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<p>In this paper, we prove three types of zero-<i>r</i> law for integrated fractional resolvent families, including <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C_{0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroups, concerning the spectrum, boundedness and scalar-type properites of their generators, respectively. By introducing a new density function, we are able to unify and generalize some existing results on zero-<i>r</i> law for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroups, cosine functions and resolvent families. We also optimize the classical results on zero–one law for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(C_0\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>-semigroups derived in [<CitationRef CitationID="CR23">23</CitationRef>].</p>

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Zero-r law for \((\alpha ,\beta )\)-resolvent families

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Chen-Yu Li,
  • Miao Li,
  • Kun-Yi Zhang

摘要

In this paper, we prove three types of zero-r law for integrated fractional resolvent families, including \(C_{0}\) C 0 -semigroups, concerning the spectrum, boundedness and scalar-type properites of their generators, respectively. By introducing a new density function, we are able to unify and generalize some existing results on zero-r law for \(C_0\) C 0 -semigroups, cosine functions and resolvent families. We also optimize the classical results on zero–one law for \(C_0\) C 0 -semigroups derived in [23].