Discrete Dynamics in Nature and Society

Volume 2009, Article ID 973714, 23 pages

http://dx.doi.org/10.1155/2009/973714

## Boundedness, Attractivity, and Stability of a Rational Difference Equation with Two Periodic Coefficients

School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece

Received 24 August 2008; Accepted 11 January 2009

Academic Editor: Yong Zhou

Copyright © 2009 G. Papaschinopoulos 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

We study the boundedness, the attractivity, and the stability of the positive solutions of the rational difference equation , , where , are positive sequences of period 2.

#### 1. Introduction

In [1], Camouzis et al. studied the global character of the positive solutions of the difference equation: where are positive parameters and the initial values are positive real numbers.

The mathematical modeling of a physical, physiological, or economical problem very often leads to difference equations (for partial review of the theory of difference equations and their applications see [2–12]). Moreover, a lot of difference equations with periodic coefficients have been applied in mathematical models in biology (see [13–15]). In addition, between others in [16–19], we can see some more difference equations with periodic coefficients that have been studied.

In this paper, we investigate the difference equation where , are positive sequences of period and the initial values are positive numbers.

Our goal in this paper is to extend some results obtained in [1]. More precisely, we study the existence of a unique positive periodic solution of (1.2) of prime period . In the sequel, we investigate the boundedness, the persistence, and the convergence of the positive solutions to the unique periodic solution of (1.2). Finally, we study the stability of the positive periodic solution and the zero solution of (1.2).

If we set , , it is easy to prove that (1.2) is equivalent to the following system of difference equations: where , are positive constants and the initial values , are positive numbers. So in order to study (1.2) we investigate system (1.3).

#### 2. Existence of the Unique Positive Equilibrium of System (1.3)

In the following proposition, we study the existence of the unique positive equilibrium of system (1.3).

Proposition 2.1. *Consider system (1.3) where , are positive
constants and the initial values , are positive
numbers. Suppose that ** are satisfied. Then system (1.3) possesses a unique positive equilibrium.*

*Proof. *Let be a positive
equilibrium of system (1.3) thenEquations (2.2) imply that is a solution
of the equation

Suppose
that Let , and
be the
solutions of (2.3). Then from (2.1), (2.3), and (2.4) we
take and so (2.3) has unique positive solution . Then from (2.2) and (2.4) we haveand so system (1.3) has a unique positive equilibrium.

Now suppose
that If , and
are the
solutions of (2.3), then from (2.3) and (2.7) we take and so (2.3) has a negative solution, but also (2.3) has a solution in
the interval , sinceMoreover, (2.3) has a solution in the interval , since therefore, we get (2.6) and so system (1.3) has a unique positive equilibrium.

Finally,
suppose that If , and
are the
solutions of (2.3), then from (2.3) and (2.11), we take We have and since , it is obvious that (2.3) has a solution in the interval . From (2.3), we get

If equation has complex
roots, then it is obvious that is the unique
solution of (2.3). Therefore, we get (2.6), and so system (1.3) has a unique
positive equilibrium.

Now, suppose
that the roots of are real numbers.

Suppose that , then it is
obvious thatand so we have that (2.3) has a unique solution

If , then it holds
that which implies that (2.3) has a unique solution

Therefore, we
can take (2.6) and so system (1.3) has a unique positive equilibrium. This
completes the proof of the proposition.

#### 3. Boundedness and Persistence of the Solutions of System (1.3)

In the following propositions we study the boundedness and the persistence of the positive solutions of system (1.3). In the sequel we will use the following result which has proved in [20].

Theorem 3.1. *Assume that all roots of the
polynomial** where have absolute
value less than 1, and let be a
nonnegative solution of the inequality** Then, the following statements are true.*

(i)
*If is a
nonnegative bounded sequence, then is also
bounded.*
(ii)
*If , then *

Proposition 3.2. *One considers the system of difference equations (1.3) where , are positive
constants and the initial values , are positive
numbers. Then the following statements are true. *

(i)
*If then every solution of (1.3) is bounded.*
(ii)
*Ifthen every solution of (1.3) is bounded and persists.*

*Proof. *Let be an arbitrary
solution of (1.3).

(i) From (3.3),
we get that one of the three following conditions holds:

We assume that
(3.5) is satisfied. We prove that there exists a positive integer such
that First, we show that if there exists a positive integer such
that then In contradiction, we assume thatUsing relations (1.3), (3.5), and (3.11), we get that and so relations (1.3) and (3.3) imply that which contradicts (3.9). So and working
inductively, we get (3.10).

If then from the
analogous relations (3.9) and (3.10), we getNow, suppose that we prove that there exists a positive integer such
that From (3.3), there exists a positive integer such
that If , then (3.16) is
true for .

Now, suppose
that Then from (1.3), (3.5), and (3.18), we get and so from
(1.3), (3.3), and (3.5), we have that If , then (3.16) is
true for .

Now, suppose
that Using (1.3), (3.3), (3.5), (3.20) and arguing as to prove (3.19) we
getWorking inductively, we get that From (3.22) for , we get which contradicts
(3.17). So which means
that (3.16) holds for .

Arguing as for , we can prove
that there exist positive integers such
that From (3.16) and (3.23), we get that there exists a positive integer such
that Finally, from (1.3) and (3.24), we get and so (3.8) is
true for .

Similarly, we
can prove that if (3.6) holds, then there exists a positive integer such
that

Finally,
suppose that (3.7) hold. From (1.3) and (3.7), we have and so, From (3.27), we get and so the subsequences either are
bounded from below by and decreasing
or bounded from above by and increasing.
Hence, is bounded and
persists. Similarly, we can prove that is bounded and
persists. This completes the proof of part (i) of the proposition.

(ii) In
statement (i), we have already proved that if (3.7) hold, then every solution of
(1.3) is bounded and persists. So, from (3.4), it remains to show that if
either orholds, then the solution persists. From
(3.3), (3.8), (3.25), (3.29), and (3.30), we get that We consider the positive number such
that Moreover, if then it is easy to see that for the functions (3.33), is increasing
with respect to for any , and is increasing
with respect to for any .

Therefore, from
(1.3), (3.31), and (3.32) we have and working inductively, we take Therefore, persists and
using statement (i), then is bounded and
persists. This completes the proof of the proposition.

Proposition 3.3. *
One considers the system of difference equations (1.3) where ,
are positive
constants, and the initial values , are positive numbers.
Then, the following statements are true. *

(i)
*If then every solution of (1.3) persists.*
(ii)
*If then every solution of (1.3) is bounded and persists.*

*Proof. *Let be an arbitrary
solution of (1.3).

(i)From (3.36),
we have or

Arguing as in
the proof of statement (i) of Proposition 3.2, we can easily prove that if
(3.38) holds, then there exists a positive integer such
that and if (3.39) holds, then there exists a positive integer such
that

(ii)From
Proposition 3.2, we have that if (3.7) holds, then every solution of (1.3) is
bounded and persists. So, from (3.37), it remains to show that if
eitheror holds, then the solution is bounded and
persists.

From (3.36),
(3.40), (3.41), (3.42), and (3.43), we get that Suppose that

From (1.3) and
(3.44), we haveWe have for the functions (3.33) that is decreasing
with respect to for any , and is decreasing
with respect to for any , . Therefore, relations (1.3), (3.44), and (3.46) imply
that and so from (1.3) and (3.46), Working inductively, we can prove thatThen from (3.42), (3.43), (3.45), and Theorem 3.1, is bounded.
Similarly, we take that is bounded.
Therefore, from (3.44), the solution is bounded and
persists.

Now, suppose
that We claim that is bounded. For
the sake of contradiction, we assume that is not bounded.
Then, there exists a subsequence such
that Moreover, from (1.3) and (3.50), we get and so from (3.51), Moreover, from (1.3) and (3.50), and so from (3.54), Working inductively, we can prove that We claim that is a bounded
sequence. Suppose on the contrary that there exists an unbounded subsequence of and without
loss of generality, we may suppose that Arguing as above, we can easily prove that Also, since from (1.3), from (3.58), we have that and so
eventually, From (1.3), (3.50), and (3.61), we have where Therefore, using (1.3) and (3.50), we get and since from (3.57) and (3.59), we have that as we can easily
prove that eventually, which contradicts to (3.52).

Therefore, is a bounded
sequence. From (1.3), (3.50), and (3.57), we get Similarly, from (1.3), (3.50) and (3.57) and (3.66) follows, Now, we prove that Otherwise, and without loss of generality, we may suppose that So, relations
(1.3), (3.50), and (3.67) imply that and so Moreover, from (1.3), (3.44), and (3.50), we get eventually and so from (3.66), which
contradicts to (3.70).

Hence, (3.68)
is true.

Similarly, we
can prove that Therefore, from (3.68) and (3.72), we have eventually where are positive
real numbers.

Hence, from
(1.3), (3.50), and (3.73) we have Then from (3.57), we can prove that eventually which contradicts to (3.52).

Therefore, is a bounded
sequence. Moreover, from (1.3), (3.50), we take that is bounded.
Therefore, the solution is bounded and
persists. This completes the proof of the proposition.

#### 4. Attractivity of the Positive Equilibrium of System (1.3)

In the following propositions, we study the convergency of the solutions of system (1.3) to its positive equilibrium.

Proposition 4.1. *
One considers the system of difference equations (1.3) where ,
are positive
constants, and the initial values , are positive
numbers. If either (3.29) or (3.30) hold, then every solution of (1.3) tents to
the positive equilibrium of (1.3). *

*Proof. *Let be an arbitrary
solution of (1.3). From Proposition 3.2, there exist From (1.3), (3.31), and the monotony of
functions (3.33), we have and hence
The third inequality of (4.3), implies thatand so from the last inequality of (4.3), we have Hence, we get or The first inequality of (4.3), implies that and so from second inequality of (4.3), we get Using (4.3), we have Therefore, from (4.5) and (4.10), we get

Using (4.9) and
(4.11), we have In Proposition 2.1, we proved that (2.3) has a unique positive solution . We can write where is defined in
(2.3) and for any . Then from (4.7), (4.12), and (4.13), we have Therefore, from (4.10) and (4.14), which implies that

In addition,
using (4.15), the first and the third inequalities of (4.3), we
have and so (4.10) implies that This completes the proof of the proposition.

Proposition 4.2. *
One considers the system of difference equations (1.3) where , are positive
constants, and the initial values , are positive
numbers. If either (3.42) or (3.43) hold, then every solution of (1.3) tents to
the positive equilibrium of (1.3).*

*Proof. *Let be an arbitrary
solution of (1.3). From Proposition 3.3, there exist , such that (4.1)
are satisfied.

From (1.3), the
monotony of functions (3.33) and (3.44), we have and hence which implies that Therefore, First suppose that (3.45) holds. Then from (3.42) or (3.43), and (3.45), we get which means
thatUsing (4.20), it is obvious that So if (3.45) holds, the proof is completed.

Now, suppose
that (3.50) hold. Then from (4.20), we have Moreover, from (4.24), it follows that In addition, from (3.50), the first and the second inequalities of (4.19), we
getTherefore, from (4.25) and (4.26), we have We may assume that there exists a positive integer such
that Moreover, from (1.3), (3.50), and (4.28), we get Since is decreasing
with respect to , for any if or , then from
(3.44), and (3.50), we get which contradicts to (4.27). So, Using the same argument, we can prove that and so Also, from
(4.24), we have Therefore, where obviously . This completes the proof of the proposition.

Proposition 4.3. *
One considers the system of difference equations (1.3) where ,
are positive
constants, and the initial values , are positive
numbers. If relations (3.7) hold, then every solution of (1.3) tents to the
positive equilibrium of (1.3).*

*Proof. *Let be an arbitrary
solution of (1.3). From the proof of Proposition 3.2, the subsequences , and are monotone
and , are bounded and
persist. So, there exist positive numbers , and such
that and from (1.3) and (3.7), we get Then, we have and hence, Therefore, we take So, Therefore, if (resp., ), we have (resp., ) and so from
(4.37), (resp., ). Hence, and from (4.37), Similarly, we
can prove that . This completes the proof of the proposition.

#### 5. Stability of System (1.3)

In this section we find conditions so that the positive equilibrium and the zero equilibrium of (1.3) are stable.

Proposition 5.1. *Consider system (1.3) where , are positive
constants and the initial values , are positive
numbers. Then, the following statements are true.*

(i)
*If then the unique positive equilibrium of (1.3) is
globally asymptotically stable.*
(ii)
*If then the zero equilibrium of (1.3) is locally asymptotically stable.*

*Proof. *(i) Since is the unique
positive positive equilibrium of (1.3), we have Then from (5.1) and (5.3), we get Without loss of generality we assume that . Then from (5.4), it results that which means that Moreover, from (5.4) and (5.6), we get In addition, from (5.3), we have and so Then the linearized system of (1.3) about the positive equilibrium is where The
characteristic equation of is According to Remark 1.3.1 of [7], all the roots of (5.12) are of modulus less
than 1 if and only if From (5.3), we get Then from (5.6), (5.7), and (5.14), inequality (5.13) is equivalent
to Using (5.9), inequality (5.15) holds if (5.1) are satisfied. Using Propositions
4.1 and 4.3, we have that the unique positive equilibrium of (1.3) is
globally asymptotically stable.

(ii) Arguing as
above, we can prove that the linearized system of (1.3) about the zero
equilibrium is where The characteristic equation of is Using [7, Remark 1.3.1], all the roots of (5.18) are of modulus less
than 1 if and only if relation (5.2) holds. This completes the proof of the
proposition.

#### 6. Conclusion

In this paper, in order to investigate (1.2), we study the equivalent system (1.3). Summarizing the results of Sections 2, 3, 4, we get the following statements, concerning (1.2).

(i)If (2.1) hold, then (1.2) has a unique positive periodic solution of period 2.(ii)If either (3.4) or (3.37) holds, then every positive solution of (1.2) is bounded and persists and tends to the unique positive periodic solution.(iii)If (5.1) hold, then the unique periodic solution of (1.2) is globally asymptotically stable and if (5.2) holds, then the zero solution of (1.2) is locally asymptotically stable.

*Open Problem*

Consider the difference equation (1.2) where , are positive
sequences of period 2, and the initial values , are positive
numbers. Prove that

(i)if are satisfied, then every positive solution of (1.2) is bounded and persists;(ii)if
relations (6.1) are satisfied, then every positive solution of (1.2) tends to
the unique positive equilibrium of (1.2) as .

#### Acknowledgment

The authors would like to thank the referees for their helpful suggestions.

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