Geometric progressions in distance spaces; applications to fixed points and coincidence points

Pub Date : 2023-01-01 DOI:10.4213/sm9773e
E. Zhukovskiy
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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.
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距离空间中的几何级数;对固定点和重合点的应用
研究了具有广义距离$\rho_X$的空间$X$上的映射可以得到类似于Banach和Nadler不动点定理和Arutyunov重合点定理的条件。如果每个比为$ 0$的几何级数存在一个包含比为$\gamma$的几何级数的$f$ -拟对称空间,且该空间不是柯西序列,则证明这是成立的。对于$f$ -拟对称空间,讨论了“0 - 1定律”,即每个比为$<1$的几何级数都是柯西数列,或者对于任何$\gamma\in (0,1)$,存在一个比为$\gamma$的几何级数不是柯西数列。参考书目:29篇。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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