A semigroup theory for a differential equation with delayed and advanced arguments is developed, with a detailed description of the infinitesimal generator. This in turn allows to study the exact controllability of the equation, by rewriting it as a classical Cauchy problem.

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy ofl1; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of{T**(t)}t≥0, which, in addition, is also characterized for abstractL- andM-spaces. As a corollary, it is proven that abstractL-spaces with no copy ofl1are finite-dimensional.

A formula for the solution of a differential-difference equation is given, and then the corresponding semigroup theory is developed, with a detailed description of the infinitesimal generator and some of its spectral properties.Results in

In this paper we study certain systems of mixed-type functional differential equations, from the point of view of the C 0 -semigroup theory. In general, this type of equations are not well-posed as initial value problems. But there are also cases where a unique differentiable solution exists. For these cases and in order to achieve our goal, we first rewrite the system as a classical Cauchy problem in a suitable Banach space. Second, we introduce the associated semigroup and its infinitesimal generator and prove important properties of these operators. As an application, we use the results to characterize the null controllability for those systems, where the control u is constrained to lie in a non-empty compact convex subset Ω of R n .

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