Abstract Cauchy Problems

Abstract

We define the convolution of Banach space valued ultradistributions in the sense of Braun, Meise, and Taylor. We then treat abstract Cauchy problems in Banach spaces as convolution equations and give a characterization of those problems that have ultradistributional fundamental solutions. Our characterization extends in the Beurling case a result due to H.A. Emamirad. We apply our result to differential operators in Banach spaces of ultradifferentiable functions with different (e.g. ultradifferential) boundary conditions.

Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

“…For more details about abstract degenerate differential equations, the reader may consult the monographs [18,[41][42][43].…”

Local and global solutions of well-posed integrated Cauchy problems

“…Particular classes of distribution semigroups have since been considered; for example quasi-distribution semigroups were introduced and studied by Wang in [21]. A brief description of distribution semigroups as well as references on the subject can be found in [1] and [17].…”

Global convoluted semigroups

“…We only note that the equality implies that the coefficient of C 2 x in the proof of [32,Proposition 2.4], in our case C = I, is equal to zero. Details are omitted; see [11] and [38] for further information. We point out that there exists a somewhat different definition of a (local) convoluted semigroup.…”

“…For further information concerning such classes of semigroups, we refer to [1]- [9], [13]- [14], [23]- [35], [47]- [51], and especially, to the monographs [3], [13], [38] and [54] where the interested reader can also find the basic material related to vector-valued Laplace transforms, degenerate semigroups, regularization methods and their importance in the theory of (higher-order) abstract Cauchy problems.…”