# zbMATH — the first resource for mathematics

Oscillation and nonoscillation of solutions of second-order difference equations involving generalized difference. (English) Zbl 1262.39017
Summary: Sufficient conditions are obtained for oscillation/nonoscillation of solutions of second-order difference equations, involving the generalized difference, of the form $\Delta_a(p(n-1)\Delta_ay(n-1))+q(n)y(n)=0, \quad n \geqslant 1,$ and $\Delta_a(p(n-1)\Delta_ay(n-1))+q(n)y(n)=f(n), \quad n \geqslant 1,$ where $$\Delta _{a}$$ is defined by $$\Delta _{a}y(n) = y(n + 1) - ay(n)$$, $$a \neq 0$$. Necessary and sufficient conditions are obtained for the oscillation of the first equation with $$a > 0$$ and $$q(n) = q$$, a non-zero constant. The signs of $$q(n)$$ and $$a$$ play an important role for the oscillation/nonoscillation of these two equations.

##### MSC:
 39A21 Oscillation theory for difference equations 39A10 Additive difference equations 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
Full Text:
##### References:
 [1] Elaydi, S.N., An introduction to difference equations, (2005), Springer-Verlag New York · Zbl 1071.39001 [2] Hartman, P.; Wintner, A., Linear differential and difference equations with monotone coefficients, Am. J. math., 75, 731-743, (1953) · Zbl 0051.07105 [3] Hartman, P., Difference equations: disconjugacy, principal solutions, greenâ€™s functions, complete monotonicity, Trans. amer. math. soc., 246, 1-30, (1978) · Zbl 0409.39001 [4] Kelley, W.G.; Peterson, A.C., Difference equations: an introduction with applications, (1991), Academic Press Inc. New York · Zbl 0733.39001 [5] Mickens, R.E., Difference equations, (1987), Van Nostrand Reinhold Company Inc. New York · Zbl 1235.70006 [6] Parhi, N.; Panda, A., Oscillation of solutions of forced nonlinear second order difference equations, (), 221-238 · Zbl 1056.39010 [7] Parhi, N., Oscillation of forced non-linear second order self-adjoint difference equations, Indian J. pure appl. math., 34, 1611-1624, (2003) · Zbl 1044.39010 [8] Popenda, J., Oscillation and non-oscillation theorems for second-order difference equations, J. math. anal. appl., 123, 34-38, (1987), zbl 0612.39002 · Zbl 0612.39002 [9] Royden, H.L., Real analysis, (1968), Macmillan Pub. Co. Inc. New york, zbl 0197.03501 · Zbl 0197.03501 [10] Rudin, W., Principles of mathematical analysis, (1976), McGraw Hill-Kogakusha Ltd. Tokyo, Japan, zbl 0346.26002 [11] Stevic, S., Growth theorems for homogeneous second order difference equations, Anziam j., 43, 559-566, (2002), Zbl 1001. 39016 · Zbl 1001.39016 [12] Stevic, S., Asymptotic behaviour of second order difference equations, Anziam j., 46, 157-170, (2004) · Zbl 1061.39007 [13] Stevic, S., Growth estimates for solutions of nonlinear second order difference equations, Anziam j., 46, 459-468, (2005) [14] Tan, M.; Yang, E., Oscillation and non-oscillation theorems for second order nonlinear difference equations, J. math. anal. appl., 276, 239-247, (2002), zbl 1033.39015 · Zbl 1033.39015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.