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 ID. 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 IN. 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 IS. 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]. (C) 2009 Elsevier Ltd. All rights reserved.