Abstract
AbstractNew fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space (which is not necessarily normal) are obtained. By adapting Assad-Kirk's method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. 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
New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space which is not necessarily normal are obtained
In 2–4, some common fixed point theorems have been proved for maps on cone metric spaces
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 particulary when dealing with cone metric spaces in which the cone needs not to be normal : p1 if u ≤ v and v w, u w, p2 if 0 ≤ u c for each c ∈ int P u 0, p3 if a ≤ b c for each c ∈ int P a ≤ b, p4 if 0 ≤ x ≤ y, and a ≥ 0, 0 ≤ ax ≤ ay, p5 if 0 ≤ xn ≤ yn for each n ∈ N, and limn → ∞xn p6 if 0 ≤ d xn, x ≤ bn and bn → 0, d xn, x a sequence and a given point in X, x, limn → ∞yn y, 0 ≤ x ≤ y, c where xn and x are, respectively, p7 if E is a real Banach space with a cone P and if a ≤ λa where a ∈ P and 0 < λ < 1, a 0, p8 if c ∈ int P , 0 ≤ an and an → 0, there exists n0 such that for all n > n0 we have an c. In 10, 11 these results were extended using complete metric spaces of hyperbolic type, instead of Banach spaces
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