Abstract and Applied Analysis

Volume 2014 (2014), Article ID 470181, 16 pages

http://dx.doi.org/10.1155/2014/470181

Research Article

On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

Zeqing Liu,¹ Heng Wu,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 14 October 2013; Accepted 16 December 2013; Published 28 January 2014

Academic Editor: Zhi-Bo Huang

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 convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

1. Introduction and Preliminaries

Recently, many researchers studied the oscillation, nonoscillation, and existence of solutions for linear and nonlinear second and third order difference equations and systems see, for example, [1–23] and the references cited therein. By means of the Reccati transformation techniques, Saker [18] discussed the third order difference equation

and presented some sufficient conditions which ensure that all solutions are to be oscillatory or tend to zero. Utilizing the Schauder fixed point theorem, Yan and Liu [22] proved the existence of a bounded nonoscillatory solution for the third order difference equation

Agarwal [2] established the oscillatory and asymptotic properties for the third order nonlinear difference equation

Andruch-Sobiło and Migda [4] studied the third order linear difference equation of neutral type

and obtained sufficient conditions which ensure that all solutions of the equation are oscillatory. Grace and Hamedani [6] discussed the difference equation

and gave some new criteria for the oscillation of all solutions and all bounded solutions.

Our goal is to discuss solvability and convergence of the Mann iterative schemes for the following third order nonlinear neutral delay difference equation:

where, , , , , , , , , and

By employing the Banach fixed point theorem and some new techniques, we establish the existence of uncountably many positive solutions of (6), conceive a few Mann iterative schemes for approximating these positive solutions, and prove their convergence and the error estimates. Six nontrivial examples are included.

Throughout this paper, we assume thatis the forward difference operator defined by, ,and denote the sets of nonnegative integers and positive integers, respectively,

and represents the Banach space of all real sequences onwith norm

It is easy to see thatis a closed and convex subset of. By a solution of (6), we mean a sequencewith a positive integersuch that (6) holds for all.

Lemma 1. Letbe a nonnegative sequence and.

(i)If then.(ii)If then.

Proof. Note that

that is,

As in the proof of (10), we infer that

which implies that

This completes the proof.

2. Uncountably Many Positive Solutions and Mann Iterative Schemes

In this section, using the Banach fixed point theorem and Mann iterative schemes, we establish the existence of uncountably many positive solutions of (6), prove convergence of the Mann iterative schemes relative to these positive solutions, and compute the error estimates between the Mann iterative schemes and the positive solutions.

Theorem 2. Assume that there exist two constantsandwithand four nonnegative sequences, , and satisfying

Then one has the following.

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

converges to a positive solutionof (6) with and has the following error estimate:

where is an arbitrary sequence insuch that

(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Firstly, we show that (a) holds. Put. It follows from (16)~(18) that there existandsatisfying

Define a mapping by

for each. In light of (14), (15), (22), (23), and (25), we obtain that for each

which yield that

which implies thatis a contraction in. The Banach fixed point theorem and (27) ensure thathas a unique fixed point; that is,

which mean that

which yields that

which gives that

which together with (24) implies thatis a positive solution of (6) in. Note that

which guarantees that. It follows from (19), (22), (24), (25), and (27) that for anyand

which implies that

That is, (20) holds. Thus Lemma 1, (20), and (21) guarantee that .

Next we show that (b) holds. Let, and. As in the proof of (a), we deduce similarly that, for each, there exist constantsandand a mappingsatisfying (22)(27), where, andare replaced by, , and, respectively, and the mappinghas a fixed point, which is a positive solution of (6) inwith. It follows that

which together with (14) and (20) means that for

which yields that

that is,This completes the proof.

Theorem 3. Assume that there exist two constantsandwithand four nonnegative sequences, , , andsatisfying (14), (15), and

Then one has the following.

(a)For any, there existandsuch that, for each, the Mann iterative sequencegenerated by the scheme

 converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Let. It follows from (38)(40) that there existandsatisfying

Define a mappingby

for each. Using (14), (15), (42), (43), and (45), we get that for eachand

which imply (27). The rest of the proof is similar to the proof of Theorem 2 and is omitted. This completes the proof.

Theorem 4. Assume that there exist three constants, , andwithand four nonnegative sequences, , andsatisfying (14), (15), (38), (39) and

Then one has the following.

(a)For any, there existandsuch that, for any, the Mann iterative sequence generated by the scheme

 converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (47) that there existandsatisfying

Define a mappingby

for each. In view of (14), (15), and (49) and (50), we obtain that for eachand

which imply (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 5. Assume that there exist constants, , andwithand four nonnegative sequences, , , and satisfying (14), (15), (38), (39), and

Then one has the following.

(a)For any, there existandsuch that, for any, the Mann iterative sequencegenerated by (48) converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (52) that there existandsatisfying

Define a mappingby (50). By virtue of (15), (50), (53), and (55), we infer that for all, and

That is, (27) holds. The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 6. Assume that there exist constants, , and withand four nonnegative sequences, , , and satisfying (14), (15), (38), (39), and

Then one has the following.

(a)For any, there existandsuch that, for any , the Mann iterative sequence generated by the scheme

 converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (57) that there existandsatisfying

Define a mappingby

for each. In view of (14), (15), and (59)(62), we obtain that for each, and

which imply (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 7. Assume that there exist constants, , andwithand four nonnegative sequences, , , and satisfying (14), (15), (38), (39) and

Then one has the following.

(a)For any, there existandsuch that, for any, the Mann iterative sequencegenerated by the scheme

 converges to a positive solution of (6) with and has the error estimate (20), where is an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (64) that there existandsatisfying

Define a mappingby

for each. Making use of (15), (66), (68), and (69), we conclude that

which yield (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

3. Examples

In this section, we suggest six examples to explain the results presented in Section 2.

Example 1. Consider the third order nonlinear neutral delay difference equation

where is fixed. Let, , and , and let andbe two positive constants withand

It is easy to see that (14), (15), and (18) are satisfied. Note that

which together with Lemma 1 yield that (16) and (17) hold. It follows from Theorem 2 that (71) possesses uncountably many positive solutions in. On the other hand, for any, there existandsuch that, for each, the Mann iterative sequencegenerated by (19) converges to a positive solutionof (71) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).

Example 2. Consider the third order nonlinear neutral delay difference equation

where is fixed. Let, , and , and let andbe two positive constants withand

It is clear that (14), (15), and (40) are fulfilled. Note that

which means that

Observe that

which yields that

Thus Theorem 3 guarantees that (74) possesses uncountably positive solutions in. On the other hand, for any, there existandsuch that the Mann iterative sequencegenerated by (41) converges to a positive solutionof (74) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).

Example 3. Consider the third order nonlinear neutral delay difference equation

where is fixed. Let, , , and , and let and be two positive constants withand

It is not difficult to verify that (14), (15), and (47) are fulfilled. Note that

which implies that

Observe that

which means that

That is, (38) and (39) hold. Consequently Theorem 4 implies that (80) possesses uncountably many positive solutions inOn the other hand, for any, there existandsuch that the Mann iterative sequencegenerated by (41) converges to a positive solutionof (80) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).

Example 4. Consider the third order nonlinear neutral delay difference equation

where is fixed. Let, , , and , and let andbe two positive constants with and

Obviously, (14), (15), and (52) are satisfied. Note that

which implies that

Notice that

which gives that

That is, (38) and (39) hold. Thus Theorem 5 shows that (86) possesses uncountably many positive solutions inOn the other hand, for any, there existandsuch that the Mann iterative sequencegenerated by (48) converges to a positive solutionof (86) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).

Example 5. Consider the third order nonlinear neutral delay difference equation

where is fixed. Let, , , and , and let and be two positive constants withand

Clearly, (14), (15), and (61) are satisfied. Note that

which means that

which implies that

That is, (38) and (39) hold. Consequently Theorem 6 implies that (92) possesses uncountably many positive solutions inOn the other hand, for any, there existandsuch that the Mann iterative sequencegenerated by (58) converges to a positive solutionof (92) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).

Example 6. Consider the third order nonlinear neutral delay difference equation

where is fixed. Let, , , and , and let and be two positive constants with and

Obviously, (14), (15), and (64) are satisfied. Note that

which means that

It is clear that

which implies that

That is, (38) and (39) hold. Consequently Theorem 7 implies that (97) possesses uncountably many positive solutions inOn the other hand, for any, there existandsuch that the Mann iterative sequence generated by (65) converges to a positive solutionof (97) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).

Conflict of Interests

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

Acknowledgment

This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).

References

M. H. Abu-Risha, “Oscillation of second-order linear difference equations,” Applied Mathematics Letters, vol. 13, no. 1, pp. 129–135, 2000.

R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.

R. P. Agarwal and J. Henderson, “Positive solutions and nonlinear eigenvalue problems for third-order difference equations,” Computers & Mathematics with Applications, vol. 36, no. 10-12, pp. 347–355, 1998.

A. Andruch-Sobiło and M. Migda, “On the oscillation of solutions of third order linear difference equations of neutral type,” Mathematica Bohemica, vol. 130, no. 1, pp. 19–33, 2005.

Z. Došlá and A. Kobza, “Global asymptotic properties of third-order difference equations,” Computers & Mathematics with Applications, vol. 48, no. 1-2, pp. 191–200, 2004.

S. R. Grace and G. G. Hamedani, “On the oscillation of certain neutral difference equations,” Mathematica Bohemica, vol. 125, no. 3, pp. 307–321, 2000.

J. Cheng, “Existence of a nonoscillatory solution of a second-order linear neutral difference equation,” Applied Mathematics Letters, vol. 20, no. 8, pp. 892–899, 2007.

L. Kong, Q. Kong, and B. Zhang, “Positive solutions of boundary value problems for third-order functional difference equations,” Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 481–489, 2002.

I. Y. Karaca, “Discrete third-order three-point boundary value problem,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 458–468, 2007.

W.-T. Li and J. P. Sun, “Existence of positive solutions of BVPs for third-order discrete nonlinear difference systems,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 53–64, 2004.

W.-T. Li and J.-P. Sun, “Multiple positive solutions of BVPs for third-order discrete difference systems,” Applied Mathematics and Computation, vol. 149, no. 2, pp. 389–398, 2004.

Z. Liu, M. Jia, S. M. Kang, and Y. C. Kwun, “Bounded positive solutions for a third order discrete equation,” Abstract and Applied Analysis, vol. 2012, Article ID 237036, 12 pages, 2012.

Z. Liu, S. M. Kang, and J. S. Ume, “Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 554–576, 2009.

Z. Liu, Y. Xu, and S. M. Kang, “Bounded oscillation criteria for certain third order nonlinear difference equations with several delays and advances,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1145–1161, 2011.

M. Migda and J. Migda, “Asymptotic properties of solutions of second-order neutral difference equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 5–7, pp. e789–e799, 2005.

N. Parhi, “Non-oscillation of solutions of difference equations of third order,” Computers & Mathematics with Applications, vol. 62, no. 10, pp. 3812–3820, 2011.

N. Parhi and A. Panda, “Nonoscillation and oscillation of solutions of a class of third order difference equations,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 213–223, 2007.

S. H. Saker, “New oscillation criteria for second-order nonlinear neutral delay difference equations,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 99–111, 2003.

S. H. Saker, “Oscillation of third-order difference equations,” Portugaliae Mathematica, vol. 61, no. 3, pp. 249–257, 2004.

S. Stević, “On a third-order system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7649–7654, 2012.

X. H. Tang, “Bounded oscillation of second-order delay difference equations of unstable type,” Computers & Mathematics with Applications, vol. 44, no. 8-9, pp. 1147–1156, 2002.

J. Yan and B. Liu, “Asymptotic behavior of a nonlinear delay difference equation,” Applied Mathematics Letters, vol. 8, no. 6, pp. 1–5, 1995.

Z. G. Zhang and Q. L. Li, “Oscillation theorems for second-order advanced functional difference equations,” Computers & Mathematics with Applications, vol. 36, no. 6, pp. 11–18, 1998.