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<p>In this article, we use the spectral theory method to find the asymptotic behavior of mild solutions of Volterra integro-differential equations in a class of spaces of polynomially bounded uniformly continuous functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(BUC_n({\mathbb {R}}_+, {\mathbb {X}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>U</mi> <msub> <mi>C</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>,</mo> <mi mathvariant="double-struck">X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We demonstrate the spectral characterizations and asymptotic behavior of mild solutions based on the approaches of resolvent operators and Laplace transform. Examples of parabolic type equations are considered as applications.</p>

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The asymptotic behavior of mild solutions for the abstract Volterra integro-differential equations based on an improved spectral theory method

  • 期刊论文

  • 发表日期：

    2025年9月3日

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  • Pham Van Hoang,
  • Vu Trong Luong

摘要

In this article, we use the spectral theory method to find the asymptotic behavior of mild solutions of Volterra integro-differential equations in a class of spaces of polynomially bounded uniformly continuous functions \(BUC_n({\mathbb {R}}_+, {\mathbb {X}})\) B U C n ( R + , X ) . We demonstrate the spectral characterizations and asymptotic behavior of mild solutions based on the approaches of resolvent operators and Laplace transform. Examples of parabolic type equations are considered as applications.