Oscillations in second order linear autonomous differential equations with distributed type deviating arguments (Q1260878)

This paper extends the characterization of oscillations in terms of the characteristic equation to the case of a class of second order scalar differential equations with deviating arguments, with possibly both retarded and advanced arguments. The most noticeable new feature is the fact that distributed arguments are allowed. The proof is based on the method of Laplace transforms, and requires as a preliminary the proving that, under the assumption that there exist nonoscillatory solutions, then there are such solutions with an at most exponential growth. Incidentally, such a fact was set as an open problem in the book by Györi and Ladas and, to the best of the reviewer's knowledge, this problem was answered positively for the first time by the reviewer for systems in a paper submitted for publication in Radov. Mat. (a journal established in Sarajevo, whose publication has been momentarily suspended). The statement given in the present paper is more general than the one in the reviewer's by not assuming discrete deviations. There are restrictions, however, notably sign conditions, which lead to several situations, namely eight distinct cases, each of them considered separately. Although a result for arbitrary distributed deviations seems to be out of reach, one is tempted to believe, based on this paper, that a more general result holds and would be desirable. Thanks to the Laplace transform method, the problem reduces to extending the property of exponential growth for nonoscillatory solutions.