Let (respectively, ) denote the set of all complex-valued infinitely differentiable functions defined on the reals whose supports are bounded on the left (respectively, right). Under addition and convolution is a commutative algebra.

Let L be a solvable Lie algebra and A be an abelian ideal of L. For a ∊ A, let da be the (right) inner derivation of L generated by a and let exp da = 1 + da . Since A is abelian, exp da is an automorphism and exp da exp db = exp da+b for all a, b ∊ A. Let I(L, A) be the subgroup of the automorphism group of L generated by exp da for all a ∊ A.

In this paper we state and prove three theorems on positive solutions of a Volterra equation of the second kind. The equation considered is

1

where K(x, t) is a Volterra type kernel, that is, K(x, t) = 0 for t > x.

Let {Xa, a ∊ A} be a family of completely regular Hausdorff spaces, {βXa } the corresponding family of their Stone-Čech compactifications and ΠaXa the usual topological product. The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1].