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Intricate Structure of the Analyticity Set for Solutions of a Class of Integral Equations

  • John Mallet-ParetEmail author
  • Roger D. Nussbaum
Article
  • 16 Downloads

Abstract

We consider a class of compact positive operators \(L:X\rightarrow X\) given by \((Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds\), acting on the space X of continuous \(2\pi \)-periodic functions x. Here \(\eta \) is continuous with \(\eta (t)\le t\) and \(\eta (t+2\pi )=\eta (t)+2\pi \) for all \(t\in \mathbf{R}\). We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem \(\kappa x=Lx\) exists for some \(\kappa >0\) (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally \(\eta \) is analytic, we study the set \({\mathcal {A}}\subseteq \mathbf{R}\) of points t at which x is analytic; in general \({\mathcal {A}}\) is a proper subset of \(\mathbf{R}\), although x is \(C^\infty \) everywhere. Among other results, we obtain conditions under which the complement \({\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}\) of \({\mathcal {A}}\) is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map \(t\rightarrow \eta (t)\).

Keywords

Integral operator Spectral radius Delay-differential equation Variable delay Analytic solution Generalized Cantor set 

Mathematics Subject Classification

Primary 26E05 26E10 37E10 34K13 47B65 Secondary 26A18 34K99 47G10 

Notes

References

  1. 1.
    Hartung, F., Krisztin, T., Walther, H.-O., Wu, J.: Functional differential equations with state-dependent delays: theory and applications. In: Handbook of Differential Equations: Ordinary Differential Equations, vol. III, pp. 435–445, Handb. Differ. Equ., Elsevier, Amsterdam (2006)Google Scholar
  2. 2.
    Krisztin, T.: Analyticity of solutions of differential equations with a threshold delay. In: Recent Advances in Delay Differential and Difference Equations (Balatonfüred, 2013), pp. 173–180, Springer Proc. Math. Stat., vol. 94. Springer, Cham (2014) (and earlier private communication)Google Scholar
  3. 3.
    Lemmens, B., Nussbaum, R.D.: Continuity of the cone spectral radius. Proc. Am. Math. Soc. 141, 2741–2754 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Mallet-Paret, J., Nussbaum, R.D.: Boundary layer phenomena for differential-delay equations with state-dependent time lags. I. Arch. Ration. Mech. Anal. 120, 99–146 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Mallet-Paret, J., Nussbaum, R.D.: Boundary layer phenomena for differential-delay equations with state dependent time lags. II. J. Reine Angew. Math. 477, 129–197 (1996)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Mallet-Paret, J., Nussbaum, R.D.: Boundary layer phenomena for differential-delay equations with state-dependent time lags. III. J. Differ. Equ. 189, 640–692 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mallet-Paret, J., Nussbaum, R.D.: Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations. J. Differ. Equ. 250, 4037–4084 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mallet-Paret, J., Nussbaum, R.D.: Analyticity and nonanalyticity of solutions of delay-differential equations. SIAM J. Math. Anal. 46, 2468–2500 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mallet-Paret, J., Nussbaum, R.D.: Analytic solutions of delay-differential equations (in preparation)Google Scholar
  10. 10.
    Mallet-Paret, J., Nussbaum, R.D.: Asymptotic homogenization for delay-differential equations and a question of analyticity (submitted)Google Scholar
  11. 11.
    Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287–289 (1977)CrossRefzbMATHGoogle Scholar
  12. 12.
    Milnor, J.: Dynamics in One Complex Variable. Annals of Mathematics Studies, vol. 160, 3rd edn. Princeton University Press, Princeton (2006)Google Scholar
  13. 13.
    Nussbaum, R.D.: Periodic solutions of analytic functional differential equations are analytic. Mich. Math. J. 20, 249–255 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Smith, H.L., Kuang, Y.: Periodic solutions of differential delay equations with threshold-type delays. In: Oscillation and Dynamics in Delay Equations (San Francisco, CA, 1991), pp. 153–176, Contemp. Math., vol. 129. Amer. Math. Soc., Providence (1992)Google Scholar
  15. 15.
    Walther, H.-O.: On a model for soft landing with state-dependent delay. J. Dyn. Differ. Equ. 19, 593–622 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

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