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A fixed point theorem for weakly inward multivalued contractions. (English) Zbl 0957.47040
Using transfinite induction the author proves the following fixed point theorem: Let $D$ be a nonempty closed subset of a Banach space $X$ and let $T$ be a mapping assigning to $x\in D$ a nonempty closed set $T(x)\subset X$. Assume that $T$ is contractive with respect to the Hausdorff metric and that $Tx$ is contained in the closure of $x+\{\lambda(z-x)\mid z\in D$, $\lambda\ge 1\}$ for each $x\in D$. Then there is an $x\in D$ such that $x\in Tx$.

MSC:
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H04Set-valued operators
47H09Mappings defined by “shrinking” properties
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References:
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[3] Lim, T. C.: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bull. amer. Math. soc. 80, 1123-1126 (1974) · Zbl 0297.47045
[4] Martin, R. H.: Differential equations on closed subsets of a Banach space. Trans. amer. Math. soc. 179, 399-414 (1973) · Zbl 0293.34092
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[6] H. K. Xu, Multivalued nonexpansive mappings in Banach spaces, Nonlinear Anal. Theory, Methods, Appl, to appear.
[7] Yi, H. W.; Zhao, Y. C.: Fixed point theorems for weakly inward multivalued mappings and their randomizations. J. math. Anal. appl. 183, 613-619 (1994) · Zbl 0815.47072