# Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

- Min Liu
^{1}and - Zhenyu Guo
^{1}Email author

**2010**:767620

**DOI: **10.1155/2010/767620

© Min Liu and Zhenyu Guo. 2010

**Received: **19 March 2010

**Accepted: **5 September 2010

**Published: **13 September 2010

## Abstract

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation , , where , , , and are integers, and are real sequences, , and is a mapping, is studied. Some sufficient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence . Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

## 1. Introduction and Preliminaries

(see [17]).

where , , , and are integers, and are real sequences, , and is a mapping. Clearly, difference equations (1.1)–(1.10) are special cases of (1.11). By using Schauder fixed point theorem and Krasnoselskii fixed point theorem, the existence of bounded nonoscillatory solutions of (1.11) is established.

Lemma 1.1 (Schauder fixed point theorem).

Let be a nonempty closed convex subset of a Banach space . Let be a continuous mapping such that is a relatively compact subset of . Then has at least one fixed point in .

Lemma 1.2 (Krasnoselskii fixed point theorem).

Let be a bounded closed convex subset of a Banach space , and let satisfy for each . If is a contraction mapping and is a completely continuous mapping, then the equation has at least one solution in .

The forward difference is defined as usual, that is, . The higher-order difference for a positive integer is defined as , . Throughout this paper, assume that , and stand for the sets of all positive integers and integers, respectively, , , , , and denotes the set of real sequences defined on the set of positive integers lager than where any individual sequence is bounded with respect to the usual supremum norm for . It is well known that is a Banach space under the supremum norm. A subset of a Banach space is relatively compact if every sequence in has a subsequence converging to an element of .

Definition 1.3 (see [5]).

whenever for any in .

Lemma 1.4 (discrete Arzela-Ascoli's theorem [5]).

A bounded, uniformly Cauchy subset of is relatively compact.

By a solution of (1.11), we mean a sequence with a positive integer such that (1.11) is satisfied for all . As is customary, a solution of (1.11) is said to be oscillatory about zero, or simply oscillatory, if the terms of the sequence are neither eventually all positive nor eventually all negative. Otherwise, the solution is called nonoscillatory.

## 2. Existence of Nonoscillatory Solutions

In this section, a few sufficient conditions of the existence of bounded nonoscillatory solutions of (1.11) are given.

Theorem 2.1.

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

- (i)
It is claimed that , for all .

- (ii)
It is declared that is continuous.

- (iii)
It can be asserted that is relatively compact.

which means that is uniformly Cauchy. Therefore, by Lemma 1.4, is relatively compact.

Therefore, is a bounded nonoscillatory solution of (1.11). This completes the proof.

Remark 2.2.

that is, .

Theorem 2.3.

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

for all .

The rest of the proof is similar to that in Theorem 2.1. This completes the proof.

Theorem 2.4.

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

- (i)
It is claimed that , for all .

- (ii)
It is declared that is a contraction mapping on .

- (iii)
Similar to (ii) and (iii) in the proof of Theorem 2.1, it can be showed that is completely continuous.

By Lemma 1.2, there exists such that , which is a bounded nonoscillatory solution of (1.11). This completes the proof.

Theorem 2.5.

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

Define two mappings as (2.25). The rest of the proof is analogous to that in Theorem 2.4. This completes the proof.

Similar to the proof of Theorem 2.5, we have the following theorem.

Theorem 2.6.

Then (1.11) has a bounded nonoscillatory solution in .

Theorem 2.7.

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

for all . The rest of the proof is analogous to that in Theorem 2.4. This completes the proof.

Similar to the proof of Theorem 2.7, we have

Theorem 2.8.

Then (1.11) has a bounded nonoscillatory solution in .

Remark 2.9.

Similar to Remark 2.2, we can also prove that the conditions of Theorems 2.3–2.8 ensure that (1.11) has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

Remark 2.10.

Theorems 2.1–2.8 extend and improve Theorem of Cheng [6], Theorems of Liu et al. [8], and corresponding theorems in [3, 4, 9–17].

## 3. Examples

In this section, two examples are presented to illustrate the advantage of the above results.

Example 3.1.

Choose and . It is easy to verify that the conditions of Theorem 2.1 are satisfied. Therefore Theorem 2.1 ensures that (3.1) has a nonoscillatory solution in . However, the results in [3, 4, 6, 8–17] are not applicable for (3.1).

Example 3.2.

Choose and . It can be verified that the assumptions of Theorem 2.5 are fulfilled. It follows from Theorem 2.5 that (3.2) has a nonoscillatory solution in . However, the results in [3, 4, 6, 8–17] are unapplicable for (3.2).

## Declarations

### Acknowledgment

The authors are grateful to the editor and the referee for their kind help, careful reading and editing, valuable comments and suggestions.

## Authors’ Affiliations

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