Oscillation in linear functional differential systems (Q1355054)

The author considers the following linear autonomous functional differential equation \[ {d\over dt} (Nx_t)+ Lx_t=0,\tag{1} \] where \(N\) and \(L\) are arbitrary linear bounded operators from the space of continuous functions \(C([-\tau,0],\mathbb{R}^n)\) into \(\mathbb{R}^n\). It is shown that the existence of an oscillatory and integrally exponentially bounded solution implies the presence of a real root of the characteristic equation. The proof is based on the theory of Laplace transforms and on a result for complex entire functions. A criterion for oscillation of all solutions of a particular class of equations of the form (1) is derived. It should be noted that equation (1) includes ordinary, delay, neutral, advanced, mixed type of functional differential equations, functional equations and difference equations and thus the results obtained are very general. Also two interesting open problems are stated at the end of the paper.