Oscillation and Non-Oscillation Theorems for a Class of Second Order Quasilinear Difference Equations

Abstract

In this paper there are established necesssary and sufficient conditions for the second order quasilinear difference equation L(p,cp(iyn)) + f(n,y,,+ i) = 0 (n E No) to have various types of non-oscillatory solutions. In addition, in the case that the equation is either strongly superlinear or strongly sublinear, there are established necessary and sufficient conditions for all solutions to oscillate.

Asymptotic results for a class of fourth order quasilinear difference equations

“…Similar problems have been studied by Thandapani et al [19,20] for the second order quasilinear difference equation D p n jDx n j a21 Dx n þ q n jx n j b21 x n ¼ 0:…”

“…and so in view of (19), there is a constant k 5 . 0 such that x n # k 5 n for all large n. Thus, a solution {x n } of equation (1) satisfying (18) is considered a "maximal" solution in the set of all eventually positive solutions that satisfy case (B).…”

“…0 be an arbitrary constant. Since (19) holds, it is possible to choose an integer N [ Nðn 0 Þ such that respectively. Similar to the arguments used above, we can show that T # S, T is continuous, and TS is relatively compact.…”

“…[19,20] for the equation (5) under the condition X 1 n¼n 0 1 p 1 a n ¼ 1 is as follows. (iii) Equation (5) has a nonoscillatory solution {x n } satisfying x n < c -0 as n !…”

On the Asymptotic Behavior of Second Order Quasilinear Difference Equations7403287

“…Further they have established some new oscillation conditions for the oscillation of solutions of Equation (3). In 1997, E. Thandapani and R. Arul [28] studied, the following quasi-linear equation…”