## Journal of the Mathematical Society of Japan

### Robustness of noninvertible dichotomies

#### Abstract

We establish the robustness of exponential dichotomies for evolution families of linear operators in a Banach space, in the sense that the existence of an exponential dichotomy persists under sufficiently small linear perturbations. We note that the evolution families may come from nonautonomous differential equations involving unbounded operators. We also consider the general case of a noninvertible dynamics, thus including several classes of functional equations and partial differential equations. Moreover, we consider the general cases of nonuniform exponential dichotomies and of dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ for some function $\rho(t)$.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 1 (2015), 293-317.

Dates
First available in Project Euclid: 22 January 2015

https://projecteuclid.org/euclid.jmsj/1421936554

Digital Object Identifier
doi:10.2969/jmsj/06710293

Mathematical Reviews number (MathSciNet)
MR3304023

Zentralblatt MATH identifier
1347.34090

#### Citation

BARREIRA, Luis; VALLS, Claudia. Robustness of noninvertible dichotomies. J. Math. Soc. Japan 67 (2015), no. 1, 293--317. doi:10.2969/jmsj/06710293. https://projecteuclid.org/euclid.jmsj/1421936554

#### References

• L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia Math. Appl., 115, Cambridge University Press, 2007.
• L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst., 22 (2008), 509–528.
• L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, 244 (2008), 2407–2447.
• L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Math., 1926, Springer-Verlag, 2008.
• L. Barreira and C. Valls, Robustness of general dichotomies, J. Funct. Anal., 257 (2009), 464–484.
• L. Barreira and C. Valls, Robustness via Lyapunov functions, J. Differential Equations, 246 (2009), 2891–2907.
• L. Barreira and C. Valls, Nonuniformly hyperbolic cocycles: admissibility and robustness, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 545–564.
• S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. Differential Equations, 120 (1995), 429–477.
• W. A. Coppel, Dichotomies and reducibility, J. Differential Equations, 3 (1967), 500–521.
• W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., 629, Springer-Verlag, 1978.
• Ju. L. Dalec$'$kiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, 43, Amer. Math. Soc., Providence, RI, 1974.
• J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., 25, Amer. Math. Soc., Providence, RI, 1988.
• D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, 1981.
• J. L. Massera and J. J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2), 67 (1958), 517–573.
• J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure Appl. Math. (Amst.), 21, Academic Press, 1966.
• R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal., 31 (1998), 559–571.
• O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703–728.
• V. A. Pliss and G. R. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 11 (1999), 471–513.
• L. H. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl., 314 (2006), 436–454.
• F. Räbiger, R. Schnaubelt, A. Rhandi and J. Voigt, Non-autonomous Miyadera perturbations, Differential Integral Equations, 13 (2000), 341–368.
• G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., 143, Springer-Verlag, 2002.
• N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332–353.