{"title":"广义ode的线性化和Hölder连续性及其在测量微分方程中的应用","authors":"Weijie Lu , Yonghui Xia","doi":"10.1016/j.jde.2025.113358","DOIUrl":null,"url":null,"abstract":"<div><div>Linearization, initiated by Poincaré (1890) <span><span>[44]</span></span>, is a bridge connecting the linear system with the nonlinear system. In this paper, we establish the linearization theorem for a new kind of differential equations defined by Kurzweil integral, so-called generalized ODEs (for short, GODEs) in a Banach space. The nonlinear GODEs are formulated as<span><span><span><math><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>=</mo><mi>D</mi><mo>[</mo><mi>A</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>x</mi><mo>+</mo><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>]</mo></math></span></span></span> on a Banach space <span><math><mi>X</mi></math></span>, where <span><math><mi>A</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a bounded linear operator on <span><math><mi>X</mi></math></span> and <span><math><mi>F</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>X</mi></math></span> is Kurzweil integrable. The letter <em>D</em> means that the GODEs are defined via its solution and <span><math><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac></math></span> is only a notation. Thus, GODEs are essentially a notational representation of a class of integral equations. Due to the differences between the theory of GODEs and ODEs, it is difficult to extend the Hartman-Grobman theorem of ODEs to the GODEs. To overcome the difficulty, we first construct the formula for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense. Afterwards, we establish a Hartman-Grobman type linearization theorem which is a bridge connecting the linear GODEs with their nonlinear perturbations. Furthermore, we show that the conjugacies are both Hölder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense). Finally, applications to the measure differential equations and impulsive differential equations, our results are effective.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113358"},"PeriodicalIF":2.3000,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Linearization and Hölder continuity of generalized ODEs with application to measure differential equations\",\"authors\":\"Weijie Lu , Yonghui Xia\",\"doi\":\"10.1016/j.jde.2025.113358\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Linearization, initiated by Poincaré (1890) <span><span>[44]</span></span>, is a bridge connecting the linear system with the nonlinear system. In this paper, we establish the linearization theorem for a new kind of differential equations defined by Kurzweil integral, so-called generalized ODEs (for short, GODEs) in a Banach space. The nonlinear GODEs are formulated as<span><span><span><math><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>=</mo><mi>D</mi><mo>[</mo><mi>A</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>x</mi><mo>+</mo><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>]</mo></math></span></span></span> on a Banach space <span><math><mi>X</mi></math></span>, where <span><math><mi>A</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a bounded linear operator on <span><math><mi>X</mi></math></span> and <span><math><mi>F</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>X</mi></math></span> is Kurzweil integrable. The letter <em>D</em> means that the GODEs are defined via its solution and <span><math><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac></math></span> is only a notation. Thus, GODEs are essentially a notational representation of a class of integral equations. Due to the differences between the theory of GODEs and ODEs, it is difficult to extend the Hartman-Grobman theorem of ODEs to the GODEs. To overcome the difficulty, we first construct the formula for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense. Afterwards, we establish a Hartman-Grobman type linearization theorem which is a bridge connecting the linear GODEs with their nonlinear perturbations. Furthermore, we show that the conjugacies are both Hölder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense). Finally, applications to the measure differential equations and impulsive differential equations, our results are effective.</div></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":\"438 \",\"pages\":\"Article 113358\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-04-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039625003857\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039625003857","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Linearization and Hölder continuity of generalized ODEs with application to measure differential equations
Linearization, initiated by Poincaré (1890) [44], is a bridge connecting the linear system with the nonlinear system. In this paper, we establish the linearization theorem for a new kind of differential equations defined by Kurzweil integral, so-called generalized ODEs (for short, GODEs) in a Banach space. The nonlinear GODEs are formulated as on a Banach space , where is a bounded linear operator on and is Kurzweil integrable. The letter D means that the GODEs are defined via its solution and is only a notation. Thus, GODEs are essentially a notational representation of a class of integral equations. Due to the differences between the theory of GODEs and ODEs, it is difficult to extend the Hartman-Grobman theorem of ODEs to the GODEs. To overcome the difficulty, we first construct the formula for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense. Afterwards, we establish a Hartman-Grobman type linearization theorem which is a bridge connecting the linear GODEs with their nonlinear perturbations. Furthermore, we show that the conjugacies are both Hölder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense). Finally, applications to the measure differential equations and impulsive differential equations, our results are effective.
期刊介绍:
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics