Motivated by a problem concerning multi-valued mappings posed by Reich [S.
Reich, Some fixed point problems, Atti Accad. Naz. Lincei Rend. Cl. Sci.
Fis. Mat. Natur. 57 (1974) 194-198] and a paper of Jleli and Samet [M.
Jleli, B. Samet, A new generalization of the Banach contraction principle,
J. Inequal. Appl. 2014:38 (2014) 1-8], we consider a new class of
multi-valued mappings that satisfy a ?-contractive condition in complete
metric spaces and prove some fixed point theorems. These results generalize
Reich?s and Mizoguchi-Takahashi?s fixed point theorems. Some examples are
given to show the usability of the obtained results.