Abstract
AbstractA lot of authors have proved various common fixed-point results for pairs of self-mappings under strict contractive conditions in metric spaces. In the case of cone metric spaces, fixed point results are usually proved under assumption that the cone is normal. In the present paper we prove common fixed point results under strict contractive conditions in cone metric spaces using only the assumption that the cone interior is nonempty. We modify the definition of property (E.A), introduced recently in the work by Aamri and Moutawakil (2002), and use it instead of usual assumptions about commutativity or compatibility of the given pair. Examples show that the obtained results are proper extensions of the existing ones.
Highlights
Introduction and PreliminariesCone metric spaces were introduced by Huang and Zhang in 1, where they investigated the convergence in cone metric spaces, introduced the notion of their completeness, and proved some fixed point theorems for contractive mappings on these spaces
In the case of cone metric spaces, fixed point results are usually proved under assumption that the cone is normal
In the present paper we prove common fixed point results under strict contractive conditions in cone metric spaces using only the assumption that the cone interior is nonempty
Summary
Cone metric spaces were introduced by Huang and Zhang in 1 , where they investigated the convergence in cone metric spaces, introduced the notion of their completeness, and proved some fixed point theorems for contractive mappings on these spaces. The following properties are often useful when dealing with cone metric spaces in which the cone may be nonnormal : p1 if 0 ≤ u c for each c ∈ int P u 0, p2 if c ∈ int P , 0 ≤ an and an → 0, there exists n0 such that an c for all n > n0. It follows from p2 that the sequence xn converges to x ∈ X if d xn, x → 0 as n → ∞. In this case, the fact that d xn, yn → d x, y if xn → x and yn → y is not applicable
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