Attractivity Properties of alpha-Contractions.
Abstract
It is known that if T: X yields X is completely continuous where X is a Banach space, then point dissipative and compact dissipative are equivalent, and imply the existence of a maximal compact invariant set which is uniformly asymptotically stable and attracts bounded sets uniformly. If T is an alpha contraction it is not known whether point dissipative and compact dissipative are equivalent. However, T is compact dissipative, then there exists a maximal compact invariant set which is uniformly asymptotically stable and attracts neighborhoods of compact sets uniformly. Since, in practice, it is much easier to verify that a map is point dissipative rather than compact dissipative, it is desirable to say more about the limiting behavior when T is only assumed to be point dissipative. In this paper, we show, with the addition of only a few general assumptions, that point dissipative and compact dissipative are equivalent. The assumptions seem to be general enough to include almost all of the practical applications. Applications are given, or referenced, to stable neutral functional differential equations, retarded functional differential equations of infinite delay, and strongly damped nonlinear wave equations. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 12, 1980
- Accession Number
- ADA091975
Entities
People
- Paul Massatt
Organizations
- Brown University