Oscillation of second-order differential equations with mixed argument
The author considers the second order differential equation with mixed argument \[ \Biggl({1\over r(t)} u'(t)\Biggr)'+ p(t) u(\tau(t))+ q(t) u(\sigma(t))= 0,\tag{1} \] where \(r, p, q, \tau, \sigma\in C([t_ 0, \infty))\), \(r(t)\) and \(q(t)\) are positive, \(p(t)\) is nonnegative, \[ \tau(t)\leq t,\;\lim_{t\to \infty} \tau(t)= \infty\quad\text{and}\quad\sigma(t)\geq t. \] A sufficient condition for oscillation of all solutions of (1) is obtained. As is customary a solution is called oscillatory if the set of its zeros is unbounded. It should be noted that despite the numerous results on oscillation of equations with deviating arguments there are comparatively few results for equations with mixed argument (they involve both advanced and delay argument). The main idea of the proof of the author's oscillation criteria is by using a comparison result (Theorem 1) and appropriate transformations to deduce the oscillatory properties of equation (1) from known oscillation criteria for simpler equations (with either delayed or advanced argument). Examples are also given that illustrate the author's results.
second order differential equation with mixed argument
scientific article; zbMATH DE number 762193