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Oscillations and nonoscillations in retarded equations. (English) Zbl 0983.45001

The author studies the oscillatory behavior of solutions of the scalar equation \[ x(t) = \int_{-1}^0 x\bigl (t-r(\theta)\bigr) d\nu(\theta), \] 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 \[ x(t) =\sum_{k=1}^\infty a_k x(t-\gamma_k). \] In addition the problem under what conditions on \(\nu\) all solutions are oscillatory for all delays \(r\) is studied as well.

MSC:

45A05 Linear integral equations
34K11 Oscillation theory of functional-differential equations
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