Abstract
We study the asymptotic behavior of positive solutions to the difference equation , where . We prove that every positive solution to this equation converges to .
1. Introduction
Recently, there has been a considerable interest in studying, the so-called, max-type difference equations, see for example, [1–21] and the references cited therein. The max-type operators arise naturally in certain models in automatic control theory (see [9, 11]). The investigation of the difference equationwhere , are real numbers such that at least one of them is different from zero and the initial values are different from zero was proposed in [6]. Some results about (1.1) and its generalizations can be found in [1, 3–5, 7, 8, 10, 12, 17, 18, 19] (see also the references therein). The study of max-type equations whose some terms contain nonconstant numerators was initiated by Stevi, see for example, [2, 14–16]. For some closely related papers, see also [20, 21].
Motivated by the aforementioned papers and by computer simulations, in this paper we study the asymptotic behavior of positive solutions to the difference equationwhere . We prove that every positive solution of this equation converges to .
2. Main Results
In this section, we will prove the following result concerning (1.2).
Theorem 2.1. Let be a positive solution to (1.2).
Then
In order to establish Theorem 2.1, we need the following lemma and its corollary which can be found in [13].
Lemma 2.2. Let be a sequence of positive numbers which satisfies the inequality where and are fixed. Then there exist such that
Corollary 2.3. Let be a sequence of positive numbers as in Lemma 2.2. Then there exists such that
Now, we are in a position to prove Theorem 2.1.
Proof. We proceed by distinguishing two
possible cases.
Case (). We prove as .
Set ,
then (1.2) becomeswhere .
Since ,
we have .
To prove as ,
it suffices to prove as .
We proceed by two cases: and .
Case . In this case (2.5) is reduced towhere .
Choose a number so that .
Let .
Then, is a solution to the difference
equationTo prove as ,
it suffices to prove as .
It can be easily proved that there is a positive
integer such that for all By simple computation, we get
that, for all ,
Since ,
(2.10) implies as .
From (2.9) and (2.11), it follows that as .
This implies .
Case . Let ,
then is a solution to the difference
equationTo prove as ,
it suffices to prove as .
If ,
then we have for all .
Next, we assume either or .
Then the following four claims are obviously true.
Claim 1. If and for some ,
then
Claim 2. If and for some ,
then .
Claim 3. If and for some ,
then .
Claim 4. If and for some ,
then
In general, we havewhere .
From (2.15) and Corollary 2.3, there exists such thatThis implies as .
Case (). We prove as .
Similar to the proof of Case 1, we set ,
then (1.2) becomeswhere .
To prove as ,
it suffices to prove as .
Let ,
then is a solution to the difference
equationTo prove as ,
it suffices to prove as .
If ,
then we have for all .
Next, we assume either or ,
then the following four claims are obviously true.
Claim 1. If and for some ,
then
Claim 2. If and for some ,
then .
Claim 3. If and for some ,
then
Claim 4. If and for some ,
then .
In general, we havewhere .
Then the rest of the proof is similar to the proof of Case 1 and will be omitted. The proof is complete.
Theorem 2.4. Every solution to the difference equation converges to .
Proof. Let , then the equation becomesFrom this and the condition , it follows that as which implies as .
3. Conclusions and Remarks
This paper examines the asymptotic behavior of positive solutions to the difference equation (1.2) with . The method used in this work may provide insight into the asymptotic behavior of positive solutions to the generic difference equationwhere . We close this work by proposing the following conjecture.
Conjecture 3.1.Assume that is a positive solution to (3.1). Then as .
Acknowledgments
The author is grateful to the anonymous referees for their huge number of valuable comments and suggestions, which considerably improved the paper. This work is supported by Natural Science Foundation of China (10771227).