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Annali di Matematica Pura ed Applicata

, Volume 104, Issue 1, pp 297–311 | Cite as

Dichotomies for linear differential equations with delays: the Carathéodory case

  • Juan Jorge Schäffer
Article

Summary

This work is concerned with differential equations with delays, on [0, ∞[, of the form\(\dot u\)+Mu=r, and the corresponding homogeneous equation\(\dot u\)+Mu=0, where u and r take values in a Banach space E, and M is a « memory ». The main results relate the existence of dichotomies or exponential dichotomies of the solutions of the homogeneous equation (a kind of conditional stability) to the admissibility of certain pairs of function spaces for the inhomogeneous equation. The « memory » M is quite general: it is a linear mapping that takes continuous E-valued functions on [−1, ∞[ into locally integrable functions on [0, ∞[ in such a way that Mu on [a, b] ⊂ [0, ∞[ depends only on u on [a −1, b], and satisfies a reasonable boundedness condition.

Keywords

Differential Equation Banach Space Linear Mapping Function Space Integrable Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1974

Authors and Affiliations

  • Juan Jorge Schäffer
    • 1
  1. 1.Pittsburgh

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