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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
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[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, green’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
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