On the boundedness character of third-order rational difference equations.

摘要

We investigate the global behavior of the solutions of rational difference equations. In particular, we study the global stability, the periodic nature, and especially the boundedness character of their solutions.; In the first manuscript we study the equation xn+1=a+bxn+g xn-1+dxn-2A+xn ,n=0,1,... and establish that every solution of this equation converges to a (not necessarily prime) period-two solution when gamma = beta + delta + A and beta + A is positive, and that unbounded solutions of this equation exist whenever gamma > beta + delta + A. In this case, we establish a very large set of initial conditions that produce unbounded solutions.; In subsequent manuscripts, we establish conditions under which every solution of the special case of the above equation with alpha = 0, A = 1, and delta = 1 converges to beta + delta.; We also present a number of open problems and conjectures related to the general third-order rational difference equation.; In all equations, the parameters are nonnegative real numbers and the initial conditions are arbitrary nonnegative real numbers such that the denominators are always positive.