Fixed point theorems for convex contraction mappings on cone metric spaces

Abstract

In 2007, Huang and Zhang [L.G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476] rediscovered normal cone metric spaces and obtained the Banach contraction principle for this setting. Later on, Rezapour and Hamlbarani [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] showed that there are non-normal cones and that the assumption of normality is redundant. In this paper, we obtain a generalization of the Banach contraction principle to the class of convex contractions on non-normal cone metric spaces. Our result includes, as special cases, the recent results of Huang and Zhang (2007) [2] and Rezapour and Hamlbarani (2008) [3].