Scholarlink

<p>The notion of mild solutions for autonomous linear retarded functional differential equations (RFDEs) has been introduced in [J. Nishiguchi, Electron. J. Qual. Theory Differ. Equ. <b>2023</b>, No.&#xa0;32, 1–77] for the purpose of defining fundamental matrix solutions and obtaining a variation of constants formula for the RFDEs. This notion gives a straightforward definition of solutions to the RFDEs under discontinuous history functions compared with previous studies in the literature. For a given autonomous linear RFDE, it holds that the fundamental matrix solutions are locally Lipschitz continuous on the interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\([0, \infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. However, it is not apparent whether a similar property is true for the mild solutions. Here we obtain a result which shows the regularity of mild solutions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\([0, \infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for autonomous linear RFDEs. The result makes clear a connection between the mild solutions and solution concepts in previous studies.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On regularity of mild solutions for autonomous linear retarded functional differential equations

  • 期刊论文

  • 发表日期：

    2026年3月31日

查看全文
  • Junya Nishiguchi

摘要

The notion of mild solutions for autonomous linear retarded functional differential equations (RFDEs) has been introduced in [J. Nishiguchi, Electron. J. Qual. Theory Differ. Equ. 2023, No. 32, 1–77] for the purpose of defining fundamental matrix solutions and obtaining a variation of constants formula for the RFDEs. This notion gives a straightforward definition of solutions to the RFDEs under discontinuous history functions compared with previous studies in the literature. For a given autonomous linear RFDE, it holds that the fundamental matrix solutions are locally Lipschitz continuous on the interval \([0, \infty )\) [ 0 , ∞ ) . However, it is not apparent whether a similar property is true for the mild solutions. Here we obtain a result which shows the regularity of mild solutions on \([0, \infty )\) [ 0 , ∞ ) for autonomous linear RFDEs. The result makes clear a connection between the mild solutions and solution concepts in previous studies.