## Nonoscillation, oscillation and growth of solutions of nonlinear difference equations of second order.(English)Zbl 0589.39003

The paper is concerned with second-order nonlinear difference equations over the set of natural numbers. Based on assumptions on nonnegativity of some coefficients or some functions as well as on some growth properties, a number of theorems consisting of sufficient conditions for the non- oscillatory resp. the oscillatory behavior of the solutions of the difference equation are derived. Furthermore, results on the growth of the solutions are proved. The theorems are proved by elementary means.
Reviewer: G.Meinardus

### MSC:

 [1] Chen, L., Sufficient conditions for nonoscillation of $$n$$-th order differential equations, Atti Accad. Naz. Lincei Rend., 60, 27-31 (1976) · Zbl 0367.34026 [2] Demidovič, V. B., A stability test for difference equations, Differentsial’nye Uravneniya, 5, 1247-1255 (1969), [in Russian] [3] Graef, J. R., Oscillation, nonoscillation, and growth of solutions of nonlinear functional differential equations of arbitrary order, J. Math. Anal. Appl., 60, 398-409 (1977) · Zbl 0377.34023 [4] Graef, J. R.; Grammatikopoulos, M. K.; Spikes, P. W., Growth and oscillatory behavior of solutions of a differential equation with a deviating argument, Funkcial. Ekvac., 23, 279-287 (1980) · Zbl 0455.34050 [5] Graef, J. R.; Spikes, P. W., Sufficient conditions for the equation $$(a(t)x$$′) + $$h(t, x, x$$′) + $$q(t)f(x, x$$′) = $$e(t, x, x$$′) to be nonoscillatory, Funkcial. Ekvac., 18, 35-40 (1975) · Zbl 0331.34030 [6] Graef, J. R.; Spikes, P. W., Asymptotic properties of solutions of functional differential equations of arbitrary order, J. Math. Anal. Appl., 60, 339-348 (1977) · Zbl 0377.34022 [7] Hooker, J. W.; Patula, W. T., Riccati type transformations for second-order linear difference equations, J. Math. Anal. Appl., 82, 451-462 (1981) · Zbl 0471.39007 [8] Kartsatos, A. G.; Manougian, M. N., Further results on oscillation of functional-differential equations, J. Math. Anal. Appl., 53, 28-37 (1976) · Zbl 0326.34087 [9] Szmanda, B., Note on the behaviour of solutions of a second order nonlinear difference equation, Atti Accad. Naz. Lincei Rend., 69, 120-125 (1980) · Zbl 0508.39004