Averaging and fixed points in Banach spaces.
Torrey M. Gallagher · D-Scholarship@Pitt (University of Pittsburgh) · 2016
We use various averaging techniques to obtain results in different aspects of functional analysis and Banach space theory, particularly in fixed point theory. Specifically, in the second chapter, we discuss the class of so-called mean nonexpansive maps, introduced in 2007 by Goebel and Japon Pineda, and we prove that mean isometries must be isometries in the usual sense. We further generalize this class of mappings to what we call the affine combination maps, give many examples, and study some preliminary properties of this class. In the third chapter, we extend Browder's and Opial's famous Demiclosedness Principles to the class of mean nonexpansive mappings in the setting of uniformly convex spaces and spaces satisfying Opial's property. Using this new demiclosedness principle, we prove that the iterates of a mean nonexpansive map converge weakly to a fixed point in the presence of asymptotic regularity at a point. In the fourth chapter, we investigate the geometry and fixed point properties of some equivalent renormings of the classical Banach space c0. In doing so, we prove that all norms on `1 which have a certain form must fail to contain asymptotically isometric copies of c0.