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# Hölder Continuous Solutions of Degenerate Differential Equations with Finite Delay

Published:2017-04-20
Printed: Jun 2018
• Shangquan Bu,
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
• Gang Cai,
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
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## Abstract

Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.
 Keywords: well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, Hölder continuous function space
 MSC Classifications: 34N05 - Dynamic equations on time scales or measure chains {For real analysis on time scales or measure chains, see 26E70} 34G10 - Linear equations [See also 47D06, 47D09] 47D06 - One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47A10 - Spectrum, resolvent 34K30 - Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]

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