Abstract
AbstractFixed 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 nonexpansive mappings? 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.
Highlights
The presence or absence of a fixed point i.e., a point which remains invariant under a map is an intrinsic property of a map
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 nonexpansive mappings? There are many properties of Banach spaces which imply weak normal structure and the FPP for single-valued mappings for example, uniform convexity, nearly uniform convexity, uniform smoothness,. . . . it is natural to consider the following problem: do these properties 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
By Metric Fixed Point Theory, we understand the branch of Fixed Point Theory concerning those results which depend on a metric and which are not preserved when this metric is replaced by another equivalent metric
Summary
The presence or absence of a fixed point i.e., a point which remains invariant under a map is an intrinsic property of a map. Domınguez and Lorenzo deduced in 26 the following partial extension of Kirk’s Theorem which, in particular, assures that nearly uniformly convex spaces have the fixed point property for multivalued nonexpansive mappings.
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