{"title":"度量空间和模态空间中 G$ 单调映射的定点","authors":"Dau Hong Quan, A. Wiśnicki","doi":"10.12775/tmna.2024.003","DOIUrl":null,"url":null,"abstract":"Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. \nIn the main theorem we show that if $T\\colon C\\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\\in C$ such that $Tc\\in [c,\\rightarrow )_{G}$, \nthen $T$ has a fixed point provided for each $a\\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. \nIn particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. \nSome counterparts of this result for modular spaces, and for commutative families of mappings are given too.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Fixed points of $G$-monotone mappings in metric and modular spaces\",\"authors\":\"Dau Hong Quan, A. Wiśnicki\",\"doi\":\"10.12775/tmna.2024.003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. \\nIn the main theorem we show that if $T\\\\colon C\\\\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\\\\in C$ such that $Tc\\\\in [c,\\\\rightarrow )_{G}$, \\nthen $T$ has a fixed point provided for each $a\\\\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. \\nIn particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. \\nSome counterparts of this result for modular spaces, and for commutative families of mappings are given too.\",\"PeriodicalId\":23130,\"journal\":{\"name\":\"Topological Methods in Nonlinear Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-03-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Topological Methods in Nonlinear Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.12775/tmna.2024.003\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topological Methods in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.12775/tmna.2024.003","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Fixed points of $G$-monotone mappings in metric and modular spaces
Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex.
In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$,
then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings.
In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces.
Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.
期刊介绍:
Topological Methods in Nonlinear Analysis (TMNA) publishes research and survey papers on a wide range of nonlinear analysis, giving preference to those that employ topological methods. Papers in topology that are of interest in the treatment of nonlinear problems may also be included.