A characterization of the positive solutions of a functional differential equation

Abstract EN

Consider the functional differential Equation (1) x’(t) = ∫ x(t + T(w))dn(w) n where t:Ω →(0, ∞) is bounded measurable and 0 < μ(Ω) < ∞, and its characteristic Equation (2) Ω= ∫eƛT(w)dµ(w).

Every real solution of (2), when it n Exists, gives rise to a non-oscillatory solutions of (1), but the size of the space of all non-oscillatory solutions has not been considered.

In this paper it is shown that the cone of all positive solutions of (1), when non-empty, is two dimensional: it is generated by exp (at) and exp ((if) where a and p are the two real roots of (2); or by exp (at) and texp (at) when a = p.