{"title":"Another approach to K-subadditivity","authors":"Eliza Jabłońska","doi":"10.1007/s00010-024-01083-z","DOIUrl":null,"url":null,"abstract":"<p>In the paper the notion of weakly <i>K</i>-subadditive set-valued maps is introduced in such a way that <i>F</i> is weakly <i>K</i>-superadditive if and only if <span>\\(-F\\)</span> is weakly <i>K</i>-subadditive. This new definition is a natural generalization of <i>K</i>-subadditive set-valued maps from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021), for which opposite set-valued maps need not be <i>K</i>-subadditive. Among others, we prove that every weakly <i>K</i>-subadditive set-valued map which is <i>K</i>–upper bounded on a “large” set has to be locally weakly <i>K</i>-upper bounded and weakly <i>K</i>-lower bounded at every point of the domain. This theorem completes an analogous result for <i>K</i>-subadditive set-valued maps which are weakly <i>K</i>-upper bounded on “large” sets from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021).</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Aequationes Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00010-024-01083-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the paper the notion of weakly K-subadditive set-valued maps is introduced in such a way that F is weakly K-superadditive if and only if \(-F\) is weakly K-subadditive. This new definition is a natural generalization of K-subadditive set-valued maps from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021), for which opposite set-valued maps need not be K-subadditive. Among others, we prove that every weakly K-subadditive set-valued map which is K–upper bounded on a “large” set has to be locally weakly K-upper bounded and weakly K-lower bounded at every point of the domain. This theorem completes an analogous result for K-subadditive set-valued maps which are weakly K-upper bounded on “large” sets from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021).
本文引入了弱 K 次正定值映射的概念，即只有当 \(-F\) 是弱 K 次正定值时，F 才是弱 K 次正定值。这个新定义是 Jabłońska 和 Nikodem (Aequ Math 95:1221-1231, 2021) 对 K-subadditive 集值映射的自然概括。其中，我们证明了在 "大 "集合上是 K 上界的每个弱 K 次相加的集值映射在域的每个点上都必须是局部弱 K 上界和弱 K 下界的。本定理完善了 Jabłońska 和 Nikodem (Aequ Math 95:1221-1231, 2021) 关于在 "大 "集合上弱 K 上界的 K 次正定值映射的类似结果。
期刊介绍：
aequationes mathematicae is an international journal of pure and applied mathematics, which emphasizes functional equations, dynamical systems, iteration theory, combinatorics, and geometry. The journal publishes research papers, reports of meetings, and bibliographies. High quality survey articles are an especially welcome feature. In addition, summaries of recent developments and research in the field are published rapidly.
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