Fixed Point Theorems for Expanding Mappings in Cone Metric Spaces

The existing literature of fixed point theory contains many results enunciating fixed point theorems for self-mappings in metric and Banach spaces. Recently, Huang and Zhang [4] introduced the concept of cone metric spaces which generalized the concept of the metric spaces, replacing the set of real numbers by an ordered Banach space, and obtained some fixed point theorems for mapping satisfying different contractive conditions. The study of fixed point theorems in such spaces is followed by some other mathematicians, see [1–2, 5–6, 8–10, 12]. In 1984, Wang et. al. [11] introduced the concept of expanding mappings and proved some fixed point theorems in complete metric spaces. In 1992, Daffer and Kaneko [3] defined an expanding condition for a pair of mappings and proved some common fixed point theorems for two mappings in complete metric spaces. In this paper, we define expanding mappings in the setting of cone metric spaces analogous to expanding mappings in complete metric spaces. We also extend a result of Daffer and Kaneko [3] for two mappings to the setting of cone metric spaces. Consistent with Huang and Zhang [4], the following definitions and results will be needed in the sequel. Let E be a real Banach space. A subset P of E is called a cone if and only if: (a) P is closed, nonempty and P = {θ}; (b) a, b ∈ R, a, b ≥ 0, x, y ∈ P implies ax+ by ∈ P ; (c) P ∩ (−P ) = {θ}.