Ordered Convex Metric Spaces

Abstract

The aim of this article is to introduce a new notion of ordered convex metric spaces and study some basic properties of these spaces. Several characterizations of these spaces are proven that allow making geometric interpretations of the new concepts.

Ordered Uniform Convexity in Ordered Convex Metric Spaces with an Application to Fixed Point Theory

“…In this study, we introduced ordered uniform convexity in ordered convex metric spaces due to Beg [13] and investigated several of its interesting properties including ordered 2-uniform convexity. As applications, we proved some fixed point theorems.…”

“…There are several examples in the existing literature [4,9,10,12] of convex metric spaces which are not embedded in any normed space. Definition 6 (see [13]). A convex metric space X is said to have property ðLÞ if for all a, b in X and λ, μ, ν in I, we have…”

“…Definition 7 (see [13]). An ordered relation ⪯ on a convex metric space X is said to be continuous if the sets fλ : ωða , b ; λÞ⪯cg and fλ : ωða, b ; λÞ ⪰ cg are closed for a, b, c in X.…”

“…Definition 8 (see [13]). A metric space ðX, dÞ with convex structure ω and continuous order relation ⪯ is called ordered convex metric space and denoted by ðX, d, ω,⪯Þ.…”

“…Afterward, Abdelhakim [12] studied convex functions on these spaces. Subsequently, Beg [13] introduced the idea of ordered convex metric spaces and proved several results on these spaces. On the other hand, in 1936, J.A.…”