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Analyticity of Solutions of Differential Equations with a Threshold Delay

  • Tibor KrisztinEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 94)

Abstract

We consider the differential equation \(\dot{x}(t) = f(x(t),x(t - r))\) where the delay r = r(x(⋅ )) is defined by the threshold condition \(\int _{t-r}^{t}a(x(s),\dot{x}(s))\,\mathrm{d}s =\rho\) with a given ρ > 0. It is shown that if f and a are analytic functions and a is positive, then the globally defined bounded solutions are analytic.

Keywords

Delay differential equation State-dependent delay Threshold condition Analyticity 

Notes

Acknowledgements

This paper is dedicated to the 70th birthday of István Győri. Supported by the Hungarian Scientific Research Fund Grant. No. K109782.

References

  1. 1.
    Dieudonné, J.: Foundations of modern analysis. Enlarged and corrected printing. Pure and Applied Mathematics, vol. 10-I. Academic Press, New York/London (1969)Google Scholar
  2. 2.
    Hartung, F.: Differentiability of solutions with respect to the initial data in differential equations with state-dependent delays. J. Dynam. Differ. Equat. 23, 843–884 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hartung, F., Krisztin, T., Walther, H.-O., Wu, J.: Functional differential equations with state-dependent delay: theory and applications. In: Canada A, Drabek P, Fonda A (eds.) Handbook of differential equations: Ordinary differential equations. vol. 3, pp. 435–545. Elsevier-North-Holland, Amsterdam (2006)Google Scholar
  4. 4.
    Hartung, F., Turi, J.: On differentiability of solutions with respect to parameters in state-dependent delays. J. Differ. Equat. 135, 192–237 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hu, Q., Wu, J., Zou, X.: Estimates of periods and global continua of periodic solutions for state-dependent delay equations. SIAM J. Math. Anal. 44, 2401–2427 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Insperger, T., Stépán, G., Turi, J.: State-dependent delay model for regenerative cutting processes. Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference, pp. 1124–1129. Eindhoven, The Netherlands (2005)Google Scholar
  7. 7.
    Krisztin, T.: A local unstable manifold for differential equations with state-dependent delay. Discrete Contin. Dyn. Syst. 9, 9930–1028 (2003)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Krisztin, T.: C 1-smoothness of center manifolds for delay differential equations with state-dependent delay. Fields Institute Communications, vol. 48, pp. 213–226. American Mathematical Society, Providence, RI (2006)Google Scholar
  9. 9.
    Mallet-Paret, J., Nussbaum, R.D.: Analyticity and nonanalyticity of solutions of delay-differential equations. SIAM J. Math. Anal. Appl. 46(4), 2468–2500 (2014)MathSciNetGoogle Scholar
  10. 10.
    Nussbaum, R.D.: Periodic solutions of analytic functional differential equations are analytic. Michigan Math. J. 20, 249–255 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Qesmi, R., Walther, H.-O.: Center-stable manifolds for differential equations with state-dependent delays. Discrete Contin. Dyn. Syst. 23, 1009–1033 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Walther, H.-O.: The solution manifold and C 1-smoothness of solution operators for differential equations with state dependent delays. J. Differ. Equat. 195, 46–65 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Walther, H.-O.: Algebraic-delay differential systems, state-dependent delay, and temporal order of reactions. J. Dynam. Differ. Equat. 21, 195–232 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Bolyai Institute, MTA-SZTE Analysis and Stochastic Research GroupUniversity of SzegedSzegedHungary

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