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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)].

MSC:
39A10 Additive difference equations
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[1] Agarwal, R.P., Difference equations and inequalities, (1992), Marcel Dekker New York · Zbl 0784.33008
[2] Lakshmikantham, V.; Trigiante, D., Difference equations with applications to numerical analysis, (1988), Academic Press New York · Zbl 0683.39001
[3] Hooker, J.W.; Patula, W.T., A second-order nonlinear difference equation: oscillation and asymptotic behavior, J. math. anal. appl., 91, 9-29, (1983) · Zbl 0508.39005
[4] Kulenovic, M.R.S.; Budincevic, Asymptotic analysis of nonlinear second order difference equations, An. stin. univ. iasi., 30, 39-52, (1984) · Zbl 0572.39001
[5] Szmanda, B., Nonoscillation, oscillation and growth of solutions of nonlinear difference equations of second order, J. math. anal. appl., 109, 22-30, (1985) · Zbl 0589.39003
[6] Thandapani, E., Oscillation theorems for perturbed nonlinear second order difference equations, Computers math. applic., 28, 1-3, 309-316, (1994) · Zbl 0807.39002
[7] Cheng, S.S.; Yan, T.C.; Li, H.J., Oscillation criteria for second order nonlinear difference equations, Funk. ekva., 34, 223-239, (1991) · Zbl 0773.39001
[8] Cheng, S.S.; Zhang, B.G., Monotone solutions of a class of nonlinear difference equations, Computers math. applic., 28, 1-3, 71-79, (1994) · Zbl 0805.39005
[9] Erbe, L.H.; Zhang, B.G., Oscillation of second order difference equations, Chinese J. math., 16, 239-252, (1988) · Zbl 0692.39001
[10] Li, H.J.; Cheng, S.S., Asymptotically monotone solutions of a nonlinear difference equation, Tamkang J. math., 24, 269-282, (1993) · Zbl 0787.39005
[11] Popenda, J., Oscillation and nonoscillation theorems for second-order difference equations, J. math. anal. appl., 123, 34-38, (1987) · Zbl 0612.39002
[12] Szmanda, B., Oscillation theorems for nonlinear second order difference equations, J. math. anal. appl., 79, 90-95, (1981) · Zbl 0455.39004
[13] Szmanda, B., Oscillation criteria for second order nonlinear difference equations, Ann. polon. math., 43, 225-235, (1983) · Zbl 0597.39001
[14] Thandapani, E., Oscillation criteria for certain second order difference equations, Zaa, 11, 425-429, (1992) · Zbl 0787.39002
[15] Thandapani, E., Asymptotic and oscillatory behavior of solutions of nonlinear second order difference equations, Indian J. pure appl. math., 24, 365-372, (1993) · Zbl 0784.39003
[16] Zhang, B.G., Oscillation and asymptotic behavior of second order difference equations, J. math. anal. appl., 173, 58-68, (1993) · Zbl 0780.39006
[17] E. Thandapani and S. Pandian, On existence of asymptotically monotone solutions of a nonlinear difference equation with damping (to appear). · Zbl 0838.39002
[18] Cheng, S.S.; Patula, W.T., An existence theorem for a nonlinear difference equation, Nonlinear analysis, 20, 193-203, (1993) · Zbl 0774.39001
[19] Li, H.J.; Yeh, C.C., Existence of positive nondecreasing solutions of nonlinear difference equations, Nonlinear analysis, 22, 1271-1284, (1994) · Zbl 0805.39004
[20] Popenda, J., The oscillation of solutions of difference equations, Computers math. applic., 28, 1-3, 271-279, (1994) · Zbl 0807.39006
[21] Moore, R.E., Computational functional analysis, (1985), Ellis Harwood · Zbl 0574.46001
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