Fixed point theorems for set-valued contractions in complete metric spaces

The fixed point theory of set-valued contractions which was initiated by Nadler [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475–488] was developed in different directions by many authors, in particular, by [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972) 26–42; N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188; Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112]. In the present paper, the concept of contraction for set-valued maps in metric spaces is introduced and the conditions guaranteeing the existence of a fixed point for such a contraction are established. One of our results essentially generalizes the Nadler and Feng–Liu theorems and is different from the Mizoguchi–Takahashi result. The second result is different from the Reich and Mizoguchi–Takahashi results. The method used in the proofs of our results is inspired by Mizoguchi–Takahashi and Feng–Liu's ideas. Comparisons and examples are given.