# zbMATH — the first resource for mathematics

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

##### MSC:
 47H10 Fixed-point theorems 47J05 Equations involving nonlinear operators (general) 54C60 Set-valued maps in general topology
##### Keywords:
fixed-point theorem; multivalued operators
Full Text: