We introduce the notion of almost-distribution cosine functions in a setting similar to that of distribution semigroups defined by Lions. We prove general results on equivalence between almost-distribution cosine functions and α-times integrated cosine functions.

We study the local well-posed integrated Cauchy problem
${v}^{\text{'}}\left(t\right)=Av\left(t\right)+\left({t}^{\alpha}\right)/\Gamma (\alpha +1)x$, v(0) = 0, t ∈ [0,κ),
with κ > 0, α ≥ 0, and x ∈ X, where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growth of solutions is given in both cases.

We investigate the weak spectral mapping property (WSMP)
$\overline{\mu \u0302\left(\sigma \right(A\left)\right)}=\sigma \left(\mu \u0302\left(A\right)\right)$,
where A is the generator of a ₀-semigroup in a Banach space X, μ is a measure, and μ̂(A) is defined by the Phillips functional calculus. We consider the special case when X is a Banach algebra and the operators ${e}^{At}$, t ≥ 0, are multipliers.

We establish an inversion formula of Post-Widder type for ${\lambda}^{\alpha}$-multiplied vector-valued Laplace transforms (α > 0). This result implies an inversion theorem for resolvents of generators of α-times integrated families (semigroups and cosine functions) which, in particular, provides a unified proof of previously known inversion formulae for α-times integrated semigroups.

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