GLOBAL ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS FOR EXPONENTIAL FORM DIFFERENCE EQUATIONS WITH THREE PARAMETERS ∗

In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.


Introduction
The study of asymptotic stability of positive solutions in difference equations is extremely useful in the behavior analysis of mathematical models in various biological systems and other applications.In recent years, the global asymptotic behavior of the difference equations of exponential form has been one of the main topics in the theory of difference equations [1][2][3][4][5][6].In particular, in [3] the authors studied the existence of the equilibrium and the boundedness of solutions of the difference equation where α, β are positive constants and the initial values x −1 , x 0 are positive numbers.Inspired by the [3], in this paper, we extend the above equation and investigate the global stability, boundedness nature of the positive solutions of the difference equation where the parameters a ∈ (0, ∞), b ∈ (0, 1), c ∈ (0, ∞), and the initial conditions x −1 , x 0 are arbitrary nonnegative numbers.Equation (1.1) could have applications in biology if we consider it as a model for the reproduction of some biennial plant.
In fact, the total amount of this plant x n+1 at the end of year n + 1 consists of three components.The first term a is a certain planting in each year, the amount remaining from the year n − 1 is bx n−1 , and the last term cx n−1 e −xn describes the plant produced in year n + 1, which is inhibited by the plant x n currently on the ground, but increased by the recycling of the previous year's plant x n−1 .

Preliminaries
Let I be an interval of real numbers, and let f : I ×I −→ I be a continuous function.Consider the difference equation where the initial values x −1 , x 0 ∈ I.
The linearized equation of (2.1) about the equilibrium point x is the linear difference equation where p = ∂f ∂x (x, x) and q = ∂f ∂y (x, x).
The characteristic equation of (2.2) is the equation The following well-known lemma, called the Linearized Stability Theorem, we just present part of results which will be used in this paper.Lemma 2.1 (Theorem A, [1]).(The Linearized Stability Theorem).The following statements are true.
(i) If both solutions of (2.3) have absolute value less than one, then the equilibrium x of (2.1) is locally asymptotically stable.
(ii) A necessary and sufficient condition for both roots of (2.3) to have absolute value less tan one is |p| < 1 + q < 2.
The main tool we will use is the following lemma.
Lemma 2.2 (Theorem D, [1]).Support that f satisfies the following conditions: (i) There exist positive number a and b with a < b such that a Before we give the main result of this paper, we establish the existence and uniqueness of equilibrium of (1.1).Proof.Suppose that ce −a < 1 − b. (2.4) Then (1.1) has a unique positive equilibrium x.
Proof.Observe that the equilibrium points of (1.1) are the solutions of the equation x = a + bx + cxe −x . (2.5) It suffices to show that g (x) < 0.
As g is continuous, there exists an ε such that for Therefore from (2.7), g is decreasing in the interval (x − ε, x + ε).Suppose that g has roots greater than the root x.Let x 1 be the smallest root of g such that x 1 > x.Similar to the argument above, we can show that there exists an ε 1 such that g is decreasing in the interval (x 1 − ε 1 , x 1 + ε 1 ).Since g(x + ε) < 0, g(x 1 − ε 1 ) > 0 and g is continuous, we see that g must have a root in the interval (x + ε, x 1 − ε 1 ).This is clearly a contradiction since x 1 is the smallest root of g such that x 1 > x.
Similarly we can prove that g has no solutions in (a, x).Therefore equation g(x) = 0 must have a unique solution in (a, +∞).So (1.1) has exactly one solution x, and furthermore x > a.

Boundedness and the asymptotic behavior of Solutions
The following theorem gives a sufficient condition for every positive solution of (1.1) to be bounded.Observe that for all n ≥ 2, We will now consider the non-homogeneous difference equations 3) is given by where r 1 , r 2 depends on the initial values y −1 , y 0 .Thus we see that relations (3.1) and (3.4) imply that y n is a bounded sequence.Now we will consider the solution y n of (3.3) such that Thus from (3.2) and (3.5) we get Therefore it follows that x n is bounded.
In the following, we will study the asymptotic behavior of the positive solutions of (1.1).
Then (1.1) has a unique positive equilibrium x is a global attractor.
Proof.It suffices to show that any positive solution x n converges to the unique positive equilibrium x of (1.1).Let x n be a solution of (1.1) with initial values x −1 , x 0 such that (3.7) Then from (1.1), we get

It follows by induction that
It is obviously that f (x, y) is decreasing in x and is increasing in y.
By lemma 2.2, it suffices to show that (1.1) has no positive solutions with prime period two.
Let x, y ∈ (a, ∞) be such that It suffices to show that x = y, x > a, y > a.
From (3.7) we get Clearly F (x) = 0.It suffices to show that F (u) has exactly one z-intercept greater than a.Let u > a be such that F (u) = 0. We claim that We have To prove (3.11), it suffices to show that The proof is complete. . (3.17) Then the equilibrium x of (1.1) is locally asymptotically stable.
Proof.The linearized equation of (1.1) at the equilibrium x is the characteristic equation is Now, we need to prove that the equilibrium x of (1.1) is locally asymptotically stable, then by Lemma 2.1, it suffices to show that the roots of (3.17 That is, combining the proof of Proposition 2.1, by direct computation, we have  .
Thus it follows immediately from Theorem 3.2 and Theorem 3.
iii) (2.1) has no solutions of prime period two in [a, b].Then there exists exactly one equilibrium solution x (2.1) which lies in [a, b].Moreover, every solution of (2.1) with initial conditions x −1 , x 0 ∈ [a, b] converges to x.

Theorem 3 . 2 .
Consider (1.1) where the initial values x −1 , x 0 are positive constants and a, b are positive constants satisfying c < e a ( −a(1

Theorem 3 . 4 .
If the relation(3.6)was hold, then the equilibrium x of (1.1) is globally asymptotically stable.