A note on cone metric fixed point theory and its equivalence

The main aim of this paper is to investigate the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of fixed point theorems in (general) metric spaces (in usual sense). We show that the Banach contraction principles in general metric spaces and in TVS-cone metric spaces are equivalent. Our theorems also extend some results in Huang and Zhang (2007) [L-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], Rezapour and Hamlbarani (2008) [Sh. Rezapour, R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719-724] and others. (C) 2009 Elsevier Ltd. All rights reserved.