International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 1, Pages 25-32
doi:10.1155/S0161171296000051
Generalized dissipativeness in a Banach space
Southeastern Louisiana University, SLU 541, Hammond 70402, LA, USA
Received 12 December 1994
Copyright © 1996 David R. Gurney. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Suppose X is a real or complex Banach space with dual X* and a semiscalar product [,]. For k a real number, a subset B of X×X will be called k-dissipative if for each pair of elements (x1,y1), (x2,y2) in B, there existsh∈{f∈X*:[x,f]=|x|2=|f|2}such thatRe[y1−y2,h]≤k|x1−x2|2.This definition extends a notion of dissipativeness which is equivalent to having k equal zero here. A number of definitions and theorems related to this original dissipative notion are generalized in the present paper to fit the k-dissipative situation, and proofs are given for the new theorems.