In this work we define Ψ -boundedness of sequences and discuss the existence of Ψ -bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of Ψ -bounded solutions for the linear nonhomogeneous difference equation x(n + 1) = A(n)x(n)+ f (n). c © 2006 Elsevier Ltd. All rights reserved.

In this paper, sufficient conditions are established for the oscillatory and asymptotic behavior of higher–order half–linear delay difference equation of the form ∆(pn(∆ (xn + qnxτn)) ) + rnx β σn = 0, n ≥ n0, where it is assumed that ∑∞ s=n0 1/p 1/α s < ∞. The main theorem improves some existing results in the literature. An example is provided to demonstrate the effectiveness of the main result.

Necessary and sufficient conditions for the asymptotic stability of a class of difference equations with three parameters are obtained. These conditions are expressed in terms of subsets of the parameter space.