{"title":"A unified characterization of convolution coefficients in nonlocal differential equations","authors":"Christopher S. Goodrich","doi":"10.1017/prm.2024.77","DOIUrl":null,"url":null,"abstract":"In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form <jats:disp-formula> <jats:alternatives> <jats:tex-math>\\[{-}M\\Big(\\big(a*(g\\circ |u|)\\big)(1)\\Big)u''(t)=\\lambda f\\big(t,u(t)\\big),\\quad t\\in(0,1), \\]</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0308210524000775_eqnU1.png\"/> </jats:alternatives> </jats:disp-formula>in the case in which <jats:inline-formula> <jats:alternatives> <jats:tex-math>$g$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000775_inline3.png\"/> </jats:alternatives> </jats:inline-formula> can satisfy very generalized growth conditions; in addition, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$M$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000775_inline4.png\"/> </jats:alternatives> </jats:inline-formula> is allowed to be both sign-changing and vanishing. Existence of at least one positive solution to this equation equipped with boundary data is considered. We demonstrate that the nonlocal coefficient <jats:inline-formula> <jats:alternatives> <jats:tex-math>$M$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000775_inline5.png\"/> </jats:alternatives> </jats:inline-formula> allows the forcing term <jats:inline-formula> <jats:alternatives> <jats:tex-math>$f$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210524000775_inline6.png\"/> </jats:alternatives> </jats:inline-formula> to be free of almost all assumptions other than continuity.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/prm.2024.77","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form \[{-}M\Big(\big(a*(g\circ |u|)\big)(1)\Big)u''(t)=\lambda f\big(t,u(t)\big),\quad t\in(0,1), \]in the case in which $g$ can satisfy very generalized growth conditions; in addition, $M$ is allowed to be both sign-changing and vanishing. Existence of at least one positive solution to this equation equipped with boundary data is considered. We demonstrate that the nonlocal coefficient $M$ allows the forcing term $f$ to be free of almost all assumptions other than continuity.
期刊介绍:
A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations.
An international journal, publishing six issues per year, Proceedings A has been publishing the highest-quality mathematical research since 1884. Recent issues have included a wealth of key contributors and considered research papers.