Global Asymptotic Stability of a Nonautonomous Difference Equation

where p ∈ [0,∞) 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:


Introduction
In [1], Xianyi and Deming studied the global asymptotic stability of positive solutions of the difference equation where  ∈ [0, ∞) 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][3][4][5][6][7][8][9][10], we can see some more difference equations with periodic coefficient that have been studied.For constant coefficient, see [1,[11][12][13][14][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  −1 ,  0 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  −1 and  0 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  −1 ,  0 ∈ , 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 V +1 = (V  ) =   V  , where and   is the Jacobian matrix of system (8) about the equilibrium point (, ).
(1) If both roots of the quadratic equation lie in the open unit disk || < 1, 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) (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)

Boundedness of Solutions to (2)
In this section, we mainly investigate the boundedness character of (2), assuming (3) is satisfied.
Proof.Suppose for the sake of contadiction that {  } is an unbounded solution of (2).Then, there exist the subsequences { 2+1 } and { 2+2 } of the solution {  } such that one of the following statements is true.
Now, we will show that the above statements are never satisfied.
From (2), we have the following inequalities: , for  ≥ 0, , for  ≥ 0, , for  ≥ 0. ( 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 ) . (17)

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   > 0 is a two periodic sequence such that Then (2) has the unique two-periodic solution.
Theorem 5. Assume that   > 0 is a two periodic sequence and Then (2) has the unique prime period-2 solution which is locally asymptotically stable.
be the unique prime period-2 solution of (2).For any  ≥ 0, set Then, ( 2) is equivalent to the system So, in order to study (2) we investigate system (26).
Let  be the function on (0, ∞) 2 defined by Then, ) . (29) The linearized system of (26) about (√, √) is the system From Theorem 2 and (23) all the roots of (33) are of modulus less than 1.So the system (26) is locally asymptotically stable.Therefore, (2) has the unique prime period-2 solution which is locally asymptotically stable.
) is a fixed point of .Clearly the unique prime period-2 solution is locally asymptotically stable when the eigenvalues of the jakobian matrix   , evaluated at (