Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term (Q5948801)

Oscillation criteria are derived for the difference equation NEWLINE\[NEWLINE\Delta\biggl( a_n\Delta \bigl( x_n+\varphi (n,x_{\tau_n}) \bigr)\biggr)+q_n f(x_{g_n})=0, \tag{1}NEWLINE\]NEWLINE where \(\{\tau_n\}\), \(\{g_n\}\) are nondecreasing and nonnegative sequences of integers such that \(\tau_n\), \(g_n\leq n\) and \(\tau_n\), \(g_n\to \infty\). The positive sequence \(\{a_n\}\) satisfies \(\sum^\infty a_n^{-1}= \infty\) and the nonnegative sequence \(\{q_n\}\) has a positive subsequence. The function \(f\) satisfies \(f(u)/u \geq\varepsilon_0\) for \(u\neq 0\) and the function \(\varphi\) satisfies \(\varphi (n,u)/u\leq p_n<1\) for \(u\neq 0\) where \(\{p_n\}\) is positive. By means of a variant of the Riccati transformation, the existence of a nonoscillatory solution of (1) implies the existence of solutions of the functional relation NEWLINE\[NEWLINE\Delta u_n+ Q_n+{A_nu^2_{n+1} \over a_{g_n} A^2_{n+1}}\leq 0,NEWLINE\]NEWLINE where \(\{A_n\}\) is a positive sequence and NEWLINE\[NEWLINEQ_n=A_n \left\{\varepsilon_0q_n(1-p_{g_n})+ a_{g_n}{(\Delta A_n)^2\over 4A_n}+\Delta \left(a_{g_{n-1}} {\Delta A_{n-1} \over 2A_{n-1}}\right) \right\}.NEWLINE\]NEWLINE Then under additional conditions NEWLINE\[NEWLINE\sum^\infty_{s=n} {A_s\over a_{g_s} A^2_{s+1}}= \infty,\;P_n\equiv \sum^\infty_{s=n} Q_s>- \infty,NEWLINE\]NEWLINE it is further shown that NEWLINE\[NEWLINEv_n\geq P_n+ \sum^\infty_{s=n} {A_sv^2_{s+1} \over a_{g_s}A^2_{s+1}}NEWLINE\]NEWLINE has a solution. The last relation is a standard Riccati type equation and therefore oscillation criteria can be obtained by straightforward manners. These criteria extend several recent results in the literature.