Abstract
We study the following nonautonomous difference equation: , , where is a period-2 sequence and the initial values . We show that the unique prime period-2 solution of the equation above is globally asymptotically stable.
1. Introduction
In [1], Xianyi and Deming studied the global asymptotic stability of positive solutions of the difference equation where and the initial conditions 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). Moreover, a lot of difference equations with periodic coefficients have been applied in mathematical models in biology. In addition, between others in [2–10], we can see some more difference equations with periodic coefficient that have been studied. For constant coefficient, see [1, 11–15] and the references cited therein.
In the present paper, we consider the following nonautonomous difference equation: where is a two periodic sequence of nonnegative real numbers and the initial conditions are arbitrary positive numbers.
The autonomous case of (2) is (1).
It is natural problem to investigate the behavior of the solutions of (1) where we replace the constant by a nonnegative periodic sequence . That is, we will consider the recursive sequence (2) where is a two-periodic sequence of nonnegative real numbers. What do the solutions of (2) with positive initial conditions and do?
Our aim in this paper is to investigate the question above and extend some results obtained in [1]. More precisely, we study the existence of a unique positive periodic solution of (2) and we investigate the boundedness character and the convergence of the positive solutions to the unique two periodic solution of (2). Finally, we study the global asyptotic stability of the positive periodic solution (2).
As far as we examine, there is no paper dealing with (2). Therefore, in this paper, we focus on (2) in order to fill the gap.
Now, assume that in (2). Then, where .
Here, we review some results which will be useful in our investigation of the behavior of positive solutions of (2).
Let be some interval of real numbers and let be a continuously differentiable function. Then for every pair of initial values , the difference equation has a unique solution .
Definition 1. Let be an equilibrium point of the map where and are continuously differentiable functions at . The linearized system of about the equilibrium point is , where and is the Jacobian matrix of system (8) about the equilibrium point .
Theorem 2 (linearized stability). Consider the following.(1)If both roots of the quadratic equation lie in the open unit disk , then the equilibrium of (6) is locally asymptotically stable.(2)If at least one of the roots of (10) has absolute value greater than one, then the equilibrium of (6) is unstable.(3)A necessary and sufficent condition for both roots of (10) to lie in the open unit disk is (4)A necessary and sufficent condition for both roots of (10) to have absolute value greater than one is (5)A necessary and sufficent condition for one root of (10) to have absolute value greater than one and for the other to have absolute value less than one is (6)A necessary and sufficent condition for a root of (10) to have absolute value equal to one is or
2. Boundedness of Solutions to (2)
In this section, we mainly investigate the boundedness character of (2), assuming (3) is satisfied.
Theorem 3. Let . Then, every positive solution of (2) is bounded.
Proof. Suppose for the sake of contadiction that is an unbounded solution of (2). Then, there exist the subsequences and of the solution such that one of the following statements is true.(i) and ( is arbitrary positive real number) for all .(ii) ( is arbitrary positive real number) and for all .(iii) and for all .
Now, we will show that the above statements are never satisfied.
From (2), we have the following inequalities:
If we get limits on both sides of the above inequalities, they contradict cases (i) and (ii). From (16) and using the induction, we obtain that
Also, if we consider the limiting case of the above inequalities, they contradict case (iii) and complete the proof.
3. Periodicity and Stability of Solutions to (2)
In this section, we show that (2) has the unique two-periodic solution and the unique prime period-2 solution of this equation is locally asymptotically stable.
Theorem 4. Suppose that is a two periodic sequence such that Then (2) has the unique two-periodic solution.
Proof. Let be a solution of (2). Using (18), is two-periodic solution if and only if and satisfy
We set and ; then from (19) we obtain the system of equations
We prove that (2) has a solution . From (20) we get
It follows that (2) has a prime period- solution if and only if , , in which case (2) has a unique two-periodic solution
Theorem 5. Assume that is a two periodic sequence and Then (2) has the unique prime period- solution which is locally asymptotically stable.
Proof. Let be a positive solution of (2) and let
be the unique prime period- solution of (2). For any , set
Then, (2) is equivalent to the system
So, in order to study (2) we investigate system (26).
Let be the function on defined by
Then,
is a fixed point of . Clearly the unique prime period- solution is locally asymptotically stable when the eigenvalues of the jakobian matrix , evaluated at (), lie inside the unit disk. We have
The linearized system of (26) about is the system
where
The characteristic equation of is
and thus
From Theorem 2 and (23) all the roots of (33) are of modulus less than . So the system (26) is locally asymptotically stable. Therefore, (2) has the unique prime period- solution which is locally asymptotically stable.
4. Global Asymptotic Stability for (2)
In this section, we give our main result. Before doing so, in the next theorem, we will provide an alternative proof for the global stability of (1). In the proof of the main theorem, we will use a similar technique.
Theorem 6. The positive equilibrium point of (1) is globally asymptotically stable.
Proof. We know from Theorem 3 that every positive solution of (1) is bounded for and we know from [1] that the positive equilibrium point of (1) is locally asymptotically stable for .
Let be a positive solution of (1) and set
Then by the boundedness of ; , are positive real numbers.
Thus, it is easy to see from (1) that
Thus, we have
and so
From (37), if , the proof is obvious. If , this case contradicts the unique equilibrium solution of (1) which is locally asymptotically stable. Consequently, it must be . This completes the proof.
Now, we are ready to give the proof of the main result.
Theorem 7. Assume that is a two-periodic sequence and ; then (2) has the unique prime period- solution which is globally asymptotically stable.
Proof. We must prove that the unique prime period- solution of (2) is both locally asymptotically stable and globally attractive. We know from Theorem 5 that the unique prime period- solution of (2) is locally asymptotically stable for . It remains to verify that every positive solution of (2) converges the unique prime period- solution. Since , by Theorem 3 every positive solution of (2) is bounded. Let be a positive solution of (2) and set
Then by the boundedness of , , , , and are positive real numbers. Moreover, it is obvious that since , then from (2)
and it is clear that if exists, then so does .
Thus, it is easy to see from (2) that
From (40), we have
From (41), we have
With cases (42) and (43), if and , the proof is obvious. If and , this case contradicts the unique prime period- solution of (2) which is locally asymptotically stable. Consequently, it must be and . The proof is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referees for their helpful suggestions. This research is supported by TUBITAK and Bulent Ecevit University Research Project Coordinatorship.