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Abstract and Applied Analysis
Volume 2012, Article ID 409237, 20 pages
http://dx.doi.org/10.1155/2012/409237
Research Article

On the Difference Equation

1Mathematical Institute of The Serbian Academy of Sciences and Arts, Knez, Mihailova 36/III, 11000 Beograd, Serbia
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic
3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
4Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia

Received 27 May 2012; Accepted 2 July 2012

Academic Editor: Jean Pierre Gossez

Copyright © 2012 Stevo Stević et al. 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.

Abstract

The behavior of well-defined solutions of the difference equation , where is fixed, the sequences , and are real, , , and the initial values are real numbers, is described.

1. Introduction

Recently there has been a huge interest in studying nonlinear difference equations and systems (see, e.g., [133] and the references therein). Here we study the difference equation where is fixed, the sequences and   are real, , , and the initial values are real numbers. Equation (1.1) is a particular case of the equation with real sequences , and . For , , the equation is trivial, and, for , it is reduced to equation (1.1) with and .

Equation where , which was treated in [32], is a particular case of equation (1.1).

As in [32], here, we employ our idea of using a change of variables in equation (1.1) which extends the one in our paper [21] and is later also used, for example, in [4]. For similar methods see also [22, 25]. Equation (1.3) in the case was also studied in [1, 2], in a different way. The case when the sequences and are two-periodic was studied in [31] (some related results are also announced in talk [3]). For related symmetric systems of difference equations, see [27, 29]. For some other recent results on difference equations and systems which can be solved, see, for example, [6, 7, 2022, 30, 31, 33]. Some classical results can be found, for example, in [11].

Equation (1.1) is a particular case of the equation where is a continuous function. Numerous particular cases of (1.4) have been investigated, for example, in [9, 21, 23]. In this paper we adopt the customary notation and .

2. Case

Here we consider the case . In this case equation (1.1) becomes ,, from which it follows that for each Using formula (2.2) the following theorem can be easily proved.

Theorem 2.1. Consider equation (1.1) with , ,. Then the following statements are true: if for some , then as ; if for each , the limits in (2.3) are greater than , then as ; if , for every and for some , then , ; if , for every and for some , then , ; if and , for some , then , as ; if for each , the limits in (2.4) belong to the interval and , then as .

3. Case

In this section we consider the case . Note that in this case equation (1.1) becomes where . If is a well-defined solution of equation (3.1) (i.e., a solution with initial values , , which implies , ), then

Hence for each

Using formula (3.3) we easily prove the next theorem.

Theorem 3.1. Consider equation (1.1) with , , . Then the following statements are true: if for some , then as ; if, for each , the limits in (3.4) are greater than , then as ; if , for every and for some , then , ; if , for every and for some , then , ; if and for some , then , as ; if, for each , and the limits in (3.5) belong to the interval , then as .

4. Case and

The case when and for every is considered in this section.

If for some , then from (1.1) we have that From (4.1) and (1.1) we have that for each From (4.1) we see that, for , (4.2) also holds. Hence Theorem 2.1 can be applied in this case. Note that if for some , then (1.1) implies that there is an such that , and by the previous consideration we have that (4.2) also holds.

If , for each , then for every well-defined solution we have for (note that there are solutions which are not well defined, that is, those for which , for some ).

Multiplying equation (1.1) by and using the transformation we obtain equation Note that from (4.3), for every well-defined solution of equation (1.1) such that , for each , it follows that , .

Since , , we have that

From (4.3) and (4.5) we have that for every .

Hence, from (4.6), we obtain that for every and each , where

5. Case

Here we consider the case , . In this case, from (4.7) we have that for each Note that this formula includes also the case when for some .

Now we formulate and prove a result in this case by using formula (5.1).

Theorem 5.1. Consider equation (1.1) with , , , , , and Then the following statements hold: if for some then as ; if (5.3) and (5.4) hold for every , then as ; if for some the sum converges, then the sequence is also convergent; if the sum in (5.5) is finite for every , then the sequences are convergent.

Proof. Let be a solution of equation (1.1). Using condition , , it is easy to see that if (5.4) holds for some , there is an such that for the terms in the product in (5.1) are positive and that the following asymptotic formula can be used with being the fraction in the limit (5.4). From (5.1) and (5.6) we have that where
Using formula (5.7), the assumptions regarding the sum and the comparison test for the series whose terms are of eventually the same sign, the results in the theorem easily follow.

6. Case

Here we consider the case , . In this case from (4.7) we have for every and each , where is defined by (4.8).

Theorem 6.1. Consider equation (1.1) with , , , and Then the following statements hold: if for some then as ; if for every , (6.3), (6.4), and (6.5) hold, then as ; if for some conditions (6.4) and (6.5) hold, and , then as ; if, for every , conditions (6.4), (6.5), and (6.6) hold, and , , then as ; if for some the sum converges and condition (6.5) holds, then the sequences and are also convergent; if, for every the sum in (6.7) converges and condition (6.5) holds, then the sequences , are convergent.

Proof. Let be a solution of equation (1.1). By (6.4) we see that irrespectively on , there is an such that for the terms in the product in (6.1) belong to the interval and that asymptotic formulae can be used with being the fraction in (6.4). From this and (6.1) we have that where
Using formula (6.9), the assumptions of the theorem and some well-known convergence tests for series, the results in (a)–(f) easily follow.

7. Case

In this section we consider equation (1.1) for the case , , , that is, when the sequences and are -periodic.

First we show the existence of -periodic solutions of equation (4.4). If is such a solution, then we have that By successive elimination, or by Kronecker theorem (note that system (7.2) is linear), we get if , where is the permutation defined by and , where Id denotes the identity.

It is easy to see that (4.4) along with periodicity of sequences and implies for every and , such that .

Since (7.5) is a linear first-order difference equation, we have that when , its general solution is

By letting in (7.6) we obtain the following corollary.

Corollary 7.1. Consider equation (4.4) with , , . Assume that Then for every solution of the equation we have that for every , that is, converges to the -periodic solution in formula (7.3).

Let

From now on we will use the following convention: if , then we regard that , if . Also if a sequence is defined by the relation , where is a real function, then we will assume that , if .

Using (7.6) and notation (7.9) in the relation (see (4.6)), for the case , we have that for every and each , and

Now we present some results, which are applications of formulae (7.10) and (7.11).

7.1. Case

If , then by (7.5) we get for ; that is, is two-periodic for each . Hence is a -periodic solution of equation (4.4), in this case.

Hence from the relation (see (4.6)), for each , we have for .

From (7.13) and by periodicity of , we get for each .

From (7.14), the behavior of solutions of equation (1.1), in this case, easily follows. For example, if for each , then the solution of (1.1) is -periodic.

7.2. Case

If and , then from (7.5) we obtain when , from which along with (4.6), it follows that

Corollary 7.2. Consider equation (1.1). Let , , and , . Then the following statements hold true. If , for some , then as . If , or and , for some , then as , if . If , for some , and , then as , if . If , for some , and , then as . If , for some , and , then the sequence is convergent. If , and , then as , if . If , and , then as . If , and , then the sequence is convergent.

Proof. The statements in (a) and (b) follow from the facts that if , if and , if , and if and .
Now assume that and let be a solution of equation (1.1). It is easy to see that there is an such that for the terms in the products in (7.17) are positive and that the following asymptotic formulae can be applied with , whenor with . Using these formulae, for the case , we have that where
Letting in (7.23), using the facts that and that the series converges, we get statements (c)–(e).
If , that is , then by using (7.22) we get where Letting in (7.26), using (7.25) and the fact that the series converges, we get statements (f)–(h), as desired.

7.3. Case

If , then from (7.6) we get where from (4.6) it follows that for , and Note that , if .

Corollary 7.3. If , , and , for every and , then the following statements hold true. If , for some , we have that as . If , and if for some , we have that as , if . If , for some , then is convergent. If , and for some , then as . If , and , then as . If , and for some , then as , if . If , and , then as , if . If , and  for some , then is constant. If , and , then is constant. If , and  for some , then . If , and , then . If , for some , then the subsequences and are convergent.

Proof. Since we have that when , when and , when , and when and , the statements in (a) and (b) easily follow from (7.30)–(7.35).
(c) If , then for , and if , then from which the statement in (c) easily follows.
(d)–(k) If , then for , and if , then from which the statements in (d)–(k) easily follow.
(l) If , then we have that for , and if , then from which the statement in (l) easily follows.

Corollary 7.4. If and , and , for every and , then the following statements hold true. If , for some , then as . If , then as . If , or , and , for some , then as , if . If , or if , and , then as , if . If , for some , then the sequence is convergent. If and for some , then as . If and , then as . If and for some , then as , if . If and , then as , if . If and for some , then the sequence is constant. If and , then the sequence is constant. If and for some , then the sequence is two-periodic. If and , then the sequence is two periodic. If , then the sequence is convergent. If , for some , then the sequences and , are convergent. If , then the sequences and , are convergent.

Proof. (a)–(d) These statements follow correspondingly from the next relations (which are derived using formulae (7.30) and (7.31)): for if , and for if , and ; if , and if , and .
(e) If , then from (7.30) we get for , from which (e) follows.
(f)–(m) If for some , then for we have while when , we have from which the statements (f)–(i) easily follow.
(n) If , then we have from which along with the assumption the statement follows.
(o) and (p) If , then for , and
From (7.50) and (7.51) the statements in (o) and (p) correspondingly follow.

Acknowledgment

The second author is supported by Grant P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government grant MSM 00 216 30519. The fourth author is supported by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.

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