An extension of Mizoguchi-Takahaashi's fixed point theorem.

link: http://arxiv.org/abs/1001.1019
Abstract

Our main theorem is an extension of the well-known Mizoguchi-Takahaashi's
fixed point theorem [N. Mizogochi and W. Takahashi, Fixed point theorems for
multi-valued mappings on complete metric space,

{\it J. Math. Anal. Appl.} 141 (1989) 177--188].