Space ofω-Periodic Limit Functions and Its Applications to an Abstract Cauchy Problem

Computable Analysis of the Abstract Cauchy Problem in a Banach Space and Its Applications (I)

We study computability of the abstract linear Cauchy problem du(t)/dt = Au(t), u(0) = x ∈ X, (1) where A is a linear operator, possibly unbounded, on a Banach space X. We give necessary and sufficient conditions for A such that the solution operator K : x → u of the problem (1) is computable. For studying computability we use the representation approach to Computable Analysis developed by Weihr...

Banach space valued ultradistributions and applications to abstract Cauchy problems

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 dif...

Integrated Semigroups and Their Applications to the Abstract Cauchy Problem

This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(A). Investigating these operators we are led to the class of "integrated semigroups". Among o...

Inhomogeneous Two-Parameter Abstract Cauchy Problem

inhomogeneous two-parameter abstract cauchy problem