Asymptotic and oscillatory properties of differential equations with deviating argument (Q1210053)

The authors consider differential equations of \(n\)th order with deviating arguments in the form \[ L_ nu(t) + p(t)| u(g(t))|^ \alpha\text{sgn }u(g(t))=0,\tag{1} \] where \(n\geq 2\), \(\alpha > 0\), \(L_ nu(t) = (r(t)\dots(r(t)(r(t)u'(t))'\dots)'\), \(p,g\in C([t_ 0,\infty))\), \(r\in C^ n([t_ 0,\infty))\), \(r(t) > 0\), \(\int^ \infty{dt\over r(t)} < \infty\), \(g(t) \to \infty\) as \(t \to \infty\). Using a technique that enables to transfer many oscillatory results from the equation \(y^{(n)}(t) + p(t)| y(g(t))|^ \alpha\text{sgn }y(g(t)) = 0\) to equation (1), the authors present seven theorems, which give sufficient conditions for oscillatory and certain asymptotic properties of solutions of (1).