 + ∑ 𝑄𝑡 + ∑ 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨) < min {𝑀 − 𝐿, 𝐿 − 𝑁} ,
𝑇 𝑖=1 𝑠=𝑇+𝑖𝜏
𝑡=𝑠
𝑡=𝑠
(20)
𝑎𝑛 = −1,
∀𝑛 ≥ 𝑇.
(21)
Define a mapping 𝑆𝐿 : 𝐴(𝑁, 𝑀) → 𝑙𝛽∞ by
𝑆𝐿 𝑥𝑛
𝑛𝐿
{
{
∞ ∞
{
{
{
{ −∑ ∑ {ℎ (𝑠, 𝑥ℎ , 𝑥ℎ , . . . , 𝑥ℎ )
{
1𝑠
2𝑠
𝑘𝑠
{
{
{
𝑖=1 𝑠=𝑛+𝑖𝜏
{
∞
={
{
−∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 ) − 𝑏𝑡 ]} ,
{
{
{
𝑡=𝑠
{
{
{
{
𝑛 ≥ 𝑇,
{
{
𝛽≤𝑛<𝑇
{𝑆𝐿 𝑥𝑇 ,
(22)
(16)
converges to an unbounded positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀) of
(6) and has the following error estimate:
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑥𝑚+1 − 𝑥󵄩󵄩󵄩 ≤ [1 − (1 − 𝜃) 𝛼𝑚 ] 󵄩󵄩󵄩𝑥𝑚 − 𝑥󵄩󵄩󵄩 + 2𝑀𝛽𝑚 ,
∀𝑚 ∈ N0 ,
(17)
where {𝛾𝑚 }𝑚∈N0 = {𝛾𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 is an arbitrary sequence
in 𝐴(𝑁, 𝑀) and {𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any sequences in
[0, 1] such that
for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀). In view of (11), (12), (19),
(20), and (22), we deduce that for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 , 𝑦 =
{𝑦𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀) and for all 𝑛 ≥ 𝑇
󵄨󵄨 𝑆 𝑥
󵄨
󵄨󵄨 𝐿 𝑛 𝑆𝐿 𝑦𝑛 󵄨󵄨󵄨
−
󵄨󵄨
󵄨
󵄨󵄨 𝑛
𝑛 󵄨󵄨󵄨
≤
󵄨
− ℎ (𝑠, 𝑦ℎ1𝑠 , 𝑦ℎ2𝑠 , . . . , 𝑦ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
∞
∑ 𝛼𝑚 = +∞,
∞
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )
𝑚=0
𝑡=𝑠
∞
∑ 𝛽𝑚 < +∞ or there exists a sequence
𝑚=0
(18)
󵄨
− 𝑓 (𝑡, 𝑦𝑓1𝑡 , 𝑦𝑓2𝑡 , . . . , 𝑦𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 ]
{𝜉𝑚 }𝑚∈N0 ⊂ [0, +∞) satisfying
𝛽𝑚 = 𝜉𝑚 𝛼𝑚 ,
𝑚 ∈ N0 ,
1 ∞ ∞
󵄨
∑ ∑ [ 󵄨󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
𝑛 𝑖=1 𝑠=𝑛+𝑖𝜏 󵄨
lim 𝜉𝑚 = 0;
𝑚→∞
(b) equation (6) possesses uncountably many unbounded
positive solutions in 𝐴(𝑁, 𝑀).
≤
1 ∞ ∞
󵄨
󵄨
∑ ∑ [𝑅𝑠 max {󵄨󵄨󵄨󵄨𝑥ℎ𝑙𝑠 − 𝑦ℎ𝑙𝑠 󵄨󵄨󵄨󵄨 : 1 ≤ 𝑙 ≤ 𝑘}
𝑛 𝑖=1 𝑠=𝑛+𝑖𝜏
∞
󵄨
󵄨
+ ∑ 𝑃𝑡 max {󵄨󵄨󵄨󵄨𝑥𝑓𝑙𝑡 − 𝑦𝑓𝑙𝑡 󵄨󵄨󵄨󵄨 : 1 ≤ 𝑙 ≤ 𝑘}]
𝑡=𝑠
4
Abstract and Applied Analysis
󵄩
󵄩󵄩
󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∞ ∞
≤󵄩
∑ ∑ [𝑅𝑠 max {ℎ𝑙𝑠 : 1 ≤ 𝑙 ≤ 𝑘}
𝑛
𝑖=1 𝑠=𝑛+𝑖𝜏
𝑥𝑛−𝜏
= (𝑛 − 𝜏) 𝐿
∞
∞
+ ∑ 𝑃𝑡 max {𝑓𝑙𝑡 : 1 ≤ 𝑙 ≤ 𝑘}]
−∑
𝑡=𝑠
∞
∑
𝑖=1 𝑠=𝑛+(𝑖−1)𝜏
󵄩
󵄩󵄩
∞
󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∞ ∞
≤󵄩
∑ ∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 )
𝑇 𝑖=1 𝑠=𝑇+𝑖𝜏
𝑡=𝑠
{ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
∞
− ∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )−𝑏𝑡 ]} ,
𝑡=𝑠
󵄩
󵄩
= 𝜃 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑛 ≥ 𝑇 + 𝜏,
(25)
󵄨󵄨 𝑆 𝑥
󵄨󵄨
󵄨󵄨 𝐿 𝑛
󵄨
− 𝐿󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑛
󵄨󵄨
󵄨󵄨 ∞ ∞
󵄨󵄨 1
= 󵄨󵄨󵄨 ∑ ∑ {ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
󵄨󵄨󵄨 𝑛 𝑖=1 𝑠=𝑛+𝑖𝜏
which imply that
󵄨󵄨
∞
󵄨
− ∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 ) − 𝑏𝑡 ]}󵄨󵄨󵄨󵄨
󵄨󵄨
𝑡=𝑠
≤
∞
𝑥𝑛 − 𝑥𝑛−𝜏
∞
= 𝜏𝐿 + ∑ {ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
∞
1
󵄨
󵄨
∑ ∑ { 󵄨󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
𝑛 𝑖=1 𝑠=𝑛+𝑖𝜏 󵄨
𝑠=𝑛
∞
− ∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 ) − 𝑏𝑡 ]} ,
∞
󵄨
󵄨 󵄨 󵄨
+ ∑ [󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨]}
𝑡=𝑠
𝑡=𝑠
≤
∞
∀𝑛 ≥ 𝑇 + 𝜏,
∞
∞
1
󵄨 󵄨
∑ ∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)]
𝑇 𝑖=1 𝑠=𝑇+𝑖𝜏
𝑡=𝑠
(26)
< min {𝑀 − 𝐿, 𝐿 − 𝑁} ,
which yields that
(23)
Δ (𝑥𝑛 − 𝑥𝑛−𝜏 ) = − ℎ (𝑛, 𝑥ℎ1𝑛 , 𝑥ℎ2𝑛 , . . . , 𝑥ℎ𝑘𝑛 )
which yield that
∞
+ ∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 ) − 𝑏𝑡 ] ,
𝑡=𝑛
𝑆𝐿 (𝐴 (𝑁, 𝑀)) ⊆ 𝐴 (𝑁, 𝑀) ,
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑆𝐿 𝑥 − 𝑆𝐿 𝑦󵄩󵄩󵄩 ≤ 𝜃 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
∀𝑥, 𝑦 ∈ 𝐴 (𝑁, 𝑀) ,
(24)
∀𝑛 ≥ 𝑇 + 𝜏,
Δ2 (𝑥𝑛 − 𝑥𝑛−𝜏 ) = − Δℎ (𝑛, 𝑥ℎ1𝑛 , 𝑥ℎ2𝑛 , . . . , 𝑥ℎ𝑘𝑛 )
− 𝑓 (𝑛, 𝑥𝑓1𝑛 , 𝑥𝑓2𝑛 , . . . , 𝑥𝑓𝑘𝑛 ) + 𝑏𝑛 ,
∀𝑛 ≥ 𝑇 + 𝜏,
which means that 𝑆𝐿 is a contraction in 𝐴(𝑁, 𝑀). It follows
from the Banach fixed point theorem that 𝑆𝐿 has a unique
fixed point 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀), that is,
which together with (21) gives that 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 is an
unbounded positive solution of (6) in 𝐴(𝑁, 𝑀). It follows
from (16), (19), (21), (22), and (24) that for any 𝑚 ∈ N0 and
𝑛≥𝑇
𝑥𝑛 = 𝑛𝐿
∞
(27)
∞
− ∑ ∑ {ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
𝑖=1 𝑠=𝑛+𝑖𝜏
∞
− ∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )−𝑏𝑡 ]} ,
𝑡=𝑠
∀𝑛 ≥ 𝑇,
󵄨󵄨 𝑥𝑚+1𝑛 𝑥𝑛 󵄨󵄨
󵄨󵄨
− 󵄨󵄨󵄨󵄨
󵄨󵄨
𝑛󵄨
󵄨 𝑛
󵄨󵄨
1 󵄨󵄨
= 󵄨󵄨󵄨 (1 − 𝛼𝑚 − 𝛽𝑚 ) 𝑥𝑚𝑛
𝑛 󵄨󵄨󵄨
Abstract and Applied Analysis
5
which together with (11) and (17) implies that for 𝑛 ≥
max{𝑇1 , 𝑇2 }
+ 𝛼𝑚 {𝑛𝐿
∞
∞
− ∑ ∑ [ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
𝑖=1 𝑠=𝑛+𝑖𝜏
∞
− ∑ (𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
󵄨󵄨 𝑧1 𝑧2 󵄨󵄨
󵄨󵄨 𝑛
󵄨
󵄨󵄨 − 𝑛 󵄨󵄨󵄨
󵄨󵄨 𝑛
𝑛 󵄨󵄨󵄨
󵄨
󵄨󵄨
󵄨
≥ 󵄨󵄨𝐿 1 −𝐿 2 󵄨󵄨󵄨
1∞ ∞
󵄨
− ∑ ∑ { 󵄨󵄨󵄨󵄨ℎ (𝑠, 𝑧ℎ11𝑠 , 𝑧ℎ12𝑠 , . . . , 𝑧ℎ1𝑘𝑠 )
𝑛 𝑖=1 𝑠=𝑛+𝑖𝜏
󵄨
− ℎ (𝑠, 𝑧ℎ21𝑠 , 𝑧ℎ22𝑠 , . . . , 𝑧ℎ2𝑘𝑠 )󵄨󵄨󵄨󵄨
𝑡=𝑠
𝑥𝑚𝑓𝑘𝑡 )−𝑏𝑡 ) ]}
󵄨󵄨
󵄨󵄨
+𝛽𝑚 𝛾𝑚𝑛 − 𝑥𝑛 󵄨󵄨󵄨
󵄨󵄨
󵄨
󵄨
󵄨󵄨
󵄨𝑥 − 𝑥𝑛 󵄨󵄨󵄨
≤ (1 − 𝛼𝑚 − 𝛽𝑚 ) 󵄨 𝑚𝑛
𝑛
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨𝑆 𝑥 − 𝑆𝐿 𝑥𝑛 󵄨󵄨
󵄨𝛾 − 𝑥𝑛 󵄨󵄨󵄨
+ 𝛼𝑚 󵄨 𝐿 𝑚𝑛
+ 𝛽𝑚 󵄨 𝑚𝑛
𝑛
𝑛
󵄩
󵄩󵄩
󵄩󵄩
󵄩
≤ (1 − 𝛼𝑚 − 𝛽𝑚 ) 󵄩󵄩𝑥𝑚 − 𝑥󵄩󵄩 + 𝜃𝛼𝑚 󵄩󵄩󵄩𝑥𝑚 − 𝑥󵄩󵄩󵄩 + 2𝑀𝛽𝑚
󵄩
󵄩
≤ [1 − (1 − 𝜃) 𝛼𝑚 ] 󵄩󵄩󵄩𝑥𝑚 − 𝑥󵄩󵄩󵄩 + 2𝑀𝛽𝑚 ,
∞
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑧𝑓11𝑡 , 𝑧𝑓12𝑡 , . . . , 𝑧𝑓1𝑘𝑡 )
𝑡=𝑠
󵄨
󵄨
≥ 󵄨󵄨󵄨𝐿 1 − 𝐿 2 󵄨󵄨󵄨
(28)
which implies that
1∞ ∞
󵄨
󵄨
− ∑ ∑ [𝑅𝑠 max {󵄨󵄨󵄨󵄨𝑧ℎ1𝑙𝑠 − 𝑧ℎ2𝑙𝑠 󵄨󵄨󵄨󵄨 : 1 ≤ 𝑙 ≤ 𝑘}
𝑛 𝑖=1 𝑠=𝑛+𝑖𝜏
∞
󵄨
󵄨
+ ∑𝑃𝑡 max {󵄨󵄨󵄨󵄨𝑧𝑓1𝑙𝑡 −𝑧𝑓2𝑙𝑡 󵄨󵄨󵄨󵄨 : 1 ≤ 𝑙 ≤ 𝑘}]
𝑡=𝑠
󵄩󵄩󵄩𝑧1 − 𝑧2 󵄩󵄩󵄩 ∞ ∞
∞
󵄩󵄩
󵄨
󵄨 󵄩
≥ 󵄨󵄨󵄨𝐿 1 − 𝐿 2 󵄨󵄨󵄨 − 󵄩
∑ ∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 )
𝑛
𝑡=𝑠
𝑖=1 𝑠=𝑛+𝑖𝜏
󵄩󵄩 1
󵄩󵄩
2
󵄩󵄩𝑧 − 𝑧 󵄩󵄩󵄩
󵄨
󵄨
≥ 󵄨󵄨󵄨𝐿 1 − 𝐿 2 󵄨󵄨󵄨 − 󵄩
max {𝑇1 , 𝑇2 }
∞
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑥𝑚+1 − 𝑥󵄩󵄩󵄩 ≤ [1 − (1 − 𝜃) 𝛼𝑚 ] 󵄩󵄩󵄩𝑥𝑚 − 𝑥󵄩󵄩󵄩 + 2𝑀𝛽𝑚 ,
∀𝑚 ∈ N0 .
∑
𝑖=1 𝑠=max{𝑇1 ,𝑇2 }+𝑖𝜏
That is, (17) holds. Thus, Lemma 1 and (17) and (18) guarantee
that lim𝑚 → ∞ 𝑥𝑚 = 𝑥.
Next we show that (b) holds. Let 𝐿 1 , 𝐿 2 ∈ (𝑁, 𝑀) and
𝐿 1 ≠ 𝐿 2 . As in the proof of (a), we deduce similarly that for
each 𝑐 ∈ {1, 2} there exist constants 𝜃𝑐 ∈ (0, 1), 𝑇𝑐 ≥ 𝑛0 + 𝜏 +
𝛽, and a mapping 𝑆𝐿 𝑐 satisfying (19)∼(24), where 𝜃, 𝐿, and 𝑇
are replaced by 𝜃𝑐 , 𝐿 𝑐 and 𝑇𝑐 , respectively, and the mapping
𝑆𝐿 𝑐 has a fixed point 𝑧𝑐 = {𝑧𝑛𝑐 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀), which is an
unbounded positive solution of (6) in 𝐴(𝑁, 𝑀), that is,
𝑧𝑛𝑐 = 𝑛𝐿 𝑐
∞
∞
×∑
(29)
󵄨
− 𝑓 (𝑡, 𝑧𝑓21𝑡 , 𝑧𝑓22𝑡 , . . . , 𝑧𝑓2𝑘𝑡 )󵄨󵄨󵄨󵄨 }
∞
(𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 )
𝑡=𝑠
󵄩
󵄩
󵄨
󵄨
≥ 󵄨󵄨󵄨𝐿 1 − 𝐿 2 󵄨󵄨󵄨 − max {𝜃1 , 𝜃2 } 󵄩󵄩󵄩󵄩𝑧1 − 𝑧2 󵄩󵄩󵄩󵄩 ,
(31)
which yields that
󵄨󵄨
󵄨
󵄨󵄨𝐿 1 − 𝐿 2 󵄨󵄨󵄨
󵄩󵄩 1
󵄩
󵄩󵄩𝑧 − 𝑧2 󵄩󵄩󵄩 ≥
󵄩
󵄩 1 + max {𝜃 , 𝜃 } > 0,
1 2
(32)
that is, 𝑧1 ≠ 𝑧2 . This completes the proof.
Theorem 4. Assume that there exist two constants 𝑀 and 𝑁
with 𝑀 > 𝑁 > 0 and four nonnegative sequences {𝑃𝑛 }𝑛∈N𝑛 ,
0
{𝑄𝑛 }𝑛∈N𝑛 , {𝑅𝑛 }𝑛∈N𝑛 , and {𝑊𝑛 }𝑛∈N𝑛 satisfying (11), (12),
0
0
0
∞
− ∑ ∑ {ℎ (𝑠, 𝑧ℎ𝑐 1𝑠 , 𝑧ℎ𝑐 2𝑠 , . . . , 𝑧ℎ𝑐 𝑘𝑠 )
1 ∞
∑ max {𝑅𝑠 𝐻𝑠 , 𝑊𝑠 } = 0,
𝑛→∞𝑛
𝑠=𝑛
(33)
1 ∞ ∞
󵄨 󵄨
∑ ∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨} = 0,
𝑛→∞𝑛
𝑠=𝑛 𝑡=𝑠
(34)
lim
𝑖=1 𝑠=𝑛+𝑖𝜏
∞
− ∑ [𝑓 (𝑡, 𝑧𝑓𝑐 1𝑡 , 𝑧𝑓𝑐 2𝑡 , . . . , 𝑧𝑓𝑐 𝑘𝑡 ) − 𝑏𝑡 ]} ,
𝑡=𝑠
∀𝑛 ≥ 𝑇𝑐 ,
(30)
lim
𝑎𝑛 = 1
𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.
(35)
6
Abstract and Applied Analysis
Then
(a) for any 𝐿 ∈ (𝑁, 𝑀), there exist 𝜃 ∈ (0, 1) and 𝑇 ≥ 𝑛0 +
𝜏 + 𝛽 such that for each 𝑥0 = {𝑥0𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀), the Mann
iterative sequence with errors {𝑥𝑚 }𝑚∈N0 = {𝑥𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽
generated by the scheme
𝑥𝑚+1𝑛
(1 − 𝛼𝑚 − 𝛽𝑚 ) 𝑥𝑚𝑛
{
{
{
{
{
{
{
{
+𝛼𝑚 {𝑛𝐿
{
{
{
{
{
{
∞ 𝑛+2𝑖𝜏−1
{
{
{
{
+
∑
∑ [ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
{
{
{
𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏
{
{
{
∞
{
{
{
{
− ∑ (𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
{
{
{
𝑡=𝑠
{
{
{
{
{
𝑥𝑚𝑓𝑘𝑡 ) − 𝑏𝑡 ) ]}
{
{
{
{
{
{
{
{
𝑛 ≥ 𝑇, 𝑚 ≥ 0,
{ +𝛽𝑚 𝛾𝑚𝑛 ,
={
{
{
(1 − 𝛼𝑚 − 𝛽𝑚 ) 𝑥𝑚𝑇
{
{
{
{
{
{
{
{
{
+𝛼𝑚 {𝑇𝐿
{
{
{
{
{
{
∞ 𝑇+2𝑖𝜏−1
{
{
{
{
+
∑
∑ [ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
{
{
{
𝑖=1 𝑠=𝑇+(2𝑖−1)𝜏
{
{
{
∞
{
{
{
−
(𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
∑
{
{
{
{
𝑡=𝑠
{
{
{
{
{
𝑥𝑚𝑓𝑘𝑡 ) − 𝑏𝑡 ) ]}
{
{
{
{
{
𝛽 ≤ 𝑛 < 𝑇, 𝑚 ≥ 0
{ +𝛽𝑚 𝛾𝑚𝑇 ,
(36)
converges to an unbounded positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀)
of (6) and has the error estimate (17), where {𝛾𝑚 }𝑚∈N0 =
{𝛾𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 is an arbitrary sequence in 𝐴(𝑁, 𝑀), and
{𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any sequences in [0, 1] satisfying
(18);
(b) equation (6) possesses uncountably many unbounded
positive solutions in 𝐴(𝑁, 𝑀).
Define a mapping 𝑆𝐿 : 𝐴(𝑁, 𝑀) → 𝑙𝛽∞ by
𝑆𝐿 𝑥𝑛
𝑛𝐿
{
{
{
∞ 𝑛+2𝑖𝜏−1
{
{
{ +∑ ∑ {ℎ (𝑠, 𝑥 , 𝑥 , . . . , 𝑥 )
{
{
ℎ1𝑠 ℎ2𝑠
ℎ𝑘𝑠
{
{
{
{ 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏 ∞
={
{
−∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )−𝑏𝑡 ]} ,
{
{
{
𝑡=𝑠
{
{
{
{
𝑛 ≥ 𝑇,
{
{
{
𝑥
,
𝛽
≤𝑛<𝑇
𝑆
{ 𝐿 𝑇
(40)
for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀). Using (11), (12), (37), (38),
and (40), we get that for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 , 𝑦 = {𝑦𝑛 }𝑛∈N𝛽 ∈
𝐴(𝑁, 𝑀) and 𝑛 ≥ 𝑇
󵄨
󵄨󵄨 𝑆 𝑥
󵄨󵄨 𝐿 𝑛 𝑆𝐿 𝑦𝑛 󵄨󵄨󵄨
−
󵄨
󵄨󵄨
𝑛 󵄨󵄨󵄨
󵄨󵄨 𝑛
≤
1 ∞ 𝑛+2𝑖𝜏−1
󵄨
∑ ∑ [ 󵄨󵄨󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
𝑛 𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏
󵄨
− ℎ (𝑠, 𝑦ℎ1𝑠 , 𝑦ℎ2𝑠 , . . . , 𝑦ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
∞
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )
𝑡=𝑠
󵄨
− 𝑓 (𝑡, 𝑦𝑓1𝑡 , 𝑦𝑓2𝑡 , . . . , 𝑦𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 ]
󵄩
󵄩󵄩
∞
󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∞ 𝑛+2𝑖𝜏−1
≤󵄩
∑ ∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 )
𝑛
𝑡=𝑠
𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏
󵄩
󵄩󵄩
∞
󵄩𝑥 − 𝑦󵄩󵄩󵄩 ∞
≤󵄩
∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 )
𝑇 𝑠=𝑇
𝑡=𝑠
󵄩
󵄩
= 𝜃 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
󵄨󵄨 𝑆 𝑥
󵄨󵄨
󵄨󵄨 𝐿 𝑛
󵄨
− 𝐿󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑛
󵄨󵄨
≤
Proof. Let 𝐿 ∈ (𝑁, 𝑀). It follows from (33)∼(35) that there
exist 𝜃 ∈ (0, 1) and 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 satisfying
1
𝑛
∞
×∑
𝑛+2𝑖𝜏−1
∑
𝑖=1 𝑠=𝑛+(2𝑖−1)𝜏
𝜃=
∞
1 ∞
∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 ) ,
𝑇 𝑠=𝑇
𝑡=𝑠
𝑡=𝑠
≤
(38)
∀𝑛 ≥ 𝑇.
∞
󵄨
󵄨 󵄨 󵄨
+ ∑ [󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨]}
(37)
∞
1 ∞
󵄨 󵄨
∑ (𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)) < min {𝑀 − 𝐿, 𝐿 − 𝑁} ,
𝑇 𝑠=𝑇
𝑡=𝑠
𝑎𝑛 = 1,
󵄨
󵄨
{ 󵄨󵄨󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
(39)
∞
∞
1
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)]
𝑇 𝑖=𝑇
𝑡=𝑠
< min {𝑀 − 𝐿, 𝐿 − 𝑁} ,
(41)
Abstract and Applied Analysis
7
which imply (24). Consequently, (24) means that 𝑆𝐿 is a
contraction in 𝐴(𝑁, 𝑀) and has a unique fixed point 𝑥 =
{𝑥𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀), which is also an unbounded positive
solution of (6) in 𝐴(𝑁, 𝑀). The rest of the proof is similar
to the proof of Theorem 3 and is omitted. This completes the
proof.
Theorem 5. Assume that there exist three constants 𝑎, 𝑀, and
𝑁 with (1 − 𝑎)𝑀 > 𝑁 > 0 and four nonnegative sequences
{𝑃𝑛 }𝑛∈N𝑛 , {𝑄𝑛 }𝑛∈N𝑛 , {𝑅𝑛 }𝑛∈N𝑛 , and {𝑊𝑛 }𝑛∈N𝑛 satisfying (11),
0
0
0
0
(12), (33), (34) and
0 ≤ 𝑎𝑛 ≤ 𝑎 < 1
𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.
𝜃=𝑎+
∞
1 ∞
∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 ) ,
𝑇 𝑠=𝑇
𝑡=𝑠
∞
1 ∞
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)] < min {𝑀 − 𝐿, 𝐿 − 𝑎𝑀 − 𝑁} ,
𝑇 𝑠=𝑇
𝑡=𝑠
0 ≤ 𝑎𝑛 ≤ 𝑎 < 1,
∀𝑛 ≥ 𝑇.
(44)
(42)
Then
(a) for any 𝐿 ∈ (𝑎𝑀 + 𝑁, 𝑀), there exist 𝜃 ∈ (0, 1)
and 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 such that for any 𝑥0 = {𝑥0𝑛 }𝑛∈N𝛽 ∈
𝐴(𝑁, 𝑀), the Mann iterative sequence with errors {𝑥𝑚 }𝑚∈N0 =
{𝑥𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 generated by the scheme
𝑥𝑚+1𝑛
(1 − 𝛼𝑚 − 𝛽𝑚 ) 𝑥𝑚𝑛
{
{
{
{
{
{
+𝛼𝑚 {𝑛𝐿 − 𝑎𝑛 𝑥𝑚𝑛−𝜏
{
{
{
{
{
∞
{
{
{
{
+ ∑ [ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
{
{
{
𝑠=𝑛
{
{
∞
{
{
{
{
−
(𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
∑
{
{
{
{
𝑡=𝑠
{
{
{
{
{
{
𝑥𝑚𝑓𝑘𝑡 ) − 𝑏𝑡 ) ]}
{
{
{
{
{
{
{ +𝛽𝑚 𝛾𝑚𝑛 ,
𝑛 ≥ 𝑇, 𝑚 ≥ 0,
={
(1
−
𝛼
−
𝛽
)
𝑥
{
𝑚
𝑚
𝑚𝑇
{
{
{
{
{
{
+𝛼𝑚 {𝑇𝐿 − 𝑎𝑇 𝑥𝑚𝑇−𝜏
{
{
{
{
{
{
∞
{
{
{
{
+
[ℎ (𝑠, 𝑥𝑚ℎ1𝑠 , 𝑥𝑚ℎ2𝑠 , . . . , 𝑥𝑚ℎ𝑘𝑠 )
∑
{
{
{
𝑠=𝑇
{
{
∞
{
{
{
{
−
(𝑓 (𝑡, 𝑥𝑚𝑓1𝑡 , 𝑥𝑚𝑓2𝑡 , . . . ,
∑
{
{
{
𝑡=𝑠
{
{
{
{
{
{
𝑥𝑚𝑓𝑘𝑡 ) − 𝑏𝑡 ) ]}
{
{
{
{
{
𝛽 ≤ 𝑛 < 𝑇, 𝑚 ≥ 0
{ +𝛽𝑚 𝛾𝑚𝑇,
Proof. Put 𝐿 ∈ (𝑎𝑀 + 𝑁, 𝑀). It follows from (33), (34), and
(42) that there exist 𝜃 ∈ (0, 1) and 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 satisfying
Define a mapping 𝑆𝐿 : 𝐴(𝑁, 𝑀) → 𝑙𝛽∞ by
𝑆𝐿 𝑥𝑛
𝑛𝐿 − 𝑎𝑛 𝑥𝑛−𝜏
{
{
{
∞
{
{
{ +∑ {ℎ (𝑠, 𝑥 , 𝑥 , . . . , 𝑥 )
{
{
ℎ1𝑠 ℎ2𝑠
ℎ𝑘𝑠
{
= { 𝑠=𝑛 ∞
{
{
{
−∑ [𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )−𝑏𝑡 ]} ,
{
{
{
𝑡=𝑠
{
{
{𝑆𝐿 𝑥𝑇 ,
𝑛 ≥ 𝑇,
𝛽 ≤ 𝑛 < 𝑇,
(45)
for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀). In view of (11), (12), and
(44) and (45), we obtain that for each 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 , 𝑦 =
{𝑦𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀) and 𝑛 ≥ 𝑇,
󵄨󵄨 𝑆 𝑥
󵄨
󵄨󵄨 𝐿 𝑛 𝑆𝐿 𝑦𝑛 󵄨󵄨󵄨
−
󵄨󵄨
󵄨
󵄨󵄨 𝑛
𝑛 󵄨󵄨󵄨
󵄨󵄨 𝑥 − 𝑦𝑛−𝜏 󵄨󵄨
󵄨󵄨
≤ 𝑎𝑛 󵄨󵄨󵄨󵄨 𝑛−𝜏
󵄨󵄨
𝑛
󵄨
󵄨
+
1 ∞ 󵄨󵄨
∑ [ 󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )
𝑛 𝑠=𝑛 󵄨
󵄨
− ℎ (𝑠, 𝑦ℎ1𝑠 , 𝑦ℎ2𝑠 , . . . , 𝑦ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
∞
󵄨
+ ∑ 󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )
(43)
𝑡=𝑠
converges to an unbounded positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀)
of (6) and has the error estimate (17), where {𝛾𝑚 }𝑚∈N0 =
{𝛾𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 is an arbitrary sequence in 𝐴(𝑁, 𝑀) and
{𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any sequences in [0, 1] satisfying
(18);
(b) equation (6) possesses uncountably many unbounded
positive solutions in 𝐴(𝑁, 𝑀).
󵄨
− 𝑓 (𝑡, 𝑦𝑓1𝑡 , 𝑦𝑓2𝑡 , . . . , 𝑦𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 ]
≤ [𝑎 +
∞
1 ∞
󵄩
󵄩
∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 )] 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩
𝑇 𝑠=𝑇
𝑡=𝑠
󵄩
󵄩
= 𝜃 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,
8
Abstract and Applied Analysis
𝑆𝐿 𝑥𝑛
≤𝐿
𝑛
+
Proof. Put 𝐿 ∈ (𝑁, (1 + 𝑎)𝑀). It follows from (33), (34), and
(47) that there exist 𝜃 ∈ (0, 1) and 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 satisfying
1 ∞ 󵄨󵄨
󵄨
∑ { 󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
𝑛 𝑠=𝑛 󵄨
𝜃 = −𝑎 +
∞
󵄨
󵄨 󵄨 󵄨
+ ∑ [󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨]}
𝑡=𝑠
≤𝐿+
∞
1 ∞
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)]
𝑇 𝑠=𝑇
𝑡=𝑠
∀𝑛 ≥ 𝑇.
(50)
Define a mapping 𝑆𝐿 : 𝐴(𝑁, 𝑀) → 𝑙𝛽∞ by (45). By virtue of
(12), (45), (48), and (50), we easily verify that
≤ 𝑀,
𝑆𝐿 𝑥𝑛
≤ 𝐿 − 𝑎𝑀
𝑛
𝑆𝐿 𝑥𝑛
≥ 𝐿 − 𝑎𝑀
𝑛
1 ∞ 󵄨󵄨
󵄨
∑ { 󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
𝑛 𝑠=𝑛 󵄨
+
1 ∞ 󵄨󵄨
󵄨
∑ { 󵄨󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
𝑛 𝑠=𝑛
∞
󵄨
󵄨 󵄨 󵄨
+ ∑ [󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨]}
∞
󵄨
󵄨 󵄨 󵄨
+ ∑ [󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨]}
𝑡=𝑠
𝑡=𝑠
≥ 𝐿 − 𝑎𝑀 −
(48)
∞
1 ∞
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)] < min {(1 + 𝑎) 𝑀 − 𝐿, 𝐿 − 𝑁} ,
𝑇 𝑠=𝑇
𝑡=𝑠
(49)
−1 < 𝑎 ≤ 𝑎𝑛 ≤ 0,
< 𝐿 + min {𝑀 − 𝐿, 𝐿 − 𝑎𝑀 − 𝑁}
−
∞
1 ∞
∑ (𝑅𝑠 𝐻𝑠 + ∑ 𝑃𝑡 𝐹𝑡 ) ,
𝑇 𝑠=𝑇
𝑡=𝑠
∞
1 ∞
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)]
𝑇 𝑠=𝑇
𝑡=𝑠
≤ 𝐿 − 𝑎𝑀 +
∞
1 ∞
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)]
𝑇 𝑠=𝑇
𝑡=𝑠
> 𝐿 − 𝑎𝑀 − min {𝑀 − 𝐿, 𝐿 − 𝑎𝑀 − 𝑁}
< 𝐿 + min {(1 + 𝑎) 𝑀 − 𝐿, 𝐿 − 𝑁}
≥ 𝑁,
≤ 𝑀,
(46)
which give (24), in turn, which implies that 𝑆𝐿 is a contraction
in 𝐴(𝑁, 𝑀) and possesses a unique fixed point 𝑥 = {𝑥𝑛 }𝑛∈N𝛽 ∈
𝐴(𝑁, 𝑀), which is an unbounded positive solution of (6) in
𝐴(𝑁, 𝑀). The rest of the proof is similar to that of Theorem 3
and is omitted. This completes the proof.
Theorem 6. Assume that there exist constants 𝑎, 𝑀, and 𝑁
with (1 + 𝑎)𝑀 > 𝑁 > 0 and four nonnegative sequences
{𝑃𝑛 }𝑛∈N𝑛 , {𝑄𝑛 }𝑛∈N𝑛 , {𝑅𝑛 }𝑛∈N𝑛 , and {𝑊𝑛 }𝑛∈N𝑛 satisfying (11),
0
0
0
0
(12), (33), (34), and
−1 < 𝑎 ≤ 𝑎𝑛 ≤ 0
𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦.
(47)
Then
(a) for any 𝐿 ∈ (𝑁, (1 + 𝑎)𝑀), there exist 𝜃 ∈ (0, 1) and
𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 such that for any 𝑥0 = {𝑥0𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀)
and the Mann iterative sequence with errors {𝑥𝑚 }𝑚∈N0 =
{𝑥𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 generated by (43) converges to an unbounded
positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀) of (6) and has the error
estimate (17), where {𝛾𝑚 }𝑚∈N0 = {𝛾𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 is an
arbitrary sequence in 𝐴(𝑁, 𝑀), {𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are
any sequences in [0, 1] satisfying (18);
(b) equation (6) possesses uncountably many unbounded
positive solutions in 𝐴(𝑁, 𝑀).
𝑆𝐿 𝑥𝑛
≥𝐿
𝑛
−
1 𝑛−1 󵄨󵄨
󵄨
∑ { 󵄨󵄨ℎ (𝑠, 𝑥ℎ1𝑠 , 𝑥ℎ2𝑠 , . . . , 𝑥ℎ𝑘𝑠 )󵄨󵄨󵄨󵄨
𝑛 𝑠=𝑇 󵄨
∞
󵄨
󵄨 󵄨 󵄨
+ ∑ [󵄨󵄨󵄨󵄨𝑓 (𝑡, 𝑥𝑓1𝑡 , 𝑥𝑓2𝑡 , . . . , 𝑥𝑓𝑘𝑡 )󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨]}
𝑡=𝑠
≥ 𝐿−
∞
∞
1
󵄨 󵄨
∑ [𝑊𝑠 + ∑ (𝑄𝑡 + 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨)]
𝑇 𝑠=𝑇
𝑡=𝑠
> 𝐿 − min {(1 + 𝑎) 𝑀 − 𝐿, 𝐿 − 𝑁}
≥ 𝑁,
(51)
which yield that 𝑆𝐿 (𝐴(𝑁, 𝑀)) ⊆ 𝐴(𝑁, 𝑀). The rest of the
proof is similar to that of Theorem 5 and is omitted. This
completes the proof.
Remark 7. Theorems 3∼6 extend and improve Theorem 1 in
[5].
3. Examples
In this section we suggest four examples to explain the results
presented in Section 2. Note that Theorem 1 in [5] is useless
for all these examples.
Abstract and Applied Analysis
9
Example 8. Consider the second-order nonlinear neutral
delay difference equation:
Δ2 (𝑥𝑛 − 𝑥𝑛−𝜏 ) + Δ (
sin2 𝑥𝑛−2
1
)+
󵄨 󵄨3
6
3
𝑛4
(𝑛 + 𝑛 + 2) (1 + 󵄨󵄨󵄨𝑥𝑛2 󵄨󵄨󵄨 )
𝑛2 − 3𝑛 + 1
,
𝑛7 + 𝑛4 + 1
=
𝑛 ≥ 3,
(52)
where 𝜏 ∈ N is fixed. Let 𝑛0 = 3, 𝑘 = 1, 𝛽 = min{3 − 𝜏, 1}, 𝑀
and 𝑁 two positive constants with 𝑀 > 𝑁 and
𝑎𝑛 = −1,
1
𝑓 (𝑛, 𝑢) =
ℎ (𝑛, 𝑢) =
sin2 𝑢
,
𝑛4
𝑃𝑛 =
(𝑛6 + 𝑛3 + 2) (1 + |𝑢|3 )
𝐹𝑛 = 𝑓1𝑛 = 𝑛2 ,
3𝑀2
(1 +
𝑛2 − 3𝑛 + 1
,
𝑛7 + 𝑛4 + 1
𝑏𝑛 =
2
𝑁3 ) 𝑛6
𝑊𝑛 =
1
,
𝑛4
Example 9. Consider the second-order nonlinear neutral
delay difference equation:
,
𝐻𝑛 = ℎ1𝑛 = 𝑛 − 2,
1
𝑄𝑛 = 6 ,
𝑛
,
which together with Lemma 2 yield that (13) and (14) hold.
It follows from Theorem 3 that (52) possesses uncountably many unbounded positive solutions in 𝐴(𝑁, 𝑀). On
the other hand, for any 𝐿 ∈ (𝑁, 𝑀), there exist 𝜃 ∈
(0, 1) and 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 such that for each 𝑥0 =
{𝑥0𝑛 }𝑛∈N𝛽 ∈ 𝐴(𝑁, 𝑀), the Mann iterative sequence with
errors {𝑥𝑚 }𝑚∈N0 = {𝑥𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 generated by (16) converges to an unbounded positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀)
of (52) and has the error estimate (17), where {𝛾𝑚 }𝑚∈N0 =
{𝛾𝑚𝑛 }(𝑚,𝑛)∈N0 ×N𝛽 is an arbitrary sequence in 𝐴(𝑁, 𝑀) and
{𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any sequences in [0, 1] satisfying
(18).
2
𝑅𝑛 = 4 ,
𝑛
Δ2 (𝑥𝑛 + 𝑥𝑛−𝜏 ) + Δ (
∀ (𝑛, 𝑢) ∈ N𝑛0 × R.
+
(53)
It is easy to see that (11), (12), and (15) are satisfied. Note that
∞
∑ 𝑠 max {𝑅𝑠 𝐻𝑠 , 𝑊𝑠 }
=
sin2 𝑥3𝑛3 +1
4
𝑛3 (𝑛2 + 2) (1 + 𝑥2𝑛
2 −3 )
)
(−1)𝑛 𝑛3 (𝑥𝑛2 −𝑛−1 + 𝑥(𝑛+1)(𝑛+2) )
2
(𝑛11 + 𝑛5 + 1) (1 + 𝑥𝑛22 −𝑛−1 + 𝑥(𝑛+1)(𝑛+2)
)
𝑛2 − ln 𝑛
,
𝑛6 + 𝑛5 + 1
(56)
𝑛 ≥ 5,
𝑠=𝑛
∞
= ∑ 𝑠 max {
𝑠=𝑛
2 (𝑠 − 2) 1
, 4}
𝑠4
𝑠
∞
2 (𝑠 − 2)
=∑
< +∞,
𝑠3
𝑠=𝑛
(54)
where 𝜏 ∈ N is fixed. Let 𝑛0 = 5, 𝑘 = 2, 𝛽 = 5 − 𝜏, 𝑀 and 𝑁
two positive constants with 𝑀 > 𝑁 and
∀𝑛 ∈ N𝑛0 ,
∞ ∞
󵄨 󵄨
∑ ∑ 𝑠 max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨}
𝑠=𝑛 𝑡=𝑠
󵄨󵄨 2
󵄨󵄨
1 󵄨󵄨󵄨𝑡 − 3𝑡 + 1󵄨󵄨󵄨
= ∑ ∑ 𝑠 max {
, 6, 7 4
}
2
𝑠=𝑛 𝑡=𝑠
(1 + 𝑁3 ) 𝑡4 𝑡 𝑡 + 𝑡 + 1
∞ ∞
3𝑀2
3𝑀2
∞ ∞
3𝑀2
∞ ∞
3𝑀2
∞
𝑎𝑛 = 1,
𝑓 (𝑛, 𝑢, 𝑣) =
𝑠
}∑∑ 4
≤ max {1,
2
(1 + 𝑁3 ) 𝑠=𝑛 𝑡=𝑠 𝑡
1
≤ max {1,
}∑∑ 3
2
3
𝑡
(1 + 𝑁 ) 𝑠=𝑛 𝑡=𝑠
3𝑀2
(1 +
2
𝑁3 )
∞
1
< +∞,
2
𝑡
𝑡=𝑛
}∑
(𝑛11
ℎ (𝑛, 𝑢, 𝑣) =
(−1)𝑛 𝑛3 (𝑢 + 𝑣)
,
+ 𝑛5 + 1) (1 + 𝑢2 + 𝑣2 )
𝐹𝑛 = 𝑓2𝑛 = (𝑛 + 1) (𝑛 + 2) ,
ℎ1𝑛 = 2𝑛2 − 3,
∀𝑛 ∈ N𝑛0 ,
(55)
𝑃𝑛 = 𝑄𝑛 =
4
,
𝑛8
𝑛2 − ln 𝑛
,
𝑛6 + 𝑛5 + 1
sin2 𝑣
,
𝑛3 (𝑛2 + 2) (1 + 𝑢4 )
𝑓1𝑛 = 𝑛2 − 𝑛 − 1,
𝑡−𝑛+1
= max {1,
}∑
2
𝑡3
(1 + 𝑁3 ) 𝑡=𝑛
≤ max {1,
𝑏𝑛 =
𝑅𝑛 = 𝑊𝑛 =
𝐻𝑛 = ℎ2𝑛 = 3𝑛3 + 1,
10
,
𝑛5
∀ (𝑛, 𝑢, 𝑣) ∈ N𝑛0 ×R2 .
(57)
10
Abstract and Applied Analysis
where 𝜏 ∈ N is fixed. Let 𝑛0 = 7, 𝑘 = 2, 𝑎 = 3/4, 𝛽 = min{7 −
𝜏, 5}, 𝑀 and 𝑁 two positive constants with 𝑀 > 4𝑁 and
It is clear that (11), (12), and (35) are fulfilled. Note that
1 ∞
∑ max {𝑅𝑠 𝐻𝑠 , 𝑊𝑠 }
𝑛→∞𝑛
𝑠=𝑛
lim
𝑎𝑛 =
10 (3𝑠3 + 1) 10
1 ∞
= lim ∑ max {
, 5 } = 0,
𝑛→∞𝑛
𝑠5
𝑠
𝑠=𝑛
𝑓 (𝑛, 𝑢, 𝑣) =
1 ∞ ∞
󵄨 󵄨
∑ ∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨}
𝑛 𝑠=𝑛 𝑡=𝑠
=
≤
ℎ (𝑛, 𝑢, 𝑣) =
1 ∞ ∞
4 (𝑡 + 1) (𝑡 + 2) 4 𝑡2 − ln 𝑡
, 8, 6 5
}
∑ ∑ max {
𝑛 𝑠=𝑛 𝑡=𝑠
𝑡8
𝑡 𝑡 +𝑡 +1
4 ∞ 1
4 ∞ ∞ 1
∑ ∑ 4 ≤ ∑ 3 󳨀→ 0
𝑛 𝑠=𝑛 𝑡=𝑠 𝑡
𝑛 𝑡=𝑛 𝑡
1∞∞
󵄨 󵄨
∑∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨} = 0.
𝑛→∞𝑛
𝑠=𝑛 𝑡=𝑠
(59)
sin (ln (1 + 𝑛2 |𝑢|))
𝑛9 − √𝑛 − 4
−
𝑛2 − (−1)𝑛(𝑛−1)/2
,
(𝑛7 + 3𝑛5 − 1) 2|𝑣|
ℎ1𝑛 = 3𝑛2 − 1,
𝐹𝑛 = 𝑓2𝑛 = 𝑛 + 4,
𝑃𝑛 = 𝑄𝑛 =
2
,
𝑛4
𝑅𝑛 = 𝑊𝑛 =
2
,
𝑛5
∀ (𝑛, 𝑢, 𝑣) ∈ N𝑛0 × R2 .
(61)
Example 10. Consider the second-order nonlinear neutral
delay difference equation:
3𝑛3 − 1
𝑥 )
4𝑛3 + 2 𝑛−𝜏
+ Δ(
𝑛 − (−1)
)
+ 3𝑛5 − 1) 2|𝑥4𝑛3 +1 |
1
(−1)𝑛
−
2 )
󵄨
󵄨
(1 + 𝑥𝑛−2
5
(𝑛 + 1) √1 + 󵄨󵄨󵄨𝑥𝑛+4 󵄨󵄨󵄨
𝑛 ≥ 7,
1 ∞ ∞
󵄨 󵄨
∑ ∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨}
𝑛→∞𝑛
𝑠=𝑛 𝑡=𝑠
lim
󵄨󵄨
󵄨󵄨
𝑡 3
1 ∞ ∞
2𝑡 + 8 2 󵄨󵄨󵄨(−1) 𝑡 − 1󵄨󵄨󵄨
= lim ∑ ∑ max { 4 , 4 , 8 3
}
𝑛→∞𝑛
𝑡
𝑡
𝑡 ln 𝑡 + 1
𝑠=𝑛 𝑡=𝑠
(62)
𝑛(𝑛−1)/2
(𝑛7
(−1)𝑛 𝑛3 − 1
,
𝑛8 ln3 𝑛 + 1
3
1 ∞ 2 (4𝑠 + 1)
= 0,
∑
𝑛→∞𝑛
𝑠5
𝑠=𝑛
= lim
1 ∞ ∞ 2𝑡 + 8
∑ ∑ 4 = 0.
𝑛→∞𝑛
𝑡
𝑠=𝑛 𝑡=𝑠
𝑛9 − √𝑛 − 4
−
3
2 (4𝑠 + 1) 2
1 ∞
, 5}
∑ max {
𝑛→∞𝑛
𝑠5
𝑠
𝑠=𝑛
= lim
= lim
󵄨
󵄨
sin (ln (1 + 𝑛2 󵄨󵄨󵄨𝑥3𝑛2 −1 󵄨󵄨󵄨))
2
1 ∞
∑ max {𝑅𝑠 𝐻𝑠 , 𝑊𝑠 }
𝑛→∞𝑛
𝑠=𝑛
lim
Thus, Theorem 4 guarantees that (56) possesses uncountably
unbounded positive solutions in 𝐴(𝑁, 𝑀). On the other
hand, for any 𝐿 ∈ (𝑁, 𝑀), there exist 𝜃 ∈ (0, 1) and
𝑇 ≥ 𝜏 + 𝑛0 + 𝛽 such that the Mann iterative sequence with
error {𝑥𝑚 }𝑚∈N0 generated by (36) converges to an unbounded
positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀) of (56) and has the error
estimate (17), where {𝛾𝑚 }𝑚∈N0 is an arbitrary sequence in
𝐴(𝑁, 𝑀) and {𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any sequences in
[0, 1] satisfying (18).
=
1
(−1)𝑛
−
,
𝑛6 (1 + 𝑢2 ) (𝑛5 + 1) √1 + |𝑣|
It is not difficult to verify that (11), (12), and (42) are fulfilled.
Note that
lim
𝑛6
(−1)𝑛 𝑛3 − 1
,
𝑛8 ln3 𝑛 + 1
𝐻𝑛 = ℎ2𝑛 = 4𝑛3 + 1,
which yields that
+
𝑏𝑛 =
𝑓1𝑛 = 𝑛 − 2,
as 𝑛 󳨀→ ∞,
(58)
Δ2 (𝑥𝑛 +
3𝑛3 − 1
,
4𝑛3 + 2
(60)
That is, (33) and (34) are satisfied. Consequently Theorem 5
implies that (60) possesses uncountably many unbounded
positive solutions in 𝐴(𝑁, 𝑀). On the other hand, for any
𝐿 ∈ ((3/4)𝑀 + 𝑁, 𝑀), there exist 𝜃 ∈ (0, 1) and 𝑇 ≥
𝑛0 + 𝜏 + 𝛽 such that the Mann iterative sequence with
error {𝑥𝑚 }𝑚∈N0 generated by (43) converges to an unbounded
positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀) of (60) and has the error
estimate (17), where {𝛾𝑚 }𝑚∈N0 is an arbitrary sequence in
𝐴(𝑁, 𝑀) and {𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any sequences in
[0, 1] satisfying (18).
Abstract and Applied Analysis
11
Example 11. Consider the second-order nonlinear neutral
delay difference equation:
1 − 2𝑛2
𝑥 )
2 + 3𝑛2 𝑛−𝜏
Δ2 (𝑥𝑛 +
+ Δ(
sin2 (𝑛3 𝑥5𝑛2 −2 )
+
=
𝑛2 − 1
)
2
(𝑛5 + 2𝑛3 − 1) (1 + 𝑥2𝑛−15
)
(63)
Acknowledgments
(𝑛 + 2)8
(−1)𝑛 𝑛4 + 𝑛3 + 3𝑛2 − 1
,
𝑛9 + 𝑛7 + 3𝑛6 + 𝑛4 + 1
𝑛 ≥ 11,
where 𝜏 ∈ N is fixed. Let 𝑛0 = 11, 𝑘 = 1, 𝑎 = −4/5, 𝛽 =
min{11 − 𝜏, 7}, 𝑀 and 𝑁 two positive constants with 𝑀 > 5𝑁
and
𝑎𝑛 =
1−2𝑛2
,
2+3𝑛2
𝑏𝑛 =
𝑓 (𝑛, 𝑢) =
ℎ (𝑛, 𝑢) =
(−1)𝑛 𝑛4 +𝑛3 +3𝑛2 −1
,
𝑛9 +𝑛7 +3𝑛6 +𝑛4 +1
sin2 (𝑛3 𝑢)
(𝑛 + 2)8
(64)
𝐹𝑛 = 𝑓1𝑛 = 5𝑛2 − 2,
𝑅𝑛 =
2
,
𝑛3
𝑊𝑛 =
𝑃𝑛 =
1
,
𝑛3
2
,
𝑛5
𝑄𝑛 =
1
,
𝑛8
∀ (𝑛, 𝑢) ∈ N𝑛0 × R.
Obviously, (11), (12), and (50) are satisfied. Note that
1 ∞
∑ max {𝑅𝑠 𝐻𝑠 , 𝑊𝑠 }
𝑛→∞𝑛
𝑠=𝑛
lim
1 ∞
4𝑠 − 30 1
, 3 } = 0,
∑ max {
𝑛→∞𝑛
𝑠3
𝑠
𝑠=𝑛
= lim
1 ∞ ∞
󵄨 󵄨
∑ ∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨}
𝑛 𝑠=𝑛 𝑡=𝑠
󵄨
󵄨
2 (5𝑡2 − 2) 1 󵄨󵄨󵄨󵄨(−1)𝑡 𝑡4 + 𝑡3 + 3𝑡2 − 1󵄨󵄨󵄨󵄨
1 ∞ ∞
= ∑ ∑ max {
, 8, 9 7
}
𝑛 𝑠=𝑛 𝑡=𝑠
𝑡5
𝑡 𝑡 + 𝑡 + 3𝑡6 + 𝑡4 + 1
≤
10 ∞ ∞ 1
∑ ∑ 󳨀→ 0
𝑛 𝑠=𝑛 𝑡=𝑠 𝑡3
as 𝑛 󳨀→ ∞,
(65)
which gives that
1 ∞ ∞
󵄨 󵄨
∑ ∑ max {𝑃𝑡 𝐹𝑡 , 𝑄𝑡 , 󵄨󵄨󵄨𝑏𝑡 󵄨󵄨󵄨} = 0.
𝑛→∞𝑛
𝑠=𝑛 𝑡=𝑠
lim
This research was supported by the Science Research Foundation of Educational Department of Liaoning Province
(L2012380) and Basic Science Research Program through
the National Research Foundation of Korea (NRF) funded
by the Ministry of Education, Science and Technology
(2012R1A1A2002165).
References
,
𝑛2 − 1
,
(𝑛5 + 2𝑛3 − 1) (1 + 𝑢2 )
𝐻𝑛 = ℎ1𝑛 = 2𝑛−15,
That is, (33) and (34) hold. Thus, Theorem 6 shows that (63)
possesses uncountably many unbounded positive solutions in
𝐴(𝑁, 𝑀). On the other hand, for any 𝐿 ∈ (𝑁, 𝑀/5), there
exist 𝜃 ∈ (0, 1) and 𝑇 ≥ 𝑛0 + 𝜏 + 𝛽 such that the Mann iterative
sequence with error {𝑥𝑚 }𝑚∈N0 generated by (43) converges to
an unbounded positive solution 𝑥 ∈ 𝐴(𝑁, 𝑀) of (63) and
has the error estimate (17), where {𝛾𝑚 }𝑚∈N0 is an arbitrary
sequence in 𝐴(𝑁, 𝑀) and {𝛼𝑚 }𝑚∈N0 and {𝛽𝑚 }𝑚∈N0 are any
sequences in [0, 1] satisfying (18).
(66)
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