Double-Composed Metric Spaces

Abstract

The double-controlled metric-type space (X,D) is a metric space in which the triangle inequality has the form D(η,μ)≤ζ1(η,θ)D(η,θ)+ζ2(θ,μ)D(θ,μ) for all η,θ,μ∈X. The maps ζ1,ζ2:X×X→[1,∞) are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form D(η,μ)≤αD(η,θ)+βD(θ,μ) for all η,θ,μ∈X. In our new space, the control functions α,β:[0,∞)→[0,∞) are composed of the metric D in the triangle inequality, where the control functions ζ1,ζ2:X×X→[1,∞) in a double-controlled metric-type space are multiplied with the metric D. We establish some fixed-point theorems along with the examples and applications.

Fredholm integral equation in composed-cone metric spaces

“…The following result is analogous to the Hardy-Rogers type fixed point theorem, discussed as an open problem in [10], when R + = P ⊂ E = R as a special case in this theorem.…”

“…In 2022, Karami et al [9] gave an extension of a type of controlled metric spaces, defined to be expanded b-metric spaces. Thereafter, in 2023, Ayoobi et al [10] gave off a new extension of the kinds of metric spaces known as double-composed metric spaces (DCMS), which represent the generalized expanded b-metric spaces. The first type depends on one controlled function (incomparable function), while the second one has two different controlled functions (see [11][12][13]).…”

“…They proved some fixed point theorems of contractive mappings on cone metric spaces. Despite all of these studies on cone metric spaces [15][16][17][18][19][20], and Meng and Cho studying algebraic cone metric spaces [21,22], there is much work concerning b-cone metric spaces, for instance, [23][24][25] Hence, by utilizing the concepts in [9,10], this study presents a generalization that represents type I composed cone metric spaces and type II composed cone metric spaces. The examples are for type I composed cone metric space, not cone metric space, and type II composed cone metric space, not for type I composed cone metric space.…”

“…Clearly, CMS is a generalization of the metric spaces, but the reverse is not true (see [14]). Subsequently, the following explanations present the concepts of type I and type II composed cone metric spaces, inspired by Karami et al [9] and Ayoobi et al [10], which are special cases of Definitions 1.2 and 1.4 at E = R, as follows. Example 1.3 Let E = R 2 , P = {t = (t 1 , t 2 ) ∈ E : t 1 , t 2 ≥ 0}, and = R. Here, d I : × → E is defined by…”