On the Existence and the Asymptotic Behavior of Nonoscillatory Solutions of Second Order Quasilinear Difference Equations

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.

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

“…Nonoscillatory solutions in M − 0;∞ is called slowly decaying solutions in literature, see [32]. The following theorems show the existence of nonoscillatory solutions in subclasses of M − given above.…”

“…If T ¼ Z, system (20) is reduced to a Emden-Fowler system of difference equations while it is reduced to a Emden-Fowler system of differential equations when T ¼ R, see Refs. [32,39,40], respectively. We also refer readers to Refs.…”

On nonoscillatory solutions of Emden-Fowler dynamic systems on time scales

“…Let M be the set of all nonoscillatory solutions of system (1). One can easily show that any nonoscillatory solution (x, y) of system (1) belongs to one of the following classes:…”

“…Let (x, y) be a solution of system (1). Then the component functions x and y are themselves nonoscillatory if (x, y) is a nonoscillatory solution of system (1).…”

Decaying solutions for discrete boundary value problems on the half line

“…Discrete boundary value problems (BVPs), associated to equations of type (1), have attracted considerable attention in the last years, especially when they are examined on unbounded domains, see, e.g., [2,4,5,10,11,18,20], the monographies [1,3] and references therein. Equation (1) appears in the discretization process for searching spherically symmetric solutions of certain nonlinear elliptic di¤erential equations with p-Laplacian, see, e.g., [13].…”