{"title":"Geometric progressions in distance spaces; applications to fixed points and coincidence points","authors":"E. Zhukovskiy","doi":"10.4213/sm9773e","DOIUrl":null,"url":null,"abstract":"Conditions on spaces $X$ with generalized distance $\\rho_X$ are investigates under which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings on such spaces. This is shown to hold if each geometric progression with ratio $<1$ (that is, each sequence $\\{ x_i\\}\\subset X$ satisfying $\\rho_X(x_{i+1},x_i)\\leq \\gamma \\rho_X(x_i,x_{i-1})$, $ i=1,2,…$, with some $\\gamma < 1$) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete $f$-quasimetric space $X$ if the distance $\\rho_X$ in it satisfies $\\rho_X(x,z) \\leq \\rho_X(x,y)+(\\rho_X(y,z))^\\eta$, $x,y,z \\in X$, for some $\\eta\\in (0,1)$, that is, if the function $f\\colon\\mathbb{R}_+^{2} \\to \\mathbb{R}_+$ is given by $f(r_1,r_2)=r_1 + r_2^{\\eta}$. Next, for $f(r_1,r_2)=\\max\\{ r_1^{\\eta}, r_2^{\\eta} \\}$, where $\\eta \\in (0,2^{-1}]$, it is shown that for any $\\gamma > 0$ there exists an $f$-quasimetric space containing a geometric progression with ratio $\\gamma$ which is not a Cauchy sequence. The ‘zero-one law’, which means that either each geometric progression with ratio $<1$ is a Cauchy sequence or, for any $\\gamma\\in (0,1)$, there exists a geometric progression with ratio $\\gamma$ that is not Cauchy, is discussed for $f$-quasimetric spaces. Bibliography: 29 titles.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4213/sm9773e","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Conditions on spaces $X$ with generalized distance $\rho_X$ are investigates under which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings on such spaces. This is shown to hold if each geometric progression with ratio $<1$ (that is, each sequence $\{ x_i\}\subset X$ satisfying $\rho_X(x_{i+1},x_i)\leq \gamma \rho_X(x_i,x_{i-1})$, $ i=1,2,…$, with some $\gamma < 1$) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete $f$-quasimetric space $X$ if the distance $\rho_X$ in it satisfies $\rho_X(x,z) \leq \rho_X(x,y)+(\rho_X(y,z))^\eta$, $x,y,z \in X$, for some $\eta\in (0,1)$, that is, if the function $f\colon\mathbb{R}_+^{2} \to \mathbb{R}_+$ is given by $f(r_1,r_2)=r_1 + r_2^{\eta}$. Next, for $f(r_1,r_2)=\max\{ r_1^{\eta}, r_2^{\eta} \}$, where $\eta \in (0,2^{-1}]$, it is shown that for any $\gamma > 0$ there exists an $f$-quasimetric space containing a geometric progression with ratio $\gamma$ which is not a Cauchy sequence. The ‘zero-one law’, which means that either each geometric progression with ratio $<1$ is a Cauchy sequence or, for any $\gamma\in (0,1)$, there exists a geometric progression with ratio $\gamma$ that is not Cauchy, is discussed for $f$-quasimetric spaces. Bibliography: 29 titles.