On oscillation of a differential equation with infinite number of delays (Q1848672)

For the scalar delay differential equation \[ \dot{x}(t)+\sum_{k=1}^{\infty}a_k(t)x(h_k(t))=0,\quad h_k(t)\leq t, \] a connection between the following four properties is established: [-] nonoscillation of this equation, [-] nonoscillation of the corresponding differential inequality, [-] positiveness of the fundamental function, [-] existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. Explicit nonoscillation and oscillation conditions, comparison theorems and a criterion for the existence of a positive solution are presented for this equation.