Oscillations of difference equations with several deviated arguments (Q744035)

The authors consider linear difference equations with delayed, resp. advanced arguments of the forms \( \Delta x(n)+\sum_{i=1}^m p_i(n)x(\tau_i(n))=0, \) \(n\geq 0\), resp. \( \nabla x(n)-\sum_{i=1}^m p_i(n)x(\sigma_i(n))=0, \) \(n\geq 1\), where \(p_i\), \(i=1,\dots,m\), are sequences of nonnegative real numbers, \(\tau_i(n)\leq n-1\) for all \(n\geq 0\), \(i=1,\dots,m\), \(\sigma_i(n)\geq n+1\) for all \(n\geq 1\), \(i=1,\dots,m\), and \(\lim_{n\to\infty}\tau_i(n)=\infty\), \(i=1,\dots,m\). As usually, \(\Delta\) denotes the forward difference operator, and \(\nabla\) denotes the backward difference operator. First, a brief historical survey of related problems is presented. The main results are various conditions which guarantee oscillation of all solutions. The proofs are given, as usually, by a contradiction. Several examples are given. For an illustration, here is one typical result: If \(\tau_i\), \(i=1,\dots,m\) are increasing, \(\limsup_{n\to\infty}\sum_{i=1}^m p_i(n)>0\), and \(\liminf_{n\to\infty}\sum_{i=1}^m\sum_{j=\tau_i(n)}^{n-1}p_i(j)>1/e\), then all solutions of the former equation oscillate.