#### Abstract

This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation . Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.

#### 1. Introduction and Preliminaries

Recently, the oscillation, nonoscillation, asymptotic behavior, and existence of solutions of different classes of linear and nonlinear second-order difference equations have been studied by many authors; see, for example, [1â€“26] and the references cited therein. Using the Banach fixed point theorem, Jinfa [5] discussed the existence of a bounded nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients: under the condition . Luo and Bainov [13] and M. Migda and J. Migda [16] considered the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima: and the second-order neutral difference equation: respectively. Meng and Yan [15] studied the existence of bounded nonoscillatory solutions for the second-order nonlinear nonautonomous neutral delay difference equation: Applying the cone compression and expansion theorem in FrÃ©chet spaces, Tian and Ge [21] established the existence of multiple positive solutions of the second-order discrete equation on the half-line: with certain boundary value conditions. But to the best of our knowledge, results on multiplicity of unbounded solutions for neutral delay difference equations are very scarce in the literature. Nothing has been done with the existence of uncountably many unbounded positive solutions for (1)~(5) and any other second-order neutral delay difference equations:

Inspired and motivated by the results in [1â€“26], in this paper we introduce and study the second-order nonlinear neutral delay difference equation: where , , , , and By means of the Banach fixed point theorem and some new techniques, we establish sufficient conditions for the existence of uncountably many unbounded positive solutions of (6), suggest a few Mann iterative schemes with errors for approximating these unbounded positive solutions, and prove their convergence and the error estimates. The results obtained in this paper extend the result in [5]. Four nontrivial examples are interested in the text to illustrate the importance of our results.

Throughout this paper, we assume that is the forward difference operator defined by , , , , , and denote the sets of all integers, nonnegative integers, and positive integers, respectively, represents the Banach space of all real sequences on with norm It is clear that is a closed and convex subset ofâ€‰â€‰ . By a solution of (6), we mean a sequence with a positive integer such that (6) holds for all .

The following lemmas play important roles in this paper.

Lemma 1 (see [27]). *Let ,, and be four nonnegative sequences satisfying the inequality
**
where , , , and . Then .*

Lemma 2 (see [11]). *Let and be a nonnegative sequence. Then*(a)*;
*(b)*. *

#### 2. Existence of Uncountably Many Unbounded Positive Solutions

Using the Banach fixed point theorem and the Mann iterative schemes with errors, we next discuss the existence of uncountably many unbounded positive solutions of (6), prove that the Mann iterative schemes with errors converge to these unbounded positive solutions, and compute the error estimates between the Mann iterative schemes with errors and the unbounded positive solutions.

Theorem 3. *Assume that there exist two constants and with and four nonnegative sequences , and satisfying
**
Then **(a) for any , there exist and such that for each , the Mann iterative sequence with errors generated by the scheme
**
converges to an unbounded positive solution of (6) and has the following error estimate:
**
where is an arbitrary sequence in and and are any sequences in such that
* (b)* equation (6) possesses uncountably many unbounded positive solutions in .*

*Proof. *First of all we show that (a) holds. Set . It follows from (13), (14), and (15) that there exist and satisfying
Define a mapping by
for each . In view of (11), (12), (19), (20), and (22), we deduce that for each and for all
which yield that
which means that is a contraction in . It follows from the Banach fixed point theorem that has a unique fixed point , that is,
which imply that
which yields that
which together with (21) gives that is an unbounded positive solution of (6) in . It follows from (16), (19), (21), (22), and (24) that for any and
which implies that
That is, (17) holds. Thus, Lemma 1 and (17) and (18) guarantee that .

Next we show that (b) holds. Let and . As in the proof of (a), we deduce similarly that for each there exist constants , , and a mapping satisfying (19)~(24), where , and are replaced by and , respectively, and the mapping has a fixed point , which is an unbounded positive solution of (6) in , that is,
which together with (11) and (17) implies that for
which yields that
that is, . This completes the proof.

Theorem 4. *Assume that there exist two constants and with and four nonnegative sequences , , , and satisfying (11), (12),
**
Then**
(a) for any , there exist and such that for each , the Mann iterative sequence with errors generated by the scheme
**
converges to an unbounded positive solution of (6) and has the error estimate (17), where is an arbitrary sequence in , and and are any sequences in satisfying (18); **
(b) equation (6) possesses uncountably many unbounded positive solutions in . *

*Proof. *Let . It follows from (33)~(35) that there exist and satisfying
Define a mapping by
for each . Using (11), (12), (37), (38), and (40), we get that for each and
which imply (24). Consequently, (24) means that is a contraction in and has a unique fixed point , which is also an unbounded positive solution of (6) in . The rest of the proof is similar to the proof of Theorem 3 and is omitted. This completes the proof.

Theorem 5. *Assume that there exist three constants , and with and four nonnegative sequences , , , and satisfying (11), (12), (33), (34) and
**
Then **
(a) for any , there exist and such that for any , the Mann iterative sequence with errors generated by the scheme
**
converges to an unbounded positive solution of (6) and has the error estimate (17), where is an arbitrary sequence in and and are any sequences in satisfying (18); **
(b) equation (6) possesses uncountably many unbounded positive solutions in .*

*Proof. *Put . It follows from (33), (34), and (42) that there exist and satisfying
Define a mapping by
for each . In view of (11), (12), and (44) and (45), we obtain that for each and ,
which give (24), in turn, which implies that is a contraction in and possesses a unique fixed point , which is an unbounded positive solution of (6) in . The rest of the proof is similar to that of Theorem 3 and is omitted. This completes the proof.

Theorem 6. *Assume that there exist constants , and with and four nonnegative sequences , , , and satisfying (11), (12), (33), (34), and
**
Then**
(a) for any , there exist and such that for any and the Mann iterative sequence with errors generated by (43) converges to an unbounded positive solution of (6) and has the error estimate (17), where is an arbitrary sequence in , and are any sequences in satisfying (18); **
(b) equation (6) possesses uncountably many unbounded positive solutions in .*

*Proof. *Put . It follows from (33), (34), and (47) that there exist and satisfying
Define a mapping by (45). By virtue of (12), (45), (48), and (50), we easily verify that
which yield that . The rest of the proof is similar to that of Theorem 5 and is omitted. This completes the proof.

*Remark 7. *Theorems 3~6 extend and improve Theorem 1 in [5].

#### 3. Examples

In this section we suggest four examples to explain the results presented in Section 2. Note that Theorem 1 in [5] is useless for all these examples.

*Example 8. *Consider the second-order nonlinear neutral delay difference equation:
where is fixed. Let , , , and two positive constants with and
It is easy to see that (11), (12), and (15) are satisfied. Note that
which together with Lemma 2 yield that (13) and (14) hold. It follows from Theorem 3 that (52) possesses uncountably many unbounded positive solutions in . On the other hand, for any , there exist and such that for each , the Mann iterative sequence with errors generated by (16) converges to an unbounded positive solution of (52) and has the error estimate (17), where is an arbitrary sequence in and and are any sequences in satisfying (18).

*Example 9. *Consider the second-order nonlinear neutral delay difference equation:
where is fixed. Let , , , and two positive constants with and
It is clear that (11), (12), and (35) are fulfilled. Note that
which yields that
Thus, Theorem 4 guarantees that (56) possesses uncountably unbounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with error generated by (36) converges to an unbounded positive solution of (56) and has the error estimate (17), where is an arbitrary sequence in and and are any sequences in satisfying (18).

*Example 10. *Consider the second-order nonlinear neutral delay difference equation:
where is fixed. Let , , , , and two positive constants with and