1. Introduction
Recently, there has been an increasing interest in the study of the qualitative analyses of rational difference equations. For example, see[1 − 8] and the references cited therein.
This work studies the boundedness character and the global asymptotic stability for the positive solutions of the difference equation
where {pn} is a two periodic sequence of nonnegative real numbers and the initial conditions x−1, x0 are arbitrary positive numbers.
As far as we can examine, this is the first work devote to the investigation of the type Eq.(1.1).
Now, we assume p2n = α and p2n+1 = β in Eq.(1.1). Then we have
and
The autonomous case of Eq.(1.1) is
where p > 0 and the initial conditions x−1, x0 are arbitrary positive numbers. We now consider the local asymptotic stability of the unique equilibrium of Eq.(1.4).
The linearized equation for Eq. (1.4) about the positive equilibrium is
The following theorem is given in [1].
Theorem A.Consider Eq. (1.4) and assume that x−1, x0, p ∈ (0,∞). Then the unique positive equilibrium of Eq. (1.4) is globally asymptotically stable.
2. Boundedness Character of Eq. (1.1)
In this section, we investigate the boundedness character of Eq. (1.1). So, we have the following result.
Theorem 2.1. Suppose that α > 1 and β > 1 with α ≠ β, then every positive solution of Eq.(1.1) is bounded.
Proof. It is clear from Eq. (1.2) and (1.3) that
Then, from (1.2) and (2.1) we obtain
and from (1.3) and (2.1) we obtain
From (2.3), (2.4) using induction we get
The result now follows.
3. Stability and Periodicity for Eq. (1.1)
In this section, we investigate the periodicity and stability character of positive solutions of Eq. (1.1). Now, we have the following result.
Proposition 3.1. Consider Eq. (1.1) when the case α ≠ β and assume that α, β ∈ (0,∞). Then there exist prime two periodic solutions of Eq. (1.1).
Proof. In order Eq. (1.1) to possess a periodic solution {xn} of prime period 2, we must find positive numbers x−1, x0 such that
Let x−1 = x, x0 = y, then from (3.1) we obtain the system of equations
We prove that (3.2) has a solution From the first relation of (3.2) we have
From (3.3) and the second relation of (3.2) we obtain
Now we consider the function
Then from (3.4) we get
Hence Eq. (3.4) has a solution Then if we have that the solution of Eq. (1.1) with initial values is a periodic solution of period two.
Theorem 3.2. Consider Eq. (1.1) when the case α ≠β and assume that α, β ∈ (0,∞). Suppose that
Then the two periodic solutions of Eq. (1.1) are locally asymptotically stable.
Proof. From equations (1:3), (1:4) and Proposition 3:1 there exist such that
We set x2n−1 = un, x2n= vn in equations (1.3), (1.4) and so we have
Then is the positive equilibrium of Eq. (3.8), and the linearised system of Eq. (3.8) about is the system
The characteristic equation of B is
Using Eq. (3.6), from Eq. (3.7), since we have
and we obtain
Then, from (3.10) and Theorem 1.3.7 of Kocic and Ladas in [4], all the roots of Eq. (3.9) are modulus less than 1. Therefore, from Proposition 3.1, system (3.8) is asymptotically stable. The proof is complete.
Theorem 3.3. Consider Eq. (1.1) when the case α ≠ β. Assume that α > 1, β > 1. Then every positive solution of Eq. (1.1) converges to a two-periodic solution of Eq. (1.1).
Proof. Since α > 1, β > 1, we know by Theorem 2.1 that every positive solution of Eq. (1.1) is bounded, it follows that there are finite
exist. Then it is easy to see from Eq. (1.2) and (1.3) that
and
Thus, we have
and
This implies that
and
Then, we get
and
Now, we shall prove that s = S and l = L. It is clear that if l = L, then by (3.11) it must be s = S. Similarly, if s = S, then by (3.12) it must be l = L.
Hence we assume that s < S and l < L. From (3.11) and (3.12) we have
then we obtain a contradiction. So, we get s = S and l = L
Moreover, it is obvious that since α ≠ β, then from Eq. (1.2) and Eq. (1.3)
Then it is clear that every positive solution of Eq. (1.1) converges to a twoperiodic solution of Eq. (1.1). The proof is complete.
Finally, using Proposition 3.1, Theorems 3.1 and 3.2, we have the following Theorem.
Theorem 3.4. Consider Eq. (1.1) when the case α ≠ β. Assume that α > 1, β > 1 and that (3.6) holds. Then two-period solutions of Eq. (1.1) are globally asymptotically stable.
References
-
R. Devault, G. Ladas and S.W. Schultz, On the recursive sequence
$x_{n+1}$ = ($A/x_n$ ) + ($1/x_{n-2}$ ), Proc. Amer. Math. Soc. 126(11), 3257-3261 (1998). https://doi.org/10.1090/S0002-9939-98-04626-7 - E.A. Grove and G. Ladas, Periodicities in nonlinear difference equations, Chapman and Hall/Crc, 2005.
- V. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, 1993.
-
M.R.S. Kulenovic, G. Ladas and C.B. Overdeep, On the dynamics of
$x_{n+1}$ =$p_n$ + ($x_{n-1}/x_n$ ), J. Difference Equ. Appl., 9(11), 1053-1056 (2003). https://doi.org/10.1080/1023619031000154644 -
M.R.S. Kulenovic, G. Ladas and C.B. Overdeep,On the dynamics of
$x_{n+1}$ =$p_n+(x_{n-1}/x_n$ with a period-two coefficient, J. Difference Equ. Appl., 10(10), 905-914 (2004). https://doi.org/10.1080/10236190410001731434 -
O. Ocalan, Dynamics of the difference equation
$x_{n+1}$ =$p_n+(x_{n-k})/(x_n)$ with a Period-two Coefficient, Appl. Math. Comput., 228, 31-37 (2014). https://doi.org/10.1016/j.amc.2013.11.020 - V.G. Papanicolaou, On the asymptotic stability of a class of linear difference equations, Mathematics Magazine, 69, 34-43 (1996). https://doi.org/10.2307/2691392
-
S. Stevic, On the recursive sequence
$x_{n+1}$ =${\alpha}_n$ + ($x_{n-1}/x_n$ ), Int. J. Math. Sci., 2(2), 237-243 (2003).
Cited by
- On the Solutions of a System of Third-Order Rational Difference Equations vol.2018, pp.1607-887X, 2018, https://doi.org/10.1155/2018/1743540
- THE DYNAMICS OF POSITIVE SOLUTIONS OF A HIGHER ORDER FRACTIONAL DIFFERENCE EQUATION WITH ARBITRARY POWERS vol.35, pp.3, 2017, https://doi.org/10.14317/jami.2017.267