On a New Class of Rational Difference Equation zn+1 = azn + bzn-k + (α + βzn−k)/ (A+ Bzn−k)

Abstract

In this article, we will investigate invariant intervals, periodic character, the character of semi cycles and global asymptotic stability of all positive solutions of a nonlinear rational difference equation of order k+1, zn+1 = azn+bzn−k+(α +β zn−k)/(A+Bzn−k), n = 0,1, ..., where the parameters a, b, α,β and A, B and the initial conditions z−k, z−k+1, z−k+2, ..., z0 are arbitrary positive real numbers. We also have studied the global stability of this equation through numerically solved examples and confirm our theoretical discussion through it.