ABSTRACT
Consider the third-order rational difference equation
xn+1 = α+ βxn + γxn−1 + δxn−2 A+Bxn + Cxn−1 +Dxn−2
, n = 0, 1, ... (3.0.1)
with nonnegative parameters α, β, γ, δ, A,B,C,D and with arbitrary nonnegative initial conditions x−2, x−1, x0, such that the denominator is always positive. This equation contains 225 special cases of equations with positive parame-
ters. It was conjectured in [69] that in 135 of these special cases, every solution of the equation is bounded and, in the remaining 90 cases, the equation has unbounded solutions in some range of their parameters and for some initial conditions. In this chapter we present several theorems on the existence of unbounded
solutions of some equations of the form of Eq.(3.0.1) and in particular we establish that in 85 special cases of Eq.(3.0.1) there exist unbounded solutions in some range of their parameters. The only five special cases of Eq.(3.0.1) where it has been conjectured that they have unbounded solutions but we are unable yet to confirm it are the following:
#28, #44, #56, #70, #120.
