数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (3): 765-777.doi: 10.1016/S0252-9602(11)60274-9

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HÖLDER CONTINUOUS SOLUTIONS FOR SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

  

  1. Department of Mathematical Science, University of Tsinghua, Beijing 100084, China
  • 收稿日期:2008-10-21 修回日期:2009-11-19 出版日期:2011-05-20 发布日期:2011-05-20
  • 基金资助:

    This work was supported by the NSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education.

HÖLDER CONTINUOUS SOLUTIONS FOR SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

  1. Department of Mathematical Science, University of Tsinghua, Beijing 100084, China
  • Received:2008-10-21 Revised:2009-11-19 Online:2011-05-20 Published:2011-05-20
  • Supported by:

    This work was supported by the NSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education.

摘要:

We study  H\"older continuous solutions for the second order integro-differential equations with infinite delay (P1): u''(t)+ cu'(t)+∫t-∞β (t-s)u'(s)ds+∫t-∞γ(t-s)u(s)ds =Au(t)-∫t-∞δ(t-s)Au(s)ds+f(t) on the line R, where 0 < α< 1, A is a closed operator in a complex Banach space Xc ∈C is a constant, f ∈Cα (R,X) and βγδ ∈L1(R+). Under suitable assumptions on the kernels β, γ and δ, we completely characterize the Cα-well-posedness of (P1) by using operator-valued Cα-Fourier multipliers.

关键词: Fourier multiplier, Cα-well-posedness, integro-differential equations

Abstract:

We study  H\"older continuous solutions for the second order integro-differential equations with infinite delay (P1): u''(t)+ cu'(t)+∫t-∞β (t-s)u'(s)ds+∫t-∞γ(t-s)u(s)ds =Au(t)-∫t-∞δ(t-s)Au(s)ds+f(t) on the line R, where 0 < α< 1, A is a closed operator in a complex Banach space Xc ∈C is a constant, f ∈Cα (R,X) and βγδ ∈L1(R+). Under suitable assumptions on the kernels β, γ and δ, we completely characterize the Cα-well-posedness of (P1) by using operator-valued Cα-Fourier multipliers.

Key words: Fourier multiplier, Cα-well-posedness, integro-differential equations

中图分类号: 

  • 45N05