| 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. |
