Global asymptotic properties of third-order difference equations (original) (raw)

2004, Computers & Mathematics with Applications

The third-order nonlinear difference equations a(p~A(~.A~.)) + q.f(~+p) = 0, p e {0,1, 2}, where (p,~), (rn), and (qn) are sequences of positive real numbers for n E N, f : R-* ~ is a continuous function such that f(u)u > 0 for u =fi 0, are investigated. All nonoscillatory solutions of these equations are classified according to the sign of their quasiditterences to classes Ni, i = 0, 1, 2, 3, and sufficient conditions ensuring N~-0, i E {1, 2, 3} are given. Special attention is paid to equation (El) for which the generalized zeros of solutions are studied and an energy function F is introduced. The relation between the class No and solutions for which Fn < 0 for n E N is established. (~ 2004 Elsevier Ltd. All rights reserved.