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Metric fixed point theory for multivalued mappings. (English) Zbl 0972.47041

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.

MSC:

47H04 Set-valued operators
47H10 Fixed-point theorems
47H40 Random nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B20 Geometry and structure of normed linear spaces
54C60 Set-valued maps in general topology
65H99 Nonlinear algebraic or transcendental equations
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