A new approach to ill-posed evolution equations : C-regularized and B- bounded semigroups.

Abstract

The theory of semigroups of linear operators forms an integral part of Functional

Analysis with substantial applications to many fields of the natural sciences. In

this study we are concerned with the application to equations of mathematical

physics. The theory of semigroups of bounded linear operators is closely related to

the solvability of evolution equations in Banach spaces that model time dependent

processes in nature.

Well-posed evolution problems give rise to a semigroup of bounded linear operators.

However, in many important and interesting cases the problem is ill-posed

making it inaccessible to the classical semigroup theory. One way of dealing

with this problem is to generalize the theory of semigroups.

In this thesis we give an outline of the theory of two such generalizations, namely,

C-regularized semigroups and B-bounded semigroups, with the inter-relations

between them and show a number of applications to ill-posed problems.