In this paper we introduce the concept of the rectangular metric like spaces, along with its topology and we prove some fixed point theorems under different contraction principles. We introduce the concept of modified metric-like space as well and prove some topological and convergence properties under the symmetric convergence. Some examples are given to illustrate the proven results and enrich the new introduced metric type spaces.
In this paper, we introduce a generalization of rectangular $b-$metric spaces, by changing the rectangular inequality as follows \begin{equation*} ρ(x,y)\le θ(x,y,u,v)[ρ(x,u)+ρ(u,v)+ρ(v,y)], \end{equation*}% for all distinct$\ x,y,u,v\in X.$ We prove some fixed-point theorems and we use our results to present a nice application in last section of this paper. Moreover, in the conclusion we present some new open questions.
Locally Convex Valued Rectangular Metric Spaces and The Kannan's Fixed Point Theorem
引入了矩形TVS-锥度量空间，并证明了Kannan的不动点定理.两种方法的证明。首先利用空间本身的结构用直接法证明了定理。其次，我们使用了杜伟世最近在[A note on cone metric不动点theory and its equivalents，{Nonlinear Analysis}，72（5），2259 -2261（2010）]中使用的非线性标量化。证明了Banach压缩原理在锥度量空间和一般度量空间中的等价性。证明过程中对局部凸拓扑向量空间的锥不作任何正规性假设，推广了已有的结果。
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.
In this paper, we introduce an extension of rectangular metric spaces called controlled rectangular metric spaces, by changing the rectangular inequality as follows: \begin{equation*} d(x, y)\leqα(x, u)d(x, u)+α(u, v)d(u, v)+α(v, y)d(v, y), \end{equation*} for all distinct $x, y, u, v\in X$ with the function $α: X\times X\rightarrow[1,\infty[$. We also establish some fixed point theorems for self-mappings defined on such spaces. Our main results extends and improves many results existing in the literature. Moreover, an illustrative example is presented to support the obtained results.
In this short note, we obtain partial quasi-metric versions of Kannan's fixed point theorem for self-mappings. Moreover, we use these fixed points results to characterize a certain type of completeness in partial quasi-metric spaces. We have reported an example to support our result.
Mustafa and Sims [12] introduced the notion of $G$-metric as a possible generalization of usual notion of a metric space. The author generalized the notion of G-metric to more than three variables and introduced the concept of Generalized $n$-metric spaces [10]. In this paper, We prove Banach fixed point theorem and a Suzuki-type fixed point theorem in Generalized $n$-metric spaces. We also discuss applications to certain functional equations arising in dynamic programming.
The concept of quasi-partial b-metric-like spaces is being introduced and studied with the help of topology. Examples are also discussed to support the results. Some fixed point theorems are proved in the setting of quasi-partial b-metric-like spaces.
The fixed-circle problem is a recent problem about the study of geometric properties of the fixed point set of a self-mapping on metric (resp. generalized metric) spaces. The fixed-disc problem occurs as a natural consequence of this problem. Our aim in this paper, is to investigate new classes of self-mappings which satisfy new specific type of contraction on a metric space. We see that the fixed point set of any member of these classes contains a circle (or a disc) called the fixed circle (resp. fixed disc) of the corresponding self-mapping. For this purpose, we introduce the notions of an $…
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland variational principle and of its equivalent - Caristi fixed point theorem. Other fixed point results having this property will be also presented in metric spaces, in quasi-metric spaces and in partial metric spaces. A discussion on topology and…
On Some Topological Concepts of TVS-Cone Metric Spaces and Fixed Point Theory Remarks
证明了每个TVS-锥度量空间（即局部凸拓扑向量空间$E$上的锥度量空间）是第一可数仿紧拓扑空间，并利用Du在“A note on cone metric不动点theory and its equivalents，{Nonlinear Analysis}，72（5），2259 -2261（2010）]"中的结果，证明了每个TVS-锥度量空间拓扑同构于拓扑度量空间.我们还展示了如何通过使用生成局部凸拓扑向量空间$E$的拓扑的范数系统来构造与TVS-锥度量拓扑可比的度量拓扑。当$E$是巴拿赫空间时，这些度量拓扑就等价于原来的TVS-锥度量拓扑.尽管如此，我们注意到在TVS-锥度量空间中仍然有一些不动点定理来处理它们。利用非线性标量化方法证明了一些具有非线性压缩条件的不动点定理。
In this present article, we etablish some existence results of $φ-$fixed point of a mapping in a $C^{\ast}$-algebra valued metric spaces and we deduce some fixed point theorems in $C^{\ast}$-algebra valued partial metric spaces. Non-trivial examples are further provided to support the hypotheses of our results.
