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Some fixed point theorems for multivalued mappings. (English) Zbl 0549.47029
Let T be a continuous operator from a nonempty closed convex set K of a Banach space E into a compact subset of E, and let F be an operator from \(\overline{TK}\times K\) into K such that \(F(x,\cdot)\) is continuous and \(F(\cdot,y)\) is a contraction. According to a fixed-point theorem of W. R. Melvin [J. Diff. Equ. 11, 335-348 (1972; Zbl 0229.47046)] there exists \(x\in K\) such that \(F(Tx,x)=x.\) In the present paper the author generalizes this result to multivalued operators F.
Reviewer: J.Appell

47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
54C60 Set-valued maps in general topology
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