Generalized Aizerman-Myshkis problem for abstract differential-delay equations

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Abstract

We consider a class of semilinear functional-differential equations in a Hilbert space. Conditions for the absolute stability are established. Moreover, it is shown that these conditions separate equations that satisfy the generalized Aizerman-Myshkis hypothesis. The suggested approach is based on a combined usage of the properties of operators on tensor products of Hilbert spaces and recent estimates for the norm of the resolvent. Our results are new even in the finite-dimensional case. We also discuss applications of the mentioned results to coupled systems of parabolic equations with delay and integro-differential equations with delay.

Original languageEnglish
Pages (from-to)771-784
Number of pages14
JournalNonlinear Analysis, Theory, Methods and Applications
Volume55
Issue number6
DOIs
StatePublished - 1 Dec 2003

Keywords

  • Absolute stability
  • Abstract nonlinear differential-delay equations
  • Tensor products
  • The Aizerman-Myshkis problem

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