On the positive solutions of the difference equation xn+1=(xn−1)/(1+xnxn−1)
Introduction
Olwaidy [2] has studied the periodic cycle of xn+2=(an+bnxn)/xn−1. Owaidy [3] has also investigated the boundedness, persistence and asymptotic behaviour of positive solutions of the equation xn+1=A/xnp+B/xn−1q+C/xn−2s. Zhang [1] has obtained some global attractivity results for the rational recursive xn+1=(a+bxn2)/(1+xn−12). Li [5] has obtained sufficient conditions for the global asymptotic stability of the difference equation xn+1=f(xn)g(xn−k). Devault [6] has investigated the global stability and periodic character of solution yn+1=(p+yn−k)/(qyn+yn−k). Cunningham [4] has investigated the global character of solutions of the nonlinear xn+1=(α+βxn)/(Bxn+Cxn−1).
Similar to the references, in this paper, we define new difference xn+1=(xn−1)/(1+xnxn−1), n=0,1,2,… and investigate the positive solutions of the difference equationwhere x−1 and x0 are the positive real numbers.
Section snippets
Main results
Theorem 2.1 Let x−1=k and x0=h be positive real numbers. Then all solution of Eq. (1) are Proof Assume that n is even. Then n−1 and n+1 are odd. If we substitute (2) in (1), then we have the following equals:
Numerical results
Example 3.1 Let xn+1=(xn−1)/(1+xnxn−1), n=0,1,2,…,999, x−1=100 and x0=30 we have the following table: Example 3.2 Let xn+1=(xn−1)/(1+xnxn−1), n=0,1,2,…,999, x−1=0.001 and x0=0.001 we have the following table: Example 3.3 Let xn+1=(xn−1)/(1+xnxn−1), nn n 1 0.3332222592e−1 749 0.1525641154e−2 2 15.00249958 750 0.8739491132 249 0.2644249062e−2 999 0.1321133966e−2 250 1.512714788 1000 0.7569252522 n n 1 0.9999990000e−3 749 0.9996251940e−3 2 0.9999990000e−3 750 0.9996251950e−3 249 0.9998750075e−3 999 0.9995003740e−3 250 0.9998750075e−3 1000 0.9995003750e−3
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