Abstract
This paper introduces the concept of the theta cone metric, studies its various topological properties, and gives some examples of it. Furthermore, it proves some lemmas and then uses them to give further generalizations of some well-known fixed point theorems. Specifically, Theorem 2 of the paper is a generalization of Reich’s fixed point theorem.
Highlights
Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced cone metric spaces as a generalization of metric spaces
If int C is the set of all interior points of C, a cone C in a normed space E induces the following ordered relations [1, 2]: u ≺ v ⟺ v − u ∈ C, u < v ⟺ (v − u ∈ C and u ≠ v), (1)
E cone C is called normal if there is a number M > 0 such that
Summary
Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced cone metric spaces as a generalization of metric spaces. Let (E, ≺ ) be an ordered normed space, where ≺ is the ordered relation induced by a cone C ⊂ E, X be a nonempty set, and θ be an ordered action on E. en, the function dθ: X × X ⟶ C is called θ-cone-metric on X if and only if dθ satisfies the following conditions: (1) dθ(x, y) Θ ⟺ x y. Fernandez et al introduced F-cone-metric spaces over Banach algebra and gave some generalization of some previous fixed point theorems.
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