{"title":"Existence and Relaxation of Solutions for a Differential Inclusion with Maximal Monotone Operators and Perturbations","authors":"A. A. Tolstonogov","doi":"10.1134/S1064562423701399","DOIUrl":null,"url":null,"abstract":"<p>A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with a convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S1064562423701399","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
A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with a convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced.