Positive Solutions and Mann Iterative Algorithms for a Second-Order Nonlinear Difference Equation

Liu, Zeqing; Li, Xin; Kang, Shin Min; Kwun, Young Chel

doi:https://doi.org/10.1155/2016/8317567

Journal of Function Spaces

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Abstract Introduction and Preliminaries Examples Acknowledgments References Copyright Related Articles

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Existence and Asymptotic Behaviour of Solutions of Differential and Integral Equations in Some Function Spaces 2016

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Research Article | Open Access

Volume 2016 | Article ID 8317567 | https://doi.org/10.1155/2016/8317567

Positive Solutions and Mann Iterative Algorithms for a Second-Order Nonlinear Difference Equation

Zeqing Liu,¹Xin Li,²Shin Min Kang,³and Young Chel Kwun⁴

Academic Editor: Jozef Banas

Received17 Dec 2015

Accepted24 Feb 2016

Published03 Apr 2016

Abstract

This paper deals with the second-order nonlinear neutral delay difference equation , . Using the Banach fixed point theorem, Mann iterative method with errors, and some new techniques, we prove the existence of uncountably many positive solutions and the convergence of the sequences generated by the Mann iterative method with errors relative to these solutions for the above equation. Six examples are included. Our results extend and improve essentially the known results in this field.

1. Introduction and Preliminaries

Consider the second-order nonlinear neutral delay difference equation with forcing term:where , , , with, , and are real sequences with for each , , and .

In the recent ten years, there have been a lot of activities concerning the oscillation, nonoscillation, asymptotic behavior, and existence of solutions, nonoscillatory solutions, and positive solutions for various types of difference equations, which are special cases of (1); see, for example, [1–21] and the references cited therein. In 2007, Cheng [3] used the Banach fixed point theorem to investigate the second-order neutral delay linear difference equation with positive and negative coefficients: and got a sufficient condition for the existence of a nonoscillatory solution of (3). In 2004, Malaguti et al. [15] discussed the second-order nonlinear difference equation and obtained sufficient and necessary conditions for the existence of bounded solutions of (4). In 2007, by using the Krasnoselskii fixed point theorem and Schauder fixed point theorem, Rath et al. [16] investigated the existence of solutions of the second-order nonlinear neutral delay difference equation In 2003, Agarwal et al. [1] used a nonlinear alternative of Leray-Schauder type to obtain the existence of a nonoscillatory solution for the following second-order neutral delay difference equation: In 2009, using the Banach fixed point theorem, Liu et al. [12] researched the global solvability for the second-order nonlinear neutral delay difference equation However, the study of the existence of bounded positive solutions for a lot of second-order neutral delay difference equations appears to be insufficient. In particular, the uncountably many bounded positive solutions for these second-order neutral delay difference equations obtain less concern.

The purpose of this paper is to study the solvability and convergence of Mann iterative approximations of (1). By means of the Banach fixed point theorem, Mann iterative method with errors, and a few new techniques, we obtain the existence of uncountably many bounded positive solutions of (1) and prove that the sequences generated by the Mann iterative method with errors converge to these bounded positive solutions. Our results generalize and improve all results in [3, 12]. Six examples are constructed in order to illustrate the superiority and applications of the results presented in this paper.

Throughout this paper, we assume that , , and denote the sets of all positive integers and integers, respectively, and stands for the forward difference operator; that is, . Let Assume that represents the Banach space of all bounded real sequences with the norm and It is easy to see that is a bounded closed and convex subset of the Banach space .

By a solution of (1), we mean a real sequence with a positive integer such that (1) holds for all .

Lemma 1 (see [22]). Let , , , and be four nonnegative real sequences satisfying the inequality where , , , and . Then, .

Lemma 2. Let , be two constants and let , be two nonnegative sequences. Then,

Proof. For each , denotes the largest integer not exceeding . Notice that Thus, (11) follows from (12). This completes the proof.

2. Main Results

According to the different ranges of the sequence , now we use the Banach fixed point theorem to study the existence of uncountably many bounded positive solutions and convergence of the Mann iterative approximations of (1).

Theorem 3. Assume that there exist constants , , , , and and two nonnegative sequences and satisfying Then, one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence with errors generated by the scheme converges to a bounded positive solution of (1) and has the following error estimate: where is an arbitrary sequence in and and are any sequences in such that (b) Equation (1) has uncountably many bounded positive solutions in .

Proof. First, we show that (a) holds. Let . It follows from (14) and (19) that there exist and sufficiently large such that Define an operator by for any . On account of (15)(18) and (23)(24), we get that, for every , and , which yield that that is, is a contraction operator in and has a unique fixed point , which means that for all which yields that for all which implies that is a bounded positive solution of (1) in .
By (20), (24), and (26), we infer that for any and which implies that That is, (21) holds. Consequently, Lemma 1 and (21)(22) imply that .
Next, we show that (b) holds. Let , denote two arbitrary constants in with . As in the proof of (a), we infer similarly that for each there exist constants , , and an operator satisfying (23), (24), and (26), where , , and take the place of , , and , respectively. Moreover, the contraction operator has a unique fixed point , which is also a bounded positive solution of (1); that is, which together with (15)(18), (23), (24), and (26) gives that for each which yields that that is, . This completes the proof.

Theorem 4. Assume that there exist constants , , , , and and two nonnegative sequences and satisfying (15)(19): Then, one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence with errors generated by (20) converges to a bounded positive solution of (1) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).
(b) Equation (1) has uncountably many bounded positive solutions in .

Proof. First, we show that (a) holds. Let . It follows from (19) and (35) that there exist and sufficiently large such that Let the operator be defined by (24). By virtue of (15)(18), (24), and (36), we obtain that, for every , , and , The rest of the proof is similar to that of Theorem 3 and is omitted. This completes the proof.

Theorem 5. Assume that there exist constants , , , , , and and two nonnegative sequences and satisfying (15)~(19): Then, one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence with errors generated by the scheme converges to a bounded positive solution of (1) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).
(b) Equation (1) has uncountably many bounded positive solutions in .

Proof. First, we show that (a) holds. Let . It follows from (19) and (39) that there exist and sufficiently large such that Define an operator as follows: for any . In view of (15)(18) and (41)(42), we get that, for every , , and , which yield that (26) holds; that is, is a contraction operator in and has a unique fixed point , which gives that for all which implies that for all which means that is a bounded positive solution of (1) in .
It follows from (26), (40), and (42) that for any and which implies that That is, (21) holds. Consequently, Lemma 1 and (21)(22) imply that .
Next, we show that (b) holds. Let with . As in the proof of (a), we conclude similarly that for each there exist constants , , and an operator satisfying (26) and (41)(42), where , , and are replaced by , , and , respectively, and the contraction operator has a unique fixed point , which is also a bounded positive solution of (1); that is, which together with (15)(18), (26), and (41)(42) guarantees that for each which yields that that is, . This completes the proof.

Theorem 6. Assume that there exist constants , , , , , and and two nonnegative sequences and satisfying (15)~(19): Then, one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence with errors generated by (40) converges to a bounded positive solution of (1) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).
(b) Equation (1) has uncountably many bounded positive solutions in .

Proof. First, we show that (a) holds. Let . It follows from (19) and (52) that there exist and satisfying Define an operator by (42). In light of (15)(19), (42), and (53), we get that, for every , , and , The rest of the proof is similar to that of Theorem 5 and is omitted. This completes the proof.

Theorem 7. Assume that there exist constants , , , and with and two nonnegative sequences and satisfying (15)(19) and Then, one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence with errors generated by the scheme converges to a bounded positive solution of (1) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).
(b) Equation (1) has uncountably many bounded positive solutions in .

Proof. First, we show that (a) holds. Let . It follows from (19) that there exist and sufficiently large such that Define an operator as follows: for each . In view of (15)(18) and (57)(58), we conclude that, for every , , and , which yield (26); that is, is a contraction operator in and has a unique fixed point , which means that for all which gives that for all which implies that for all which means that is a bounded positive solution of (1) in .
Using (20), (26), and (58), we deduce that for any and which implies that That is, (21) holds. Consequently, Lemma 1 and (21)(22) ensure that .
Next, we show that (b) holds. Let with . As in the proof of (a), we deduce similarly that for each there exist constants , , and an operator satisfying (26) and (57)(58), where , , and are replaced by , , and , respectively, and the contraction operator has a unique fixed point , which is also a bounded positive solution of (1); that is, which together with (15)(18), (26), and (57)(58) means that for each which yields that which ensures that . This completes the proof.

Theorem 8. Assume that there exist constants , , , and with and two nonnegative sequences and satisfying (15)~(18): Then, one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence with errors generated by the scheme converges to a bounded positive solution of (1) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).
(b) Equation (1) has uncountably many bounded positive solutions in .

Proof. First, we show that (a) holds. Let . It follows from Lemma 2 and (69) that which guarantees that there exist and sufficiently large such that Define an operator by for each . By virtue of (15)(18) and (72)(73), we get that, for every , , and , which yield (26); that is, is a contraction operator in and has a unique fixed point with which yields that which implies that which means that is a bounded positive solution of (1) in .
It follows from (20), (26), and (73) that for any and which implies that That is, (21) holds. Thus, Lemma 1 and (21)(22) ensure that .
Next, we show that (b) holds. Let with . As in the proof of (a), we deduce that for each there exist constants , , and an operator satisfying (26) and (72)(73), where , , and are replaced by , , and , respectively, and the contraction operator has a unique fixed point , which is also a bounded positive solution of (1); that is, which together with (15)(18), (26), and (72)(73) gives that for each which yields that that is, . This completes the proof.

Remark 9. Theorems 3~8 extend and unify Theorem in [3] and Theorems 2.1~2.7 in [12], respectively.

3. Examples

In what follows, we construct six examples to demonstrate that Theorems 3~8 extend properly the corresponding results in [3, 12], respectively. Note that none of the known results can be applied to the examples.

Example 1. Consider the second-order nonlinear neutral delay difference equation with forcing term where is fixed. Let , , , , , , It is easy to show that (13)(18) are satisfied. Note that that is, (19) holds. It follows from Theorem 3 that (83) has uncountably many bounded positive solutions in . On the other hand, for each , there exist and such that the Mann iterative sequence with errors generated by (20) converges to a bounded positive solution of (83) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).

Example 2. Consider the second-order nonlinear neutral delay difference equation with forcing term where is fixed. Let , , , , , It is clear that (15)(18), (34), and (35) hold. Notice that which implies (19). It follows from Theorem 4 that (86) has uncountably bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with errors generated by (20) converges to a bounded positive solution of (86) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).

Example 3. Consider the second-order nonlinear neutral delay difference equation with forcing term where is fixed. Let , , , , , , , It is easy to show that (15)(18), (38), and (39) are satisfied. Note that which means (19). It follows from Theorem 5 that (89) has uncountably bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with errors generated by (40) converges to a bounded positive solution of (89) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).

Example 4. Consider the second-order nonlinear neutral delay difference equation with forcing term where is fixed. Let , , , , , , , It is easy to show that (15)(18), (51), and (52) are satisfied. Note that that is, (19) holds. It follows from Theorem 6 that (92) has uncountably bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with errors generated by (40) converges to a bounded positive solution of (92) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).

Example 5. Consider the second-order nonlinear neutral delay difference equation with forcing term where is fixed. Let , , , , , It is easy to show that (15)(18) and (55) hold. Note that that is, (19) holds. It follows from Theorem 7 that (95) has uncountably bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with errors generated by (56) converges to a bounded positive solution of (95) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).

Example 6. Consider the second-order nonlinear neutral delay difference equation with forcing term where is fixed. Let , , , , , It is easy to show that (15)(18) and (68) are satisfied. Note that that is, (69) holds. It follows from Theorem 8 that (98) has uncountably bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with errors generated by (70) converges to a bounded positive solution of (98) and has the error estimate (21), where is an arbitrary sequence in and and are any sequences in satisfying (22).

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This study was supported by research funds from Dong-A University.

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Copyright © 2016 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.

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