Caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications

Abstract

In this paper, we obtain the existence and uniqueness of the solution for three self mappings in a complete bipolar metric space under a new Caristi type contraction with an example. We also provide applications to homotopy theory and nonlinear integral equations. MSC: 54H25; 47H10; 54E50 Keywords: Bipolar metric space; Covariant and contravariant map; Compatible mapping and common fixed point 2 Methods/experimental Definition 2.1 ([1]) Let A and B be a two non-empty sets. Suppose that d : A × B → [0, ∞) is a mapping satisfying the following properties: (B 1 ) d(a, b) = 0 if and only if a = b for all (a, b) ∈ A × B, © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Some fixed point theorems in fuzzy bipolar metric spaces

“…The new notions of generalization and improvement of metric spaces leads to give existence and development of fixed point theorems. As in bipolar metric spaces a lot of significant work has been done by researchers (see, [5,[10][11][12]) and in fuzzy metric spaces, Grabeic [2] extend the well known fixed point theorem of Banach to fuzzy metric spaces in the sense of Karamosil and Michalek [6]. After that, Gregori and Sapena [3] extended the fuzzy banach contraction theorem to fuzzy metric space in the sense George and Veeramani's [2].…”

On Two Banach-Type Fixed Points in Bipolar Metric Spaces

“…Finding fixed points for different type of contractions has become the focus of fruitful research activity, and numerous fixed point results in that theory can be found in the literature. Recently, many investigators have published various papers on the fixed point in bipolar metric spaces, and these can be read in [4][5][6][7][8][9][10][11][12][13] and the references therein, just to name a few. Bartwal et al [14] even initiated the concept of fuzzy bipolar metric spaces.…”

α , BK -Contractions in Bipolar Metric Spaces

“…Some basic examples are distance between lines and points in an Euclidean space, distance between points and sets in a metric spaces, affinity between a class of students and a set of activities, lifetime mean distances between people and places, and many more. Many investigators have published various papers on fixed point in bipolar metric spaces, and these can be read in [2][3][4][5][6] and the references therein, just to name a few. is new notion of generalization and improvement of metric spaces leads to give existence and development of fixed-point theorems.…”

“…is new notion of generalization and improvement of metric spaces leads to give existence and development of fixed-point theorems. In bipolar metric spaces, a lot of significant work has been done (see [3,[7][8][9]). More recently, Kishore et al [7] established the existence and uniqueness of common coupled fixed-point results for three covariant mappings in bipolar metric spaces, while Mutlu [8] extended certain coupled fixed-point theorems, which can be considered as generalization of Banach fixed-point theorem, to bipolar metric spaces.…”