Maps Which Are Continuously Differentiable in the Sense of Michal and Bastiani but not of Fre´chet

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Abstract

We construct examples of nonlinear maps on function spaces which are continuously differentiable in the sense of Michal and Bastiani but not in the sense of Fre´chet. The search for such examples is motivated by studies of delay differential equations with the delay variable and not necessarily bounded.

About the authors

Hans-Otto Walther

Mathematisches Institut, Universita¨t Gießen

Author for correspondence.
Email: Hans-Otto.Walther@math.uni-giessen.de
Arndtstr. 2, D 35392 Gießen, Germany

References

  1. Bastiani A. Applications diffe´rentiables et variete´s de dimension infinie// J. Anal. Math. - 1964. - 13.- С. 1-114.
  2. Diekmann O., van Gils S. A., Verduyn Lunel S. M., Walther H. O. Delay equations: functional-, complexand nonlinear analysis. - New York: Springer, 1995.
  3. Glo¨ckner H. Implicit functions from topological vector spaces to Banach spaces// Israel J. Math. - 2006. - 155. - С. 205-252.
  4. Glo¨ckner H. Finite order differentiability properties, fixed points and implicit functions over valued fields// http://arxiv.org/pdf/math/0511218. - 2007.
  5. Hale J. K. Functional differential equations. - New York: Springer, 1971.
  6. Hale J. K., Verduyn Lunel S. M. Introduction to functional differential equations. - New York: Springer, 1993.
  7. Hamilton R. S. The inverse function theorem of Nash and Moser// Bull. Am. Math. Soc. (N. S.). - 1982. - 7. - С. 65-222.
  8. Hartung F., Krisztin T., Walther H. O., Wu J. Functional differential equations with state-dependent delays: theory and applications// Handb. Differ. Equ. - 2006. - 3. - С. 435-545.
  9. Krisztin T., Walther H. O. Smoothness issues in differential equations with state-dependent delay// Rend. Istit. Mat. Univ. Trieste. - 2017. - 49. - С. 95-112.
  10. Michal A. D. Differential calculus in linear topological spaces// Proc. Natl. Acad. Sci. USA. - 1938. - 24. - С. 340-342.
  11. Szilasi J., Lovas R. L. Some aspects of differential theories// В сб.: Handbook of global analysis. - Amsterdam: Elsevier, 2007. - С. 1071-1116.
  12. Walther H. O. The solution manifold and C1-smoothness of solution operators for differential equations with state dependent delay// J. Differ. Equ. - 2003. - 195. - С. 46-65.
  13. Walther H. O. Smoothness properties of semiflows for differential equations with state dependent delay// J. Math. Sci. (N. Y.). - 2004. - 124. - С. 5193-5207.
  14. Walther H. O. Differential equations with locally bounded delay// J. Differ. Equ. - 2012. - 252. - С. 3001- 3039.
  15. Walther H. O. Evolution systems for differential equations with variable time lags// J. Math. Sci. (N. Y.). - 2014. - 202. - С. 911-933.
  16. Walther H. O. Semiflows for differential equations with locally bounded delay on solution manifolds in the space C1((-∞, 0], Rn)// Topol. Methods Nonlinear Anal. - 2016. - 48. - С. 507-537.
  17. Walther H. O. Local invariant manifolds for delay differential equations with state space in C1((-∞, 0], Rn)// Electron. J. Qual. Theory Differ. Equ. - 2016. - 85. - С. 1-29.
  18. Walther H. O. Fre´chet differentiability in Fre´chet spaces, and differential equations with unbounded variable delay// Preprint, 2016.

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