Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 975740 | https://doi.org/10.1155/2010/975740

Stevo Stević, Bratislav D. Iričanin, "On a Max-Type Difference Inequality and Its Applications", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 975740, 8 pages, 2010. https://doi.org/10.1155/2010/975740

On a Max-Type Difference Inequality and Its Applications

Academic Editor: Leonid Berezansky
Received01 Nov 2009
Accepted31 Mar 2010
Published23 Jun 2010

Abstract

We prove a useful max-type difference inequality which can be applied in studying of some max-type difference equations and give an application of it in a recent problem from the research area. We also give a representation of solutions of the difference equation .

1. Introduction

The investigation of max-type difference equations attracted some attention recently; see, for example, [120] and the references therein. In the beginning of the study of these equations the following difference equation was investigated: where ,  , are real sequences (mostly constant or periodic), and the initial values are different from zero (see, e.g., monograph [6] or paper [19] and the references therein).

The study of the next difference equation where are natural numbers such that , , , and , was proposed by the first author in numerous talks; see, for example, [11, 13]. For some results in this direction see [1, 4, 5, 7, 8, 12, 1418, 20].

A particular case of the difference equation arises naturally in certain models in automatic control (see [9]). By the change the equation is transformed into the equation which is a special case of (1.2) and which is a natural prototype for the equation.

The following result, which extends the main result from the study in [18] was proved by the first author in [17] (see also [16]).

Theorem A. Every positive solution to the difference equation where , converges to

Here we continue to study (1.5) by considering the cases when some of 's are equal to one. We also give a representation of well-defined solutions of the difference equation where ,

2. Main Results

In this section we prove the main results of this note. Before this we formulate the following very useful auxiliary result which can be found in [10] and give a definition.

Lemma A. Let be a sequence of positive numbers which satisfy the inequality where and are fixed. Then there exists an such that

Definition 2.1. For a sequence, , we say that it converges to zero geometrically if there is a and such that for .

Now we are in a position to formulate and prove the main results of this note.

Proposition 2.2. Assume that is a sequence of nonnegative numbers satisfying the difference inequality where , , , , and if, for some , , then . Then the sequence converges geometrically to zero as .

Proof. Let be chosen such that where Then from (2.4) and using the fact that are nonnegative numbers, we have that where .
From (2.7), (2.5) and since , we have that
Now assume that , for Then from (2.4) we get Inequalities (2.8) and (2.9) along with the method of induction show that
Now note that from (2.10) we have that
From (2.7), (2.11) and the choice of , it follows that for
Applying Lemma A in inequality (2.12) with , the result follows.

Remark 2.3. Note that the constant in the proof of Proposition 2.2 depends on initial conditions of solutions to difference equation (2.4), so that this is not a uniform constant.

Lemma 2.4. Consider the difference equation where ,  ,  , and there is such that Then

Proof. If all terms in the right-hand side of (2.13) are nonnegative then clearly , so that Otherwise, the set of all indices for which the terms in (2.13) are negative is nonempty, so that
From this and since for such , must be positive, it follows that From (2.15) and (2.16) inequality (2.14) easily follows.

By Proposition 2.2 and Lemma 2.4 we obtain the following theorem.

Theorem 2.5. Consider the difference equation where , for each , and for at least one Then every positive solution of (2.17) converges to one.

Proof. Taking the logarithm of (2.17) and using the change , we obtain that Now note that for those such that , since , and there is an such that when By Lemma 2.4 we have that for every From (2.19), noticing that if and , then so that and by applying Proposition 2.2 we obtain that as , from which it follows that as , as desired.

Remark 2.6. Recently Gelişken and Çinar in the paper: “On the global attractivity of a max-type difference equation,’’ Discrete Dynamics in Nature and Society, vol. 2009, Article ID 812674, 5 pages, 2009, have studied the asymptotic behavior to positive solutions of the difference equation where and . They claim that if , then every positive solution to (2.20) converges to one. However the proof given there cannot be regarded as complete one. Namely, they first formulated the following lemma.
Lemma 2.7. Let be a solution to the difference equation Then for all , the following inequality holds: Then they tried to show that as . Note that (2.21) is obtained by the change from (2.20), so that if it is proved that as then as from which the claim follows. In the beginning of the proof of the theorem they choose a number such that but do not say if these inequalities hold for all or not, which is a bit confusing. Note that for different the chosen number can be different, which means that in this case might be a function of . Hence it is important that these inequalities hold for every which was not proved. This motivated us to prove Proposition 2.2 which, among others, removes the gap.

Now we present a representation of solutions of a particular case of (1.5). The first author would like to express his sincere thanks to Professor L. Berg for a nice communication regarding this [2].

Theorem 2.8. Consider the equation where , , Then every well-defined solution of equation (2.23) has the following form: where , and where is equal to one of the initial values
Moreover, if , then as

Proof. The case is well known and simple. Just note that . Hence assume that We prove the result by induction. For we have Note that can be equal to one of the numbers and that which is nothing but formula (2.24) in this case. From this we also have that which is (2.25) in this case.
Now assume that we have proved (2.24) and (2.25) for Then where is the Kronecker symbol and for Thus for and Hence the first statement follows by induction.
Now assume that . From this and (2.25) we have Inequality (2.32), the assumption , and (2.24) imply that tends to 1 as finishing the proof of the theorem.

Remark 2.9. Note that formula (2.24) holds for each value of parameters , and for all solutions whose initial values are different from zero if one of these exponents is negative.

Remark 2.10. The second statement in Theorem 2.8 follows easily also from Lemma A.

References

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Copyright © 2010 Stevo Stević and Bratislav D. Iričanin. 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.


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