This book provides an overview of recent advances in fixed-point theory for pointwise Lipschitzian semigroups of nonlinear operators, with emphasis on the asymptotic approach. It consolidates otherwise fragmented, inconsistent, and incomplete, publications surrounding the foundations of the theory of common fixed points for semigroups of nonlinear, pointwise Lipschitzian mappings acting in Banach spaces, with some pointers to the parallel results in other settings, including metric and modular spaces. The main focus of the proposed book will be on the following aspects: (1) existence results, (2) construction algorithms convergence in the strong and the weak topology, (3) stability of such algorithms, (4) applications to differential equations, dynamical systems and stochastic processes.

The main feature of this work can be described as the introduction of the common, very general and yet relatively elementary (using basic notions of the Banach space geometry) framework, which will allow the reader to comprehend the whole story, including the inner interdependencies, behind the theory of such common fixed points. As the sub-title suggests, we will use the lenses of asymptotic and pointwise asymptotic variants of nonexpansiveness. This approach, when used in a consistent way, assures generality of the results, illustrate in relatively simple terms the current stage of the research, while allowing the readers to start or continue work on further extensions and generalizations. The value of and the need for the use of the asymptotic approach will be explained from the theoretical point of view and illustrated by examples.

While the main benefit the readers should expect form this work is to get a guidebook for the fixed point theory for the asymptotic pointwise Lipschitzian semigroups, the book can be also used as a brief compendium of the common fixed point results for more classical semigroups of nonexpansive mappings, being a special case in our much more general settings. Also, and importantly, the results discussed in this work are generally proved for semigroups parametrized by any additive sub-semigroups of the set of all nonnegative real numbers, and hence can be also applied to discrete cases, including the fixed point results for asymptotic pointwise nonexpansive mapping, generalizing in this way classical results of Goebel, Kirk, Xu, and others.

Autorentext

Wojciech M. (Walter) Kozlowski has been actively involved in research activities in mathematics and applications since the 1980s with a particular interest in functional analysis, fixed point theory and applications. A Fulbright scholar, an author of a monographic book "Modular Function Spaces" (Marcel Dekker 1988), a co-author of a monograph "Fixed Point Theory in Modular Spaces" (Springer 2015), and author of numerous scientific papers, he has held several academic posts at the universities around the world, currently as an Adjunct Associate Professor at the School of Mathematics and Statistics, University of New South Wales in Sydney. In parallel, he has been pursuing ICT professional career within leading global organizations. He has been active in the international professional organisations, including positions of a member of Linux Foundation Networking Governing Board, member of TheOpenGroup Certification Board, and Chair of the GSMA Open Infrastructure Group.