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Zbl 0972.47041
Xu, Hong-Kun
Metric fixed point theory for multivalued mappings. (English)
[J] Diss. Math. 389, 39 p. (2000). ISSN 0012-3862; ISSN 1730-6310/e

The paper provides some new results in the fixed point theory of non-expansive and contractive set-valued mappings defined on subsets of a metric or Banach space with a special attention paid to the maps satisfying suitable inwardness conditions. In particular, the Kirk-Massa theorem on the existence of fixed points of a nonexpansive self-map is extended to the case of non-self inward maps. The precise discussion of the Reich problem concerning the existence of fixed point of generalized contractions of the Browder type is included. Finally, the author studies random contractive and nonexpansive set-valued maps and discusses in detail the existence of their fixed points. The paper may be also considered as a well-prepared survey in the field; it provides a number of examples and counterexamples, open problems as well as extensive bibliography in the subject.
[Wojciech Kryszewski (Toruń)]

MSC 2000:: *47H04 Set-valued operators
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H40 Random operators
47H09 Mappings defined by "shrinking" properties
46B20 Geometry and structure of normed spaces
54C60 Set-valued maps
65H99 Nonlinear algebraic or transcendental equations

Keywords: fixed point theory; non-expansive and contractive set-valued mappings; Kirk-Massa theorem; Reich problem

Cited in: Zbl 1105.47047 Zbl 1086.47009

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