Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain
Academic Editor: Mohamed A. Khamsi
Copyright © 2010 T. Domínguez Benavides and B. Gavira. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fixed Point Theory for multivalued mappings has many useful applications in Applied
Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural
to try of extending the known fixed point results for single-valued mappings to the setting
of multivalued mappings.
Some theorems of existence of fixed points of single-valued mappings have already been
extended to the multivalued case. However, many other questions remain still open, for
instance, the possibility of extending the well-known Kirk's Theorem, that is: do Banach
spaces with weak normal structure have the fixed point property (FPP) for multivalued
There are many properties of Banach spaces which imply weak normal structure and
consequently the FPP for single-valued mappings (for example, uniform convexity, nearly
uniform convexity, uniform smoothness,…). Thus, it is natural to consider the following
problem: do these properties also imply the FPP for multivalued mappings? In this way,
some partial answers to the problem of extending Kirk's Theorem have appeared, proving that those properties imply the existence of fixed point for multivalued nonexpansive mappings.
Here we present the main known results and current research directions in this subject.
This paper can be considered as a survey, but some new results are also shown.