Mizoguchi–Takahashi’s type fixed point theorem

Abstract

Recently, Eldred [A.A. Eldred, J. Anuradha, P. Veeramani, On equivalence of generalized multi-valued contactions and Nadler’s fixed point theorem, J. Math. Anal. Appl. (2007) doi:10.1016/j.jmaa.2007.01.087] claimed that Nadler’s [S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475–488] fixed point theorem is equivalent to Mizoguchi–Takahashi’s [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric space, J. Math. Anal. Appl. 141 (1989) 177–188] fixed point theorem. Very recently, Suzuki [T. Suzuki, Mizoguchi–Takahashi’s fixed point theorem is a real generalization of Nadler’s, J. Math. Anal. Appl. (2007) doi:10.1016/j.jmaa.2007.08.022] produced an example to disprove their claim and showed that Mizoguchi–Takahashi’s fixed point theorem is a real generalization of Nadler’s fixed point theorem. We refine/generalize Mizoguchi–Takahashi’s fixed point theorem. Our result improves a recent result by Klim and Wadowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] and extends Hicks and Rhoades [T.L. Hicks, B.E. Rhoades, A banach type fixed point theorem, Math. Japonica 24 (1979) 327–330] fixed point theorem to multivalued maps.