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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1199–1219 | Cite as

On Exponential Decay and the Markus–Yamabe Conjecture in Infinite Dimensions with Applications to the Cima System

  • Hildebrando M. Rodrigues
  • Marco A. Teixeira
  • Marcio Gameiro
Article
  • 136 Downloads

Abstract

In this work we extend to infinite dimensions a previous result of Lyascenko given in finite dimensions for a class of Lipschitz functions. In particular, this extension substitutes the class of Lipschitz functions by a class of Hölder functions. Then we consider applications of these results to analyze its relation with the Markus–Yamabe Conjecture in infinite dimensions. We discuss in detail a celebrated example of Cima et al. ( Adv Math 131(2): 453–457, 1997) of a nonlinear system in dimension three whose Jacobian has a unique eigenvalue \(-1\) of multiplicity 3, and yet an explicit unbounded solution as \(t \rightarrow \infty \) exists. We also present explicit solutions of the same equation that tends to 0 as \(t \rightarrow \infty \). Then we look at this conjecture by considering not the properties of the whole system, but instead the properties of some solutions. Finally we present an application in an infinite dimensional Hilbert space, where we use different techniques to study the local and global asymptotic stabilities.

Keywords

Markus-Yamabe conjecture in infinite dimensions Extension of a result of Lyascenko Cima system Global asymptotic stability 

Notes

Acknowledgements

Hildebrando M. Rodrigues was partially supported by FAPESP Grant 2015/19165-5. Marco A. Teixeira was partially supported by FAPESP Grant 2012/18780-0 and CNPq Grant 300596/2009-0. Marcio Gameiro was partially supported by FAPESP Grants 2013/07460-7, 2016/08704-5, and 2016/21032-6, and by CNPq Grants 305860/2013-5 and 310740/2016-9, Brazil.

References

  1. 1.
    Cima, A., van den Essen, A., Gasull, A., Hubbers, E., Mañosas, F.: A polynomial counterexample to the Markus–Yamabe conjecture. Adv. Math. 131(2), 453–457 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Rodrigues, H.M.: Invariância para sistemas de equações diferenciais com retardamento e aplicações. Universidade de São Paulo, Tese de Mestrado (1970)Google Scholar
  3. 3.
    Rodrigues, H.M., Solà-Morales, J.: Linearization of class \(C^1\) for contractions on Banach spaces. J. Differ. Equ. 201(2), 351–382 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Rodrigues, H.M., Solà-Morales, J.: On the Hartman-Grobman theorem with parameters. J. Dyn. Differ. Equ. 22(3), 473–489 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Rodrigues, H.M., Solà-Morales, J.: Invertible contractions and asymptotically stable ODE’s that are not \(C^1\)-linearizable. J. Dyn. Differ. Equ. 18(4), 961–974 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Rodrigues, H.M., Solà-Morales, J.: Smooth linearization for a saddle on Banach spaces. J. Dyn. Differ. Equ. 16(3), 767–793 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rodrigues, H.M., Ruas-Filho, J.G.: Evolution equations: dichotomies and the Fredholm alternative for bounded solutions. J. Differ. Equ. 119(2), 263–283 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rodrigues, H.M., Silveira, M.: Properties of bounded solutions of linear and nonlinear evolution equations: homoclinics of a beam equation. J. Differ. Equ. 70(3), 403–440 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rodrigues, H.M., Silveira, M.: On the relationship between exponential dichotomies and the Fredholm alternative. J. Differ. Equ. 73(1), 78–81 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kloeden, P.E., Rodrigues, H.M.: Dynamics of a class of ODEs more general than almost periodic. Nonlinear Anal. 74(7), 2695–2719 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. D. C. Heath and Co., Boston (1965)zbMATHGoogle Scholar
  12. 12.
    Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, Vol. 629. Springer, Berlin (1978)Google Scholar
  13. 13.
    Markus, L., Yamabe, H.: Global stability criteria for differential systems. Osaka Math. J. 12, 305–317 (1960)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Neves, A.F., Ribeiro, H.S., Lopes, O.: On the spectrum of evolution operators generated by hyperbolic systems. J. Funct. Anal. 67(3), 320–344 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaashoek, M.A., Lunel, S.M.V.: An integrability condition on the resolvent for hyperbolicity of the semigroup. J. Differ. Equ. 112(2), 374–406 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Henry, Daniel: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics. vol. 840. Springer, Berlin (1981)Google Scholar
  17. 17.
    Vinograd, R.E.: The inadequacy of the method of characteristic exponents when applied to non-linear equations. Dokl. Akad. Nauk SSSR (N.S.) 114, 239–240 (1957)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Silva, Elves A.B., Teixeira, Marco A.: Global injectivity and asymptotic stability via minimax method. In: Progress in Nonlinear Analysis (Tianjin, 1999), Nankai Ser. Pure Appl. Math. Theoret. Phys., vol. 6, pp. 339–358. World Sci. Publ., River Edge (2000)Google Scholar
  19. 19.
    Silva, E.A.B., Teixeira, M.A.: Global asymptotic stability on Euclidean spaces. Nonlinear Anal. 50, 91–114 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bernat, J., Llibre, J.: Counterexample to Kalman and Markus–Yamabe conjectures in dimension larger than \(3\). Dyn. Contin. Discrete Impuls. Syst. 2(3), 337–379 (1996)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hartman, P., Olech, C.: On global asymptotic stability of solutions of differential equations. Trans. Am. Math. Soc. 104, 154–178 (1962)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gasull, A., Llibre, J., Sotomayor, J.: Global asymptotic stability of differential equations in the plane. J. Differ. Equ. 91(2), 327–335 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gutiérrez, C.: A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 12(6), 627–671 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976). Grundlehren der Mathematischen Wissenschaften, Band 132zbMATHGoogle Scholar
  25. 25.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Daleckiĭ, Ju. L., Kreĭn, M.G.: Stability of solutions of differential equations in Banach space. American Mathematical Society, Providence, R.I., Translated from the Russian by S. Smith, Translations of Mathematical Monographs, vol. 43 (1974)Google Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil
  2. 2.Instituto de Matemática Estatística e Ciência da Computação, IMECC-UNICAMPUniversidade Estadual de CampinasCampinasBrazil

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