No abstract

Abstract. For the second-order quasilinear di¤erence equation Dð p n jDx n j aÀ1 Dx n Þ þwe establish a necessary and su‰cient condition for the existence of slowly growing and slowly decaying positive solution. In particular, when p n ¼ 1, the precise asymptotic forms of its slowly growing positive solutions are obtained.

Positive solutions Regularly varying solutions Slowly varying solutions Asymptotic behavior of solutions a b s t r a c tThis paper is concerned with asymptotic analysis of positive solutions of the second-order nonlinear differential equationwhere q : [a, ∞) → (0, ∞) is a continuous function which is regularly varying and φ : (0, ∞) → (0, ∞) is a continuous increasing function which is regularly varying of index γ ∈ (0, 1). An application of the theory of regular variation gives the possibility of determining precise information about the asymptotic behavior at infinity of intermediate solutions of Eq. (A).

Some new criteria for the oscillation of fourth-order nonlinear functional differential equations of the formare established.

No abstract