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Oscillation theorems and existence of positive monotone solutions for second order nonlinear difference equations. (English) Zbl 0820.39002
The authors establish sufficient conditions for the oscillation of all solutions of the perturbed difference equation $\Delta(a_{n- 1}(\Delta y_{n- 1})^ \sigma)+ F(n, y_ n)= G(n, y_ n, \Delta y_ n),\quad n\geq 1$ as well as for the existence of a positive monotone solution of the damped difference equation $\Delta(a_ n(\Delta y_ n)^ \sigma)+ b_ n(\Delta y_ n)^ \sigma+ H(n, y_ n, \Delta y_ n)= 0,\quad n\geq 0,$ where $$0< \sigma= p/q$$ with $$p$$, $$q$$ odd integers, or $$p$$ even and $$q$$ odd integer.
For related results see the reviewer [Comput. Math. Appl. 28, No. 1-3, 309-316 (1994; Zbl 0807.39002)] and H. J. Li and S. S. Cheng [Tamkang J. Math. 24, No. 3, 269-282 (1993; Zbl 0787.39005)].