ABSTRACT
It was known from the classical books by A.D.Myshkis and J.Hale that the delay equation x′′(t) + px(t − g) = 0 with constant coefficient p and delay g, in contrast with the corresponding ordinary differential equation x′′ + px – 0, can have solutions unbounded on the semi-axis. They proved that there exists a positive roots among roots of the characteristic equation of this delay equation. Their results used a technique of analysis of roots of quasi-polynomials. In this Chapter we answer question formulated by A.D.Myshkis about existence of unbounded solutions of the delay equation x′′(t) + p(t)x(t − g(t)) = 0 in the case of the variable coefficient p(t) and delay g(t). We develop results about nondecreasing Wronskian of the fundamental solutions and construct inequality estimating the Wronskian from below. On the basis of the estimates obtained on this basis, results on existence of unbounded solutions are obtained. It is obtain that necessary and sufficient condition for boundedness of all solutions of the equation x′′(t) + px(t Ȓ g(t)) = 0 with the constant coefficient p is the fact that the delay g(t) is a summable on the semiaxis. Results on existence of unbounded solutions to delay differential equations with a damping term have been obtained also on the basis of other ideas. One of them is positivity of the Cauchy function (fundamental solution in another terminology). Existence of unbounded solutions is obtained for delay differential equations with asymptotically small coefficients and equations with negative damping terms.
