Global Dynamics of a Rational Difference Equation and Its Solutions to Several Conjectures

In this paper, we investigate the global attractivity of a higher-order rational difference equation in the form x n + 1 = p + q x n 1 + r x n + s x n − k , where p , q , r , s ≥ 0 , k is a positive integer, and the initial conditions are nonnegative. This equation generalizes several well-known rational difference equations studied in the literature. By employing a combination of advanced mathematical techniques, including the use of key lemmas and intricate computations, we establish that the unique nonnegative equilibrium point of the equation is globally attractive under specific parameter conditions. Our results not only extend and improve upon existing findings but also resolve several conjectures posed by previous researchers, including those by G. Ladas and colleagues. The methods involve transforming the higher-order equation into a first-order difference equation and analyzing the properties of the resulting function, particularly its Schwarzian derivative. The findings demonstrate that the equilibrium point is globally attractive when certain inequalities involving the parameters are satisfied. This work contributes to the broader understanding of the dynamics of rational difference equations and has potential applications in various fields such as biology, physics, and cybernetics.