Abstract and Applied Analysis

Volume 2012 (2012), Article ID 240378, 57 pages

doi:10.1155/2012/240378

Research Article

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

Zeqing Liu,¹ Shin Min Kang,² and Young Chel Kwun³

¹Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China

²Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

³Department of Mathematics, Dong-A University, Busan 614-714, Republic of Korea

Received 16 March 2012; Accepted 30 May 2012

Academic Editor: Josef Diblík

Copyright © 2012 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper studies the nonlinear two-dimensional difference system with multiple delays . Using the Banach fixed point theorem and a few new analysis techniques, we show the existence of uncountably many bounded positive solutions for the system, suggest Mann iterative algorithms with errors, and discuss the error estimates between the positive solutions and iterative sequences generated by the Mann iterative algorithms. Examples to illustrates the results are included.

1. Introduction and Preliminaries

In recent years, there has been an increasing interest in the study of oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, positive solutions, nonoscillatory solutions, and periodic solutions, respectively, for various difference equations and systems, see for example, [1–34] and the references cited therein.

Jiang and Tang [18] and Graef and Thandapani [14] studied the oscillation of the linear two-dimensional difference system

and nonlinear two-dimensional difference system

where , , and are nonnegative sequences, with and for all . Jiang and Tang [19] also gave some necessary and sufficient conditions for all solutions of System (1.2) to be oscillatory. Agarwal et al. [2] discussed the two-dimensional nonlinear difference system of the form

where , and are positive sequences, are increasing with and for all , and they provided a classification scheme of positive solutions for System (1.3) and established conditions for the existence of solutions with designated asymptotic behavior. Li [21] introduced the two-dimensional nonlinear difference system:

where , and are nonnegative sequences, with , and for all , he showed both classification schemes for nonoscillatory solutions of System (1.4) and gave necessary and sufficient conditions for the existence of these solutions. Huo and Li [15, 16] considered the nonlinear two-dimensional difference system

and the Emden-Fowler difference system

where , is a real sequence, is a nonnegative sequence, , with , and for all , they proved some oscillation results for Systems (1.5) and (1.6). Jiang and Li [17] investigated the nonlinear two-dimensional difference system:

where , is a nonnegative sequence, with for all , and and for all , they obtained some necessary and sufficient conditions for all solutions of System (1.7) to be oscillatory. Thandapani and Kumar [31] studied the oscillation for the nonlinear two-dimensional difference system of the neutral type

where , is a positive sequence, and are nonnegative sequences with , , are nondecreasing with and for all . Wu and Liu [33] established the existence and multiplicity of periodic solutions for the first-order neutral difference system

where , with , , and are positive -periodic sequences, , and are positive -periodic integer sequences. Tang [30] proved the existence results of a bounded nonoscillatory solution for the second-order linear delay difference equation

where and is a nonnegative sequence. Cheng [20] utilized the Banach fixed point theorem to discuss the existence of a nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients:

where , , and are nonnegative sequences with ,. M. Migda and J. Migda [29] obtained the asymptotic behavior of the second-order neutral difference equation:

where , , and . Liu et al. [27] studied the global existence of uncountably many bounded nonoscillatory solutions for the second-order nonlinear neutral delay difference equation:

where is a positive sequence, is a real sequence, with for , and is a mapping.

The purpose of this paper is to study the below nonlinear two-dimensional difference system with multiple delays

where , , and for , with for and with for . It is easy to see that the system (1.14) includes Systems (1.1)–(1.9), (1.10)–(1.13), and a lot of the first- and second-order nonlinear, half-linear, and quasilinear difference equations as special cases. Using the Banach fixed point theorem and some analysis techniques, we prove the existence of uncountably many bounded positive solutions for System (1.14), establish their iterative approximations, and discuss the error estimates between the positive solutions and iterative approximations. Our results sharp, and improve [14, Theorem 1] and [27, Theorems 2.1–2.7]. To illustrate our results, fifteen examples are also included.

Throughout this paper, we assume that is the forward difference operator defined by , , and stand for the sets of all integers, positive integers, and nonnegative integers, respectively,

denotes the Banach space of all bounded sequences on with norm

Let

for any and . It is easy to see that and are bounded closed and convex subsets of the Banach space with norm

By a solution of System (1.14), we mean a sequence with a positive integer such that System (1.14) is satisfied for all . A solution of System (1.14) is said to be positive if both components are positive.

Lemma 1.1 (see [35]). Let , and be four nonnegative real sequences satisfying the inequality

where , and . Then .

Lemma 1.2 (see [36]). Let , and be a nonnegative sequence. Then

2. Main Results

In this section, we investigate the existence of uncountably many bounded positive solutions for System (1.14) and their iterative approximations and the error estimates between the positive solutions and iterative approximations.

Theorem 2.1. Assume that there exist constants , , and four nonnegative sequences , and satisfying

Then,

(a) for each , there exist and such that for any , the Mann iterative sequence with errors generated by the schemes

 converges to a bounded positive solution of System (1.14) and has the error estimate:

 where and =,⊂ are arbitrary sequences with

(b) System (1.14) has uncountably many bounded positive solutions in .

Proof. (a) Let . Now we construct a contraction mapping and prove that its fixed point is a bounded positive solution of System (1.14). It follows from (2.3) and (2.4) that there exist and satisfying

Define three mappings by

for all . It follows from (2.1), (2.3), (2.12), (2.13), and Lemma 1.2 that and are well defined.

In view of (2.1), (2.2), and (2.10)–(2.14), we know that for any and ,

which give that

Observe that . It is easy to see that (2.12)–(2.17) give that is a contraction. Consequently, possesses a unique fixed point , which implies that

which yield that

which give that

that is, is a bounded positive solution of System (1.14) in .

In light of (2.5), (2.6), and (2.11)–(2.16), we arrive at

that is, (2.7) holds. It follows from (2.7)–(2.9) and Lemma 1.1 that .

(b) Let , and . As in the proof of (a), we similarly infer that for each , there exist a constant , a positive integer , and mappings satisfying (2.10)–(2.14), where and are replaced by and , respectively, and the contraction mappings and have the unique fixed points , respectively, and are bounded positive solutions of System (1.14) in . In order to show that System (1.14) possesses uncountably many bounded positive solutions in , we prove only that . By means of (2.10) and (2.12)–(2.16), we conclude that

which implies that

that is, . This completes the proof.

Theorem 2.2. Assume that there exist constants , , and four nonnegative sequences , and satisfying (2.1), (2.2),

Then,

(a) for each , there exist and such that for any , the Mann iterative sequence with errors generated by the schemes

 converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in .

Proof. (a) Let . Now we construct a contraction mapping and prove that its fixed point is a bounded positive solution of System (1.14). Note that (2.24) and (2.25) guarantee that there exist and satisfying

Define three mappings by (2.14),

for all .

Using (2.1), (2.2), (2.14), and (2.28)–(2.31), we deduce that for any and ,

which imply (2.16) and (2.17) and which ensure that is a self-mapping from into itself and is a contraction by . The Banach fixed point theorem means that has a unique fixed point , that is,

which reveal that

which yield that

that is, is a bounded positive solution of System (1.14) in .

It follows from (2.14), (2.16), (2.26)–(2.28), (2.30), and (2.31) that

which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) mean that .

(b) Let and . As in the proof of (a), we conclude that for each , there exist a constant , a positive integer , and mappings satisfying (2.14) and (2.28)–(2.31), where , and are replaced by , and , respectively, and the contraction mappings and have the unique fixed points =,,,, respectively, and are bounded positive solutions of the system (1.14) in . In order to show that System (1.14) possesses uncountably many bounded positive solutions in , we prove only that . By virtue of (2.14) and (2.28)–(2.31), we infer that

which yields that

that is, . This completes the proof.

Theorem 2.3. Assume that there exist constants , , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.24), and

Then,

(a) for each , there exist and such that for any , the Mann iterative sequence with errors generated by the schemes

 converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in .

Proof. (a) Let . Now we construct a contraction mapping and prove that its fixed point is a bounded positive solution of the system (1.14). Notice that (2.24) and (2.39) mean that there exist and satisfying

Define three mappings by (2.14),

for all .

It follows from (2.1), (2.2), (2.14), and (2.42)–(2.45) that for any and ,

which give (2.16) and (2.17), which together with guarantee that is a contraction. Thus the Banach fixed point theorem ensures that has a unique fixed point , that is,

which yield that

that is, is a bounded positive solution of System (1.14) in .

In light of (2.14), (2.16), (2.40), (2.41), (2.44), and (2.45), we deduce that

which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) imply that .

(b) Let and . Similar to the proof of (a), we know that for each , there exist a constant , a positive integer and mappings satisfying (2.14) and (2.42)–(2.45), where , and are replaced by and , respectively, and the contraction mappings and have the unique fixed points , which are bounded positive solutions of System (1.14) in , respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in , we prove only that . In terms of (2.14) and (2.42)–(2.45), we get that

which yields that

that is, . This completes the proof.

Theorem 2.4. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.24), and

Then,

(a) for each , there exist and such that for any , the Mann iterative sequence with errors generated by the schemes

 converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in .

Proof. (a) Let . Now we construct a contraction mapping and prove that its fixed point is a bounded positive solution of the system (1.14). Observe that (2.24) and (2.52) guarantee that there exist and satisfying

Define three mappings by (2.14),

for all .

It follows from (2.1), (2.2), (2.14), and (2.55)–(2.58) that for any and

which yield (2.16) and (2.17), which together with ensure that is a contraction in . By the Banach fixed point theorem, we deduce that has a unique fixed point , that is,

which yield that

that is, is a bounded positive solution of the system (1.14) in .

In light of (2.14), (2.16), (2.53)–(2.55), (2.57), and (2.58), we get that

which implies (2.7). Thus Lemma 1.1 and (2.7)–(2.9) imply that .

(b) Let and . Similar to the proof of (a), we conclude that for each , there exist a constant , a positive integer , and mappings satisfying (2.14) and (2.55)–(2.58), where , and are replaced by , and , respectively, and the contraction mappings and have the unique fixed points , which are bounded positive solutions of System (1.14) in , respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in , we prove only that . By virtue of (2.14) and (2.55)–(2.58), we get that

which yields that

that is, . This completes the proof.

Theorem 2.5. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.24), and

Then,

(a) for each , there exist and such that for any , the Mann iterative sequence with errors generated by the schemes

 converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b) System (1.14) has uncountably many bounded positive solutions in .

Proof. (a) Let . Now we construct a contraction mapping and prove that its fixed point is a bounded positive solution of System (1.14). Observe that (2.24) and (2.65) imply that there exist and satisfying

Define three mappings by (2.14),

for all .

Using (2.1), (2.2), (2.14), (2.65), and (2.68)–(2.71), we deduce that for any and ,

which yield (2.16) and (2.17), and which guarantee that is a self-mapping from into itself and is a contraction by . Thus the Banach fixed point theorem means that has a unique fixed point , that is,

which yield that

that is, is a bounded positive solution of the system (1.14) in .

In view of (2.14), (2.16), (2.65)–(2.67), (2.70), and (2.71), we gain that

which implies (2.7). Thus, Lemma 1.1 and (2.7)–(2.9) imply that .

(b) Let and . Similar to the proof of (a), we conclude that for each , there exist a constant , a positive integer , and mappings satisfying (2.14) and (2.68)–(2.71), where , and are replaced by , and , respectively, and the contraction mappings and have the unique fixed points , which are bounded positive solutions of System (1.14) in , respectively. In order to show that System (1.14) possesses uncountably many bounded positive solutions in , we prove only that . In light of (2.14) and (2.68)–(2.71), we have

which yields that

that is, . This completes the proof.

Remark 2.6. Let and be two arbitrary intervals in . It is easy to see that

From Remark 2.6 and Theorems 2.1–2.5, we can obtain the following.

Theorem 2.7. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2),

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.5) and (2.27) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.8. Assume that there exist constants , , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.79)–(2.81),

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.5) and (2.41) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.9. Assume that there exist constants , and four nonnegative sequences and , satisfying (2.1), (2.2), (2.79)–(2.81),

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.5) and (2.54) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.10. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2) with , (2.79)–(2.81),

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.5) and (2.67) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)the system (1.14) has uncountably many bounded positive solutions in .

Theorem 2.11. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.24), (2.83) and

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors =, generated by Schemes (2.26) and (2.41) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.12. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.24), (2.84), (2.85), and (2.90). Then,

(a)for each , there exist and such that for any =,, the Mann iterative sequence with errors =, generated by Schemes (2.26) and (2.54) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.13. Assume that there exist , and four nonnegative sequences , , , and satisfying (2.1), (2.2) with , (2.24), and (2.86)–(2.90). Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.26) and (2.67) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.14. Assume that there exist constants , and four nonnegative sequences , and satisfying (2.1), (2.2), (2.24), (2.84),

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.41) and (2.54) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.15. Assume that there exist constants and four nonnegative sequences ,, , and satisfying (2.1), (2.2) with , (2.24), (2.86)–(2.88), (2.91), and

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.41) and (2.67) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9)(b)System (1.14) has uncountably many bounded positive solutions in .

Theorem 2.16. Assume that there exist constants and four nonnegative sequences ,, , and satisfying (2.1), (2.2) with , (2.24), (2.86)–(2.88),

Then,

(a)for each , there exist and such that for any , the Mann iterative sequence with errors generated by Schemes (2.54) and (2.67) converges to a bounded positive solution of System (1.14) and has the error estimate (2.7), where , and =,⊂ are arbitrary sequences with (2.8) and (2.9),(b)System (1.14) has uncountably many bounded positive solutions in .

Remark 2.17. Theorems 2.1–2.5 extend and improve [14, Theorem 1] and [27, Theorems 2.1–2.7].

3. Examples

In this section we construct fifteen examples to explain the results presented in Section 2.

Example 3.1. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

It is easy to see that (2.1)–(2.4) are satisfied. Thus Theorem 2.1 implies that System (3.1) possesses uncountably many bounded positive solutions in . But [14, Theorem 1] and [27, Theorem 2.1] are not valid for System (3.1).

Example 3.2. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

It is easy to verify that (2.1), (2.2), (2.24), and (2.25) are fulfilled. Thus Theorem 2.2 implies that System (3.3) possesses uncountably many bounded positive solutions in . But in [14, Theorem 1] and [27, Theorem 2.2] are unapplicable for System (3.3).

Example 3.3. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

It is easy to see that (2.1), (2.2), (2.24), and (2.39) hold. Thus, Theorem 2.3 ensures that System (3.5) possesses uncountably many bounded positive solutions in . But [14, Theorem 1] and [27, Theorems 2.3, 2.5, and 2.6] are not valid for System (3.5).

Example 3.4. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

Clearly (2.1), (2.2), (2.24), and (2.52) hold. Thus, Theorem 2.4 means that System (3.7) possesses uncountably many bounded positive solutions in . But [14, Theorem 1] and [27, Theorem 2.4] are inapplicable for System (3.7).

Example 3.5. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

Obviously (2.1), (2.2), (2.24), and (2.65) hold. Thus, Theorem 2.5 implies that System (3.9) possesses uncountably many bounded positive solutions in . But [14, Theorem 1] and [27, Theorem 2.7] are useless for System (3.9).

Example 3.6. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

It is easy to see that (2.1), (2.2), and (2.79)–(2.82) are satisfied. Thus, Theorem 2.7 implies that System (3.12) possesses uncountably many bounded positive solutions in .

Example 3.7. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

Clearly (2.1), (2.2), (2.79)–(2.81), and (2.83) are satisfied. Thus, Theorem 2.8 guarantees that System (3.15) possesses uncountably many bounded positive solutions in .

Example 3.8. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

Obviously (2.1), (2.2), (2.79)–(2.81), (2.84), and (2.85) hold. Thus, Theorem 2.9 implies that System (3.18) possesses uncountably many bounded positive solutions in .

Example 3.9. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

It is easy to verify that (2.1), (2.2) with , (2.79)–(2.81), and (2.86)–(2.89) are satisfied. Thus, Theorem 2.10 reveals that System (3.21) possesses uncountably many bounded positive solutions in .

Example 3.10. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

Obviously (2.1), (2.2), (2.24), (2.83), and (2.90) hold. Thus, Theorem 2.11 gives that System (3.24) possesses uncountably many bounded positive solutions in .

Example 3.11. Consider the nonlinear difference system with multiple delays:

where are fixed and

Let

Clearly (2.1), (2.2), (2.24), (2.84), (2.85), and (2.90) hold. Thus, Theorem 2.12 ensures that System (3.27) possesses uncountably many bounded positive solutions in .

Example 3.12. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

It is easy to verify that (2.1), (2.2) with , (2.24), and (2.86)–(2.90) hold. Thus, Theorem 2.13 guarantees that System (3.30) possesses uncountably many bounded positive solutions in .

Example 3.13. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

Obviously (2.1), (2.2), (2.24), (2.84), (2.91), and (2.92) hold. Thus, Theorem 2.14 means that System (3.32) possesses uncountably many bounded positive solutions in .

Example 3.14. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

It is easy to see that (2.1), (2.2) with , (2.24), (2.86)–(2.88), (2.91), and (2.93) hold. Thus, Theorem 2.15 implies that System (3.34) possesses uncountably many bounded positive solutions in .

Example 3.15. Consider the nonlinear difference system with multiple delays:

where are fixed. Let

Clearly (2.1), (2.2) with , (2.24), (2.86)–(2.88), (2.94), and (2.95) hold. Thus, Theorem 2.16 implies that System (3.36) possesses uncountably many bounded positive solutions in .

Acknowledgments

The authors thank the editor and reviewers for useful comments and suggestions. This study was supported by research funds from Dong-A University.

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