Oscillations and nonoscillations in retarded equations (Q5946164)

The author studies the oscillatory behavior of solutions of the scalar equation NEWLINE\[NEWLINE x(t) = \int_{-1}^0 x\bigl (t-r(\theta)\bigr) d\nu(\theta), NEWLINE\]NEWLINE where \(\nu\) is a function of bounded variation on \([-1,0]\) and \(r\) is a positive continuous real function on \([-1,0]\). Starting from the fact that every solution is oscillatory, i.e., has an infinite number of zeros if and only if the characteristic equation \(1-\int_{-1}^0 e^{-\lambda r(\theta)} d\nu(\theta)\) has no real roots, the author studies conditions on \(\nu\) and \(r\) under which this is the case. Special consideration is given to equations with discrete delays of the form NEWLINE\[NEWLINE x(t) =\sum_{k=1}^\infty a_k x(t-\gamma_k). NEWLINE\]NEWLINE In addition the problem under what conditions on \(\nu\) all solutions are oscillatory for all delays \(r\) is studied as well.