Oscillation and monotone solutions of second order quasilinear difference equations. (English) Zbl 0871.39005

Wong, Patricia J. Y.; Agarwal, Ravi P.

Funkc. Ekvacioj, Ser. Int. 39, No. 3, 491-517 (1996).

The authors consider the nonlinear difference equations \[ \Delta \bigl(a_{n-1} |\Delta y_{n-1} |^{\alpha-1} \Delta y_{n-1} \bigr)+ F(n,y_n)= G(n,y_n, \Delta y_n),\;n\geq 1 \tag{1} \] and \[ \Delta \bigl(a_n|\Delta y_n|^{\alpha-1} \Delta y_n\bigr) +b_n|\Delta y_n |^{\alpha-1} \Delta y_n+ H(n,y_n, \Delta y_n) =0,\;n\geq 0, \tag{2} \] where \(\Delta y_n= y_{n+1} -y_n\), \(\alpha>0\), \(\{a_n\}\) is an eventually positive real sequence. A solution \(\{y_n\}\) is said to be oscillatory if it is neither eventually positive nor negative, and nonoscillatory otherwise. Sufficient conditions for all solutions of (1) and (2) to be oscillatory are given. Existence criteria of an eventually positive monotone solution of (2) are established.

Reviewer: V.I.Tkachenko (Kiev)

MSC:

39A12	Discrete version of topics in analysis
39A10	Additive difference equations

Keywords:

oscillation; second order quasilinear difference equations; positive monotone solution