Metric fixed point theory encompasses the branch of fixed point
theory which metric conditions on the underlying space and/or on
the mappings play a fundamental role. In some sense the theory is a
far-reaching outgrowth of Banach's contraction mapping principle. A
natural extension of the study of contractions is the limiting case
when the Lipschitz constant is allowed to equal one. Such mappings
are called nonexpansive. Nonexpansive mappings arise in a variety
of natural ways, for example in the study of holomorphic mappings
and hyperconvex metric spaces. Because most of the spaces studied
in analysis share many algebraic and topological properties as well
as metric properties, there is no clear line separating metric
fixed point theory from the topological or set-theoretic branch of
the theory. Also, because of its metric underpinnings, metric fixed
point theory has provided the motivation for the study of many
geometric properties of Banach spaces. The contents of this
Handbook reflect all of these facts. The purpose of the Handbook is
to provide a primary resource for anyone interested in fixed point
theory with a metric flavor. The goal is to provide information for
those wishing to find results that might apply to their own work
and for those wishing to obtain a deeper understanding of the
theory. The book should be of interest to a wide range of
researchers in mathematical analysis as well as to those whose
primary interest is the study of fixed point theory and the
underlying spaces. The level of exposition is directed to a wide
audience, including students and established researchers.
General
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