RESIDUAL ν-METRIC SPACE AND BANACH CONTRACTION PRINCIPLE

2019

Very recently, the notion of extended b-metric spaces was introduced by replacing the modified triangle inequality with a functional triangle inequality and the analog of the renowned Banach fixed point theorem was proved in this new structure. In this paper, continuing in this direction, we further refine the functional inequality and establish some fixed point results for nonlinear contractive mappings in the new setting. A nontrivial example for the new extended b-metric space is given.

Nonlinear Analysis: Modelling and Control

The notion of extended b-metric space plays an important role in the field of applied analysis to construct new theorems in the field of fixed point theory. In this paper, we construct and prove new theorems in the filed of fixed point theorems under some new contractions. Our results extend and modify many existing results in the literature. Also, we provide an example to show the validity of our results. Moreover, we apply our result to solve the existence and uniqueness of such equations.

Abstract and Applied Analysis, 2017

A novel class ofα-β-contraction for a pair of mappings is introduced in the setting ofb-metric spaces. Existence and uniqueness of coincidence and common fixed points for such kind of mappings are investigated. Results are supported with relevant examples. At the end, results are applied to find the solution of an integral equation.

The Journal of Nonlinear Sciences and Applications, 2017

The work presented in this paper extends the idea of α-β-contractive mappings in the framework of b-metric spaces. Fixed points are investigated for such kind of mappings. An example is given to show the superiority of our results. As applications we discuss Ulam-Hyres stability, well-posedness and limit shadowing of fixed point problem.

International publication, 2024

The root of metric fixed point theory is Stefen Banach's contraction mapping, a research source for shrinking the distance between two points in space. As a source, many authors have introduced many contraction mappings as extensions and generalizations of Banach contraction and established fixed point theorems under the property that each such mapping in complete metric and Menger space has a unique fixed point. This article presents updated results of Banach contraction generalization and extension forms in metric and Menger space which helps the comparative and interrelationship study in these spaces.

Advances in Difference Equations

This paper provides two wide classes of contractions, which are obtained by using notions of $\alpha _{s^{p}} $ α s p -admissibility and the rich set of C-class functions in the setting of a complete b-metric-like space under more general contractive conditions. An application is provided and many known results in the literature can be derived.

The concept of rectangular b-metric space is introduced as a generalization of metric space, rectangular metric space and b-metric space. An analogue of Banach contraction principle and Kannan's fixed point theorem is proved in this space. Our result generalizes many known results in fixed point theory.

Journal of Fixed Point Theory and Applications, 2018

Using some combinatorial arguments and, in particular, the pigeonhole principle, we prove that the generalized Banach contraction conjecture in b-metric spaces is true if every b-metric space (X, D, K) is a metric-type space and M ∈ (0, 1 K). Within M ∈ (0, 1 K), we also prove that the generalized Banach contraction conjecture in b-metric space (X, D, K) is true for the case J = 2; and for the case J = 3 if the map T : X −→ X is continuous. These results are generalizations of corresponding results in Jachymski et al.

2018

In 1922, Banach [1] proved a fundamental result in fixed point theory which is one of the important tools in the field of nonlinear analysis and its applications. Many authors extended Banach Theorem in different metric spaces; for example see [20]-[29]. In 1989, Bakhtin [2] generalized Banach’s contraction principle. Actually, he introduced the concept of a b-metric space and proved some fixed point theorems for some contractive mappings in bmetric spaces. Later, in 1993, Czerwik [15] extended the results of b-metric spaces. In 2012, Samet [5] introduced the α− ψ-contraction on the Banach contraction principle. Recently, many research was conducted on b-metric space under different contraction conditions, [7][14]. After that many authors used α − ψ-contraction mapping on different metric spaces [16]-[18]. The notion of extended b-metric space has been introduced recently by Kamran et al [6]. In this paper, we generalize the results of Mehmet [19] and Mukheimer [4] by introducing th...

Journal of Nonlinear Sciences and Applications

The object of this paper is to establish a generalized form of Banach contraction principle for a cone metric space which is not necessarily normal. This happens to be a generalization of all different forms of Banach contraction Principle, which have been arrived at in L.