# 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]).

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.

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.

Theorem 2.3.

Then (1.11) has a bounded nonoscillatory solution in .

Proof.

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.

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

## References

- Agarwal RP:
*Difference Equations and Inequalities: Theory, Methods, and Application, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 228*. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar - Agarwal RP, Grace SR, O'Regan D:
*Oscillation Theory for Difference and Functional Differential Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:viii+337.View ArticleMATHGoogle Scholar - Agarwal RP, Thandapani E, Wong PJY:
**Oscillations of higher-order neutral difference equations.***Applied Mathematics Letters*1997,**10**(1):71-78. 10.1016/S0893-9659(96)00114-0MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Grace SR:
**Oscillation of higher-order nonlinear difference equations of neutral type.***Applied Mathematics Letters*1999,**12**(8):77-83. 10.1016/S0893-9659(99)00126-3MathSciNetView ArticleMATHGoogle Scholar - Cheng SS, Patula WT:
**An existence theorem for a nonlinear difference equation.***Nonlinear Analysis: Theory, Methods & Applications*1993,**20**(3):193-203. 10.1016/0362-546X(93)90157-NMathSciNetView ArticleMATHGoogle Scholar - Cheng J:
**Existence of a nonoscillatory solution of a second-order linear neutral difference equation.***Applied Mathematics Letters*2007,**20**(8):892-899. 10.1016/j.aml.2006.06.021MathSciNetView ArticleMATHGoogle Scholar - Győri I, Ladas G:
*Oscillation Theory of Delay Differential Equations, Oxford Mathematical Monographs*. The Clarendon Press Oxford University Press, New York, NY, USA; 1991:xii+368.MATHGoogle Scholar - Liu Z, Xu Y, Kang SM:
**Global solvability for a second order nonlinear neutral delay difference equation.***Computers & Mathematics with Applications*2009,**57**(4):587-595. 10.1016/j.camwa.2008.09.050MathSciNetView ArticleMATHGoogle Scholar - Meng Q, Yan J:
**Bounded oscillation for second-order nonlinear neutral difference equations in critical and non-critical states.***Journal of Computational and Applied Mathematics*2008,**211**(2):156-172. 10.1016/j.cam.2006.11.008MathSciNetView ArticleMATHGoogle Scholar - Migda M, Migda J:
**Asymptotic properties of solutions of second-order neutral difference equations.***Nonlinear Analysis: Theory, Methods & Applications*2005,**63**(5–7):e789-e799. 10.1016/j.na.2005.02.005View ArticleMATHGoogle Scholar - Thandapani E, Manuel MMS, Graef JR, Spikes PW:
**Monotone properties of certain classes of solutions of second-order difference equations.***Computers & Mathematics with Applications*1998,**36**(10–12):291-297. 10.1016/S0898-1221(98)80030-8MathSciNetView ArticleMATHGoogle Scholar - Yang FJ, Liu JC:
**Positive solution of even-order nonlinear neutral difference equations with variable delay.***Journal of Systems Science and Mathematical Sciences*2002,**22**(1):85-89.MathSciNetMATHGoogle Scholar - Zhang BG, Yang B:
**Oscillation in higher-order nonlinear difference equations.***Chinese Annals of Mathematics*1999,**20**(1):71-80.MathSciNetMATHGoogle Scholar - Zhang Z, Li Q:
**Oscillation theorems for second-order advanced functional difference equations.***Computers & Mathematics with Applications*1998,**36**(6):11-18. 10.1016/S0898-1221(98)00157-6MathSciNetView ArticleMATHGoogle Scholar - Zhou Y:
**Existence of nonoscillatory solutions of higher-order neutral difference equations with general coefficients.***Applied Mathematics Letters*2002,**15**(7):785-791. 10.1016/S0893-9659(02)00043-5MathSciNetView ArticleMATHGoogle Scholar - Zhou Y, Huang YQ:
**Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations.***Journal of Mathematical Analysis and Applications*2003,**280**(1):63-76. 10.1016/S0022-247X(03)00017-9MathSciNetView ArticleMATHGoogle Scholar - Zhou Y, Zhang BG:
**Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients.***Computers & Mathematics with Applications*2003,**45**(6–9):991-1000. 10.1016/S0898-1221(03)00074-9MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.