Abstract and Applied Analysis

Volume 2014 (2014), Article ID 954182, 23 pages

http://dx.doi.org/10.1155/2014/954182

Research Article

Positive Solutions and Mann Iterative Algorithms for a Nonlinear Three-Dimensional Difference System

Zeqing Liu,¹ Yan Lu,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, Pusan 614-714, Republic of Korea

Received 31 July 2013; Accepted 14 October 2013; Published 8 January 2014

Academic Editor: Ondřej Došlý

Copyright © 2014 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.

The existence of uncountably many positive solutions and Mann iterative approximations for a nonlinear three-dimensional difference system are proved by using the Banach fixed point theorem. Four illustrative examples are also provided.

1. Introduction

In recent years, the oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, bounded solutions, unbounded solutions, positive solutions, and nonoscillatory solutions for some two- and three-dimensional difference systems have been studied by many authors and a significant number of important results have been found; see [1–12] and the references therein.

In order to solve the problem that the Picard iteration fails to converge under some conditions, Mann [13] introduced a modified iteration, which is now called Mann iterative scheme. It is well known that the Mann iterative schemes are often used in the fields of nonlinear differential equations, nonlinear equations, nonlinear mappings, optimization, variational inequalities, nonlinear analysis, and so forth.

Note that the difference systems in [1–12] are as follows:

where ,  ,   and are nonnegative sequences, and is a real sequence with :

where , ,   and are real sequences with , and    is a nonnegative sequence:

where ,  ,   , , , for , and ,  ,  ,      with

However, to the best of our knowledge, there exists no result in the literature dealing with the following nonlinear three-dimensional difference system:

which is abbreviated as, for convenience,

where ,  ,  ,  , , and ,  with

The main purpose of this paper is to study solvability and convergence of the Mann iterative schemes for the system (6). Sufficient conditions for the existence of uncountably many positive solutions of the system (6) and convergence of the Mann iterative schemes relative to these positive solutions are provided by utilizing the Banach fixed point theorem. Four illustrative examples are given.

2. Preliminaries

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

Let denote the Banach space of all sequences in with norm

and put

It is easy to see that for each , is a nonempty closed convex subset of the Banach space with norm for each .

By a solution of the system (6), we mean a three-dimensional sequence with a positive integer such that (6) holds for all .

Let be a nonempty convex subset of a Banach space and let be a mapping. For any given , the sequence defined by

where is a sequence in with certain condition, is called the Mann iterative scheme.

Lemma 1. Let be a nonnegative sequence and .

(i)If , then

(ii)If , then

Proof. It is easy to see that

which yields (12). Note that

where denotes the largest integral number not exceeding . Hence (13) holds. This completes the proof.

3. Uncountably Many Positive Solutions and Mann Iterative Schemes

Our main results are as follows.

Theorem 2. Assume that there exist constants , and nonnegative sequences and satisfying

Then one has the following.

(a) For any , there exist and  +  such that, for each , the Mann iterative sequence generated by the scheme

converges to a positive solution of the system (6) with for each and has the following error estimate:

where is an arbitrary sequence in such that

(b) The system (6) has uncountably many positive solutions in .

Proof. Firstly, we prove that (a) holds. Put . It follows from (19) that there exist and satisfying

where

Define mappings and by

for each .

Now we assert that

Using (18), (25), and (26), we get that, for any and ,

which implies (28). Thus (29) follows from (27) and (28).

Next we claim that

for all , . In view of (17), (23), (26), and (27), we deduce that, for any , , and ,

which yield (31) and (32). Clearly, (29) and (32) ensure that is a contraction mapping in and it has a unique fixed point , that is,

which guarantee that

which together with (16) means that is a positive solution of the system (6) in . Observe that (18) and (19) give that, for each ,

which implies that

By means of (20), (26), and (31), we infer that

which imply that

that is, (21) holds. It follows from (21) and (22) that .

Secondly, we show that holds. Let , with . As in the proof of , we infer similarly that, for each , there exist constants , and mappings and satisfying (23)(32), where , , , , , , , , , , , and are replaced by , , , ,  , , , , , , , and , respectively, and the contraction mapping has a unique fixed point , which is a positive solution of the system (6); that is,

On account of (17), (23), and (40), we conclude that

which means that

which gives that

that is, . Hence the system (6) possesses uncountably many positive solutions in . This completes the proof.

Theorem 3. Assume that there exist constants , and nonnegative sequences , , and satisfying (17), (18), and (22) as follows:

Then one has the following.

(a) For any , there exist and such that, for each , the Mann iterative sequence generated by the scheme

converges to a positive solution of the system (6) with  for each and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

Proof. Firstly, we prove that (a) holds. Set . It follows from (45) that there exist and satisfying

where

Define mappings and by (27) and

for each . It follows from (18), (49), and (50) that, for any and ,

which means (28). It is easy to see that (27) and (28) yield (29). By virtue of (17), (47), and (50), we know that for any , and ,

which yield (31) and (32). Thus (29) and (32) guarantee that is a contraction mapping in and it has a unique fixed point , that is,

which mean that

which implies that

which together with (44) ensures that is a positive solution of the system (6) in . Obviously, (18) and (45) yield that

which means that

In light of (31), (46), and (50), we get that

which yield that

that is, (21) holds. It follows from (21) and (22) that .

Secondly, we show that (b) holds. Put , with . Similar to the proof of , we conclude that, for each , there exist constants , , , ,  + and mappings and satisfying (27)(32) and (47)(50), where , , , ,   ,   , , ,   , , , and are replaced by , , , ,  ,, , , , , , and , respectively, and the contraction mapping has a unique fixed point , which is a positive solution of the system (6); that is,

Using (17), (47), and (60), we infer that, for each and ,

which implies that

which gives that

that is, . Hence the system (6) possesses uncountably many positive solutions in . This completes the proof.

Theorem 4. Assume that there exist constants ,   and nonnegative sequences , , and satisfying (17), (18), and (22) as follows:

Then one has the following.

(a) For any , there exist and such that, for each , the Mann iterative sequence generated by the scheme

converges to a positive solution of the system (6) with for each and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

Proof. Firstly, we show that (a) holds. Let . Observe that (65) implies that there exist and satisfying

where

Define mappings and by (27) and

for each . It follows from (18), (69), and (70) that, for any ,

which mean (28). Consequently, (29) follows from (27) and (28). By virtue of (17), (67), and (70), we infer that, for any , hh , and ,

which yield (31) and (32). Thus (29) and (32) ensure that is a contraction mapping in and it has a unique fixed point , that is,

which yields that

which implies that

which together with (64) means that is a positive solution of the system (6) in . Note that (18) and (65) give that, for each ,

which means that

Making use of (31), (66), and (70), we get that

which yield that

that is, (21) holds. It follows from (21) and (22) that .

Secondly, we show that (b) holds. Let ,     with . Similarly we infer that, for each , there exist constants ,   and mappings and satisfying (27)(32) and (67)(70), where , , , , , , , ,  , , , and are replaced by , , , , , , , , , , , and , respectively, and the contraction mapping has a unique fixed point , which is a positive solution of the system (6); that is,

Using (17), (67), (70), and (80), we know that, for each ,

which implies that

which gives that

that is, . Hence the system (6) possesses uncountably many positive solutions in . This completes the proof.

Theorem 5. Assume that there exist constants , , and nonnegative sequences , , and satisfying (17), (18), (22), (65), and

Then one has the following.

(a) For any , there exist and such that, for each , the Mann iterative sequence generated by (66) converges to a positive solution of the system (6) with for each and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

Proof. Let . It follows from (65) that there exist and satisfying (67) and

Let , , , and be defined by (27) and (70), respectively. It follows from (18), (70), and (85) that, for any ,

which lead to (28). The rest of the proof is similar to that of Theorem 4 and is omitted. This completes the proof.

Similar to the proofs of Theorems 25, we have the following results and omit their proofs.

Theorem 6. Assume that there exist constants , and nonnegative sequences , , and  for each satisfying (17), (18), and (22) as follows:

Then one has the following.

(a)  For any      , there exist and such that, for each , the Mann iterative sequence generated by the scheme

converges to a positive solution of the system (6) with

and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

Theorem 7. Assume that there exist constants , ,   and nonnegative sequences , , and for each satisfying (17), (18), (22), (88), and

Then one has the following.

(a)  For any   ×  , there exist and such that, for each , the Mann iterative sequence generated by the scheme (89)(91) converges to a positive solution of the system (6) with

and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

Theorem 8. Assume that there exist constants , ,  and nonnegative sequences , , and for each satisfying (17), (18), and (22) as follows:

Then one has the following.

(a) For any , , there exist and    such that, for each , the Mann iterative sequence generated by the scheme (89), (91), and

converges to a positive solution of the system (6) with

and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

Theorem 9. Assume that there exist constants , ,   and nonnegative sequences , , and for each satisfying (17), (18), and (22) as follows:

Then one has the following.

(a) For any , , there exist and   such that, for each  , the Mann iterative sequence generated by the scheme (91), (96), and

converges to a positive solution of the system (6) with

and has the error estimate (21).

(b) The system (6) has uncountably many positive solutions in .

4. Examples

Now we suggest four examples to explain the main results in Section 3.

Example 1. Consider the nonlinear three-dimensional difference system

where and are fixed. Let , , , , , , , , , and

It is clear that (16)(18) are satisfied. Note that Lemma 1 implies that

which yield that (19) holds. Thus Theorem 2 implies that the system (102) possesses uncountably many positive solutions in .

Example 2. Consider the nonlinear three-dimensional difference system

where and are fixed. Let ,  , , , , , , , , and

Obviously, (17) and (18) are satisfied. Note that Lemma 1 yields that

which implies that (45) holds. Thus Theorem 3 implies that the system (105) possesses uncountably many positive solutions in .

Example 3. Consider the nonlinear three-dimensional difference system

where and are fixed. Let , , , , , , , , , , , , and

It is easy to see that (17), (18), and (64) are satisfied. Note that Lemma 1 yields that

which implies that (65) holds. Thus Theorem 4 implies that the system (108) possesses uncountably many positive solutions in .

Example 4. Consider the nonlinear three-dimensional difference system

where and are fixed. Let , , , , , , , , , , , , and

Clearly, (17), (18), and (84) hold. Observe that Lemma 1 means that

that is, (65) holds. Thus Theorem 5 implies that the system (111) possesses uncountably many positive solutions in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are indebted to the referee for carefully reading the paper and for making useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).

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