Two generalizations of some fixed point theorems

Abstract

The aim of this paper is to generalize two classical fixed point theorems given by Bogin [J. Bogin, A generalization of a fixed point theorem of Goebel, Kirk and Shimi, Canad. Math. Bull. 19 (1976) 7–12] and Greguš [M. Greguš, A fixed point theorem in Banach spaces, Boll. Un. Math. Ital. A (5) 17 (1980) 193–198]. We also complement and extend some very recent results proved by Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861–1869].