Existence of bounded positive solutions for a system of difference equations

Applied Mathematics and Computation 218 (2011) 2889–2912

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Applied Mathematics and Computation

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

Existence of bounded positive solutions for a system of difference equations

Zeqing Liu a, Liangshi Zhao a, Shin Min Kang b, Young Chel Kwun c,⇑a Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, People’s Republic of Chinab Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Koreac Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

a r t i c l e i n f o

Keywords:Second order nonlinear neutral delaysystem of difference equationsUncountably many bounded positivesolutionsContraction mapping

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.08.033

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

a b s t r a c t

This paper deals with the second order nonlinear neutral delay system of differenceequations

. All righ

hizhao8

D½anDðxn þ bnxn�sÞ� þ f ðn; xh1n; . . . ; xhkn

; yw1n; . . . ; ywkn

Þ ¼ cn; n P n0;

D½pnDðyn þ qnyn�rÞ� þ gðn; xs1n ; . . . ; xskn; yt1n

; . . . ; ytknÞ ¼ rn; n P n0:

(

By virtue of the Banach fixed point theorem and some natural modifications of the tech-niques in the literature, we prove the existence results of uncountably many bounded po-sitive solutions for the system of difference equations. Seven examples are constructed toexplain the main results presented in this paper.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

We consider the second order nonlinear neutral delay system of difference equations of the form

D½anDðxn þ bnxn�sÞ� þ f ðn; xh1n; . . . ; xhkn

; yw1n; . . . ; ywkn

Þ ¼ cn; n P n0;

D½pnDðyn þ qnyn�rÞ� þ gðn; xs1n ; . . . ; xskn; yt1n

; . . . ; ytknÞ ¼ rn; n P n0;

(ð1:1Þ

where s;r; k 2 N;n0 2 N0; fangn2Nn0; fbngn2Nn0

; fcngn2Nn0; fpngn2Nn0

; fqngn2Nn0and frngn2Nn0

� R with anpn – 0 for eachn 2 Nn0 ; f ; g 2 C Nn0 � R2k;R

� �and

Sl2f1;2;...;kgfhln;wln; sln; tlngn2Nn0

� Z with

limn!1

hln ¼ limn!1

wln ¼ limn!1

sln ¼ limn!1

tln ¼ þ1; l 2 f1;2; . . . ; kg:

Note that a few special cases of the system (1.1) were studied in [2,6,12,16,18]. Chen [6] studied the second-order self-ad-joint difference equation

DðcnDxnÞ þ ðan þ fnÞxnþ1 ¼ 0; n P 0; ð1:2Þ

as a perturbation of the equation

DðcnDznÞ þ anznþ1 ¼ 0; n P 0; ð1:3Þ

where fcngn2N0; fangn2N0

and ffngn2N0are real sequences with cn – 0 for each n 2 N0, and he found conditions on fn such that

solutions of Eqs. (1.2) and (1.3) behave asymptotically the same as n ?1. Some results on the growth of solutions ofEq. (1.3) and related second-order difference equations can be found, for example, in [4,5,21–23] (see also the references

ts reserved.

[email protected] (L. Zhao), [email protected] (S.M. Kang), [email protected] (Y.C. Kwun).

http://dx.doi.org/10.1016/j.amc.2011.08.033

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http://dx.doi.org/10.1016/j.amc.2011.08.033

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2890 Z. Liu et al. / Applied Mathematics and Computation 218 (2011) 2889–2912

therein). Using the Banach contraction principle, Jinfa [12] obtained the existence of a nonoscillatory solution for the secondorder neutral delay difference equation with positive and negative coefficients

D2ðxn þ pxn�mÞ þ pnxn�k � qnxn�l ¼ 0; n P n0; ð1:4Þ

where p 2 R n f�1g; m; k; l 2 N, fpngn2Nn0and fqngn2Nn0

are nonnegative sequences withP1

n¼n0npn < þ1 andP1

n¼n0nqn < þ1. Li and Yeh [16] established some oscillation criteria of the second-order delay difference equation

D½an�1Dðxn�1 þ pn�1xn�1�rÞ� þ qnf ðxn�sÞ ¼ 0; n P 1; ð1:5Þ

where r; s 2 N0; f 2 CðR;RÞ; fangn2N; fpngn2N and fqngn2N are nonnegative sequences. Agarwal et al. [2] investigated theoscillatory behavior of solutions for the neutral difference equation

D½pnDðyn þ hnyn�kÞ� þ qnþ1f ðynþ1�lÞ ¼ 0; n P 0; ð1:6Þ

where k, l are fixed nonnegative integers, fpngn2N0; fhngn2N0

and fqngn2N0are nonnegative sequences. With the help of the

Banach contraction principle, Liu et al. [18] discussed the global existence of nonoscillatory solutions for the second ordernonlinear neutral delay difference equation

D½anDðxn þ bxn�sÞ� þ f ðn; xn�d1n; xn�d2n

; . . . ; xn�dknÞ ¼ cn; n P n0; ð1:7Þ

where b 2 R; s; k 2 N; n0 2 N0; fangn2Nn0and fcngn2Nn0

are real sequences with an > 0 for n 2 Nn0 ;Sk

l¼1fdlngn2Nn0# Z, and

f : Nn0 � Rk ! R is a mapping. Recently, Jiang and Tang [11] obtained some sufficient conditions of the oscillation for thelinear two-dimensional difference system

Dxn ¼ pnyn; n P 1;Dyn ¼ �qnxn; n P 1;

�ð1:8Þ

where fpngn2N and fqngn2N are nonnegative sequences. Jiang and Tang [10] got some necessary and sufficient conditions forthe oscillation of all solutions for the nonlinear two-dimensional difference system

Dxn ¼ bngðynÞ; n P n0;

Dyn ¼ �anf ðxnÞ; n P n0;

�ð1:9Þ

where n0 2 N0; f ; g 2 CðR;RÞ; fangn2Nn0and fbngn2Nn0

are nonnegative sequences. Li [14] got classification schemes for

nonoscillatory solutions of the system (1.9) in terms of their asymptotic magnitudes and provided necessary as well as suf-ficient conditions for the existence of these solutions of the system (1.9). Graef and Thandapani [7] gave sufficient conditionsfor the oscillation of all solutions for the nonlinear two-dimensional difference system


Dyn�1 ¼ �anf ðxnÞ; n P n0;

�ð1:10Þ


are nonnegative sequences. Agarwal et al. [1] studied the nonlineartwo-dimensional difference system

Dxn ¼ anf ðynÞ; n P n0;

Dyn ¼ bngðxnÞ; n P n0;

�ð1:11Þ


are nonnegative sequences. They classified these solutions accordingto asymptotic behavior and gave some necessary and sufficient conditions for the existence of solutions of such classes.Huo and Li [8] established oscillation criteria for two-dimensional nonlinear difference system


Dyn ¼ �anf ðxnÞ þ rn; n P n0;

�ð1:12Þ


are nonnegative sequences, frngn2Nn0is real sequence withP1

i¼n0jrij <1. Huo and Li [9] proved the existence and uniqueness of oscillation solutions for the Emden–Fowler difference

system


Dyn�1 ¼ �anf ðxnÞ þ rn; n P n0;

�ð1:13Þ

where n0 2 N0; f ; g 2 CðR;RÞ; fangn2Nn0and frngn2Nn0

are real sequences withP1

i¼n0jrij <1, fbngn2Nn0

is nonnegative

sequence. Except above mentioned, there are numerous, very different, methods for establishing the existence of non-oscillatory and/or positive solutions of difference equations (see, for example, [3,13,24], as well as the references therein).

However, to the best of our knowledge, nothing has been done with the solvability for the system (1.1). This paperattempts to fill this gap in the literature. Motivated and inspired by the research work in [1–24], we establish the existence

Z. Liu et al. / Applied Mathematics and Computation 218 (2011) 2889–2912 2891

results of uncountably many bounded positive solutions for the system (1.1). Seven examples are provided to show theimportance and superiority of the main results presented in this paper.

2. Preliminaries

Throughout this paper, we assume that D is the forward difference operator defined by Dxn ¼ xnþ1 � xn; D2xn ¼DðDxnÞ; R ¼ ð�1;þ1Þ; Rþ ¼ ½0;þ1Þ; R� ¼ ð�1;0Þ; Z; N and N0 stand for the sets of all integers, positive integers andnonnegative integers, respectively,

Nn0 ¼ fn : n 2 N0 with n P n0g; n0 2 N0;

b ¼minfn0 � s;n0 � r; inffhln;wln; sln; tln : n 2 Nn0 ; l ¼ 1;2; . . . ; kgg;

Zb ¼ fn : n 2 Z with n P bg:

Let l1b denote the Banach space of all bounded sequences in Zb with norm

kxk ¼ supn2Zb

jxnj for x ¼ fxngn2Zb2 l1b

and

Xðd;DÞ ¼ fx ¼ fxngn2Zb2 l1b : kx� dk 6 Dg

for each D > 0 and d ¼ fdgn2Zb2 l1b . It is easy to see that for each i 2 f1;2g; Di 2 Rþ n f0g and di ¼ fdign2Zb

2 l1b ; Xðd1;D1Þ�Xðd2;D2Þ is a nonempty bounded closed convex subset of the Banach space l1b � l1b with norm k(x,y)k1 = max{kxk,kyk} foreach ðx; yÞ 2 l1b � l1b .

By a solution of the system (1.1), we mean a sequence fðxn; ynÞgn2Zbwith a positive integer T P n0 + s + r + jbj such that

(1.1) holds for all n P T.

Lemma 2.1. Let n0 2 N0; s 2 N; fangn2Nn0and fbngn2Nn0

be nonnegative sequences. Then

X1i¼n0þ1

iai

Xi�1

j¼n0

bj

!< þ1 ()

X1s¼1

X1i¼n0þss

ai

Xi�1

j¼n0

bj

!< þ1:

Proof. For each t 2 R; ½t� denotes for the largest integer not exceeding t. Notice that

X1s¼1

X1i¼n0þss

ai

Xi�1

j¼n0

bj

!¼X1

i¼n0þs

ai

Xi�1

j¼n0

bj

!þ

X1i¼n0þ2s

ai

Xi�1

j¼n0

bj

!þ

X1i¼n0þ3s

ai

Xi�1

j¼n0

bj

!þ � � �

¼X1

i¼n0þs

1þ i� n0 � ss

� �� ai

Xi�1

j¼n0

bj

( )

and

limi!þ1

1þ i�n0�ss

h ii�n0�s

s

¼ 1;

which yield that

X1s¼1

X1i¼n0þss

ai

Xi�1

j¼n0

bj

!< þ1 ()

X1i¼n0þs

i� n0 � ss

� �ai

Xi�1

j¼n0

bj

( )< þ1 ()

X1i¼n0þs

iai

Xi�1

j¼n0

bj

!

< þ1 ()X1

i¼n0þ1

iai

Xi�1

j¼n0

bj

!< þ1:

This completes the proof. h

3. Existence of uncountably many bounded positive solutions

According to the different ranges of fbngn2Nn0and fqngn2Nn0

, in this section we study the existence of uncountably boundedpositive solutions for the system (1.1). Theorem 3.1 below considers the case bn = qn = �1.


Theorem 3.1. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g and four nonnegative sequencesfUngn2Nn0

; fVngn2Nn0; fFngn2Nn0

and fGngn2Nn0satisfying

d1 > D1; bn ¼ �1; 8n P n1; ð3:1Þd2 > D2; qn ¼ �1; 8n P n1; ð3:2Þ

jf ðn;u1; . . . ;uk;v1; . . . ;vkÞ � f ðn; �u1; . . . ; �uk; �v1; . . . ; �vkÞj 6 Un maxfjui � �uij; jv i � �v ij : i ¼ 1;2; . . . ; kg;jgðn;u1; . . . ;uk; v1; . . . ;vkÞ � gðn; �u1; . . . ; �uk; �v1; . . . ; �vkÞj 6 Vn maxfjui � �uij; jv i � �v ij : i ¼ 1;2; . . . ; kg;8ðn;ul; �ul; v l; �v lÞ 2 Nn0 � ½d1 � D1;d1 þ D1�2 � ½d2 � D2;d2 þ D2�2; l ¼ 1;2; . . . ; k;

ð3:3Þ

jf ðn;u1; . . . ;uk;v1; . . . ;vkÞj 6 Fn; jgðn; u1; . . . ;uk;v1; . . . ;vkÞj 6 Gn;

8ðn;ul;v lÞ 2 Nn0 � ½d1 � D1;d1 þ D1� � ½d2 � D2; d2 þ D2�; l ¼ 1;2; . . . ; k;ð3:4Þ

X1i¼n0þ1

ijaij

Xi�1

j¼n0

maxfFj; jcjj;Ujg < þ1 ð3:5Þ

and

X1i¼n0þ1

ijpij

Xi�1

j¼n0

maxfGj; jrjj;Vjg < þ1: ð3:6Þ

Then the system (1.1) has uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).

Proof. Let (L1,L2) 2 (d1 � D1,d1 + D1) � (d2 � D2,d2 + D2). It follows from (3.5), (3.6) and Lemma 2.1 that there exist h 2 (0,1)and T P 1 + n0 + n1 + s + r + jbj satisfying

h ¼maxX1s¼1

X1i¼Tþs

1jaij

Xi�1

j¼n0

Uj;X1s¼1

X1i¼Tþs

1jpij

Xi�1

j¼n0

Vj

( ); ð3:7Þ

X1s¼1

X1i¼Tþss

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ < D1 � jL1 � d1j; ð3:8Þ

X1s¼1

X1i¼Tþsr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ < D2 � jL2 � d2j: ð3:9Þ

Put

h� ¼X1s¼1

X1i¼Tþss

1jaij

Xi�1

j¼n0

Uj ð3:10Þ

and

h� ¼X1s¼1

X1i¼Tþsr

1jpij

Xi�1

j¼n0

Vj: ð3:11Þ

Define three mappings AL1 ; BL2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1 ;L2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b by

AL1 ðxn; ynÞ ¼L1 �

P1s¼1

P1i¼nþss

1ai

Pi�1

j¼n0

½f ðj; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

Þ � cj�; n P T;

AL1 ðxT ; yTÞ; b 6 n < T;

8><>: ð3:12Þ

BL2 ðxn; ynÞ ¼L2 �

P1s¼1

P1i¼nþsr

1pi

Pi�1

j¼n0

gðj; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

Þ � rj

h i; n P T;

BL2 ðxT ; yTÞ; b 6 n < T

8><>: ð3:13Þ

and

SL1 ;L2 ðx; yÞ ¼ ðAL1 ðx; yÞ;BL2 ðx; yÞÞ ð3:14Þ

for each ðx; yÞ ¼ fðxn; ynÞgn2Zb2 Xðd1;D1Þ �Xðd2;D2Þ.


Now we claim that

AL1 ðXðd1;D1Þ �Xðd2;D2ÞÞ# Xðd1;D1Þ; BL2 ðXðd1;D1Þ �Xðd2;D2ÞÞ# Xðd2;D2Þ ð3:15Þ

and

SL1 ;L2 ðXðd1;D1Þ �Xðd2;D2ÞÞ# Xðd1;D1Þ �Xðd2;D2Þ: ð3:16Þ

It follows from (3.4), (3.8), (3.9), (3.12) and (3.13) that for any ðx; yÞ ¼ fðxn; ynÞgn2Zb2 Xðd1;D1Þ �Xðd2;D2Þ and n P T

jAL1 ðxn; ynÞ � d1j ¼ L1 � d1 �X1s¼1

X1i¼nþss

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i��

6 jL1 � d1j þX1s¼1

X1i¼nþss

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ 6 jL1 � d1j þX1s¼1

X1i¼Tþss

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

6 jL1 � d1j þ D1 � jL1 � d1j ¼ D1

and

jBL2 ðxn; ynÞ � d2j ¼ L2 � d2 �X1s¼1

X1i¼nþsr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i��

6 jL2 � d2j þX1s¼1

X1i¼nþsr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ 6 jL2 � d2j þX1s¼1

X1i¼Tþsr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

6 jL2 � d2j þ D2 � jL2 � d2j ¼ D2;

which imply that (3.15) holds. It is easy to see that (3.14) and (3.15) yield (3.16).Next we prove that

kAL1 ðx; yÞ � AL1 ð�x; �yÞk 6 h� maxfkx� �xk; ky� �ykg;kBL2 ðx; yÞ � BL2 ð�x; �yÞk 6 h� maxfkx� �xk; ky� �ykg

ð3:17Þ

and

kSL1 ;L2 ðx; yÞ � SL1 ;L2 ð�x; �yÞk1 6 hkðx; yÞ � ð�x; �yÞk1 ð3:18Þ

for all ðx; yÞ; ð�x; �yÞ 2 Xðd1;D1Þ �Xðd2;D2Þ. In terms of (3.3), (3.7) and (3.10)–(3.14), we infer that for anyðx; yÞ ¼ fðxn; ynÞgn2Zb

; ð�x; �yÞ ¼ fð�xn; �ynÞgn2Zb2 Xðd1;D1Þ �Xðd2;D2Þ and n P T

jAL1 ðxn; ynÞ � AL1 ð�xn; �ynÞj ¼X1s¼1

X1i¼nþss

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� f j; �xh1j

; . . . ; �xhkj; �yw1j

; . . . ; �ywkj

h i��

6

X1s¼1

X1i¼Tþss

1jaij

Xi�1

j¼n0

Uj maxfjxhlj� �xhlj

j; jywlj� �ywlj

j : 1 6 l 6 kg 6 h� max kx� �xk; ky� �ykf g;

jBL2 ðxn; ynÞ � BL2 ð�xn; �ynÞj ¼X1s¼1

X1i¼nþsr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� g j; �xs1j

; . . . ; �xskj; �yt1j

; . . . ; �ytkj

h i��

6

X1s¼1

X1i¼Tþsr

1jpij

Xi�1

j¼n0

Vj maxfjxslj� �xslj

j; jytlj� �ytlj

j : 1 6 l 6 kg 6 h� maxfkx� �xk; ky� �ykg

and

kSL1 ;L2 ðx; yÞ � SL1 ;L2 ð�x; �yÞk1 ¼ kðAL1 ðx; yÞ; BL2 ðx; yÞÞ � ðAL1 ð�x; �yÞ;BL2 ð�x; �yÞÞk1

¼maxfkAL1 ðx; yÞ � AL1 ð�x; �yÞk; kBL2 ðx; yÞ � BL2 ð�x; �yÞkg

¼max supn2Zb

jAL1 ðxn; ynÞ � AL1 ð�xn; �ynÞj; supn2Zb

jBL2 ðxn; ynÞ � BL2 ð�xn; �ynÞj( )

6 maxfh� maxfkx� �xk; ky� �ykg; h� maxfkx� �xk; ky� �ykgg¼maxfh�; h�gmaxfkx� �xk; ky� �ykg 6 hkðx; yÞ � ð�x; �yÞk1;


which imply (3.17) and (3.18). Thus (3.16) and (3.18) ensure that SL1 ;L2 is a contraction mapping in X(d1,D1) �X(d2,D2) and ithas a unique fixed point ðx; yÞ ¼ fðxn; ynÞgn2Zb

2 Xðd1;D1Þ �Xðd2;D2Þ, that is,

xn ¼ L1 �X1s¼1

X1i¼nþss

1ai

Xi�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj�; 8n P T;

xn�s ¼ L1 �X1s¼1

X1i¼nþðs�1Þs

1ai

Xi�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj�; 8n P T þ s;

yn ¼ L2 �X1s¼1

X1i¼nþsr

1pi

Xi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj

h i; 8n P T

and

yn�r ¼ L2 �X1s¼1

X1i¼nþðs�1Þr

1pi

Xi�1

j¼n0

½gðj; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

Þ � rj�; 8n P T þ r;

which yield that

Dðxn � xn�sÞ ¼ �1an

Xn�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj�; 8n P T þ sþ r

and

Dðyn � yn�rÞ ¼ �1pn

Xn�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; 8n P T þ sþ r;

which imply that

D½anDðxn � xn�sÞ� ¼ �f ðn; xh1n; . . . ; xhkn

; yw1n; . . . ; ywkn

Þ þ cn; 8n P T þ sþ r

and

D½pnDðyn � yn�rÞ� ¼ �gðn; xs1n ; . . . ; xskn; yt1n

; . . . ; ytknÞ þ rn; 8n P T þ sþ r;

which together with (3.1) and (3.2) mean that ðx; yÞ ¼ fðxn; ynÞgn2Zbis a bounded positive solution of the system (1.1) in

X(d1,D1) �X(d2,D2).Let (L11,L21), (L12,L22) 2 (d1 � D1,d1 + D1) � (d2 � D2,d2 + D2) with max{jL11 � L12j, jL21 � L22j} > 0. Similarly we infer that

for each l 2 {1,2}, there exist constants hl; h�l ; h�0 2 ð0;1Þ, Tl P 1þ n0 þ n1 þ sþ rþ jbj and mappings AL1l; BL2l

: Xðd1;D1Þ�Xðd2;D2Þ ! l1b and SL1l ;L2l

: Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b satisfying (3.11)–(3.18), where h, h⁄, h⁄, T, L1, L2, AL1 ; BL2 and SL1 ;L2

are replaced by hl; h�l ; h�l; Tl; L1l; L2l; AL1l; BL2l

and SL1l ;L2l, respectively, and SL1l ;L2l

has a fixed point ðxl; ylÞ ¼ fðxln; y

lnÞgn2Zb

2Xðd1;D1Þ �Xðd2;D2Þ, which is a bounded positive solution of the system (1.1), that is,

xln ¼ L1l �

X1s¼1

X1i¼nþss

1ai

Xi�1

j¼n0

f j; xlh1j; . . . ; xl

hkj; yl

w1j; . . . ; yl

wkj

� cj

h i; 8n P Tl; l 2 f1;2g ð3:19Þ

and

yln ¼ L2l �

X1s¼1

X1i¼nþsr

1pi

Xi�1

j¼n0

g j; xls1j; . . . ; xl

skj; yl

t1j; . . . ; yl

tkj

� rj

h i; 8n P Tl; l 2 f1;2g: ð3:20Þ

Using (3.3), (3.7), (3.10), (3.11), (3.19) and (3.20), we know that for each n P max{T1,T2}

jx1n � x2

nj ¼ L11 � L12 �X1s¼1

X1i¼nþss

1ai

Xi�1

j¼n0

f j; x1h1j; . . . ; x1

hkj; y1

w1j; . . . ; y1

wkj

� f j; x2

h1j; . . . ; x2

hkj; y2

w1j; . . . ; y2

wkj

h i��

P jL11 � L12j �X1s¼1

X1i¼nþss

1jaij

Xi�1

j¼n0

Uj max jx1hlj� x2

hljj; jy1

wlj� y2

wljj : 1 6 l 6 k

n o

P jL11 � L12j �X1s¼1

X1i¼maxfT1 ;T2gþss

1jaij

Xi�1

j¼n0

Uj

!max kx1 � x2k; jy1 � y2k

� P jL11 � L12j �maxfh�1; h

�2gkðx1; y1Þ � ðx2; y2Þk1 ð3:21Þ


and

jy1n � y2

nj ¼ L21 � L22 �X1s¼1

X1i¼nþsr

1pi

Xi�1

j¼n0

g j; x1s1j; . . . ; x1

skj; y1

t1j; . . . ; y1

tkj

� g j; x2

s1j; . . . ; x2

skj; y2

t1j; . . . ; y2

tkj

h i��

P jL21 � L22j �X1s¼1

X1i¼nþsr

1jpij

Xi�1

j¼n0

Vj max jx1slj� x2

sljj; jy1

tlj� y2

tljj : 1 6 l 6 k

n o

P jL21 � L22j �X1s¼1

X1i¼maxfT1 ;T2gþsr

1jpij

Xi�1

j¼n0

Vj

!maxfkx1 � x2k; ky1 � y2kg

P jL21 � L22j �maxfh�1; h�2gkðx1; y1Þ � ðx2; y2Þk1: ð3:22Þ

Notice that (3.21) and (3.22) imply that

kðx1;y1Þ� ðx2;y2Þk1 ¼max supn2Zb

jx1n� x2

nj;supn2Zb

jy1n�y2

nj( )

P max supn2NmaxfT1 ;T2g

jx1n� x2

nj; supn2NmaxfT1 ;T2g

jy1n�y2

nj( )

P maxfjL11� L12j�maxfh�1;h�2gkðx1;y1Þ� ðx2;y2Þk1; jL21� L22j�maxfh�1;h�2gkðx1;y1Þ� ðx2;y2Þk1g

P maxfjL11� L12j; jL21� L22jg�maxfh1;h2gkðx1;y1Þ� ðx2;y2Þk1;

which gives that

kðx1; y1Þ � ðx2; y2Þk1 PmaxfjL11 � L12j; jL21 � L22jg

1þmaxfh1; h2g> 0;

that is, (x1,y1) – (x2,y2). Hence the system (1.1) possesses uncountably many bounded positive solutions inX(d1,D1) �X(d2,D2). This completes the proof. h

Theorem 3.2 below talks over the case bn = qn = 1.



and fGngn2Nn0satisfying (3.3) and (3.4),

d1 > D1; bn ¼ 1; 8n P n1; ð3:23Þ

d2 > D2; qn ¼ 1; 8n P n1; ð3:24Þ

X1i¼n0þ1

1jaij

Xi�1

j¼n0

maxfFj; jcjj;Ujg < þ1 ð3:25Þ

and

X1i¼n0þ1

1jpij

Xi�1

j¼n0

maxfGj; jrjj;Vjg < þ1: ð3:26Þ


Proof. Let (L1,L2) 2 (d1 � D1,d1 + D1) � (d2 � D2,d2 + D2). It follows from (3.25) and (3.26) that there exist h 2 (0,1) andT P 1 + n0 + n1 + s + r + jbj satisfying

h ¼maxX1i¼T

1jaij

Xi�1

j¼n0

Uj;X1i¼T

1jpij

Xi�1

j¼n0

Vj

( ); ð3:27Þ

supn2NT

X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ( )

< D1 � jL1 � d1j; ð3:28Þ

supn2NT

X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ( )

< D2 � jL2 � d2j: ð3:29Þ


Let

h� ¼ supn2NT

X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1jaij

Xi�1

j¼n0

Uj

( )ð3:30Þ

and

h� ¼ supn2NT

X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1jpij

Xi�1

j¼n0

Vj

( ): ð3:31Þ

Define three mappings AL1 ; BL2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1 ;L2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b by (3.14),

AL1 ðxn; ynÞ ¼L1 þ

P1s¼1

Pnþ2ss�1

i¼nþð2s�1Þs

1ai

Pi�1

j¼n0


; yw1j; . . . ; ywkj


AL1 ðxT ; yTÞ; b 6 n < T

8><>: ð3:32Þ

and

BL2 ðxn; ynÞ ¼L2 þ

P1s¼1

Pnþ2sr�1

i¼nþð2s�1Þr

1pi

Pi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; n P T;


8><>: ð3:33Þ

for all ðx; yÞ ¼ fðxn; ynÞgn2Zb2 Xðd1;D1Þ �Xðd2;D2Þ.

Now we claim that (3.15) and (3.16) hold. It follows from (3.4), (3.28), (3.29), (3.32) and (3.33) that for anyðx; yÞ ¼ fðxn; ynÞgn2Zb

2 Xðd1;D1Þ �Xðd2;D2Þ and n P T

jAL1 ðxn; ynÞ � d1j ¼ L1 � d1 þX1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i��

6 jL1 � d1j þX1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ 6 jL1 � d1j þ D1 � jL1 � d1j ¼ D1

and

jBL2 ðxn; ynÞ � d2j ¼ L2 � d2 þX1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i��

6 jL2 � d2j þX1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ 6 jL2 � d2j þ D2 � jL2 � d2j ¼ D2;

which imply (3.15). Obviously (3.16) follows from (3.14) and (3.15).Next we assert that (3.17) and (3.18) hold. In view of (3.3), (3.27) and (3.30)–(3.33), we obtain that for any

ðx; yÞ ¼ fðxn; ynÞgn2Zb; ð�x; �yÞ ¼ fð�xn; �ynÞgn2Zb


jAL1 ðxn; ynÞ � AL1 ð�xn; �ynÞj ¼X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1ai

Xi�1

j¼n0

f ðj; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

Þ � f j; �xh1j; . . . ; �xhkj

; �yw1j; . . . ; �ywkj

h i��

6

X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1jaij

Xi�1

j¼n0

Uj max jxhlj� �xhlj

j; jywlj� �ywlj

j : 1 6 l 6 kn o

6 h� max kx� �xk; ky� �ykf g;

jBL2 ðxn; ynÞ � BL2 ð�xn; �ynÞj ¼X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� g j; �xs1j

; . . . ; �xskj; �yt1j

; . . . ; �ytkj

h i��

6

X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1jpij

Xi�1

j¼n0

Vj max jxflj� �xflj j; jyglj

� �ygljj : 1 6 l 6 k

n o6 h� maxfkx� �xk; ky� �ykg

and

kSL1 ;L2 ðx; yÞ � SL1 ;L2 ð�x; �yÞk1 ¼max supn2Zb



6 max h� maxfkx� �xk; ky� �ykg; h� maxfkx� �xk; ky� �ykgf g 6 hkðx; yÞ � ð�x; �yÞk1;


which mean that (3.17) and (3.18) hold. Therefore SL1 ;L2 is a contraction mapping in X(d1,D1) �X(d2,D2) by (3.16) and (3.18)and it has a unique fixed point ðx; yÞ ¼ fðxn; ynÞgn2Zb


xn ¼ L1 þX1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1ai

Xi�1

j¼n0

ðj; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

Þ � cj

h i; 8n P T;

xn�s ¼ L1 þX1s¼1

Xnþð2s�1Þs�1

i¼nþ2ðs�1Þs

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i; 8n P T þ s;

yn ¼ L2 þX1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i; 8n P T

and

yn�r ¼ L2 þX1s¼1

Xnþð2s�1Þr�1

i¼nþ2ðs�1Þr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i; 8n P T þ r;

which yield that

Dðxn þ xn�sÞ ¼ �1an

Xn�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj�; 8n P T þ sþ r

and

Dðyn þ yn�rÞ ¼ �1pn

Xn�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; 8n P T þ sþ r;

which imply that

D½anDðxn þ xn�sÞ� ¼ �f ðn; xh1n; . . . ; xhkn

; yw1n; . . . ; ywkn


and

D½pnDðyn þ yn�rÞ� ¼ �gðn; xs1n ; . . . ; xskn; yt1n

; . . . ; ytknÞ þ rj; 8n P T þ sþ r;

which together with (3.23) and (3.24) mean that ðx; yÞ ¼ fðxn; ynÞgn2Zbis a bounded positive solution of the system (1.1) in

X(d1,D1) �X(d2,D2).Let (L11,L21), (L12,L22) 2 (d1 � D1,d1 + D1) � (d2 � D2, d2 + D2) with max{jL11 � L12j, jL21 � L22j} > 0. Similarly we conclude

that for each l 2 {1,2}, there exist constants hl; h�l ; h�l 2 ð0;1Þ; Tl P 1þ n0 þ n1 þ sþ rþ jbj and mappings AL1l; BL2l

:

Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1l ;L2l: Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b satisfying (3.14) and (3.27)–(3.33), where

h; h�; h�; T; L1; L2; AL1 ; BL2 and SL1 ;L2 are replaced by hl; h�l ; h�l; Tl; L1l; L2l, AL1l; BL2l



lnÞgn2Zb

2 Xðd1;D1Þ �Xðd2;D2Þ, which is a bounded positive solution of the system (1.1),that is,

xln ¼ L1l þ

X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1ai

Xi�1

j¼n0

f j; xlh1j; . . . ; xl

hkj; yl

w1j; . . . ; yl

wkj

� cj

h i; 8n P Tl; l 2 f1;2g ð3:34Þ

and

yln ¼ L2l þ

X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1pi

Xi�1

j¼n0

g j; xls1j; . . . ; xl

skj; yl

t1j; . . . ; yl

tkj

� rj

h i; 8n P Tl; l 2 f1;2g: ð3:35Þ


Using (3.3), (3.27), (3.30), (3.31), (3.34) and (3.35), we deduce that for all n P max{T1,T2}

jx1n � x2

nj ¼ L11 � L12 þX1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1ai

Xi�1

j¼n0

f j; x1h1j; . . . ; x1

hkj; y1

w1j; . . . ; y1

wkj

� f j; x2

h1j; . . . ; x2

hkj; y2

w1j; . . . ; y2

wkj

h i��

P jL11 � L12j �X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1jaij

Xi�1

j¼n0

Uj max jx1hlj� x2

hljj; jy1

wlj� y2

wljj : 1 6 l 6 k

n o

P jL11 � L12j �X1s¼1

Xnþ2ss�1

i¼nþð2s�1Þs

1jaij

Xi�1

j¼n0

Uj max kx1 � x2k; ky1 � y2k�

P jL11 � L12j �maxfh�1; h�2gkðx1; y1Þ � ðx2; y2Þk1 ð3:36Þ

and

jy1n � y2

nj ¼ L21 � L22 þX1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1pi

Xi�1

j¼n0

g j; x1s1j; . . . ; x1

skj; y1

t1j; . . . ; y1

tkj

� g j; x2

s1i; . . . ; x2

skj; y2

t1j; . . . ; y2

tkj

h i��

P jL21 � L22j �X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1jpij

Xi�1

j¼n0

Vj max jx1slj� x2

sljj; jy1

tlj� y2

tljj : 1 6 l 6 k

n o

P jL21 � L22j �X1s¼1

Xnþ2sr�1

i¼nþð2s�1Þr

1jpij

Xi�1

j¼n0

Vj max kx1 � x2k; ky1 � y2k�


Thus (3.36) and (3.37) lead to

kðx1;y1Þ� ðx2;y2Þk1 P max supn2NmaxfT1 ;T2g

jx1n� x2


jy1n�y2

nj( )



which gives that

kðx1; y1Þ � ðx2; y2Þk1 PmaxfjL11 � L12j; jL21 � L22jg

1þmaxfh1; h2g> 0;

that is, (x1,y1) – (x2,y2). Hence the system (1.1) possesses uncountably many bounded positive solutions inX(d1,D1) �X(d2,D2). This completes the proof. h

Theorem 3.3 below deals with the case jbnj 6 b and jqnj 6 q.

Theorem 3.3. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g, b; q 2 ð0; 12Þ and four nonnegative

sequences fUngn2Nn0; fVngn2Nn0

; fFngn2Nn0and fGngn2Nn0

satisfying (3.3), (3.4), (3.25) and (3.26)

1 <d1

D1<

1� bb

; jbnj 6 b; 8n P n1 ð3:38Þ

and

1 <d2

D2<

1� qq

; jqnj 6 q; 8n P n1: ð3:39Þ


Proof. Let (L1,L2) 2 (d1 � (1 � b)D1 + bd1, d1 + (1 � b)D1 � bd1) � (d2 � (1 � q)D2 + bd2, d2 + (1 � q)D2 � qd2). It follows from(3.25) and (3.26) that there exist h 2 (0,1) and T P 1 + n0 + n1 + s + r + jbj satisfying

h ¼max jbj þX1i¼T

1jaij

Xi�1

j¼n0

Uj; jqj þX1i¼T

1jpij

Xi�1

j¼n0

Vj

( ); ð3:40Þ

X1i¼T

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ < ð1� bÞD1 � bd1 � jL1 � d1j; ð3:41Þ

X1i¼T

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ < ð1� qÞD2 � qd2 � jL2 � d2j: ð3:42Þ


Put

h� ¼ jbj þX1i¼T

1jaij

Xi�1

j¼n0

Uj ð3:43Þ

and

h� ¼ jqj þX1i¼T

1jpij

Xi�1

j¼n0

Vj: ð3:44Þ

Define three mappings AL1 ; BL2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1 ;L2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b by (3.14),

AL1 ðxn; ynÞ ¼L1 � bnxn�s þ

P1i¼n

1ai

Pi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i; n P T;


8><>: ð3:45Þ

and

BL2 ðxn; ynÞ ¼L2 � qnyn�r þ

P1i¼n

1pi

Pi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i; n P T;


8><>: ð3:46Þ

for any ðx; yÞ ¼ fðxn; ynÞgn2Zb2 Xðd1;D1Þ �Xðd2;D2Þ.

Now we prove (3.15) and (3.16). It follows from (3.4), (3.41), (3.42), (3.45) and (3.46) that for each ðx; yÞ ¼fðxn; ynÞgn2Zb


jAL1 ðxn;ynÞ�d1j ¼ L1�d1�bnxn�sþX1i¼n

1ai

Xi�1

j¼n0

f j;xh1j; . . . ;xhkj

;yw1j; . . . ;ywkj

� cj

h i��

6 jL1�d1jþ jbnjjxn�sjþX1i¼n

1ai

Xi�1

j¼n0

jf j;xh1j; . . . ;xhkj

;yw1j; . . . ;ywkj

jþ jcjj

h i

6 jL1�d1j þbðd1þD1ÞþX1i¼T

1jaijXi�1

j¼n0

ðFjþjcjjÞ6 jL1�d1j þbðd1þD1Þþ ð1�bÞD1�bd1�jL1�d1j ¼D1

and

jBL2 ðxn;ynÞ�d2j ¼ L2�d2�qnyn�rþX1i¼n

1pi

Xi�1

j¼n0

g j;xs1j; . . . ;xskj

;yt1j; . . . ;ytkj

� rj

h i��

6 jL2�d2jþ jqnjjyn�rj þX1i¼T

1jpij

Xi�1

j¼n0

½jgðj;xs1j; . . . ;xskj

;yt1j; . . . ;ytkj

Þjþ jrjj�

6 jL2�d2j þqðd2þD2ÞþX1i¼T

1jpij

Xi�1

j¼n0

ðGjþjrjjÞ6 jL2�d2j þqðd2þD2Þþ ð1�qÞD2�qd2�jL2�d2j ¼D2;

which give (3.15). It is easy to see that (3.14) and (3.15) mean (3.16).Next we show (3.17) and (3.18). By virtue of (3.3), (3.40) and (3.43)–(3.46), we conclude that for each



jAL1 ðxn; ynÞ � AL1 ð�xn; �ynÞj ¼ bnðxn�s � �xn�sÞ �X1i¼n

1ai

Xi�1

j¼n0


; yw1j; . . . ; ywkj

Þ � f j; �xh1j; . . . ; �xhkj

; �yw1j; . . . ; �ywkj

h i��

6 bkx� �xk þX1i¼T

1jaij

Xi�1

j¼n0

Uj max jxhlj� �xhlj

j; jywlj� �ywlj

j : 1 6 l 6 kn o

6 h� maxfkx� �xk; ky� �ykg;

jBL2 ðxn; ynÞ � BL2 ð�xn; �ynÞj ¼ qnðyn�r � �yn�rÞ �X1i¼n

1pi

Xi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � g j; �xs1j; . . . ; �xskj

; �yt1j; . . . ; �ytkj

h i��

6 qky� �yk þX1i¼T

1jpij

Xi�1

j¼n0

Vj max jxslj� �xslj

j; jytlj� �ytlj

j : 1 6 l 6 kn o

6 h� max kx� �xk; ky� �ykf g


and




6 maxfh� maxfkx� �xk; ky� �ykg; h� maxfkx� �xk; ky� �ykgg ¼ hkðx; yÞ � ð�x; �yÞk1;

which yield (3.17) and (3.18). It follows from (3.16) and (3.18) that SL1 ;L2 is a contraction mapping in X(d1,D1) �X(d2,D2) andit has a unique fixed point ðx; yÞ ¼ fðxn; ynÞgn2Zb


xn ¼ L1 � bnxn�s þX1i¼n

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i; 8n P T

and

yn ¼ L2 � qnyn�r þX1i¼n

1pi

Xi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; 8n P T;

which yield that

Dðxn þ bnxn�sÞ ¼ �1an

Xn�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj

h i; 8n P T

and

Dðyn þ qnyn�rÞ ¼ �1pn

Xn�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; 8n P T;

which imply that

D½anDðxn þ bnxn�sÞ� ¼ �f ðn; xh1n; . . . ; xhkn

; yw1n; . . . ; ywkn

Þ þ cn; 8n P T

and

D½pnDðyn þ qnyn�rÞ� ¼ �gðn; xs1n ; . . . ; xskn; yt1n

; . . . ; ytknÞ þ rj; 8n P T;

which give that ðx; yÞ ¼ fðxn; ynÞgn2Zbis a bounded positive solution of the system (1.1) in X(d1,D1) �X(d2,D2).

Let (L11,L21), (L12,L22) 2 (d1 � D1,d1 + D1) � (d2 � D2, d2 + D2) with max{jL11 � L12j, jL21 � L22j} > 0. Similarly we concludethat for each l 2 {1,2}, there exist constants hl; h�l ; h�l 2 ð0;1Þ; Tl P 1þ n0 þ n1 þ sþ rþ jbj and mappingsAL1l

; BL2l: Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1l ;L2l

: Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b satisfying (3.14) and (3.40)–(3.44), whereh; h�; h�; T; L1; L2; AL1 ; BL2 and SL1 ;L2 are replaced by hl; h�l ; h�l; Tl; L1l; L2l, AL1l

; BL2land SL1l ;L2l

, respectively, and SL1l ;L2lhas a

fixed point ðxl; ylÞ ¼ fðxln; y

lnÞgn2Zb

2 Xðd1;D1Þ �Xðd2;D2Þ, which is a bounded positive solution of the system (1.1), that is,

xln ¼ L1l � bnxl

n�s þX1i¼n

1ai

Xi�1

j¼n0

f j; xlh1j; . . . ; xl

hkj; yl

w1j; . . . ; yl

wkj

� cj

h i; 8n P Tl; l 2 f1;2g ð3:47Þ

and

yln ¼ L2l � qnyl

n�r þX1i¼n

1pi

Xi�1

j¼n0

g j; xls1j; . . . ; xl

skj; yl

t1j; . . . ; yl

tkj

� rj

h i; 8n P Tl; l 2 f1;2g: ð3:48Þ

In view of (3.3), (3.43), (3.44), (3.47), (3.48), we infer that for any n P max{T1,T2}

jx1n � x2

nj ¼ L11 � L12 � bnðx1n�s � x2

n�sÞ þX1i¼n

1ai

Xi�1

j¼n0

f ðj; x1h1j; . . . ; x1

hkj; y1

w1j; . . . ; y1

wkjÞ � f j; x2

h1j; . . . ; x2

hkj; y2

w1j; . . . ; y2

wkj

h i��

P jL11 � L12j � bkx1 � x2k �X1i¼n

1jaij

Xi�1

j¼n0


P jL11 � L12j �maxfh�1; h�2gkðx1; y1Þ � ðx2; y2Þk1 ð3:49Þ


and

jy1n � y2

nj ¼ L21 � L22 � qnðy1n�r � y2

n�rÞ þX1i¼n

1pi

Xi�1

j¼n0

g j; x1s1j; . . . ; x1

skj; y1

t1j; . . . ; y1

tkj

� g j; x2

s1j; . . . ; x2

skj; y2

t1j; . . . ; y2

tkj

h i��

P jL21 � L22j � qky1 � y2k �X1i¼n

1jpij

Xi�1

j¼n0



It follows from (3.3), (3.49) and (3.50)

kðx1;y1Þ� ðx2;y2Þk1 P max supn2NmaxfT1 ;T2g

jx1n� x2


jy1n�y2

nj( )



which gives that

kðx1; y1Þ � ðx2; y2Þk1 Pmax jL11 � L12j; jL21 � L22jf g

1þmaxfh1; h2g> 0;

that is, (x1,y1) – (x2,y2). Consequently the system (1.1) possesses uncountably many bounded positive solutions inX(d1,D1) �X(d2,D2). This completes the proof. h

Theorem 3.4 below takes into account the case 0 6 bn 6 b and 0 6 qn 6 q.

Theorem 3.4. Assume that there exist constants n1 2 Nn0 , d1; d2;D1;D2 2 Rþ n f0g; b; q 2 ð0;1Þ and four nonnegative sequencesfUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.25), (3.26),

1 <d1

D1<

2� bb

; 0 6 bn 6 b; 8n P n1 ð3:51Þ

and

1 <d2

D2<

2� qq

; 0 6 qn 6 q; 8n P n1: ð3:52Þ


Proof. Let (L1,L2) 2 (b(d1 + D1) + d1 � D1, d1 + D1) � (q(d2 + D2) + d2 � D2,d2 + D2). It follows from (3.25) and (3.26) that thereexist h 2 (0,1) and T P 1 + n0 + n1 + s + r + jbj satisfying (3.40),

X1i¼T

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ < minfd1 þ D1 � L1; L1 � bðd1 þ D1Þ � ðd1 � D1Þg ð3:53Þ

and

X1i¼T

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ < minfd2 þ D2 � L2; L2 � qðd2 þ D2Þ � ðd2 � D2Þg: ð3:54Þ

Let h�; h�; AL1 ; BL2 and SL1 ;L2 be defined by (3.14) and (3.43)–(3.46), respectively. It follows from (3.4), (3.45), (3.46), (3.53)and (3.54) that for any ðx; yÞ ¼ fðxn; ynÞgn2Zb


AL1 ðxn; ynÞ ¼ L1 � bnxn�s þX1i¼n

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i6 L1 þ

X1i¼n

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

6 L1 þmin d1 þ D1 � L1; L1 � bðd1 þ D1Þ � ðd1 � D1Þf g 6 d1 þ D1;


1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h iP L1 � bðd1 þ D1Þ �

X1i¼n

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

P L1 � bðd1 þ D1Þ �min d1 þ D1 � L1; L1 � bðd1 þ D1Þ � ðd1 � D1Þf gP d1 � D1;


BL2 ðxn; ynÞ ¼ L2 � qnyn�r þX1i¼n

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i6 L2 þ

X1i¼n

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

6 L2 þmin d2 þ D2 � L2; L2 � qðd2 þ D2Þ � ðd2 � D2Þf g 6 d2 þ D2

and


1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h iP L2 � qðd2 þ D2Þ �

X1i¼n

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

P L2 � qðd2 þ D2Þ �min d2 þ D2 � L2; L2 � qðd2 þ D2Þ � ðd2 � D2Þf gP d2 � D2;

which imply (3.15). The rest of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof. h

Theorem 3.5 below involves the case b 6 bn 6 0 and q 6 qn 6 0.

Theorem 3.5. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; b; q 2 R� and four nonnegative sequencesfUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.25) and (3.26),

1 <d1

D1<

bþ 2�b

; b 6 bn 6 0; 8n P n1 ð3:55Þ

and

1 <d2

D2<

qþ 2�q

; q 6 qn 6 0; 8n P n1: ð3:56Þ


Proof. Let (L1,L2) 2 (d1 � D1, (1 + b)(d1 + D1)) � (d2 � D2, (1 + q)(d2 + D2)). It follows from (3.25) and (3.26) that there existh 2 (0,1) and T P 1 + n0 + n1 + s + r + jbj satisfying (3.40),

X1i¼T

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ < minfð1þ bÞðd1 þ D1Þ � L1; L1 � ðd1 � D1Þg ð3:57Þ

and
X1i¼T
1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ < minfð1þ qÞðd2 þ D2Þ � L2; L2 � ðd2 � D2Þg: ð3:58Þ

Let h�; h�; AL1 ; BL2 and SL1 ;L2 be defined by (3.14) and (3.43)–(3.46), respectively. It follows from (3.4), (3.45), (3.46), (3.57)and (3.58) that for any x ¼ fxngn2Zb

2 Xðd1;D1Þ; y ¼ fyngn2Zb2 Xðd2;D2Þ and n P T


1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i6 L1 � bðd1 þ D1Þ þ

X1i¼n

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

6 L1 � bðd1 þ D1Þ þminfð1þ bÞðd1 þ D1Þ � L1; L1 � ðd1 � D1Þg 6 d1 þ D1;


1ai

Xi�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj

h iP L1 �

X1i¼n

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

P L1 �minfð1þ bÞðd1 þ D1Þ � L1; L1 � ðd1 � D1ÞgP d1 � D1;


1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i6 L2 � qðd2 þ D2Þ þ

X1i¼n

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

6 L2 � qðd2 þ D2Þ þmin ð1þ qÞðd2 þ D2Þ � L2; L2 � ðd2 � D2Þf g 6 d2 þ D2

and


1pi

Xi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj

h iP L2 �

X1i¼n

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

P L2 �min ð1þ qÞðd2 þ D2Þ � L2; L2 � ðd2 � D2Þf gP d2 � D2;

which lead to (3.15). The rest of the proof is similar to that of Theorem 3.3 and is omitted. This completes the proof. h

Theorem 3.6 below deals with the case 1 < b⁄ 6 bn 6 b⁄ and 1 < q⁄ 6 qn 6 q⁄.


Theorem 3.6. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b�; b�; q�; q� 2 Rþ n f0g and four nonnegativesequences fUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.25) and (3.26),

d1 > D1; b2�b� þ b�b

�2 � b�2 � b2�

D1 > b�2 � b2

� � b2�b� þ b�b

�2

d1; 1 < b� 6 bn 6 b�; 8n P n1 ð3:59Þ

and

d2 > D2; q2�q� þ q�q

�2 � q�2 � q2�

� �D2 > q�2 � q2

� � q2�q� þ q�q

�2� �d2; 1 < q� 6 qn 6 q�; 8n P n1: ð3:60Þ


Proof. Let ðL1;L2Þ 2 b�

b�ðd1þD1Þþb�ðd1�D1Þ;b�ðd1þD1Þþ b�

b� ðd1�D1Þ

� q�

q�ðd2þD2Þþq�ðd2�D2Þ;q�ðd2þD2Þþ q�

q� ðd2�D2Þ

.It follows from (3.25) and (3.26) that there exist h 2 (0,1) and T P 1 + n0 + n1 + s + r + jbj satisfying

h ¼max1jb�j

1þX1i¼T

1jaij

Xi�1

j¼n0

Uj

!;

1jq�j

1þX1i¼T

1jpij

Xi�1

j¼n0

Vj

!( ); ð3:61Þ

X1i¼T

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ < min b�ðd1 þ D1Þ þb�b�ðd1 � D1Þ � L1;

b�L1

b�� ðd1 þ D1Þ � b�ðd1 � D1Þ

� �ð3:62Þ

and

X1i¼T

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ < min q�ðd2 þ D2Þ þq�q�ðd2 � D2Þ � L2;

q�L2

q�� ðd2 þ D2Þ � q�ðd2 � D2Þ

� �: ð3:63Þ

Let

h� ¼ 1jb�j

1þX1i¼T

1jaij

Xi�1

j¼n0

Uj

!ð3:64Þ

and

h� ¼1jq�j

1þX1i¼T

1jpij

Xi�1

j¼n0

Vj

!: ð3:65Þ

Define three mappings AL1 ;BL2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1 ;L2 : Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1b by (3.14),

AL1 ðxn; ynÞ ¼L1

bnþs� xnþs

bnþsþ 1

bnþs

P1i¼nþs

1ai

Pi�1

j¼n0


; yw1j; . . . ; ywkj



8><>: ð3:66Þ

and

BL2 ðxn; ynÞ ¼L2

qnþr� ynþr

qnþsþ 1

qnþr

P1i¼nþr

1pi

Pi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; n P T;


8><>: ð3:67Þ

for any ðx; yÞ ¼ fðxn; ynÞgn2Zb2 Xðd1;D1Þ �Xðd2;D2Þ.

Now we prove that (3.15) and (3.16) hold. It follows from (3.4), (3.59), (3.60), (3.62), (3.63), (3.66) and (3.67) that for eachðx; yÞ ¼ fðxn; ynÞgn2Zb


AL1 ðxn; ynÞ ¼L1

bnþs� xnþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i

6L1

b�� d1 � D1

b�þ 1

b�

X1i¼nþs

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

6L1

b�� d1 � D1

b�þ 1

b�min b�ðd1 þ D1Þ þ

b�b�ðd1 � D1Þ � L1;

b�L1


� �6 d1 þ D1;


AL1 ðxn; ynÞ ¼L1

bnþs� xnþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i

PL1

b�� d1 þ D1

b�� 1

b�

X1i¼nþs

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

PL1

b�� d1 þ D1

b�� 1

b�min b�ðd1 þ D1Þ þ

b�b�ðd1 � D1Þ � L1;

b�L1


� �P d1 � D1;

BL2 ðxn; ynÞ ¼L2

qnþr� ynþr

qnþrþ 1

qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i

6L2

q�� d2 � D2

q�þ 1

q�

X1i¼nþr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

6L2

q�� d2 � D2

q�þ 1

q�min q�ðd2 þ D2Þ þ

q�q�ðd2 � D2Þ � L2;

q�L2


� �6 d2 þ D2

and

BL2 ðxn; ynÞ ¼L2

qnþr� ynþr

qnþrþ 1

qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i

PL2

q�� d2 þ D2

q�� 1

q�

X1i¼nþr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

PL2

q�� d2 þ D2

q�� 1

q�min q�ðd2 þ D2Þ þ

q�q�ðd2 � D2Þ � L2;

q�L2


� �P d2 � D2;

which yield (3.15). Clearly (3.16) follows from (3.14) and (3.15).Next we prove that (3.18) holds. In light of (3.3), (3.61) and (3.64)–(3.67), we infer that for any



jAL1 ðxn;ynÞ�AL1 ð�xn;�ynÞj ¼ �xnþs� �xnþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j;xh1j; . . . ;xhkj

;yw1j; . . . ;ywkj

� f j;�xh1j

; . . . ;�xhkj;�yw1j

; . . . ;�ywkj

h i��

6jxnþs��xnþsjjbnþsj

þ 1jbnþsj

X1i¼nþs

1jaijXi�1

j¼n0

Uj max jxhlj��xhlj

j; jywlj� �ywlj

j : 16 l6 kn o

6kx� �xk

b�þ 1

b�

X1i¼nþs

1jaijXi�1

j¼n0

Uj max kx� �xk;ky� �ykf g6 h�max kx��xk;ky� �ykf g

and

jBL2 ðxn;ynÞ�BL2 ð�xn;�ynÞj ¼ �ynþr� �ynþr

qnþrþ 1

qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j;xs1j; . . . ;xskj

;yt1j; . . . ;ytkj

� gðj;�xs1j

; . . . ;�xskj;�yt1j

; . . . ;�ytkjÞ

h i��

6jynþr� �ynþrjjqnþrj

þ 1jqnþrj

X1i¼nþr

1jpij

Xi�1

j¼n0

Vj max jxslj��xslj

j; jytlj� �ytlj

j : 16 l6 kn o

6ky� �yk

q�þ 1

q�

X1i¼nþr

1jpij

Xi�1

j¼n0

Vj max kx� �xk;ky� �ykf g6 h�max kx��xk;ky� �ykf g;

which together with (3.15) give that




6 max h� maxfkx� �xk; ky� �ykg; h� maxfkx� �xk; ky� �ykgf g¼ hkðx; yÞ � ð�x; �yÞk1; 8ðx; yÞ; ð�x; �yÞ 2 Xðd1;D1Þ �Xðd2;D2Þ;


which leads to (3.18). It follows from (3.16) and (3.18) that SL1 ;L2 is a contraction mapping in X(d1,D1) �X(d2,D2) and it has aunique fixed point ðx; yÞ ¼ fðxn; ynÞgn2Zb


xn ¼L1

bnþs� xnþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj�; 8n P T

and

yn ¼L2

qnþr� ynþr

qnþrþ 1

qnþr

X1i¼nþr

1pi

Xi�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; 8n P T;

which yield that

Dðxn þ bnxn�sÞ ¼ �1an

Xn�1

j¼n0


; yw1j; . . . ; ywkj

Þ � cj�; 8n P T þ s

and

Dðyn þ qnyn�rÞ ¼ �1pn

Xn�1

j¼n0


; yt1j; . . . ; ytkj

Þ � rj�; 8n P T þ r;

which imply that

D½anDðxn þ xn�sÞ� ¼ �f ðn; xh1n; . . . ; xhkn

; yw1n; . . . ; ywkn


and

D½pnDðyn þ yn�rÞ� ¼ �gðn; xs1n ; . . . ; xskn; yt1n

; . . . ; ytknÞ þ rj; 8n P T þ sþ r;

that is, ðx; yÞ ¼ fðxn; ynÞgn2Zbis a bounded positive solution of the system (1.1) in X(d1,D1) �X(d2,D2).

Let (L11,L21), (L12,L22) 2 (d1 � (1 � b)D1 + bd1,d1 + (1 � b)D1 � bd1) � (d2 � (1 � q)D2 + bd2,d2 + (1 � q)D2 � qd2) withmax{jL11 � L12j, jL21 � L22j} > 0. Similarly we conclude that for each l 2 {1,2}, there exist constants hl; h�l ; h�l 2 ð0;1Þ;Tl P 1þ n0 þ n1 þ sþ rþ jbj and mappings AL1l

; BL2l: Xðd1;D1Þ �Xðd2;D2Þ ! l1b and SL1l ;L2l

: Xðd1;D1Þ �Xðd2;D2Þ ! l1b � l1bsatisfying (3.14) and (3.61)–(3.67), where h; h�; h�; T; L1; L2; AL1 ; BL2 and SL1;L2 are replaced by hl; h�l ; h�l; Tl; L1l;

L2l; AL1l; BL2l



lnÞgn2Zb

2 Xðd1;D1Þ �Xðd2;D2Þ, which is abounded positive solution of the system (1.1), that is,

xln ¼

L1l

bnþs� xl

nþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j; xlh1j; . . . ; xl

hkj; yl

w1j; . . . ; yl

wkj

� cj

h i; 8n P Tl; l 2 f1;2g ð3:68Þ

and

yln ¼

L2l

qnþr� yl

nþrqnþr

þ 1qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j; xls1j; . . . ; xl

skj; yl

t1j; . . . ; yl

tkj

� rj

h i; 8n P Tl; l 2 f1;2g: ð3:69Þ

Using (3.3), (3.59)–(3.61), (3.64), (3.65), (3.68) and (3.69), we derive that for every n P max{T1,T2}

jx1n � x2

nj ¼L11 � L12

bnþs� x1

nþs � x2nþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j; x1h1j; . . . ; x1

hkj; y1

w1j; . . . ; y1

wkj

� f j; x2

h1j; . . . ; x2

hkj; y2

w1j; . . . ; y2

wkj

h i��

PjL11 � L12jjbnþsj

�jx1

nþs � x2nþsj

jbnþsj� 1jbnþsj

X1i¼nþs

1jaij

Xi�1

j¼n0

Uj max jx1hlj� x2

hljj; jy1

wlj� y2

wljj : 1 6 l 6 k

n o

PjL11 � L12j

b�� kx

1 � x2kb�

� 1b�

X1i¼nþs

1jaij

Xi�1

j¼n0


PjL11 � L12j

b�� 1þ 1

b�

X1i¼maxfT1 ;T2g

1jaij

Xi�1

j¼n0

Uj

!maxfkx1 � x2k; ky1 � y2kg

PjL11 � L12j

b��maxfh�1; h

�2gkðx1; y1Þ � ðx2; y2Þk1


and

jy1n � y2

nj ¼L21 � L22

qnþr�

y1nþs � y2

nþsqnþr

þ 1qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j; x1s1j; . . . ; x1

skj; y1

t1j; . . . ; y1

tkj

� g j; x2

s1j; . . . ; x2

skj; y2

t1j; . . . ; y2

tkj

h i��

PjL21 � L22jjqnþrj

� jy1nþs � y2

nþsjjqnþrj

� 1jqnþrj

X1i¼nþr

1jpij

Xi�1

j¼n0

Vj max jx1slj� x2

sljj; jy1

tlj� y2

tljj : 1 6 l 6 k

n o

PjL21 � L22j

q�� ky

1 � y2kq�

� 1q�

X1i¼nþr

1jpij

Xi�1

j¼n0


PjL21 � L22j

q�� 1þ 1

q�

X1i¼maxfT1 ;T2g

1jpij

Xi�1

j¼n0

Vj

!max kx1 � x2k; ky1 � y2k

�

PjL21 � L22j

q��maxfh�1; h�2gkðx1; y1Þ � ðx2; y2Þk1;

which yield that

kðx1;y1Þ�ðx2;y2Þk1 P max supn2NmaxfT1 ;T2g

jx1n�x2


jy1n�y2

nj( )

P maxjL11�L12j

b��maxfh�1;h

�2gkðx1;y1Þ�ðx2;y2Þk1;

jL21�L22jq�

�maxfh�1;h�2gkðx1;y1Þ�ðx2;y2Þk1

� �

P maxjL11�L12j

b�;jL21�L22j

q�

� ��maxfh1;h2gkðx1;y1Þ�ðx2;y2Þk1;

which means that

kðx1; y1Þ � ðx2; y2Þk1 Pmax jL11�L12 j

b� ; jL21�L22 jq�

n o1þmaxfh1; h2g

> 0;

that is, (x1,y1) – (x2,y2). Therefore the system (1.1) possesses uncountably many bounded positive solutions inX(d1,D1) �X(d2,D2). This completes the proof. h

Theorem 3.7 below investigates the case b⁄ 6 bn 6 b⁄ < �1 and q⁄ 6 qn 6 q⁄ < �1.

Theorem 3.7. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; b�; b�; q�; q� 2 R� and four nonnegativesequences fUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.25), (3.26),

d1 > D1; D1ð2þ b� þ b�Þ < d1ðb� � b�Þ; b� 6 bn 6 b� < �1; 8n P n1 ð3:70Þ

and

d2 > D2; D2ð2þ q� þ q�Þ < d2ðq� � q�Þ; q� 6 qn 6 q� < �1; 8n P n1: ð3:71Þ


Proof. Let (L1,L2) 2 ((1 + b⁄)(d1 + D1), (1 + b⁄)(d1 � D1)) � ((1 + q⁄)(d2 + D2), (1 + q⁄)(d2 � D2)). It follows from (3.25) and (3.26)that there exist h 2 (0,1) and T P 1 + n0 + n1 + s + r + jbj satisfying (3.61),

X1i¼T

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ < min L1 � ð1þ b�Þðd1 þ D1Þ; b� 1þ 1b�

� �ðd1 � D1Þ �

L1

b�

� �� ð3:72Þ

and

X1i¼T

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ < min L2 � ð1þ q�Þðd2 þ D2Þ; q� 1þ 1q�

� �ðd2 � D2Þ �

L2

q�

� �� : ð3:73Þ


Let h�; h�; AL1 ; BL2 and SL1 ;L2 be defined by (3.14) and (3.64)–(3.67), respectively. It follows from (3.4), (3.66), (3.67), (3.72)and (3.73) that for any ðx; yÞ ¼ fðxn; ynÞgn2Zb


AL1 ðxn; ynÞ ¼L1

bnþs� xnþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i

6L1

b�� d1 þ D1

b�� 1

b�X1

i¼nþs

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

6L1

b�� d1 þ D1

b�� 1

b�min L1 � ð1þ b�Þðd1 þ D1Þ; b� 1þ 1

b�

� �ðd1 � D1Þ �

L1

b�

� �� 6 d1 þ D1;

AL1 ðxn; ynÞ ¼L1

bnþs� xnþs

bnþsþ 1

bnþs

X1i¼nþs

1ai

Xi�1

j¼n0

f j; xh1j; . . . ; xhkj

; yw1j; . . . ; ywkj

� cj

h i

PL1

b�� d1 � D1

b�þ 1

b�X1

i¼nþs

1jaij

Xi�1

j¼n0

ðFj þ jcjjÞ

PL1

b�� d1 � D1

b�þ 1

b�min L1 � ð1þ b�Þðd1 þ D1Þ; b� 1þ 1

b�

� �ðd1 � D1Þ �

L1

b�

� �� P d1 � D1;

BL2 ðxn; ynÞ ¼L2

qnþr� ynþr

qnþrþ 1

qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i

6L2

q�� d2 þ D2

q�� 1

q�X1

i¼nþr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

6L2

q�� d2 þ D2

q�� 1

q�min L2 � ð1þ q�Þðd2 þ D2Þ; q� 1þ 1

q�

� �ðd2 � D2Þ �

L2

q�

� �� 6 d2 þ D2

and

BL2 ðxn; ynÞ ¼L2

qnþr� ynþr

qnþrþ 1

qnþr

X1i¼nþr

1pi

Xi�1

j¼n0

g j; xs1j; . . . ; xskj

; yt1j; . . . ; ytkj

� rj

h i

PL2

q�� d2 � D2

q�þ 1

q�m1

i¼nþr

1jpij

Xi�1

j¼n0

ðGj þ jrjjÞ

PL2

q�� d2 � D2

q�þ 1

q�min L2 � ð1þ q�Þðd2 þ D2Þ; q� 1þ 1

q�

� �ðd2 � D2Þ �

L2

q�

� �� P d2 � D2;

which yields (3.15). The rest of the proof is similar to that of Theorem 3.6 and is omitted. This completes the proof. h



and fGngn2Nn0satisfying (3.1), (3.3)–(3.5), (3.24) and (3.26). Then the system (1.1) has uncount-

ably many bounded positive solutions in X(d1,D1) �X(d2,D2).

Theorem 3.9. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; q 2 Rþ n f0g and four nonnegative sequencesfUngn2Nn0

, fVngn2Nn0; fFngn2Nn0







Theorem 3.11. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; q 2 R� and four nonnegative sequencesfUngn2Nn0





Theorem 3.12. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; q�; q� 2 Rþ n f0g and four nonnegative sequencesfUngn2Nn0

; fVngn2Nn0, fFngn2Nn0



Theorem 3.13. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; q�; q� 2 R� and four nonnegativesequences fUngn2Nn0


and fGngn2Nn0satisfying (3.1), (3.3)–(3.5), (3.26) and (3.71). Then the system (1.1)

has uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).



and fGngn2Nn0satisfying (3.3), (3.4), (3.23), (3.25), (3.26) and (3.39). Then the system (1.1)



, fVngn2Nn0; fFngn2Nn0

and fGngn2Nn0satisfying (3.3), (3.4), (3.23), (3.25), (3.26) and (3.52). Then the system (1.1) has

uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).

Theorem 3.16. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; q 2 R� and four nonnegative sequencesfUngn2Nn0




Theorem 3.17. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; q�; q� 2 Rþ n f0g and four nonnegative sequencesfUngn2Nn0




Theorem 3.18. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; q�; q� 2 R� and four nonnegativesequences fUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.23), (3.25), (3.26) and (3.71). Then the system

(1.1) has uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).

Theorem 3.19. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b; q 2 Rþ n f0g and four nonnegative sequencesfUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.25), (3.26), (3.38) and (3.52). Then the system (1.1)


Theorem 3.20. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b 2 Rþ n f0g; q 2 R� and four nonnegativesequences fUngn2Nn0




Theorem 3.21. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b; q�; q� 2 Rþ n f0g and four nonnegativesequences fUngn2Nn0




Theorem 3.22. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b 2 Rþ n f0g; q�; q� 2 R� and four nonnegativesequences fUngn2Nn0




Theorem 3.23. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b 2 Rþ n f0g; q 2 R� and four nonnegativesequences fUngn2Nn0




Theorem 3.24. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b; q�; q� 2 Rþ n f0g and four nonnegativesequences fUngn2Nn0




Theorem 3.25. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b 2 Rþ n f0g; q�; q� 2 R� and four nonnegativesequences fUngn2Nn0





Theorem 3.26. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; q�; q� 2 Rþ n f0g; b 2 R� and four nonnegativesequences fUngn2Nn0




Theorem 3.27. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2 2 Rþ n f0g; b; q�; q� 2 R� and four nonnegativesequences fUngn2Nn0




Theorem 3.28. Assume that there exist constants n1 2 Nn0 ; d1; d2; D1; D2; b�; b� 2 Rþ n f0g; q�; q� 2 R� and four nonnega-tive sequences fUngn2Nn0


and fGngn2Nn0satisfying (3.3), (3.4), (3.25), (3.26), (3.59) and (3.71). Then the sys-

tem (1.1) has uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).

Remark 3.1. Theorems 3.1–3.7 generalize, improve and unify Theorem 1 in [12] and Theorems 2.1–2.7 in [18], respectively.The examples in Section 4 reveal that Theorems 3.1–3.7 extend authentically Theorem 1 in [12] and Theorems 2.1–2.7 in[18].

4. Examples and applications

Now we construct seven examples as applications of the main results presented in Section 3. Note that Theorem 1 in [12]and Theorems 2.1–2.7 in [18] are invalid for Examples 4.1–4.7, respectively.

Example 4.1. Consider the second order nonlinear neutral delay system of difference equations

D ð�1Þn�1n5Dðxn � xn�sÞh i

þ nx2nþ6y2n�3þð1�nÞx5n2�1y3n�9

n7þ5n3þ1 ¼ 2nþð�1Þn ln nn5þ3n4þ5nþ1 ; n P 2;

D ð�1Þnn3 1þ 1n

� �nDðyn � yn�rÞ

h iþ

2yn2þ9�nx23n2�7

y38n�3

n6þx26n3�9

¼ n�sin nn4þ3n2þ2n2þ5 ; n P 2;

8><>: ð4:1Þ

where n0 = 2 and s;r 2 N are fixed. Let n1 = k = 2, d1 = 6, D1 = 2, d2 = 10, D2 = 6, b = min{2 � s,2 � r,�3},

an ¼ ð�1Þn�1n5; pn ¼ ð�1Þnn3 1þ 1n

� �n

; bn ¼ qn ¼ �1; cn ¼2nþ ð�1Þn ln n

n5 þ 3n4 þ 5nþ 1;

rn ¼n� sin n

n4 þ 3n2 þ 2n2 þ 5; h1n ¼ 2nþ 6; h2n ¼ 5n2 � 1; w1n ¼ 2n� 3;w2n ¼ 3n� 9;

s1n ¼ 3n2 � 7; s2n ¼ 6n3 � 9; t1n ¼ n2 þ 9; t2n ¼ 8n� 3;

f ðn;u;v ; �u; �vÞ ¼ nu�uþ ð1� nÞv �vn7 þ 5n3 þ 1

; gðn;u;v ; �u; �vÞ ¼ 2�u� nu2 �v3

n6 þ v2 ; Fn ¼256n6 ;

Gn ¼262144nþ 32

n6 ; Un ¼48n6 ; Vn ¼

106496n7 þ 2n6 þ 8388608nþ 640

ðn6 þ 16Þ2;8ðn;u;v ; �u; �vÞ 2 Nn0 � R4:

Obviously (3.1)–(3.6) hold. Hence Theorem 3.1 yields that the system (4.1) possesses uncountably many bounded positivesolutions in X(d1,D1) �X(d2,D2).


D ðnþ 1Þðnþ 2Þffiffiffiffiffiffiffiffiffiffiffiffinþ 3p

Dðxn þ xn�sÞ� �

þ n2�ðnþ3Þx23n�10

n8þy44n�9

¼ ð�1Þnffiffiffiffiffiffiffiffi3n�5p

n3þ3n�1 ; n P 3;

D n2ð1� 2nÞDðyn þ yn�rÞ� �

þ4nx3

n2�5�y2

nþ1

n5þ1 ¼ n cosðn3�2nÞn4þ2n3þ1 ; n P 3;

8><>: ð4:2Þ

where n0 = 3 and s;r 2 N are fixed. Let n1 = 3, k = 1, d1 = 3, D1 = 1, d2 = 4, D2 = 2, b = min{3 � s,3 � r,�1},

an ¼ ðnþ 1Þðnþ 2Þffiffiffiffiffiffiffiffiffiffiffiffinþ 3

p; pn ¼ n2ð1� 2nÞ; bn ¼ qn ¼ 1; cn ¼

ð�1Þnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3n� 5p

n3 þ 3n� 1;

rn ¼n cosðn3 � 2nÞn4 þ 2n3 þ 1

; h1n ¼ 3n� 10; w1n ¼ 4n� 9; s1n ¼ 2n2 � 5; t1n ¼ nþ 1;


f ðn;u;vÞ ¼ n2 � ðnþ 3Þu2

n8 þ v4 ; gðn; u;vÞ ¼ 4nu3 � v2

n5 þ 1; Fn ¼

nþ 32n7 ; Gn ¼

300n4 ;

Un ¼ð8n8 þ 24192Þðnþ 3Þ þ 864n2

ðn8 þ 1296Þ2; Vn ¼

192nþ 12n5 ; 8ðn;u;vÞ 2 Nn0 � R2:

It is easy to show that (3.3), (3.4) and (3.23)–(3.26) hold. It follows from Theorem 3.2 that the system (4.2) possessesuncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).


D ð�1Þn�1n2D xn þ 2n cosðn3�6Þ5nþ3 xn�s

h iþ

ffiffiffiffiffiffiffiffiffiffiffiffin2�n�1p

n3þjxn�2ynþ3 j¼ ð�1Þnn3

n7þ1 ; n P 1;

D nðnþ 1ÞD yn þ 3n2 sinðn5�nþ1Þ15n2þ1 yn�r

h iþ n�100

n4þx2n�5y6

nþ6¼ n3 � 5n2 þ 1� �

sin ð�1Þnn5þ1 ; n P 1;

8><>: ð4:3Þ

where n0 = 1 and s; r 2 N are fixed. Let n1 ¼ k¼ 1; d1 ¼ 4; D1 ¼ 3; d2 ¼ 3; D2 ¼ 1; b¼minf1� s;1�r;�4g; b¼ 25 ; q¼ 1

5,

an ¼ ð�1Þn�1n2; pn ¼ nðnþ 1Þ; bn ¼2n cosðn3 � 6Þ

5nþ 3; qn ¼

3n2 sinðn5 � nþ 1Þ15n2 þ 1

;

cn ¼ð�1Þnn3

n7 þ 1; rn ¼ n3 � 5n2 þ 1

� �sinð�1Þn

n5 þ 1; h1n ¼ n� 2; w1n ¼ nþ 3; s1n ¼ n� 5;

t1n ¼ nþ 6; f ðn; u;vÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 � nþ 1p

n3 þ juv j ; gðn; u;vÞ ¼ n� 100n4 þ u2v6 ; Fn ¼

1n2 ; Gn ¼

nþ 100n4 ;

Un ¼11

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 � nþ 1p

ðn3 þ 2Þ2; Vn ¼

358400ðnþ 100Þðn4 þ 64Þ2

; 8ðn;u;vÞ 2 Nn0 � R2:

Clearly (3.3), (3.4), (3.25), (3.26), (3.38) and (3.39) hold. Consequently Theorem 3.3 gives that the system (4.3) possessesuncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).


D nðn3 � 2ÞD xn þ 3n2

4n2þ6n�5 xn�s

h iþ 1�nyn2þ7

n5þðnþ1Þx22n�9¼ n6�7n5�3n3þ1

n9þ5n4þ1 ; n P 2;

D ð�1Þn�1nffiffiffiffiffiffiffiffiffiffiffiffinþ 3p

D yn þ2n6 sin 1

n2þ13n4þ1 yn�r

� �� þ

ffiffinp

xn2�2nþny2nþ1

n6þx2n2�2n

þ2y22nþ1¼ ð�1Þn

nln2n; n P 2;

8>><>>: ð4:4Þ

where n0 = 2 and s; r 2 N are fixed. Let n1 = 2, k ¼ 1; d1 ¼ 4; D1 ¼ 3; d2 ¼ 9; D2 ¼ 6; b ¼minf2� s;2� r;�5g;b ¼ 3

4 ; q ¼ 23,

an ¼ nðn3 � 2Þ; pn ¼ nffiffiffiffiffiffiffiffiffiffiffiffinþ 3

p; bn ¼

3n2

4n2 þ 6n� 5; qn ¼

2n6 sin 1n2þ1

3n4 þ 1;

cn ¼n6 � 7n5 � 3n3 þ 1

n9 þ 5n4 þ 1; rn ¼

ð�1Þn

nln2n; h1n ¼ 2n� 9; w1n ¼ n2 þ 7;

s1n ¼ n2 � 2n; t1n ¼ 2nþ 1; f ðn;u; vÞ ¼ 1� nvn5 þ ðnþ 1Þu2 ; gðn;u;vÞ ¼

ffiffiffinp

uþ nvn6 þ u2 þ 2v2 ;

Fn ¼15nþ 1

n5 ; Gn ¼15nþ 7

ffiffiffinp

n6 ; Un ¼n6 þ 14nþ 273

n10 ; Vn ¼n7 þ n6

ffiffiffinpþ 709nþ 815n12 ;

8ðn;u;vÞ 2 Nn0 � R2:


Obviously (3.3), (3.4), (3.25), (3.26), (3.51) and (3.52) hold. Thus Theorem 3.4 guarantees that the system (4.4) possessesuncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).


D nð1� 2nÞD xn � 4n3þ15n3þ2 xn�s

h iþ 1�3nx2

3n�10yn�7

n5þ1 ¼ ð�1Þnn3þn�1n8þ3n5þ2 ; n P 2;

D ðn2 � 1ÞD yn � 5n9þ16n9þ5 yn�r

h iþ 3n2�sin2ðn�1Þ

n4þx2n�6þnjy5n�4 j

¼ n4�n3 sin n�4n3þ1n7þn5þ3n2þ1 ; n P 2;

8><>: ð4:5Þ

where n0 = 2 and s; r 2 N are fixed. Let n1 ¼ 3; k ¼ 1; d1 ¼ 4; D1 ¼ 3; d2 ¼ 6; D2 ¼ 5; b ¼minf2� s;2� r;�5g;b ¼ � 4

5 ; q ¼ � 56,

an ¼ nð1� 2nÞ; pn ¼ ðn2 � 1Þ; bn ¼ �4n3 þ 15n3 þ 2

; qn ¼ �5n9 þ 16n9 þ 5

; cn ¼ð�1Þnn3 þ n� 1

n8 þ 3n5 þ 2;

rn ¼n4 � n3 sin n� 4n3 þ 1

n7 þ n5 þ 3n2 þ 1; h1n ¼ 3n� 10; w1n ¼ n� 7; s1n ¼ n� 6; t1n ¼ 5n� 4;

f ðn;u;vÞ ¼ 1� 3nu2vn5 þ 1

; gðn; u;vÞ ¼ 3n2 � sin2ðn� 1Þn4 þ u2 þ njvj ; Fn ¼

1617nþ 1n5 þ 1

;

Gn ¼3n2 þ 1

n4 þ nþ 1; Un ¼

609nn5 þ 1

; Vn ¼ð3n2 þ 1Þðnþ 14Þðn4 þ nþ 1Þ2

; 8ðn;u;vÞ 2 Nn0 � R2:

It is easy to show that (3.3), (3.4), (3.25), (3.26), (3.55) and (3.56) hold. Hence Theorem 3.5 ensures that the system (4.5) pos-sesses uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).


D nln3nD xn þ 5n2

n2�n�6 xn�s

h iþ

n3�x22n2 y2n�9

n7þ7n6þ1 ¼ð�1Þnn5�nþ1

n8þ4n3þ1 ; n P 4;

D ð�1Þn2�1ðn� 2Þðn� 3ÞD yn þ 25n13þ6

5n13þ1 yn�r

h iþ n ln 1þx2

3ny4n�3ð Þ

n6þ3n4þ1 ¼ n3ln3nn9þ5n6þ3 ; n P 4;

8><>: ð4:6Þ

where n0 = 4 and s; r 2 N are fixed. Let n1 = 10, k = 1, d1 = 2, D1 = 1,d2 = 3, D2 = 2, b = min{4 � s,4 � r,�1}, b⁄ = 5, b⁄ = 6,q⁄ = 5, q⁄ = 6,

an ¼ nln3n; pn ¼ ð�1Þn2�1ðn� 2Þðn� 3Þ; bn ¼

5n2

n2 � n� 6; qn ¼

25n13 þ 65n13 þ 1

;

cn ¼ð�1Þnn5 � nþ 1

n8 þ 4n3 þ 1; rn ¼

n3ln3nn9 þ 5n6 þ 3

; h1n ¼ 2n2; w1n ¼ 2n� 9;

s1n ¼ 3n; t1n ¼ n� 3; f ðn;u; vÞ ¼ n3 � u2vn7 þ 7n6 þ 1

; gðn;u;vÞ ¼n ln 1þ u2v4

� �n6 þ 3n4 þ 1

;

Fn ¼n3 þ 45

n7 ; Gn ¼ln 5626

n5 ; Un ¼39n7 ; Vn ¼

8250n5 ; 8ðn;u;v ; Þ 2 Nn0 � R2:

It is easy to prove that (3.3), (3.4), (3.25), (3.26), (3.59) and (3.60) hold. Therefore Theorem 3.6 yields that the system (4.6)possesses uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).


D ð�1Þnn2ðnþ 3ÞD xn � 3þsinðn3þ1Þ2þsinðn3þ1Þ

h iþ 1�nyn�3

n3þx2n�5¼ n4�ð�1Þn�1n3þ5nþ1

n8þ3n5þ1 ; n P 1;

D ð�1Þn�1ðnþ 2Þðnþ 5ÞD yn � 3n2þ4n2þ2 yn�r

h iþ

x3n2�n

�n2yn�1

n4þ3n3þ1 ¼n3�5n2þ1n4ln3ðnþ1Þ

; n P 1;

8><>: ð4:7Þ


where n0 = 1 and s; r 2 N are fixed. Let n1 = 1, k = 1, d1 = 2, D1 = 1, d2 = 4, D2 = 2, b = min{1 � s,1 � r,�4} andb� ¼ � 4

3 ; b� ¼ �3; q� ¼ �2; q� ¼ �3,

an ¼ ð�1Þnn2ðnþ 3Þ; pn ¼ ð�1Þn�1ðnþ 2Þðnþ 5Þ; bn ¼ �3þ sinðn3 þ 1Þ2þ sinðn3 þ 1Þ ;

qn ¼ �3n2 þ 4n2 þ 2

; cn ¼n4 � ð�1Þn�1n3 þ 5nþ 1

n8 þ 3n5 þ 1; rn ¼

n3 � 5n2 þ 1

n4ln3ðnþ 1Þ;

h1n ¼ n� 5; w1n ¼ n� 3; s1n ¼ n2 � n; t1n ¼ n� 1; f ðn;u; vÞ ¼ 1� nvn3 þ u2 ;

gðn;u;vÞ ¼ u3 � n2vn4 þ 3n3 þ 1

; Fn ¼6nþ 1n3 þ 1

; Gn ¼6n2 þ 27

n4 þ 3n3 þ 1; Un ¼

n4 þ 51ðn3 þ 1Þðn3 þ 2Þ ;

Vn ¼n2 þ 27

n4 þ 3n3 þ 1; 8ðn;u; vÞ 2 Nn0 � R2:

It is easy to verify that (3.3), (3.4), (3.25), (3.26), (3.70) and (3.71) hold. Thus Theorem 3.7 means that the system (4.7) pos-sesses uncountably many bounded positive solutions in X(d1,D1) �X(d2,D2).

Acknowledgements

The authors are indebted to the referee for carefully reading the paper and making useful comments and suggestions. Thisstudy was supported by research funds from Dong-A University.

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Existence of bounded positive solutions for a system of difference equations