Multi-valued nonlinear contraction mappings

Two concepts of nonlinear contractions for multi-valued mappings in complete metric spaces are introduced and three fixed point theorems are proved. Presented theorems are generalizations of very recent fixed point theorems due to Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139] and Feng and Liu [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], as well as a fixed point theorem of Mizoguchi and Takahashi [N. Mizoguchi, W. Takahash, Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188] and several others. An example is given to show that presented results are genuine generalizations. Our results are also some contribution to an open problem raised by Reich [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972), 26-42; S. Reich, Some fixed point problems, Atti Acad. Naz. Lincei 57 (1974), 194-198; S. Reich, Some problems and results in fixed point theory, Contemp. Math. 21 (1983), 179–187].