Series expansions and small solutions for Volterra equations of convolution type

Abstract In general, retarded functional differential equations have an infinite dimensional character, in the sense that there exist an infinite number of linear independent characteristic solutions p j ( t ) e λ j t , where λ j denotes a zero of a transcendental equation and p j a polynomial. In this paper we use Laplace transform methods to study the asymptotic behaviour of the solutions of this type of differential equations. Furthermore, we present necessary conditions such that a solution can be represented as a series of characteristic solutions. With these results we then can study the geometric structure of the strongly continuous semigroup T ( t ) associated with a retarded functional differential equation. The main result will be a characterization of the closure of the system of generalized eigenfunctions of the infinitesimal generator A of T ( t ).

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