Fixed point theorems for convex contraction mappings on cone metric spaces

In 2007, Huang and Zhang (2007) rediscovered normal cone metric spaces and obtained the Banach contraction principle for this setting. Later on, Rezapour and Hamlbarani (2008) 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) and Rezapour and Hamlbarani (2008).