Fixed Point of Multi-valued Mizoguchi-Takahashi’s Type Mappings and Answer to the Rouhani-Moradi’s Open Problem

The fixed point theorem of Nadler (1966) was extended by Mizoguchi and Takahashi in 1989 and then for multi-valued contraction mappings, the existence of fixed point was demonstrated by Daffer and Kaneko (1995). Their results generalized the Nadler’s theorem. In 2009 Kamran generalized Mizoguchi-Takahashi’s theorem. His theorem improve Klim and Wadowski results (2007), and extended Hicks and Rhoades (1979) fixed point theorem. Recently Rouhani and Moradi (2010) generalized Daffer and Kaneko’s results for two mappings. The results of the current work, extend the previous results given by Kamram (2009), as well as by Rouhani and Moradi (2010), Nadler (1969), Daffer and Kaneko (1995), and Mizoguchi and Takahashi (1986) for tow multi-valued mappings. We also give a positive answer to the Rouhani-Moradi’s open problem.