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

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.