Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations

Applied Mathematics and Computation 213 (2009) 554–576

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Existence of uncountably many bounded nonoscillatory solutionsand their iterative approximations for second order nonlinearneutral delay difference equations

Zeqing Liu a, Shin Min Kang b,*, Jeong Sheok Ume c

a Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian, Liaoning 116029, People’s Republic of Chinab Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Koreac Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Keywords:Second order nonlinear neutral delaydifference equationUncountably many bounded nonoscillatorysolutionsContraction mappingMann iterative sequence with errors

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.03.050

* Corresponding author.E-mail addresses: [email protected] (Z. Liu), smkan

In this paper, the Banach fixed-point theorem is employed to establish several existenceresults of uncountably many bounded nonoscillatory solutions for the second order nonlin-ear neutral delay difference equation

. All righ

g@nong

D aðnÞD xðnÞ þ bðnÞxðn� sÞð Þ½ � þ Dhðn; xðh1ðnÞÞ; xðh2ðnÞÞ; . . . ; xðhkðnÞÞÞþ f ðn; xðf1ðnÞÞ; xðf2ðnÞÞ; . . . ; xðfkðnÞÞÞ ¼ cðnÞ; n P n0;

where s; k 2 N, n0 2 N0, a; b; c : Nn0 ! R with aðnÞ > 0 for n 2 Nn0 , h; f : Nn0 � Rk ! R andhl; fl : Nn0 ! Z with

limn!1

;hlðnÞ ¼ limn!1

flðnÞ ¼ þ1; l 2 f1;2; . . . ; kg:

A few Mann type iterative approximation schemes with errors are suggested, and the er-rors estimates between the iterative approximations and the nonoscillatory solutions arediscussed. Seven nontrivial examples are given to illustrate the advantages of our results.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

Within the past 20 years or so, many authors used the fixed point theorems, critical point theory, coincidence degree the-ory, Riccati transformation technique, upper and lower solution method and other nonlinear analysis methods to study theexistence, boundedness, monotonicity, oscillatory, nonoscillatory, asymptotic behavior of solutions or periodic solutions forsome classes of second order neutral delay linear/nonlinear difference equations [1–10,12–23,25].

Li and Saker [8], Li and Zhu [9], Zhang and Li [23] and Zhang and Zhang [25] presented several oscillation criteria for thesecond order advanced functional difference equation

D aðnÞDxðnÞð Þ þ pðnÞxðgðnÞÞ ¼ 0; n P 0; ð1:1Þ

the second order nonlinear difference equation

D aðnÞDxðnÞð Þ þ qðnÞðDxðnÞÞb � pðnÞxaðnÞ ¼ eðnÞ; n P 1; ð1:2Þ

ts reserved.

ae.gsnu.ac.kr (S.M. Kang), [email protected] (J.S. Ume).

mailto:[email protected]



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Z. Liu et al. / Applied Mathematics and Computation 213 (2009) 554–576 555

where b and a are ratios of odd positive integers and the second order nonlinear difference equation

D rðn� 1ÞDðxðn� 1Þ þ pðnÞxðn� sÞÞð Þ þ qðnÞxcðn� rÞ ¼ 0; n P 0; ð1:3Þ

where 0 < c < 1 is a quotient of odd positive integers, respectively. Applying the matrix theory and coincidence degree the-ory, Ma et al. [13] established the existence conditions of periodic solutions for the below second order nonlinear differenceequation

D rðnÞDxðn� 1Þð Þ þ f ðn; xðnÞÞ ¼ 0; jnjP 0: ð1:4Þ

Utilizing the Schauder fixed-point theorem, Thandapani et al. [19] investigated the asymptotic behavior of nonoscillatorysolutions for the second order difference equation

D aðnÞDxðnÞð Þ ¼ qðnÞf ðxðnþ 1ÞÞ; n P 0: ð1:5Þ

Recently, Luo and Bainov [12] and Migda and Migda [15] discussed the asymptotic behaviors of nonoscillatory solutions forthe second order neutral difference equation with maxima

D2 xðnÞ þ pðnÞxðn� kÞð Þ þ qðnÞmax xðsÞ : n� l 6 s 6 nf g ¼ 0; n P 1 ð1:6Þ

and the second order neutral difference equation

D2 xðnÞ þ pxðn� kÞð Þ þ f ðn; xðnÞÞ ¼ 0; n P 1; ð1:7Þ

respectively. Very recently, Meng and Yan [14] investigated the existence of bounded nonoscillatory solutions for the secondorder nonlinear and nonautonomous neutral delay difference equation

D2 xðnÞ � pxðn� sÞð Þ ¼Xm

n¼1

qixðn� riÞ þ f ðn; xðn� g1ðnÞÞ; . . . ; xðn� glðnÞÞÞ; n P n0: ð1:8Þ

Jinfa [5] considered the second order neutral delay difference equation with positive and negative coefficients

D2 xðnÞ þ pxðn�mÞð Þ þ pðnÞxðn� kÞ � qðnÞxðn� lÞ ¼ 0; n P n0 ð1:9Þ

and investigated the existence of a nonoscillatory solution of Eq. (1.9) under the condition p–� 1 by using the Banach fixed-point theorem. Employing the cone compression and expansion theorem in Fréchet spaces, Tian and Ge [20] established theexistence of multiple positive solutions of second order discrete equation on the half-line

D2xðn� 1Þ � pDxðn� 1Þ � qxðn� 1Þ þ f ðn; xðnÞÞ ¼ 0; n P 1 ð1:10Þ

with certain boundary value conditions. However, to the best of our knowledge, no work dealing with the existence ofuncountably many nonoscillatory solutions for Eqs. (1.1)–(1.10) and any other second order difference equations has beendone.

In this paper we study the more general second order nonlinear neutral delay difference equation

D aðnÞD xðnÞ þ bðnÞxðn� sÞð Þ½ � þ Dh n; xðh1ðnÞÞ; xðh2ðnÞÞ; . . . ; xðhkðnÞÞð Þþ f n; xðf1ðnÞÞ; xðf2ðnÞÞ; . . . ; xðfkðnÞÞð Þ ¼ cðnÞ; n P n0; ð1:11Þ

where s; k 2 N, n0 2 N0, a; b; c : Nn0 ! R with aðnÞ > 0 for n 2 Nn0 , h; f : Nn0 � Rk ! R and hl; fl : Nn0 ! Z with

limn!1

hlðnÞ ¼ limn!1

flðnÞ ¼ þ1; l 2 f1;2; . . . ; kg:

Applying the Banach fixed-point theorem, we establish sufficient conditions of the existence of uncountably many boundednonoscillatory solutions for Eq. (1.11), suggest a few Mann type iterative approximation methods with errors for thesebounded nonoscillatory solutions and study error estimates between the approximation sequences and the bounded nonos-cillatory solutions.

Throughout this paper, we assume that D is the forward difference operator defined by DxðnÞ ¼ xðnþ 1Þ � xðnÞ,R ¼ ð�1;þ1Þ;Rþ ¼ ½0;þ1Þ, Z and N denote the sets of all integers and positive integers, respectively,

Zt ¼ fn : n 2 Z with n P tg; Nt ¼ fn : n 2 N with n P tg; t 2 Z;

a ¼ inffhlðnÞ; flðnÞ : 1 6 l 6 k;n 2 Nn0g; b ¼minfn0 � s;ag;

l1b represents the Banach space of all bounded sequences on Zb with norm

kxk ¼ supn2Zb

jxðnÞj for x ¼ fxðnÞgn2Zb2 l1b

and

XðN;MÞ ¼ x ¼ fxðnÞgn2Zb2 l1b : N 6 xðnÞ 6 M; n 2 Zb

n ofor M > N > 0:

556 Z. Liu et al. / Applied Mathematics and Computation 213 (2009) 554–576

It is easy to see that XðN;MÞ is a bounded closed and convex subset of the Banach space l1b . By a solution of Eq. (1.11), wemean a sequence fxðnÞgn2Zb

with a positive integer T P n0 þ sþ jaj such that Eq. (1.11) is satisfied for all n P T. As is cus-tomary, a solution of Eq. (1.11) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise,it is said to be nonoscillatory.

The following lemmas play important roles in this paper.

Lemma 1.1 [11]. Let fangnP0; fbngnP0; fcngnP0 and ftngnP0 be four nonnegative real sequences satisfying the inequality

anþ1 6 ð1� tnÞan þ tnbn þ cn; 8n P 0;

where ftngnP0 � ½0;1�,P1

n¼0tn ¼ þ1, limn!1bn ¼ 0 andP1

n¼0cn < þ1. Then limn!1an ¼ 0.

Lemma 1.2. Let s 2 N, n0 2 N0 and B : Nn0 ! Rþ be a mapping. Then

(a) [24]P1

i¼0

P1s¼n0þisBðsÞ < þ1()

P1n¼n0

nBðnÞ < þ1;

(b)P1

i¼0

P1s¼n0þis

P1t¼sBðtÞ < þ1()

P1s¼n0

P1t¼ssBðtÞ < þ1.

Proof. We only prove (b). For each t 2 R, [t] stands for the largest integer not exceeding t. Note that

X1i¼0

X1s¼n0þis

X1t¼s

BðtÞ ¼X1s¼n0

X1t¼s

BðtÞ þX1

s¼n0þs

X1t¼s

BðtÞ þX1

s¼n0þ2s

X1t¼s

BðtÞ þ � � �

¼X1s¼n0

X1t¼s

1þ s� n0

s

h in oBðtÞ

and

lims!þ1

1þ s�n0s

� �ss

¼ 1:

It follows that (b) holds. This completes the proof. h

The paper is organized as follows. In Section 2, by using the Banach fixed-point theorem, we establish the existence, iter-ative approximations and errors estimates of uncountably many bounded nonoscillatory solutions for Eq. (1.11). The resultspresented in this paper extend properly Theorem 1 in [5]. In Section 3, seven examples are constructed to illustrate ourresults.

2. Existence of uncountably bounded nonoscillatory solutions

Now we study those conditions under which Eq. (1.11) possesses uncountably bounded nonoscillatory solutions. More-over, we suggest a few Mann iterative approximation schemes with errors for these bounded nonoscillatory solutions. Undersuitable conditions, some error estimates between the bounded nonoscillatory solutions and the approximate solutions arediscussed.

Theorem 2.1. Let M and N be two positive constants with M > N and

bðnÞ ¼ �1; eventually: ð2:1Þ

Suppose that there exist four mappings PN;M;Q N;M, RN;M ;WN;M : Nn0 ! Rþ satisfying

jf ðn;u1;u2; . . . ; ukÞ � f ðn; �u1; �u2; . . . ; �ukÞj 6 PN;MðnÞmaxfjul � �ulj : 1 6 l 6 kg;

jhðn;u1;u2; . . . ; ukÞ � hðn; �u1; �u2; . . . ; �ukÞj 6 RN;MðnÞmaxfjul � �ulj : 1 6 l 6 kg

for n 2 Nn0 ; ul; �ul 2 ½N;M� and 1 6 l 6 k; ð2:2Þ

jf ðn;u1;u2; . . . ; ukÞj 6 QN;MðnÞ and jhðn;u1;u2; . . . ;ukÞj 6WN;MðnÞ for n 2 Nn0 ; ul 2 ½N;M� and 1 6 l 6 k; ð2:3Þ

X1n¼n0

naðnÞ max RN;MðnÞ;WN;MðnÞ

� �< þ1; ð2:4Þ

X1s¼n0

X1t¼s

saðsÞ max PN;MðtÞ;Q N;MðtÞ; jcðtÞj

� �< þ1: ð2:5Þ


Then

(a) For any L 2 ðN;MÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for each x0 2 XðN;MÞ, the Mann iterative sequencewith errors fxmgmP0 generated by the scheme:

xmþ1ðnÞ ¼

ð1� am � bmÞxmðnÞ þ am L�P1i¼1

P1s¼nþis

1aðsÞ

(

��

hðs; xmðh1ðsÞÞ; xmðh2ðsÞÞ; . . . ; xmðhkðsÞÞÞ

�P1t¼s

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; . . . ; xmðfkðtÞÞÞ þP1t¼s

cðtÞ�)

þ bmcmðnÞ; n P T; m P 0;

ð1� am � bmÞxmðTÞ þ am L�P1i¼1

P1s¼Tþis

1aðsÞ

(

��


�P1t¼s


cðtÞ�)

þbmcmðTÞ; b 6 n < T; m P 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð2:6Þ

converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (1.11) and has the following error estimate:

kxmþ1 � xk 6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm; m P 0; ð2:7Þ

where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in ½0;1� such that

X1m¼0

am ¼ þ1 ð2:8Þ

and

X1m¼0

bm < þ1 or there exists a sequence fnmgmP0 # ½0;þ1Þ satisfying bm ¼ nmam; m P 0 and limm!1

nm ¼ 0; ð2:9Þ

(b) Eq. (1.11) possesses uncountably bounded nonoscillatory solutions.

Proof. First of all we show that (a) holds. Let L 2 ðN;MÞ. It follows from (2.1), (2.4), (2.5) and Lemma 1.2 that there existh 2 ð0;1Þ and T P n0 þ sþ jaj satisfying

bðnÞ � �1; n P T; ð2:10Þ

h ¼X1i¼1

X1s¼Tþis

1aðsÞ RN;MðsÞ þ

X1t¼s

PN;MðtÞ" #

; ð2:11Þ

X1i¼1

X1s¼Tþis

1aðsÞ WN;MðsÞ þ

X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 min M � L; L� Nf g: ð2:12Þ

Define a mapping SL : XðN;MÞ ! l1b by

SLxðnÞ ¼

L�P1i¼1

P1s¼nþis

1aðsÞ

�hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ

�P1t¼s

f ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ

þP1t¼s

cðtÞ�; n P T; x 2 XðN;MÞ;

SLxðTÞ; b 6 n < T; x 2 XðN;MÞ:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð2:13Þ


Employing (2.2) and (2.11), we infer that for x; y 2 XðN;MÞ and n P T

jSLxðnÞ � SLyðnÞj 6X1i¼1

X1s¼nþis

1aðsÞ

�jhðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ � hðs; yðh1ðsÞÞ; yðh2ðsÞÞ; . . . ; yðhkðsÞÞÞj

þX1t¼s

jf ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ � f ðt; yðf1ðtÞÞ; yðf2ðtÞÞ; . . . ; yðfkðtÞÞÞj#

6

X1i¼1

X1s¼nþis

1aðsÞ

�RN;MðsÞmaxfjxðhlðsÞÞ � yðhlðsÞÞj : 1 6 l 6 kg

þX1t¼s

PN;MðtÞmaxfjxðflðtÞÞ � yðflðtÞÞj : 1 6 l 6 kg#

6

X1i¼1

X1s¼nþis


X1t¼s

PN;MðtÞ" #

kx� yk

6

X1i¼1

X1s¼Tþis


X1t¼s

PN;MðtÞ" #

kx� yk ¼ hkx� yk:

This leads to

kSLx� SLyk 6 hkx� yk; x; y 2 XðN;MÞ: ð2:14Þ

In view of (2.3), (2.12) and (2.13), we conclude that for any x 2 XðN;MÞ and n P T

SLxðnÞ 6 LþX1i¼1

X1s¼nþis

1aðsÞ jhðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞj þ

X1t¼s

jf ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞj þX1t¼s

jcðtÞj" #

6 LþX1i¼1

X1s¼nþis


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 LþX1i¼1

X1s¼Tþis


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 Lþmin M � L; L� Nf g 6 M

and

SLxðnÞP L�X1i¼1

X1s¼nþis


X1t¼s


jcðtÞj" #

P L�X1i¼1

X1s¼nþis


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

P L�X1i¼1

X1s¼Tþis


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

P L�min M � L; L� Nf gP N;

which give that SLðXðN;MÞÞ# XðN;MÞ. Consequently, (2.14) ensures that SL is a contraction mapping and it has a unique fixedpoint x 2 XðN;MÞ, that is,

xðnÞ ¼ L�X1i¼1

X1s¼nþis

1aðsÞ hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ �

X1t¼s

f ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ þX1t¼s

cðtÞ" #

; n P T

ð2:15Þ

and

xðn� sÞ ¼ L�X1i¼1

X1s¼nþði�1Þs


X1t¼s

f ðt; xðf1ðtÞÞ;"

xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ þX1t¼s

cðtÞ#; n P T þ s: ð2:16Þ

Thus (2.10), (2.15) and (2.16) lead to

xðnÞ � xðn� sÞ ¼X1s¼n


X1t¼s


cðtÞ" #

;

n P T þ s;


which yields that

DðxðnÞ � xðn� sÞÞ ¼ � 1aðnÞ hðn; xðh1ðnÞÞ; xðh2ðnÞÞ; . . . ; xðhkðnÞÞÞ �

X1t¼n

f ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ þX1t¼n

cðtÞ" #

;

n P T þ s;
that is,
D aðnÞDðxðnÞ � xðn� sÞÞð Þ ¼ �Dhðn; xðh1ðnÞÞ; xðh2ðnÞÞ; . . . ; xðhkðnÞÞÞ � f ðn; xðf1ðnÞÞ; xðf2ðnÞÞ; . . . ; xðfkðnÞÞÞ þ cðnÞ;n P T þ s;

which means that x a bounded nonoscillatory solution of Eq. (1.11). It follow from (2.6), (2.10), (2.11), (2.13) and (2.14) thatfor any m P 0 and n P T ,

jxmþ1ðnÞ � xðnÞj ¼ ð1� am � bmÞxmðnÞ þ am L�X1i¼1

X1s¼nþis

1aðsÞ hðs; xmðh1ðsÞÞ; xmðh2ðsÞÞ; . . . ; xmðhkðsÞÞÞ½

(��X1t¼s

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; . . . ; xmðfkðtÞÞÞ þX1t¼s

cðtÞ#þ bmcmðnÞ � xðnÞ

��)

6 ð1� am � bmÞjxmðnÞ � xðnÞj þ amjSLxmðnÞ � SLxðnÞj þ bmjcmðnÞ � xðnÞj6 ð1� am � bmÞkxm � xk þ amhkxm � xk þ 2Mbm 6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm;

which gives that

kxmþ1 � xk 6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm; m P 0:

That is, (2.7) holds. Thus Lemma 1.1, (2.1), (2.8) and (2.9) guarantee that limm!1xm ¼ x.Now we show that (b) holds. By (2.1), (2.4) and (2.5), we know that for any distinct L;K 2 ðN;MÞ, there exist h and

T P n0 þ sþ jaj satisfying (2.10), (2.11), (2.12) and

X1i¼1

X1s¼Tþis


X1t¼s

PN;MðtÞ" #

<jL� Kj

2M: ð2:17Þ

Let the mapping SL be defined by (2.13), and define a mapping SK : XðN;MÞ ! l1b by (2.13) with L replaced by K. As in theproof of (a), we conclude easily that the mappings SL and SK have the unique fixed points x; y 2 XðN;MÞ, respectively. Thatis, x and y are bounded nonoscillatory solutions of Eq. (1.11) in XðN;MÞ. In light of (2.13), we deduce that for n P T

xðnÞ ¼ L�X1i¼1

X1s¼nþis


X1t¼s


cðtÞ" #

; n P T

ð2:18Þ
and " #
yðnÞ ¼ K �X1i¼1

X1s¼nþis

1aðsÞ hðs; yðh1ðsÞÞ; yðh2ðsÞÞ; . . . ; yðhkðsÞÞÞ �

X1t¼s

f ðt; yðf1ðtÞÞ; yðf2ðtÞÞ; . . . ; yðfkðtÞÞÞ þX1t¼s

cðtÞ ; n P T:

ð2:19Þ

It follows from (2.2) and (2.17)–(2.19) that for each n P T

jxðnÞ � yðnÞjP jL� Kj �X1i¼1

X1s¼nþis

1aðsÞ


þX1t¼s


P jL� Kj �X1i¼1

X1s¼nþis

1aðsÞ

�RN;MðsÞmaxfjxðhlðsÞÞ � yðhlðsÞÞj : 1 6 l 6 kg

þX1t¼s

PN;MðtÞmaxfjxðflðtÞÞ � yðflðtÞÞj : 1 6 l 6 kg#


X1s¼nþis


X1t¼s

PN;MðtÞ" #

kx� yk

P jL� Kj � 2MX1i¼1

X1s¼Tþis


X1t¼s

PN;MðtÞ" #

> 0;

which yields that x – y. This completes the proof. h


Theorem 2.2. Assume that there exists a constant b 2 ½0;1Þ satisfying

0 6 bðnÞ 6 �b; eventually: ð2:20Þ

Let M and N be two positive constants with ð1� �bÞM > N and there exist four mappings PN;M ;QN;M, RN;M;WN;M : Nn0 ! Rþ satis-fying (2.2) and (2.3),

X1n¼n0

1aðnÞ max RN;MðnÞ;WN;MðnÞ

� �< þ1 ð2:21Þ

and

X1s¼n0

X1t¼s

1aðsÞ max PN;MðtÞ;Q N;MðtÞ; jcðtÞj

� �< þ1: ð2:22Þ

Then

(a) For any L 2 ðN;MÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for any x0 2 XðN;MÞ, the Mann iterative sequencewith errors fxmgmP0 generated by the scheme:

xmþ1ðnÞ ¼

ð1� am � bmÞxmðnÞ þ am L� bðnÞxmðn� sÞf

�Pn�1

s¼T

1aðsÞ

�hðs; xmðh1ðsÞÞ; xmðh2ðsÞÞ; . . . ; xmðhkðsÞÞÞ

�P1t¼s


cðtÞ�

þbmcmðnÞ; n P T þ 1; m P 0;

ð1� am � bmÞxmðT þ 1Þ þ am L� bðT þ 1ÞxmðT þ 1� sÞf

� 1aðTÞ

�hðT; xmðh1ðTÞÞ; xmðh2ðTÞÞ; . . . ; xmðhkðTÞÞÞ

�P1t¼T

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; . . . ; xmðfkðtÞÞÞ þP1t¼T

cðtÞ�

þbmcmðT þ 1Þ; b 6 n 6 T; m P 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð2:23Þ

converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (1.11) and satisfies the error estimate (2.7), wherefcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying (2.8) and (2.9);


Proof. Set L 2 ðN;MÞ. In view of (2.20)–(2.22), we choice h 2 ð0;1Þ and T P n0 þ sþ jaj such that

0 6 bðnÞ 6 �b; n P T; ð2:24Þ

h ¼ �bþX1s¼T


X1t¼s

PN;MðtÞ" #

; ð2:25Þ

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6min M � L; L� �bM � N� �

: ð2:26Þ


SLxðnÞ ¼

L� bðnÞxðn� sÞ �Pn�1

s¼T

1aðsÞ


�P1t¼s


þP1t¼s

cðtÞ�; n P T þ 1; x 2 XðN;MÞ;

SLxðT þ 1Þ; b 6 n 6 T; x 2 XðN;MÞ:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð2:27Þ


By virtue of (2.2), (2.24), (2.25) and (2.27), we gain that for x; y 2 XðN;MÞ and n P T þ 1

jSLxðnÞ � SLyðnÞj 6 bðnÞjxðn� sÞ � yðn� sÞj

þXn�1

s¼T

1aðsÞ


þX1t¼s


6�bkx� yk þ

Xn�1

s¼T

1aðsÞ

"RN;MðsÞmax jxðhiðsÞÞ � yðhiðsÞÞj : 1 6 i 6 kf g

þX1t¼s

PN;MðtÞmax jxðfiðtÞÞ � yðfiðtÞÞj : 1 6 i 6 kf g#

6�bþ

Xn�1

s¼T


X1t¼s

PN;MðtÞ !" #

kx� yk 6 hkx� yk;

which gives (2.14). Applying (2.3), (2.24), (2.26) and (2.27), we derive that for any x 2 XðN;MÞ and n P T þ 1

SLxðnÞ 6 LþXn�1

s¼T


X1t¼s


jcðtÞj" #

6 LþXn�1

s¼T


X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

6 LþX1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 Lþmin M � L; L� �bM � N� �

6 M

and

SLxðnÞP L� �bM �Xn�1

s¼T


X1t¼s


jcðtÞj" #

P L� �bM �Xn�1

s¼T


X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

P L� �bM �X1s¼T


X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

P L�min M � L; L� �bM � N� �

P N;

which imply that SLðXðN;MÞÞ# XðN;MÞ. Consequently, (2.14) means that SL is a contraction mapping and it has a uniquefixed point x 2 XðN;MÞ, which is a bounded nonoscillatory solution of Eq. (1.11). Using (2.14), (2.23), (2.25) and (2.27),we infer that for any m P 0 and n P T þ 1,

jxmþ1ðnÞ � xðnÞj ¼ ð1� am � bmÞxmðnÞ þ am L� bðnÞxðn� sÞ �Xn�1

s¼T


(��X1t¼s

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; � � � ; xmðfkðtÞÞÞ þX1t¼s

cðtÞ#)þ bmcmðnÞ � xðnÞ

��6 ð1� am � bmÞjxmðnÞ � xðnÞj þ amjSLxmðnÞ � SLxðnÞj þ bmjcmðnÞ � xðnÞj6 ð1� am � bmÞkxm � xk þ amhkxm � xk þ 2Mbm 6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm;

which yields that

kxmþ1 � xk 6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm; m P 0:

That is, (2.7) holds. Consequently, Lemma 1.1, (2.8) and (2.9) imply that limm!1xm ¼ x. That is, (a) holds.Next we show that (b) holds. By (2.20)–(2.22), we know that for any distinct L;K 2 ðN;MÞ, there exist h 2 ð0;1Þ and

T P n0 þ sþ jaj satisfying (2.24)–(2.26) and

X1s¼T


X1t¼s

PN;MðtÞ" #

<jL� Kj

2M: ð2:28Þ

Let the mapping SL be defined by (2.27), and define a mapping SK : XðN;MÞ ! l1b by (2.27) with L replaced by K. It is easy tocheck that the mappings SL and SK have the unique fixed points x; y 2 XðN;MÞ, respectively. That is, x and y are boundednonoscillatory solutions of Eq. (1.11) in XðN;MÞ. In view of (2.27), we deduce that


xðnÞ ¼ L� bðnÞxðn� sÞ �Xn�1

s¼T


X1t¼s


cðtÞ" #

;

n P T þ 1 ð2:29Þ
and
yðnÞ ¼ K � bðnÞyðn� sÞ �Xn�1

s¼T


X1t¼s


cðtÞ" #

;

n P T þ 1: ð2:30Þ

Notice that (2.2) together with (2.28)–(2.30) ensures that

jxðnÞ � yðnÞ þ bðnÞðxðn� sÞ � yðn� sÞÞj

P jL� Kj �Xn�1

s¼T

1aðsÞ

"jhðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ � hðs; yðh1ðsÞÞ; yðh2ðsÞÞ; . . . ; yðhkðsÞÞÞj

þX1t¼s



s¼T

1aðsÞ RN;MðsÞmaxfjxðhlðsÞÞ � yðhlðsÞÞj : 1 6 l 6 kg þ

X1t¼s

PN;MðtÞmaxfjxðflðtÞÞ � yðflðtÞÞj : 1 6 l 6 kg" #


s¼T


X1t¼s

PN;MðtÞ" #

kx� yk

P jL� Kj � 2MX1s¼T


X1t¼s

PN;MðtÞ" #

> 0; n P T þ 1;

that is, x – y. This completes the proof. h

Theorem 2.3. Assume that there exists a constant b 2 ð�1;0� satisfying

b 6 bðnÞ 6 0; eventually: ð2:31Þ

Let M and N be two positive constants with M > N and there exist four mappings PN;M;Q N;M, RN;M ;WN;M : Nn0 ! Rþ satisfying (2.2),(2.3), (2.21) and (2.22). Then

(a) For any L 2 ðð1þ bÞN; ð1þ bÞMÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for any x0 2 XðN;MÞ, the Mann iter-ative sequence with errors fxmgmP0 generated by (2.23) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq.(1.11) and satisfies the error estimate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 areany sequences in [0,1] satisfying (2.8) and (2.9);


Proof. Put L 2 ðð1þ bÞN; ð1þ bÞMÞ. It follows from (2.21), (2.22) and (2.31) that there exist h 2 ð0;1Þ and T P n0 þ sþ jajsatisfying

b 6 bn 6 0; n P T; ð2:32Þ

h ¼ jbj þX1s¼T


X1t¼s

PN;MðtÞ" #

; ð2:33Þ

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6min ð1þ bÞM � L; L� ð1þ bÞNf g: ð2:34Þ

Define a mapping SL : XðN;MÞ ! l1b by (2.27). Similar to the proof of Theorem 2.2, we conclude that (2.14) holds.Now (2.3), (2.27), (2.32) and (2.34) lead to

SLxðnÞ 6 L� bM þXn�1

s¼T


X1t¼s


jcðtÞj" #

6 L� bM þXn�1

s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 L� bM þX1s¼T


X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

6 L� bM þmin ð1þ bÞM � L; L� ð1þ bÞNf g 6 M; x 2 XðN;MÞ;n P T þ 1


and

SLxðnÞP L� bN �Xn�1

s¼T


X1t¼s


jcðtÞj" #

P L� bN �Xn�1

s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

P L� bN �X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

P L� bN �min ð1þ bÞM � L; L� ð1þ bÞNf gP N; x 2 XðN;MÞ; n P T þ 1;

which imply that SLðXðN;MÞÞ# XðN;MÞ. Consequently, (2.14) means that SL is a contraction mapping and it has a uniquefixed point x 2 XðN;MÞ, which is a bounded nonoscillatory solution of Eq. (1.11). The rest of the proof is similar to that ofTheorem 2.2, and is omitted. This completes the proof. h

Theorem 2.4. Assume that there exist two constants �b and b satisfying

b 6 bðnÞ 6 �b; eventually ð2:35Þ
and
1 < b and �b < b2: ð2:36Þ

Let M and N be arbitrary positive constants with M�b b2 � �b �

> Nb �b2 � b �

and there exist four mappings PN;M;Q N;M,RN;M;WN;M : Nn0 ! Rþ satisfying (2.2), (2.3), (2.21) and (2.22). Then

(a) For any L 2 �bN þ �bMb ; bM þ bN

�b

� , there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for each x0 2 XðN;MÞ, the Mann iter-

ative sequence with errors fxmgmP0 generated by the scheme:

xmþ1ðnÞ ¼

ð1� am � bmÞxmðnÞ þ amL

bðnþsÞ �xmðnþsÞbðnþsÞ � 1

bðnþsÞPnþs�1

s¼Tþs

1aðsÞ

�

��


�P1t¼s

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; . . . ; xmðfkðtÞÞÞ

þP1t¼s

cðtÞ�þ bmcmðnÞ; n P T þ 1; m P 0;

ð1� am � bmÞxmðT þ 1Þ þ am

�L

bðTþ1þsÞ �xmðTþ1þsÞbðTþ1þsÞ � 1

bðTþ1þsÞaðTþsÞ

��

hðT þ s; xmðh1ðT þ sÞÞ; xmðh2ðT þ sÞÞ; . . . ; xmðhkðT þ sÞÞÞ

�P1

t¼Tþsf ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; . . . ; xmðfkðtÞÞÞ

þP1

t¼TþscðtÞ

�þ bmcmðT þ 1Þ; b 6 n 6 T; m P 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð2:37Þ

converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (1.11) and has the error estimate (2.7), where fcmgmP0 isan arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying (2.8) and (2.9);


Proof. Let L 2 �bN þ �bMb ; bM þ bN

�b

� . In terms of (2.21), (2.22), (2.35) and (2.36), we select h 2 ð0;1Þ and T P n0 þ sþ jaj such

that

b 6 bðnÞ 6 �b; n P T; ð2:38Þ

h ¼ 1b

1þX1s¼T


X1t¼s

PN;MðtÞ" #( )

; ð2:39Þ

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6min bM � Lþ bN�b;bL�b�M � bN

� : ð2:40Þ



SLxðnÞ ¼

LbðnþsÞ �

xðnþsÞbðnþsÞ � 1

bðnþsÞPnþs�1

s¼Tþs

1aðsÞ

��

hðs; xðh1ðbÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ

�P1t¼s


þP1t¼s

cðtÞ�; n P T þ 1; x 2 XðN;MÞ;

SLxðT þ 1Þ; b 6 n 6 T; x 2 XðN;MÞ:

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

ð2:41Þ

Thus (2.2), (2.36), (2.38), (2.39) and (2.41) imply that for x; y 2 XðN;MÞ and n P T þ 1

jSLxðnÞ � SLyðnÞj 6 jxðnþ sÞ � yðnþ sÞjbðnþ sÞ þ 1

bðnþ sÞXnþs�1

s¼Tþs

1aðsÞ

��

hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ � hðs; yðh1ðsÞÞ; yðh2ðsÞÞ; . . . ; yðhkðsÞÞÞj j

þX1t¼s

f ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ � f ðt; yðf1ðtÞÞ; yðf2ðtÞÞ; . . . ; yðfkðtÞÞÞj j#

6kx� yk

bþ 1

b

Xnþs�1

s¼Tþs

1aðsÞ

"RN;MðsÞmax jxðhlðsÞÞ � yðhlðsÞÞj : 1 6 l 6 kf g

þX1t¼s

PN;MðtÞmax jxðflðtÞÞ � yðflðtÞÞj : 1 6 l 6 kf g#

61b

1þX1s¼T


X1t¼s

PN;MðtÞ" #( )

kx� yk ¼ hkx� yk;

that is, (2.14) holds. It follows from (2.3), (2.36), (2.38), (2.40) and (2.41) that for any x 2 XðN;MÞ and n P T þ 1

SLxðnÞ 6 Lb� N

�bþ 1

b

Xnþs�1

s¼Tþs

1aðsÞ hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞj j þ

X1t¼s

f ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞj j þX1t¼s

cðtÞj j" #

6Lb� N

�bþ 1

b

Xnþs�1

s¼Tþs


X1t¼s

QN;MðtÞ þX1t¼s

cðtÞj j" #

6Lb� N

�bþ 1

b

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

6Lb� N

�bþ 1

bmin bM � Lþ bN

�b;bL�b�M � bN

� 6 M

and

SLxðnÞP L�b�M

b� 1

b

Xnþs�1

s¼Tþs


X1t¼s


cðtÞj j" #

PL�b�M

b� 1

b

Xnþs�1

s¼Tþs


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

PL�b�M

b� 1

b

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

PL�b�M

b� 1

bmin bM � Lþ bN

�b;bL�b�M � bN

� P N;

which imply that SLðXðN;MÞÞ# XðN;MÞ. Eq. (2.14) gives that SL is a contraction mapping and hence it has a unique fixed pointx 2 XðN;MÞ, which is a bounded nonoscillatory solution of Eq. (1.11). It follows from (2.14), (2.37) and (2.41) that for anym P 0 and n P T þ 1


jxmþ1ðnÞ � xðnÞj ¼ ð1� am � bmÞxmðnÞ þ amL

bðnþ sÞ �xmðnþ sÞbðnþ sÞ

�� 1

bðnþ sÞXnþs�1

s¼Tþs

1aðsÞ

�hðs; xmðh1ðsÞÞ; xmðh2ðsÞÞ; . . . ; xmðhkðsÞÞÞ

þX1t¼s

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; � � � ; xmðfkðtÞÞÞ þX1t¼s


��6 ð1� am � bmÞjxmðnÞ � xðnÞj þ amjSLxmðnÞ � SLxðnÞj þ bmjcmðnÞ � xðnÞj6 ð1� am � bmÞkxm � xk þ amhkxm � xk þ 2Mbm

6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm;

which yields (2.7). Thus Lemma 1.1, (2.8) and (2.9) ensure that limm!1xm ¼ x.Now we show that (b) holds. In terms of (2.21), (2.22) and (2.35), we derive that for any distinct L;K 2 �bN þ �bM

b ; bM þ bN�b

� ,

there exist h 2 ð0;1Þ and T P n0 þ sþ jaj satisfying (2.38)–(2.40) and

X1s¼T


X1t¼s

PN;MðtÞ" #

<bjL� Kj

2M�b: ð2:42Þ

Let the mapping SL be defined by (2.41), and define a mapping SK : XðN;MÞ ! l1b by (2.41) with L replaced by K. Obviously,the mappings SL and SK have the unique fixed points x; y 2 XðN;MÞ, respectively. That is, x and y are bounded nonsocillatorysolutions of Eq. (1.11) in XðN;MÞ. In view of (2.41), we conclude that

xðnÞ ¼ Lbðnþ sÞ �

xðnþ sÞbðnþ sÞ �

1bðnþ sÞ

Xnþs�1

s¼Tþs

1aðsÞ

"hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ

�X1t¼s


cðtÞ#; n P T þ 1 ð2:43Þ

and

yðnÞ ¼ Kbðnþ sÞ �

yðnþ sÞbðnþ sÞ �

1bðnþ sÞ

Xnþs�1

s¼Tþs

1aðsÞ

"hðs; yðh1ðsÞÞ; yðh2ðsÞÞ; . . . ; yðhkðsÞÞÞ

�X1t¼s


cðtÞ#; n P T þ 1: ð2:44Þ

Notice that (2.2), (2.36), (2.38) and (2.42), (2.43), (2.44) mean that for n P T þ 1

xðnÞ � yðnÞ þ xðnþ sÞ � yðnþ sÞbðnþ sÞ

��

PjL� Kj

bðnþ sÞ �1

bðnþ sÞXnþs�1

s¼Tþs

1aðsÞ

�hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ � hðs; yðh1ðsÞÞ; yðh2ðsÞÞ; . . . ; yðhkðsÞÞÞj j

þX1t¼s

f ðt; xðf1ðtÞÞ; xðf2ðtÞÞ; . . . ; xðfkðtÞÞÞ � f ðt; yðf1ðtsÞÞ; yðf2ðtÞÞ; . . . ; yðfkðtÞÞÞj j#

PjL� Kj

�b� 1

b

Xnþs�1

s¼Tþs

1aðsÞ

�RN;MðsÞmax jxðhlðsÞÞ � yðhlðsÞÞj : 1 6 l 6 kf g

þX1t¼s


PjL� Kj

�b� 1

b

Xnþs�1

s¼Tþs


X1t¼s

PN;MðtÞ" #

kx� yk

PjL� Kj

�b� 2M

b

X1s¼T


X1t¼s

PN;MðtÞ" #

> 0;

that is, x – y. This completes the proof. h

Theorem 2.5. Assume that there exist two constants �b and b satisfying (2.35) and

�b < �1: ð2:45Þ

Let M and N be arbitrary positive constants with MN >

1þb1þ�b

and there exist four mappings PN;M ;Q N;M, RN;M;WN;M : Nn0 ! Rþ satisfying(2.2), (2.3), (2.21) and (2.22). Then


(a) For any L 2 Mð1þ �bÞ;Nð1þ bÞ �

, there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for each x0 2 XðN;MÞ, the Mann iter-ative sequence with errors fxmgmP0 generated by (2.37) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq.(1.11) and has the error estimate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are anysequences in [0,1] satisfying (2.8) and (2.9);


Proof. First of all we show (a). Taking L 2 Mb 1þ 1�b

� ;N�b 1þ 1

b

� � , from (2.21), (2.35) and (2.45), we infer that there exist

h 2 ð0;1Þ and T P n0 þ sþ jaj satisfying (2.38),

h ¼ �1�b

1þX1s¼T


X1t¼s

PN;MðtÞ" #( )

ð2:46Þ

and
X1s¼T

X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

6 min L�Mð1þ �bÞ; �b N 1þ 1b

� �� L

b

� �� : ð2:47Þ

Let the mapping SL : XðN;MÞ ! l1b be defined by (2.41). It follows from (2.2), (2.38), (2.41), (2.45) and (2.46) that forx; y 2 XðN;MÞ and n P T þ 1

jSLxðnÞ � SLyðnÞj 6 � jxðnþ sÞ � yðnþ sÞjbðnþ sÞ � 1

bðnþ sÞ

�Xnþs�1

s¼Tþs

1aðsÞ


þX1t¼s


6 �kx� yk�b

� 1�b

Xnþs�1

s¼Tþs

1aðsÞ


þX1t¼s


6 �1�b

1þX1s¼T


X1t¼s

PN;MðtÞ" #( )

kx� yk ¼ hkx� yk:

Hence (2.14) holds. By means of (2.3), (2.38), (2.41), (2.45) and (2.47), we gain that for any x 2 XðN;MÞ and n P T þ 1

SLxðnÞ 6 L�b�M

�b� 1

�b

Xnþs�1

s¼Tþs

1aðsÞ

�hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞj j

þX1t¼s


cðtÞj j#

6L�b�M

�b� 1

�b

Xnþs�1

s¼Tþs


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

6L�b�M

�b� 1

�b

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

6L�b�M

�b� 1

�bmin L�Mð1þ �bÞ; �b N 1þ 1

b

� �� L

b

� �� 6 M

and

SLxðnÞP Lb� N

bþ 1

�b

Xnþs�1

s¼Tþs


X1t¼s


cðtÞj j" #

PLb� N

bþ 1

�b

Xnþs�1

s¼Tþs


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

PLb� N

bþ 1

�b

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

cðtÞj j" #

PLb� N

bþ 1

�bmin L�Mð1þ �bÞ; �b N 1þ 1

b

� �� L

b

� �� P N;


which imply that SLðXðN;MÞÞ# XðN;MÞ. (2.14) ensures that SL is a contraction mapping and hence it has a unique fixed pointx 2 XðN;MÞ, which is a bounded nonoscillatory solution of Eq. (1.11). As in the proof of Theorem 2.4, we infer that (2.7) holdsand limm!1xm ¼ x.

Next we prove (b). By (2.21), (2.22), (2.35) and (2.45), we get that for any distinct L;K 2 Mð1þ �bÞ;Nð1þ bÞ �

, there existh 2 ð0;1Þ and T > n0 þ sþ jaj satisfying (2.38), (2.46), (2.47) and
X1
s¼T


X1t¼s

PN;MðtÞ" #

<�bjL� Kj

2Mb: ð2:48Þ

Let the mapping SL be defined by (2.41), and define a mapping SK : XðN;MÞ ! l1b by (2.41) with L replaced by K. It is easy toverify that the mappings SL and SK have the unique fixed points x; y 2 XðN;MÞ, which satisfy (2.43) and (2.44), respectively,that is, x and y are bounded oscillatory solutions of Eq. (1.11) in XðN;MÞ. Thus (2.35), (2.43)–(2.45) and (2.48) lead to

xðnÞ � yðnÞ þ xðnþ sÞ � yðnþ sÞbðnþ sÞ

��

P � jL� Kjbðnþ sÞ þ

1bðnþ sÞ

Xnþs�1

s¼Tþs

1aðsÞ


þX1t¼s


P � jL� Kjbþ 1

�b

Xnþs�1

s¼Tþs

1aðsÞ


þX1t¼s


P � jL� Kjbþ 1

�b

Xnþs�1

s¼Tþs


X1t¼s

PN;MðtÞ" #

kx� yk

P � jL� Kjbþ 2M

�b

X1s¼T


X1t¼s

PN;MðtÞ" #

> 0

for n P T þ 1. That is, x–y. This completes the proof. h

Theorem 2.6. Assume that

bðnÞ ¼ 1; eventually: ð2:49Þ
Let M and N be two positive constants with M > N and there exist four mappings PN;M;Q N;M, RN;M;WN;M : Nn0 ! Rþ satisfying (2.2),(2.3), (2.21) and (2.22). Then
(a) For any L 2 ðN;MÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for each x0 2 XðN;MÞ, the Mann iterative sequencewith errors fxmgmP0 generated by the scheme:

xmþ1ðnÞ ¼

ð1� am � bmÞxmðnÞ þ am LþP1i¼1

Pnþ2is�1

s¼nþð2i�1Þs

1aðsÞ

(

��


�P1t¼s


cðtÞ�

þbmcmðnÞ; n P T; m P 0;

ð1� am � bmÞxmðTÞ þ am LþP1i¼1

PTþ2is�1

s¼Tþð2i�1Þs

1aðsÞ

(

��


�P1t¼s


cðtÞ�

þbmcmðTÞ; b 6 n < T; m P 0

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð2:50Þ

converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (1.11) and has the error estimate (2.7), where fcmgmP0 isan arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying (2.8) and (2.9);



Proof. First of all we show that (a) holds. Let L 2 ðN;MÞ. It follows from (2.21), (2.22), (2.49) that there exist h 2 ð0;1Þ andT P n0 þ sþ jaj satisfying

bðnÞ � 1; n P T ð2:51Þ

h ¼X1s¼T


X1t¼s

PN;MðtÞ" #

; ð2:52Þ

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6min M � L; L� Nf g: ð2:53Þ


SLxðnÞ ¼

LþP1i¼1

Pnþ2is�1

s¼nþð2i�1Þs

1aðsÞ


�P1t¼s


þP1t¼s

cðtÞ�; n P T; x 2 XðN;MÞ;

SLxðTÞ; b 6 n < T; x 2 XðN;MÞ:

8>>>>>>>>>><>>>>>>>>>>:

ð2:54Þ

On account of (2.2), (2.52) and (2.54), we derive that for x; y 2 XðN;MÞ and n P T

jSLxðnÞ � SLyðnÞj 6X1i¼1

Xnþ2is�1

s¼nþð2i�1Þs

1aðsÞ


þX1t¼s


6

X1i¼1

Xnþ2is�1

s¼nþð2i�1Þs

1aðsÞ


þX1t¼s


6

X1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s

PN;MðtÞ" #

kx� yk

6

X1s¼T


X1t¼s

PN;MðtÞ" #

kx� yk ¼ hkx� yk;

which yields (2.14). By means of (2.3), (2.53) and (2.54), we gain that for any x 2 XðN;MÞ and n P T

SLxðnÞ 6 LþX1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s


jcðtÞj" #

6 LþX1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

6 LþX1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 Lþmin M � L; L� Nf g 6 M

and

SLxðnÞP L�X1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s


jcðtÞj" #

P L�X1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

P L�X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

P L�min M � L; L� Nf gP N;

which yield that SLðXðN;MÞÞ# XðN;MÞ. Consequently, (2.14) ensures that SL is a contraction mapping and it has a uniquefixed point x 2 XðN;MÞ, that is,


xðnÞ ¼ LþX1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s


cðtÞ" #

;

n P T ð2:55Þ

and

xðn� sÞ ¼ LþX1i¼1

Xnþð2i�1Þs�1

s¼nþð2i�2Þs


X1t¼s


cðtÞ" #

;

n P T þ s: ð2:56Þ

Adding (2.55) and (2.56), we have

xðnÞ þ xðn� sÞ ¼ 2LþX1i¼1

Xnþ2is�1

s¼nþð2i�1Þsþ

Xnþð2i�1Þs�1

s¼nþ2ði�1Þs

" #

� 1aðsÞ hðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞ �

X1t¼s


cðtÞ" #

¼ 2LþX1s¼n


X1t¼s


cðtÞ" #

;

n P T þ s:

This together with (2.51) implies that x a bounded nonoscillatory solution of Eq. (1.11). It follow from (2.14), (2.52) and(2.54) that for any m P 0 and n P T ,

xmþ1ðnÞ � xðnÞj ¼ ð1� am � bmÞxmðnÞ þ am LþX1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


(��X1t¼s

f ðt; xmðf1ðtÞÞ; xmðf2ðtÞÞ; . . . ; xmðfkðtÞÞÞ þX1t¼s


��6 ð1� am � bmÞjxmðnÞ � xðnÞj þ amjSLxmðnÞ � SLxðnÞj þ bmjcmðnÞ � xðnÞj6 ð1� am � bmÞkxm � xk þ amhkxm � xk þ 2Mbm 6 ð1� ð1� hÞamÞkxm � xk þ 2Mbm;

which guarantees that (2.7) holds. Thus Lemma 1.1, (2.8) and (2.9) yield that limm!1xm ¼ x.Now we show that (b). In view of (2.21), (2.22) and (2.49), we know that for any distinct L;K 2 ðN;MÞ, there exist h 2 ð0;1Þ

and T > n0 þ sþ jaj satisfying (2.51)–(2.53) and

X1s¼T


X1t¼s

PN;MðtÞ" #

<jL� Kj

2M: ð2:57Þ

Let the mapping SL be defined by (2.54), and define a mapping SK : XðN;MÞ ! l1b by (2.54) with L replaced by K. It is clear thatthe mappings SL and SK have the unique fixed points x; y 2 XðN;MÞ, respectively. That is, x and y are bounded nonoscillatorysolutions of Eq. (1.11) in XðN;MÞ. By virtue of (2.54), we deduce that for n P T

xðnÞ ¼ LþX1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s


cðtÞ" #

ð2:58Þ

and

yðnÞ ¼ K þX1i¼1

Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s


cðtÞ" #

:

ð2:59Þ


It follows from (2.2) and (2.57)–(2.59) that for n P T

jxðnÞ � yðnÞjP jL� Kj �X1i¼1

Xnþ2is�1

s¼nþð2i�1Þs

1aðsÞ


þX1t¼s



Xnþ2is�1

s¼nþð2i�1Þs

1aðsÞ


þX1t¼s



Xnþ2is�1

s¼nþð2i�1Þs


X1t¼s

PN;MðtÞ" #

kx� yk

P jL� Kj � 2MX1s¼T


X1t¼s

PN;MðtÞ" #

> 0;

which yields that x – y. This completes the proof. h

Theorem 2.7. Assume that there exists a constant �b 2 0; 12

� �satisfying

jbðnÞj 6 �b; eventually: ð2:60Þ

Let M and N be two positive constants with Mð1� 2�bÞ > N and there exist four mappings PN;M ;QN;M, RN;M;WN;M : Nn0 ! Rþ sat-isfying (2.2), (2.3), (2.21) and (2.22). Then

(a) For any L 2 ðN þ �bM;Mð1� �bÞÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ jaj such that for any x0 2 XðN;MÞ, the Mann iter-ative sequence with errors fxmgmP0 generated by (2.23) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq.(1.11) and satisfies the error estimate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 areany sequences in [0,1] satisfying (2.8) and (2.9);


Proof. Let L 2 ðN þ �bM;Mð1� �bÞÞ. It follows from (2.21), (2.22) and (2.60) that there exist h 2 ð0;1Þ and T P n0 þ sþ jaj sat-isfying (2.25),

jbðnÞj 6 �b; n P T ð2:61Þ

and

X1s¼T


X1t¼s

Q N;MðtÞ þX1t¼s

jcðtÞj" #

6 min Mð1� �bÞ � L; L� �bM � N� �

: ð2:62Þ

Let the mapping SL : XðN;MÞ ! l1b be defined by (2.27). In terms of (2.2), (2.25), (2.27) and (2.61), we gain that forx; y 2 XðN;MÞ and n P T þ 1

jSLxðnÞ � SLyðnÞj 6 jbðnÞjjxðn� sÞ � yðn� sÞj

þXn�1

s¼T

1aðsÞ


þX1t¼s


6�bkx� yk þ

Xn�1

s¼T

1aðsÞ

�RN;MðsÞmax jxðhiðsÞÞ � yðhiðsÞÞj : 1 6 i 6 kf g

þX1t¼s

PN;MðtÞmax jxðfiðtÞÞ � yðfiðtÞÞj : 1 6 i 6 kf g#

6�bþ

Xn�1

s¼T


X1t¼s

PN;MðtÞ !" #

kx� yk 6 hkx� yk;


which implies (2.14). It follows from (2.3), (2.27), (2.61) and (2.62) that for any x 2 XðN;MÞ and n P T þ 1

SLxðnÞ 6 Lþ jbðnÞjM þXn�1

s¼T

1aðsÞ

"jhðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞj

þX1t¼s


jcðtÞj#

6 Lþ �bM þX1s¼T


X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

6 Lþ �bM þmin Mð1� �bÞ � L; L� �bM � N� �

6 M

and

SLxðnÞP L� jbðnÞjM �Xn�1

s¼T

1aðsÞ

"jhðs; xðh1ðsÞÞ; xðh2ðsÞÞ; . . . ; xðhkðsÞÞÞj

þX1t¼s


jcðtÞj#

P L� �bM �X1s¼T

1aðtÞ WN;MðsÞ þ

X1t¼s

QN;MðtÞ þX1t¼s

jcðtÞj" #

P L� �bM �min Mð1� �bÞ � L; L� �bM � N� �

P N;

which imply that SLðXðN;MÞÞ# XðN;MÞ. Consequently, (2.14) means that SL is a contraction mapping and it has a uniquefixed point x 2 XðN;MÞ, which is a bounded nonoscillatory solution of Eq. (1.11). The rest of the proof is similar to that ofTheorem 2.2, and is omitted. This completes the proof. h

Remark 2.1. Theorems 2.1–2.5 extend and improve Theorem 1 in [5].

3. Applications

Now we display seven examples as applications of the results presented in Section 2 and note that Theorem 1 in [5] cannot deal with all these examples.

Example 3.1. Consider the second order nonlinear neutral delay difference equation

D ðnþ 1Þ3D xðnÞ � xðn� sÞð Þ�

þ Dðffiffiffinp

x2ðn3ÞÞ þ ln n

ðnþ 1Þ2x2ðn� 2Þ ¼ ð�1Þn

2n2 þ sin n; n P n0 ¼ 1; ð3:1Þ

where s 2 N is fixed. Let k ¼ 1, M and N be two positive constants with M > N and

aðnÞ ¼ ðnþ 1Þ3; bðnÞ ¼ �1; cðnÞ ¼ ð�1Þn

2n2 þ sin n; f ðn;uÞ ¼ u2 ln n

ðnþ 1Þ2;

hðn;uÞ ¼ffiffiffinp

u2; f 1ðnÞ ¼ n� 2; h1ðnÞ ¼ n3; PN;MðnÞ ¼2M ln n

ðnþ 1Þ2;

Q N;MðnÞ ¼M2 ln n

ðnþ 1Þ2; RN;MðnÞ ¼ 2M

ffiffiffinp

; WN;MðnÞ ¼ M2 ffiffiffinp

; n P 1; u 2 R:

It is easy to verify that the conditions (2.1)–(2.3) are satisfied. Note that

X1n¼n0

naðnÞ max RN;MðnÞ;WN;MðnÞ

� �¼X1n¼n0

Mn32

ðnþ 1Þ3max 2;Mf g < þ1

and

X1s¼n0

X1t¼s

saðsÞ max PN;MðtÞ;Q N;MðtÞ; jcðtÞj

� �¼X1s¼n0

X1t¼s

s

ðsþ 1Þ3max

2M ln t

ðt þ 1Þ2;

M2 ln t

ðt þ 1Þ2;

12t2 þ sin t

( )< þ1:

It follows from Theorem 2.1 that Eq. (3.1) possesses uncountably bounded nonoscillatory solutions in XðN;MÞ. On the otherhand, for any L 2 ðN;MÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ 1 such that the Mann iterative sequence with error fxmgmP0

generated by (2.6) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.1) and has the error estimate (2.7),where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in ½0;1� satisfying (2.8) and (2.9).



D n4D xðnÞ þ 2n� 43nþ 2

xðn� sÞ� ��

þ D ðn2 � 2ffiffiffinp� 4Þxð3n2 � 2Þx2ðn5 � 1Þ

�þ 1

n2 x2 nðn� 1Þðn� 2Þ3

� �x3 nðnþ 1Þ

2

� �

¼ ð�1Þnffiffiffiffiffiffiffiffiffiffiffiffiffiffin5 þ 1p ; n P n0 ¼ 1; ð3:2Þ

where s 2 N is fixed. Let k ¼ 2, �b ¼ 23, M and N be two positive constants with M > N and

aðnÞ ¼ n4; bðnÞ ¼ 2n� 43nþ 2

; cðnÞ ¼ ð�1Þnffiffiffiffiffiffiffiffiffiffiffiffiffiffin5 þ 1p ; f ðn;u; vÞ ¼ u2v3

n2 ;

hðn;u; vÞ ¼ ðn2 � 2ffiffiffinp� 4Þuv2; f 1ðnÞ ¼

nðn� 1Þðn� 2Þ3

; f 2ðnÞ ¼nðnþ 1Þ

2;

h1ðnÞ ¼ 3n2 � 2; h2ðnÞ ¼ n5 � 1; PN;MðnÞ ¼5M4

n2 ; Q N;MðnÞ ¼M5

n2 ;

RN;MðnÞ ¼ 3M2ðn2 þ 2ffiffiffinpþ 4Þ; WN;MðnÞ ¼ M3ðn2 þ 2

ffiffiffinpþ 4Þ; n P 1; u; v 2 R:

It is clear that the conditions (2.2), (2.3) and (2.20) are fulfilled. Note that

X1n¼n0


� �¼X1n¼n0

M2

n4 ðn2 þ 2

ffiffiffinpþ 4Þmax 3;Mf g < þ1

and
X1s¼n0
X1t¼s


� �¼X1s¼n0

X1t¼s

1s4 max

5M4

t2 ;M5

t2 ;1ffiffiffiffiffiffiffiffiffiffiffiffiffi

t5 þ 1p

( )< þ1:

It follows from Theorem 2.2 that Eq. (3.2) possesses uncountably bounded nonoscillatory solutions in XðN;MÞ. On the otherhand, for any L 2 ðN;MÞ, there exist h 2 ð0;1Þ and T P n0 þ s such that the Mann iterative sequence with error fxmgmP0 gen-erated by (2.23) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.2) and has the error estimate (2.7),where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying (2.8) and (2.9).


D ðn3 � n2 þ 1ÞD xðnÞ þ 1� n2

1þ 2n2 xðn� sÞ� ��

þ Dffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

pxðn2 � 2Þ � nx2ðn2 � nÞ

� þ

ffiffiffinp

n2 þ 1x3ðn� 3Þ

� cosðn2 � ln nÞn2 x2ðn3 � 2nÞ

¼ sinðn2 � nÞn2 � nþ 1

; n P n0 ¼ 1; ð3:3Þ

where s 2 N is fixed. Let k ¼ 2, b ¼ � 23, M and N be two positive constants with M > N and

aðnÞ ¼ n3 � n2 þ 1; bðnÞ ¼ 1� n2

1þ 2n2 ; cðnÞ ¼ sinðn2 � nÞn2 � nþ 1

;

f ðn;u; vÞ ¼ffiffiffinp

n2 þ 1u3 � cosðn2 � ln nÞ

n2 v2; hðn; u;vÞ ¼ uffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

p� nv2;

f1ðnÞ ¼ n� 3; f 2ðnÞ ¼ n3 � 2n; h1ðnÞ ¼ n2 � 2; h2ðnÞ ¼ n2 � n;

PN;MðnÞ ¼3M2 ffiffiffi

np

n2 þ 1þ 2M

n2 ; Q N;MðnÞ ¼M3 ffiffiffi

np

n2 þ 1þM2

n2 ;

RN;MðnÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

pþ 2Mn; WN;MðnÞ ¼ M

ffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

pþM2n; n P 1; u;v 2 R:

It is not difficult to verify that the conditions (2.2), (2.3) and (2.31) are fulfilled. Note that

X1n¼n0


� �¼X1n¼n0

1n3 � n2 þ 1

maxffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

pþ 2Mn;M

ffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 1

pþM2n

n o< þ1

and

X1s¼n0

X1t¼s


� �¼X1s¼n0

X1t¼s

1s3 � s2 þ 1

max3M2 ffiffi

tp

t2 þ 1þ 2M

t2 ;M3 ffiffi

tp

t2 þ 1þM2

t2 ;j sinðt2 � tÞj

t2 � t þ 1

( )< þ1:


Thus Theorem 2.3 ensures that Eq. (3.3) possesses uncountably bounded nonoscillatory solutions in XðN;MÞ. On the otherhand, for any L 2 N

3 ;M3

�, there exist h 2 ð0;1Þ and T P n0 þ sþ 2 such that the Mann iterative sequence with error

fxmgmP0 generated by (2.23) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.3) and has the error esti-mate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying(2.8) and (2.9).


D n2D xðnÞ þ 2nþ 3nþ 1


þ D sinðln nÞx2ðn� 3Þ �

þ 1

nln2nx4ðn� 5Þ ¼ ð�1Þnn

n3 þ 1; n P n0 ¼ 1; ð3:4Þ

where s 2 N is fixed. Let k ¼ 1;a ¼ 1, b ¼ 2; �b ¼ 3, M and N be two positive constants with M > 14N3 and

aðnÞ ¼ n2; bðnÞ ¼ 2nþ 3nþ 1

; cðnÞ ¼ ð�1Þnnn3 þ 1

; f ðn;uÞ ¼ u4

nln2n;

hðn;uÞ ¼ u2 sinðln nÞ; f 1ðnÞ ¼ n� 5; h1ðnÞ ¼ n� 3; PN;M ¼4M3

nln2n;

Q N;MðnÞ ¼M4

nln2n; RN;MðnÞ ¼ 2M; WN;MðnÞ ¼ M2; n P 1; u 2 R:

Obviously, the conditions (2.2), (2.3), (2.35) and (2.36) are satisfied. Note that

X1n¼n0


� �¼X1n¼n0

Mn2 max M;2f g < þ1

and

X1s¼n0

X1t¼s


� �¼X1s¼n0

X1t¼s

1s2 max

4M3

tln2t;

M4

tln2t;

tt3 þ 1

( )< þ1:

Consequently Theorem 2.4 implies that Eq. (3.4) possesses uncountably bounded nonoscillatory solutions in XðN;MÞ. On theother hand, for any L 2 3N þ 3

2 N;2M þ 23 N

�, there exist h 2 ð0;1Þ and T P n0 þ sþ 4 such that the Mann iterative sequence

with error fxmgmP0 generated by (2.37) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.4) and has theerror estimate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] sat-isfying (2.8) and (2.9).


D nln2nD xðnÞ � 4nþ ð�1Þnnnþ 1


þ Dx4ðn� 2Þ

1þ nx2ðn� 2Þ

� �þ x2ðn� 3Þ

n2 þ 1� nx3ðn� 3Þ

n3 þ nþ 1

¼ cosðn�ffiffiffinpþ 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n3 þ 1p ; n P n0 ¼ 1; ð3:5Þ

where s 2 N is fixed. Let k ¼ 1, b ¼ �5; �b ¼ �2, M and N be two positive constants with M > 4N and

aðnÞ ¼ nln2n; bðnÞ ¼ �4nþ ð�1Þnnnþ 1

; cðnÞ ¼ cosðn�ffiffiffinpþ 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n3 þ 1p ;

f ðn;uÞ ¼ u2

n2 þ 1� nu3

n3 þ nþ 1; hðn;uÞ ¼ u4

1þ nu2 ; f 1ðnÞ ¼ n� 3; h1ðnÞ ¼ n� 2;

PN;MðnÞ ¼2M

n2 þ 1þ 3M2n

n3 þ nþ 1; QN;MðnÞ ¼

M2

n2 þ 1þ M3n

n3 þ nþ 1;

RN;MðnÞ ¼2M3ð2þM2nÞð1þ N2nÞ2

; WN;MðnÞ ¼M4

1þ N2n; n P 1; u 2 R:

It is clear that the conditions (2.2), (2.3), (2.35) and (2.45) are fulfilled. Note that

X1n¼n0


� �¼X1n¼n0

1

nln2nmax

2M3ð2þM2nÞð1þ N2nÞ2

;M4

1þ N2n

( )< þ1


and

X1s¼n0

X1t¼s


� �

¼X1s¼n0

X1t¼s

1

sln2smax

2Mt2 þ 1

þ 3M2tt3 þ t þ 1

;M2

t2 þ 1þ M3t

t3 þ t þ 1;j cosðt �

ffiffitpþ 1Þjffiffiffiffiffiffiffiffiffiffiffiffiffi

t3 þ 1p

( )< þ1:

Thus Theorem 2.5 yields that Eq. (3.5) possesses uncountably bounded nonoscillatory solutions in XðN;MÞ. On the otherhand, for any L 2 ð�M;�4NÞ, there exist h 2 ð0;1Þ and T P n0 þ sþ 2 such that the Mann iterative sequence with errorfxmgmP0 generated by (2.37) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.5) and has the error esti-mate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying(2.8) and (2.9).


D n32D xðnÞ þ xðn� sÞð Þ

� þ D

1� x3ðn3 � 2nþ 1Þnþ x2ðn3 � 2nþ 1Þ

� �þðð�1Þn � 1

nÞx4ðn2 � nÞn2 þ x2ðn2 � nÞ ¼ sin n

nln3n; n P n0 ¼ 1; ð3:6Þ

where s 2 N is fixed. Let k ¼ 1, M and N be two positive constants with M > N and

aðnÞ ¼ n32; bðnÞ ¼ 1; cðnÞ ¼ sin n

nln3n; f ðn;uÞ ¼

ðð�1Þn � 1nÞu4

n2 þ u2 ;

hðn;uÞ ¼ 1� u3

nþ u2 ; f 1ðnÞ ¼ n2 � n; h1ðnÞ ¼ n3 � 2nþ 1;

PN;MðnÞ ¼4M3ðM2 þ 2n2Þðn2 þ N2Þ2

; Q N;MðnÞ ¼2M4

n2 þ N2 ;

RN;MðnÞ ¼Mð2þM3 þ 3MnÞðn2 þ N2Þ2

; WN;MðnÞ ¼1þM3

nþ N2 ; n P 1; u 2 R:

It is clear that the conditions (2.2), (2.3) and (2.49) are fulfilled. Notice that

X1n¼n0


� �¼X1n¼n0

1

n32

maxMð2þM3 þ 3MnÞðn2 þ N2Þ2

;1þM3

nþ N2

( )< þ1

and

X1s¼n0

X1t¼s


� �¼X1s¼n0

X1t¼s

1

s32

max4M3ðM2 þ 2t2Þðt2 þ N2Þ2

;2M4

t2 þ N2 ;j sin tjtln3t

( )< þ1:

Therefore we can invoke Theorem 2.6 to show that Eq. (3.6) possesses uncountably bounded nonoscillatory solutions inXðN;MÞ. On the other hand, for any L 2 ðN;MÞ, there exist h 2 ð0;1Þ and T P n0 þ s such that the Mann iterative sequencewith error fxmgmP0 generated by (2.50) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.6) and hasthe error estimate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in[0,1] satisfying (2.8) and (2.9).

Example 3.7. Consider the following second order nonlinear neutral delay difference equation:

D ðn2 � nþ 2ÞD xðnÞ þ ð�1Þnn3nþ 1


þ Dnx4ðn2 � 2Þ

n2 þ jxðn2 � 2Þj

� �þð�1Þn�1x5 nðnþ1Þ

2

� nln2nþ x2 nðnþ1Þ

2

�

¼ lnðffiffiffiffiffiffi3npÞ sin n3

n32

; n P n0 ¼ 1; ð3:7Þ


where s 2 N is fixed. Let k ¼ 1, �b ¼ 13, M and N be two positive constants with M > 3N and

aðnÞ ¼ n2 � nþ 2; bðnÞ ¼ ð�1Þnn3nþ 1

Þ; cðnÞ ¼ lnðffiffiffiffiffiffi3npÞ sin n3

n32

;

f ðn;uÞ ¼ ð�1Þn�1u5

nln2nþ u2; hðn; uÞ ¼ nu4

n2 þ juj ; f 1ðnÞ ¼nðnþ 1Þ

2; h1ðnÞ ¼ n2 � 2;

PN;MðnÞ ¼M4ð3M2 þ 5nln2nÞðN2 þ nln2nÞ2

; QN;MðnÞ ¼M5

N2 þ nln2n;

RN;MðnÞ ¼M3nð3M þ 4n2ÞðN þ n2Þ2

; WN;MðnÞ ¼M4n

N þ n2 ; n P 1; u 2 R:

Clearly, the conditions (2.2), (2.3) and (2.60) hold. Notice that

X1n¼n0


� �¼X1n¼n0

1n2 � nþ 2

maxM3nð3M þ 4n2ÞðN þ n2Þ2

;M4n

N þ n2

( )< þ1

and

X1s¼n0

X1t¼s

1aðsÞ max PN;MðtÞ;QN;MðtÞ; jcðtÞj

� �

¼X1s¼n0

X1t¼s

1s2 � sþ 2

maxM4ð3M2 þ 5tln2tÞðN2 þ tln2tÞ2

;M5

N2 þ tln2t;lnð

ffiffiffiffiffi3tpÞj sin t3jt

32

( )< þ1:

It follows from Theorem 2.7 that Eq. (3.7) possesses uncountably bounded nonoscillatory solutions in XðN;MÞ. On the otherhand, for any L 2 N þ 1

3 M; 23 M

�, there exist h 2 ð0;1Þ and T P n0 þ sþ 2 such that the Mann iterative sequence with error

fxmgmP0 generated by (2.23) converges to a bounded nonoscillatory solution x 2 XðN;MÞ of Eq. (3.7) and has the error esti-mate (2.7), where fcmgmP0 is an arbitrary sequence in XðN;MÞ, famgmP0 and fbmgmP0 are any sequences in [0,1] satisfying(2.8) and (2.9).

Acknowledgements

The authors would like to express their thanks to the anonymous referee for her/his valuable suggestions and comments.This work was supported by the Science Research Foundation of Educational Department of Liaoning Province (2008352)and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund)(KRF-2008-313-C00042).

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Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations