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On a Max-Type Difference Equation

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Abstract

We prove that every positive solution of the max-type difference equation , converges to where are positive integers, , and .

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, [120] 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.

In [20], Yang et al. proved that every positive solution of the difference equation

(11)

converges to or eventually periodic with period 4, where and

In [9], We proved that every positive solution of the difference equation

(12)

converges to or eventually periodic with period 2, where and

In [17], Sun proved that every positive solution of the difference equation

(13)

converges to where , and

The following difference equation is more general than (1.3):

(14)

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

It is easy to see that by the change

(21)

Equation (1.4) is transformed into the difference equation

(22)

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.

Let be a solution of (2.2). If , then

(23)

Proof.

Clearly, (2.2) implies the following difference equation:

(24)

From (2.4), we get the following statements.

  1. (i)
  2. (ii)
  3. (iii)
  4. (iv)

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

Lemma 2.2.

Let be a solution of (2.2). If , then

(25)

Proof.

Assume that . Then (2.2) implies the following difference equation:

(26)

From (2.6), we get the following statements.

  1. (i)
  2. (ii)
  3. (iii)
  4. (iv)

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 Lemma 2.1, we have that

(27)

Let Immediately, we have that the following inequality

(28)

From (2.8) and by induction, we get

(29)

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

Now, we assume that From Lemma 2.2, we have that

(210)

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

It is easy to see that by the change

(211)

Equation (1.4) is transformed into the difference equation:

(212)

where initial conditions are real numbers.

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

Lemma 2.4.

Let be a solution of (2.12). Then

(213)

Proof.

From (2.12), we get the following statements.

  1. (i)
  2. (ii)
  3. (iii)
  4. (iv)

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 Lemma 2.4, we have that

(214)

From (2.14) and by induction, we get

(215)

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

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Acknowledgment

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

Author information

Correspondence to Ali Gelisken.

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Keywords

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