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Fixed point theorems for certain classes of multifunctions


Author: R. E. Smithson
Journal: Proc. Amer. Math. Soc. 31 (1972), 595-600
DOI: https://doi.org/10.1090/S0002-9939-1972-0288750-4
MathSciNet review: 0288750
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Abstract | References | Additional Information

Abstract: The following two fixed point theorems for multi-functions are proved: Theorem. If $ X$ is a tree and if $ F:X \to X$ is a lower semicontinuous multifunction such that $ F(x)$ is connected for each $ x \in X$, then $ F$ has a fixed point. Theorem. Let $ X$ be a topologically chained, acyclic space in which every nest of topological chains is contained in a topological chain. If $ F:X \to X$ is a point closed multi-function such that the image of a topological chain is chainable and such that $ {F^{ - 1}}(x)$ is either closed or chainable for each $ x \in X$, then $ F$ has a fixed point.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0288750-4
Keywords: Fixed point theorems for multifunctions, tree, arcwise connected spaces, topologically chained spaces, lower semicontinuous multifunctions
Article copyright: © Copyright 1972 American Mathematical Society

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