ABSTRACT
Let x be a positive equilibrium of Eq(e). Then except possibly for the first semicycle, every oscillatory solution of Eq(f} has semicycles of length one.
Proof. Assume for the sake of contradiction that {xn} is an oscillatory solution with two consecutive terms XN-1 and XN in a positive semicycle
with at least one of the inequalities being strict. The proof in the case of negative semicycle is similar and is ommited. Then by using the increasing character of f we obtain
XN+t = j(XN,XN-1) > j(x,x) =X which shows that the next term XN+t also belongs to the positive semicycle. It follows by induction that all terms Xn for n ~ N of this solution belong to a positive semicycle, which is a contradiction. D
When a{J,..,AB = 0
Eq{l} contains 20 special difference equations with positive coefficients. Of these cases, eight equations are linear, five are of the Riccati type and the remaining seven equations are the following:
