Oscillation criteria for second-Order nonlinear difference equations with "summation small" coefficient

Bull. Korean Math. Soc. 2005 Vol. 42, No. 2, 245-256 Printed June 1, 2005

Kang Guolian Institute of System Science

Abstract : We consider the second-order nonlinear difference\break equation $$\Delta( a_nh(x_{n+1})\Delta x_n)+p_{n+1}f(x_{n+1})=0, \,\,n\geq n_0,\tag1$$where $\{a_n\},\{p_n\}$ are sequences of integers with $a_n>0, \{p_n\}$ is a real sequence without any restriction on its sign. $h$ and $f$ are real-valued functions. We obtain some necessary conditions for (1) existing nonoscillatory solutions and sufficient conditions for (1) being oscillatory.