WELL-POSEDNESS OF DEGENERATE DIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN HÖLDER CONTINUOUS FUNCTION SPACES

Shangquan Bu; Gang Cai

doi:10.11948/2019.187

2019 Volume 9 Issue 1

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Shangquan Bu, Gang Cai. WELL-POSEDNESS OF DEGENERATE DIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN HÖLDER CONTINUOUS FUNCTION SPACES[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 187-199. doi: 10.11948/2019.187

Citation:

Shangquan Bu, Gang Cai. WELL-POSEDNESS OF DEGENERATE DIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN HÖLDER CONTINUOUS FUNCTION SPACES[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 187-199. doi: 10.11948/2019.187

WELL-POSEDNESS OF DEGENERATE DIFFERENTIAL EQUATIONS WITH INFINITE DELAY IN HÖLDER CONTINUOUS FUNCTION SPACES

Shangquan Bu^1,,
Gang Cai^2, ,

1.
Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, China
2.
School of Mathematical Sciences, Chongqing Normal University, 401331 Chongqing, China

Author Bio: Email address:sbu@math.tsinghua.edu.cn(S. Bu)
Corresponding author: Email address:caigang-aaaa@163.com(G. Cai)
Fund Project: The authors were supported by the NSF of China (Grant No. 11571194, 11731010, 11771063), the Natural Science Foundation of Chongqing(Grant No. cstc2017jcyjAX0006), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJ1703041, KJZDM201800501), the University Young Core Teacher Foundation of Chongqing (Grant No. 020603011714), Talent Project of Chongqing Normal University (Grant No. 02030307-00024)

Abstract

Abstract

Using operator-valued $ \dot{C}^\alpha $-Fourier multiplier results on vector- valued Hölder continuous function spaces, we give a characterization for the $ C^\alpha $-well-posedness of the first order degenerate differential equations with infinite delay $ (Mu)'(t) = Au(t) + \int_{-\infty}^t a(t-s)Au(s)ds + f(t) $ ($ t\in{\mathbb R} $), where $ A, M $ are closed operators on a Banach space $ X $ such that $ D(A)\cap D(M)\neq \{0\} $, $ a\in L^1_{\rm{loc}}({\mathbb R}_+)\cap L^1(\mathbb{R}_+; t^\alpha dt) $.
- ${C}^\alpha$-well-posedness /
- degenerate differential equations /
- infinite delay /
- $\dot{C}^\alpha$-Fourier multiplier /
- Hö /
- lder continuous function spaces
MSC: 34G10, 47D06, 47A10, 34K30

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References

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