Suppose ${G₁\left(t\right)}_{t\ge 0}$ and ${G₂\left(t\right)}_{t\ge 0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators for...

This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^{\text{'}}\left(t\right)=Au\left(t\right)+\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in ${(\lambda -A)}^{-1}C\left(X\right)$.