multi-valued map; Mann iterates; Ishikawa iterates; fixed points
Existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi. An alternative proof of this theorem was given by Peter Z. Daffer and H. Kaneko. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map $T$ with a fixed point $p$ and prove that these iterates converge to a fixed point $q$ of $T$ under certain conditions. This fixed point $q$ may be different from $p$. To illustrate this phenomenon, an example is given.
 H. Kaneko: Generalized contractive multi-valued mappings and their fixed points
. Math. Japon. 33 (1988), 57–64. MR 0935499