In this chapter we consider various extensions of the results in the former chapters. In particular, we develop a general approach to the problem of constructing pairs of Banach spaces whose admissibility property can be used to characterize an exponential dichotomy. This generalizes and unifies some of the results in the former chapters. Moreover, we discuss what we call Pliss type theorems. These results deal with a weaker form of admissibility on the line not requiring the uniqueness condition and guarantee the existence of exponential dichotomies on both the positive and negative half-lines. Finally, we introduce the more general notion of a nonuniform exponential dichotomy and again we characterize it in terms of an appropriate admissibility property also for maps and flows.
Bitte loggen Sie sich ein, um Zugang zu diesem Inhalt zu erhalten
L. Barreira, Y. Pesin,
Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and its Applications, vol. 115 (Cambridge University Press, Cambridge, 2007)
L. Barreira, C. Valls, Admissibility in the strong and weak senses, Preprint IST, 2017
L. Barreira, D. Dragičević, C. Valls, Nonuniform hyperbolicity and admissibility. Adv. Nonlinear Stud.
14, 791–811 (2014)
L. Barreira, D. Dragičević, C. Valls, Strong and weak (
Lq)-admissibility. Bull. Sci. Math.
138, 721–741 (2014)
L. Barreira, D. Dragičević, C. Valls, Nonuniform hyperbolicity and one-sided admissibility. Rend. Lincei Mat. Appl.
27, 235–247 (2016)
L. Barreira, D. Dragičević, C. Valls, A version of a theorem of Pliss for nonuniform and noninvertible dichotomies. Proc. R. Soc. Edinburgh Sect. A.
147, 225–243 (2017)
L. Barreira, D. Dragičević, C. Valls, Admissibility on the half line for evolution families. J. Anal. Math.
132, 157–176 (2017)
A. Ben-Artzi, I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators, in
Time-Variant Systems and Interpolation. Operator Theory: Advances and Applications, vol. 56 (Birkhäuser, Basel, 1992), pp. 90–119
A. Ben-Artzi, I. Gohberg, M. Kaashoek, Invertibility and dichotomy of differential operators on a half-line. J. Dyn. Differ. Equ.
5, 1–36 (1993)
D. Bylov, R. Vinograd, D. Grobman, V. Nemyckii,
Theory of Lyapunov Exponents and Its Application to Problems of Stability (Izdat. “Nauka”, Moscow, 1966) [in Russian]
C. Chicone, Y. Latushkin,
Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70 (American Mathematical Society, Providence, 1999)
Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629 (Springer, New York, 1981)
J. Dalec’kiı̆, M. Kreı̆n,
Stability of Solutions of Differential Equations in Banach Space. Translations of Mathematical Monographs, vol. 43 (American Mathematical Society, Providence, RI, 1974)
D. Dragičević, Admissibility, a general type of Lipschitz shadowing and structural stability. Commun. Pure Appl. Anal.
14, 861–880 (2015)
Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840 (Springer, Berlin, 1981)
N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal.
235, 330–354 (2006)
Y. Latushkin, A. Pogan, R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations. J. Oper. Theory
58, 387–414 (2007)
The General Problem of the Stability of Motion (Taylor & Francis, Ltd, London, 1992)
J. Massera, J. Schäffer, Linear differential equations and functional analysis. I. Ann. of Math. (2)
67, 517–573 (1958)
J. Massera, J. Schäffer,
Linear Differential Equations and Function Spaces. Pure and Applied Mathematics, vol. 21 (Academic, New York, 1966)
K. Palmer, Exponential dichotomies and Fredholm operators. Proc. Am. Math. Soc.
104, 149–156 (1988)
O. Perron, Die Ordnungszahlen linearer Differentialgleichungssyteme. Math. Z.
31, 748–766 (1930)
Y. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv.
40, 1261–1305 (1976)
Y. Pesin, Geodesic flows on closed Riemannian manifolds without focal points. Math. USSR-Izv.
11, 1195–1228 (1977)
Shadowing in Dynamical Systems. Lecture Notes Mathematics, vol. 1706 (Springer, Berlin, 1999)
S. Pilyugin, Generalizations of the notion of hyperbolicity. J. Differ. Equ. Appl.
12, 271–282 (2006)
S. Pilyugin, S. Tikhomirov, Lipschitz shadowing implies structural stability. Nonlinearity
23, 2509–2515 (2010)
V. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations, in
Problems of the Asymptotic Theory of Nonlinear Oscillations (Naukova Dumka, Kiev, 1977), pp. 168–173 [in Russian]
P. Preda, M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces. Bull. Aust. Math. Soc.
27, 31–52 (1983)
C. Preda, O. Onofrei, Discrete Schäffer spaces and exponential dichotomy for evolution families. Monatsh. Math.
185, 507–523 (2018)
A. Sasu, Integral equations on function spaces and dichotomy on the real line. Integr. Equ. Oper. Theory
58, 133–152 (2007)
A. Sasu, Exponential dichotomy and dichotomy radius for difference equations. J. Math. Anal. Appl.
344, 906–920 (2008)
A. Sasu, Pairs of function spaces and exponential dichotomy on the real line Adv. Differ. Equ.
2010, 347670, 15 pp. (2010)
A. Sasu, B. Sasu, Integral equations, dichotomy of evolution families on the half-line and applications. Integr. Equ. Oper. Theory
66, 113–140 (2010)
A. Sasu, B. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line. Discrete Contin. Dyn. Syst.
33, 3057–3084 (2013)
D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory. Discrete Contin. Dyn. Syst.
33, 4187–4205 (2013)
N. Van Minh, F. Räbiger, R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integr. Equ. Oper. Theory
32, 332–353 (1998)
W. Zhang, The Fredholm alternative and exponential dichotomies for parabolic equations. J. Math. Anal. Appl.
191, 180–201 (1985)