Fixed point theorems for certain classes of multifunctions

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.