C-Cosine Functions and the Abstract Cauchy Problem, I

Inhomogeneous Two-Parameter Abstract Cauchy Problem

inhomogeneous two-parameter abstract cauchy problem

A singular abstract cauchy problem.

In this note a singular abstract Cauchy problem is considered. A solution is obtained for this problem in terms of a regular abstract Cauchy problem. As an application we obtain a new solution of the initial value problem for a class of singular partial differential equations.

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...

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...