In this paper, we introduce the notion of topologically Banach contraction mapping defined on an arbitrary topological space X with the help of a continuous function $g:X\times X\rightarrow \mathbb{R}$ and investigate the existence of fixed points of such mapping. Moreover, we introduce two types of mappings defined on a non-empty subset of X and produce sufficient conditions which will ensure the existence of best proximity points for these mappings. Our best proximity point results also extend some existing results from metric spaces or Banach spaces to topological spaces. More precisely, our newly introduced mappings are more general than that of the corresponding notions introduced by Bunlue and Suantai [Arch. Math. (Brno), 54(2018), 165-176]. We present several examples to validate our results and justify its motivation. To study best proximity point results, we introduce the notions of g-closed, g-sequentially compact subsets of X and produce examples to show that there exists a non-empty subset of X which is not closed, sequentially compact under usual topology but is g-closed and g-sequentially compact.
In the year 2011, S.Basha \cite{BS} introduced the notion of proximal contraction in a metric space $X$ and study the existence and uniqueness of best proximity point for this class of mappings. Also, the author gave an algorithm to achieve this best proximity point. In this paper, we show that the best proximity point theorem can be proved by Banach contraction principle.
Recently, In the year 2020, Altun et al. \cite{AL} introduced the notion of $p$-proximal contractions and discussed about best proximity point results for this class of mappings. Then in the year 2021, Gabeleh and Markin \cite{GB} showed that the best proximity point theorem proved by Altun et al. in \cite{AL} follows from the fixed point theory. In this short note, we show that if the $p$-proximal contraction constant $k<\frac{1}{3}$ then the existence of best proximity point for $p$-proximal contractions follows from the celebrated Banach contraction principle.
In the year 2021, Beg et al. \cite{beg} [J. Fixed Point Theory Appl.(2021)] introduced two classes of non-self mappings namely, generalized $F$-proximal contraction of the first kind and generalized $F$-proximal contraction of the second kind. Then authors studied the existence and uniqueness of best proximity points for this two classes of mappings. In this short note, we show that the existence of best proximity point for generalized $F$-proximal contraction of the first kind follows from the same conclusion in fixed point theory.
Banach Contraction Principle for Cyclical Mappings on Partial Metric Spaces
本文证明了S. G. Matthews在1994年关于0-完备部分度量空间的结果可以推广到循环映射.然而，D. Ilić，V. Pavlović and V. Rakoucević in“Some new extensions of Banach's constraction principle to partial度量空间，Appl. Math. Lett. 24（2011），1326--1330”在完备的部分度量空间上不能推广到循环映射。最后通过一些例子说明了所得结果的有效性.此外，我们推广了W. A.柯克，P.S. Srinivasan和P. Veeramani，“满足循环压缩条件的映射的不动点，不动点理论4（1）（2003），79 - 89”.最后，Edelstein型定理也被推广到循环分解中的一个集合是0-紧的情况。
Arithmetic-geometric mean, additive, and multiplicative contractions: New generalizations of the Banach contraction principle
Manipur University（曼尼普尔大学）；Manipur Institute of Technology（曼尼普尔理工学院）；Institute of Applied Mathematics and Mechanics of the NAS of Ukraine（乌克兰国家科学院应用数学与力学研究所）；Institute of Mathematics of the NAS of Ukraine（乌克兰国家科学院数学研究所）
It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems …
We first introduce a notion of convex structure in generalized metric spaces, then we introduce tripartite contractions, tripartite semi-contractions, tripartite coincidence points, as well as tripartite best proximity points for a given triple $(K;S;T)$ of nonlinear mappings defined on the union $A\cup B\cup C$ of closed subsets of a generalized metric space. We prove theorems on the existence and convergence of tripartite coincidence-best proximity points.
An Entourage Approach to the Contraction Principle in Uniform Spaces Endowed with a Graph
本文研究了赋图的一致空间中的Banach压缩，给出了映射是Picard算子的几个充分条件.我们的主要结果推广了[J. Jachymski，“The contraction principle for mappings on a度量空间with a graph”，Proc. Amer.Math.Soc.136（2008）1359-1373]中利用一致空间的基本随行的一些结果.
Let $U$ be a Banach Lie group and $G\le U$ a compact subgroup. We show that closed Lie subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ are compact, and conjugate to subgroups of $G$ by elements close to $1\in U$; this generalizes a well-known result of Montgomery and Zippin's from finite- to infinite-dimensional Lie groups. Along the way, we also prove an approximate counterpart to Jordan's theorem on finite subgroups of general linear groups: finite subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ have normal abelian subgroups of inde…
The Banach contraction principle is the most celebrated fixed point theorem, it has been generalized in various directions. In this paper, inspired by the concept of $(ϕ, F)-$contraction in metric spaces, introduced by Wardowski. We present the notion of $(ϕ, F)-$contraction in $b-$rectangular metric spaces to study the existence and uniqueness of fixed point for the mappings in this spaces. Our results improve many existing results.
A $ϕ$ - contraction Principle in Partial Metric Spaces with Self-distance Terms
在完备部分度量空间中证明了一个带控制函数的广义压缩原理。使用的收缩类型条件允许出现自距离项。所得结果推广了以前的一些结果，如最近的“D。Ilić，V. Pavlović and V. Rakoucević，Banach压缩原理在部分度量空间上的一些新扩展，应用数学快报. 24（2011），1326- 1330”。最后通过一个例子说明了这种推广及其正确性。我们给出的例子并不验证S。Romaguera在“不动点定理for generalized contractions on partial度量空间，拓扑应用159（2012），194-199”和I. Altun，F. Sola和H. Simsek在“Generalized contractions on partial度量空间，拓扑及其应用157（18）（2010），2778- 2785”中。因此，我们的结果比以前得到的有优势。
Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for…
In this paper, we introduce the $\mathcal{F}$-metric space concept, which generalizes the metric space notion. We define a natural topology $τ_{\mathcal{F}}$ in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of $\mathcal{F}$-metric spaces. Several examples are presented to illustrate our study.
