Using operator-valued <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\dot{C}^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>C</mi> <mo>˙</mo> </mover> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation>-Fourier multiplier results on Hölder continuous function spaces with values in Banach spaces and the Carleman transform methods, we completely characterize the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation>-well-posedness of the vector-valued first order degenerate differential equations: <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu * (Mu)'(t) + \beta * u(t) - \gamma *(Au)(t)= f(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mrow /> <mo>∗</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>β</mi> <mrow /> <mo>∗</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>γ</mi> <mrow /> <mo>∗</mo> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>, where A, M are two closed linear operators in a complex Banach space X such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$D(A)\subset D(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mu ,\ \beta ,\ \gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>,</mo> <mspace width="4pt" /> <mi>β</mi> <mo>,</mo> <mspace width="4pt" /> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation> are complex Borel measures on <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {R}_+$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation>.

$C^\alpha $-well-posedness of first order degenerate differential equations in banach spaces

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发表日期：

2026年4月1日

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Shangquan Bu,
Gang Cai

摘要

Using operator-valued $\dot{C}^{\alpha }$ ${\dot{C}}^{α}$ -Fourier multiplier results on Hölder continuous function spaces with values in Banach spaces and the Carleman transform methods, we completely characterize the $C^{\alpha }$ $C^{α}$ -well-posedness of the vector-valued first order degenerate differential equations: $\mu * (Mu)'(t) + \beta * u(t) - \gamma *(Au)(t)= f(t)$ $μ * {(M u)}^{'} (t) + β * u (t) - γ * (A u) (t) = f (t)$ on $\mathbb {R}$ $R$ , where A, M are two closed linear operators in a complex Banach space X such that $D(A)\subset D(M)$ $D (A) \subset D (M)$ , $0<\alpha <1$ $0 < α < 1$ and $\mu ,\ \beta ,\ \gamma $ $μ, β, γ$ are complex Borel measures on $\mathbb {R}_+$ $R_{+}$ .

\(C^\alpha \)-well-posedness of first order degenerate differential equations in banach spaces

摘要