On the period five trichotomy of all positive solutions of xn+1 =( p+ xn−2 )/ xn

Abstract

We study the behavior of all positive solutions of the difference equation in the title, where p is a positive real parameter and the initial conditions x −2, x −1, x 0 are positive real numbers. For all the values of the positive parameter p there exists a unique positive equilibrium x̄ which satisfies the equation x ̄ 2= x ̄ +p. We show that if 0< p<1 or p⩾2 every positive bounded solution of the equation in the title converges to the positive equilibrium x̄. When 0< p<1 we show the existence of unbounded solutions. When p⩾2 we show that the positive equilibrium is globally asymptotically stable. Finally we conjecture that when 1< p<2, the positive equilibrium is globally asymptotically stable.