- Research Article
- Open Access

# On a Max-Type Difference Equation

- Ali Gelisken
^{1}Email author, - Cengiz Cinar
^{1}and - Ibrahim Yalcinkaya
^{1}

**2010**:584890

https://doi.org/10.1155/2010/584890

© Ali Gelisken et al. 2010

**Received: **8 December 2009

**Accepted: **23 April 2010

**Published: **30 May 2010

## Abstract

## Keywords

- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis

## 1. Introduction

Recently, the study of max-type difference equations attracted a considerable attention. Although max-type difference equations are relatively simple in form, it is unfortunately extremely difficult to understand thoroughly the behavior of their solutions; see, for example, [1–20] and the relevant references cited therein. The max operator arises naturally in certain models in automatic control theory (see [13, 14]). Furthermore, difference equation appear naturally as a discrete analogue and as a numerical solution of differential and delay differential equations having applications and various scientific branches, such as in ecology, economy, physics, technics, sociology, and biology.

converges to or eventually periodic with period 4, where and

converges to or eventually periodic with period 2, where and

where are positive integers, , , and initial conditions are positive real numbers.

In this paper, we investigate the asymptotic behavior of the positive solutions of (1.4). We prove that every positive solution of (1.4) converges to Clearly, we can assume that without loss of generality.

## 2. Main Results

### 2.1. The Case

In this section, we consider the asymptotic behavior of the positive solutions of (1.4) in the case

where and the initial conditions are real numbers. Since we have

We need the following two lemmas in order to prove the main result of this section.

Lemma 2.1.

Proof.

From the above statements, we have for all Therefore, the proof is complete.

Lemma 2.2.

Proof.

From the above statements, we have for all Therefore, the proof is complete.

Theorem 2.3.

Let be a solution of (1.4) where Then converges to

Proof.

Assume that is a solution of (2.2). If it is proved that converges to zero as , then converges to

From (2.9), it is clear that converges to zero as

Then, the rest of proof is similar to the case and will be omitted. Therefore, the proof is complete.

### 2.2. The Case

In this section, we consider the asymptotic behavior of the positive solutions of (1.4) in the case

where initial conditions are real numbers.

We need the following lemma in order to prove the main result of this section.

Lemma 2.4.

Proof.

From the above statements, we have for all Therefore, the proof is complete.

Theorem 2.5.

Let be a solution of (1.4) where Then converges to

Proof.

Let be a solution of (2.12). To prove the desired result, it suffices to prove that converges to zero.

From (2.15), it is clear that converges to zero as Therefore, the proof is complete.

## Declarations

### Acknowledgment

The authors are grateful to the anonymous referees for their valuable suggestions that improved the quality of this study.

## Authors’ Affiliations

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## 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.