Abstract and Applied Analysis

Volume 2012 (2012), Article ID 409237, 20 pages

doi:10.1155/2012/409237

Research Article

On the Difference Equation

Stevo Stević,¹ Josef Diblík,2,3 Bratislav Iričanin,⁴ and Zdeněk Šmarda³

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., [1–33] 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, 20–22, 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.

References

A. Andruch-Sobiło and M. Migda, “Further properties of the rational recursive sequence ,” Opuscula Mathematica, vol. 26, no. 3, pp. 387–394, 2006.

A. Andruch-Sobiło and M. Migda, “On the rational recursive sequence ,” Tatra Mountains Mathematical Publications, vol. 43, pp. 1–9, 2009.

A. Andruch-Sobiło and M. Migda, “On the rational difference equation with period-two coefficient,” in Proceedings of the 16th International Conference on Difference Equations and Applications, p. 39, Riga, Latvia, July 2010.

I. Bajo and E. Liz, “Global behaviour of a second-order nonlinear difference equation,” Journal of Difference Equations and Applications, vol. 17, no. 10, pp. 1471–1486, 2011.

L. Berezansky and E. Braverman, “On impulsive Beverton-Holt difference equations and their applications,” Journal of Difference Equations and Applications, vol. 10, no. 9, pp. 851–868, 2004.

L. Berg and S. Stević, “On some systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1713–1718, 2011.

B. Iričanin and S. Stević, “On some rational difference equations,” Ars Combinatoria, vol. 92, pp. 67–72, 2009.

G. L. Karakostas, “Asymptotic 2-periodic difference equations with diagonally self-invertible responses,” Journal of Difference Equations and Applications, vol. 6, no. 3, pp. 329–335, 2000.

C. M. Kent, “Convergence of solutions in a nonhyperbolic case,” Nonlinear Analysis, vol. 47, no. 7, pp. 4651–4665, 2001.

W. Kosmala, “A period 5 difference equation,” International Journal of Nonlinear Analysis and Applications, vol. 2, no. 1, pp. 82–84, 2011.

H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.

E. Liz and J. B. Ferreiro, “A note on the global stability of generalized difference equations,” Applied Mathematics Letters, vol. 15, no. 6, pp. 655–659, 2002.

G. Papaschinopoulos, M. Radin, and C. J. Schinas, “Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5310–5318, 2012.

G. Papaschinopoulos and C. J. Schinas, “On a system of two nonlinear difference equations,” Journal of Mathematical Analysis and Applications, vol. 219, no. 2, pp. 415–426, 1998.

G. Papaschinopoulos and C. J. Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations,” Communications on Applied Nonlinear Analysis, vol. 5, no. 2, pp. 47–59, 1998.

G. Papaschinopoulos and C. J. Schinas, “Invariants for systems of two nonlinear difference equations,” Differential Equations and Dynamical Systems, vol. 7, no. 2, pp. 181–196, 1999.

G. Papaschinopoulos and C. J. Schinas, “Invariants and oscillation for systems of two nonlinear difference equations,” Nonlinear Analysis, vol. 46, no. 7, pp. 967–978, 2001.

G. Papaschinopoulos, C. J. Schinas, and V. Hatzifilippidis, “Global behavior of the solutions of a max-equation and of a system of two max-equations,” Journal of Computational Analysis and Applications, vol. 5, no. 2, pp. 237–254, 2003.

G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On the nonautonomous difference equation ,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5573–5580, 2011.

G. Papaschinopoulos and G. Stefanidou, “Asymptotic behavior of the solutions of a class of rational difference equations,” International Journal of Difference Equations, vol. 5, no. 2, pp. 233–249, 2010.

S. Stević, “More on a rational recurrence relation,” Applied Mathematics E-Notes, vol. 4, pp. 80–85, 2004.

S. Stević, “A short proof of the Cushing-Henson conjecture,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 37264, 5 pages, 2006.

S. Stević, “On positive solutions of a th order difference equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427–431, 2006.

S. Stević, “On the recursive sequence ,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791–796, 2008.

S. Stević, “Global stability of a max-type difference equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354–356, 2010.

S. Stević, “On a nonlinear generalized max-type difference equation,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 317–328, 2011.

S. Stević, “On a system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372–3378, 2011.

S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562–9566, 2011.

S. Stević, “On a third-order system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7649–7654, 2012.

S. Stević, “On some solvable systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5010–5018, 2012.

S. Stević, “On the difference equation ,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4507–4513, 2011.

S. Stević, “On the difference equation ,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6291–6296, 2012.

S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, “On a third-order system of difference equations with variable coefficients,” Abstract and Applied Analysis, vol. 2012, Article ID 508523, 22 pages, 2012.