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Non-analytic growth bounds and delay semigroups

  • Shard Rastogi
  • Sachi SrivastavaEmail author
Research Article
  • 16 Downloads

Abstract

The non-analytic growth bound of a \(C_{0}\)-semigroup measures the extent to which the semigroup may be approximated via holomorphic functions and may be thought of as a non-analytic analogue of the exponential growth bound. In this paper, we study the non-analytic growth bound \( \zeta (\mathcal {T}) \) of the delay semigroup \( \{\mathcal {T}(t)\}_{t\ge 0} \) associated with the delay equation \( u^{\prime }(t) = Bu(t) + \Phi u_{t} \), where \( u_{t} \) is the history function, \( \Phi \in \mathcal {L}(W^{1,p}([-1,0],X),X) \) is the delay operator, and (BD(B)) is the generator of a \( C_{0} \)-semigroup \( \{T(t)\}_{t\ge 0} \) on Banach space X. In particular, we obtain conditions under which \(\zeta (T) < \alpha \) implies that \( \zeta (\mathcal {T}) < \alpha \).

Keywords

Non-analytic growth bound Critical growth bound Delay equation Delay semigroup Fourier multipliers 

Notes

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia
  2. 2.Department of MathematicsUniversity of DelhiNew DelhiIndia

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