Fixed Points of Semigroups of Pointwise Lipschitzian Operators

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.