March  2019, 18(2): 845-868. doi: 10.3934/cpaa.2019041

An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation

Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, São Carlos, SP, Brazil

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: The author was partially supported by Grant: 2014/02899-3, São Paulo Research Foundation (FAPESP), Brazil

In this paper we prove versions, in Fréchet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces. To be more specific, here we define a kind of exponential dichotomy in Fréchet spaces, which extends the former one in Banach spaces, establish necessary conditions for its existence and provide sufficient conditions for its stability under perturbation.

We apply the conclusions by providing an example of a semigroup of bounded linear operators, on a Fréchet space, which has this new exponential dichotomy but does not in Banach spaces, namely, $\{e^{mΔ}:\; m∈ \mathbb{N}\}$, where $Δ$ is the Laplace operator on the unbounded domain $\mathbb{R}^{n}\setminus \{0\}$.

Also, we show how these new concepts allow us to study a hyperbolic equilibrium point of a backwards heat equation with nonlinearity involving convolution products, which cannot be obtained from the knowledge of exponential dichotomy in Banach spaces.

Citation: Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041
References:
[1]

E. R. Aragão-CostaT. CaraballoA. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, J. Nonlinearity, 24 (2011), 2099-2117.  doi: 10.2307/2152750.  Google Scholar

[2]

E. R. Aragão-Costa and A. P. Silva, On the generation of groups of bounded linear operators on Fréchet spaces, preprint. Google Scholar

[3]

L. BarreiraD. Dragičević and C. Valls, A Perron-type theorem for nonautonomous difference equations, Nonlinearity, 26 (2013), 855-870.  doi: 10.2307/2152750.  Google Scholar

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L. BarreiraD. Dragičević and C. Valls, Nonuniform spectrum on Banach spaces, Advances in Mathematics, 321 (2017), 547-591.  doi: 10.2307/2152750.  Google Scholar

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A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182 Springer, New York, 2013. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., Springer-Verlag, Berlin, 1978. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[7]

J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Trans. Math. Monographs, vol. 43, Amer. Math. Soc., Providence, 1974. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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G. B. Folland, Real Analysis, Modern Techniques and Their Aplications, 2$^{nd}$ edition, Wiley Intercience, 1999. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

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J. L. Massera and J. J. Schfer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[11]

M. MeganA. L. Sasu and B. Sasu, Theorems of Perron type for uniform exponential stability of linear skew-product semiflows, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 23-43.  doi: 10.2307/2152750.  Google Scholar

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1963. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[13]

O. Perron, Die stabilitätsfrage bei differentialgleichungen, Math. Z., 32 (1930), 703-728.   Google Scholar

[14]

Preda Petre and Mureȿan Raluca, Uniform exponential stability for evolution families on the half-line, J. Nonlinear Sci. Appl., 6 (2013), 68-73.  doi: 10.2307/2152750.  Google Scholar

[15]

W. Rudin, Functional Analysis, 2$^{nd}$ edition, MacGraw-Hill, New York, 1991. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[16]

A. L. Sasu, Exponential instability and complete admissibility for semigroups in Banach spaces, Rend. Sem. Mat. Univ. Politec. Torino, 63 (2005), 141-151.   Google Scholar

[17]

L. Ta, Die stabilitätsfrage bei differenzengleichungen, Acta Math., 63 (1934), 99-141.   Google Scholar

[18]

F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York 1967. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[19]

F. Treves, Study of a model in the theory of complexes of pseudodifferential operators, Ann. of Math. (2) 104 (1976), 269–324. Google Scholar

[20]

H. O. Walther, Semiflows for differential equations with locally bounded delay on solution manifolds in the space $C^{1}((-∞, 0], \mathbb{R}^{n})$, Topol. Methods Nonlinear Anal., 48 (2016), 507-537.   Google Scholar

[21]

H. O. Walther, Local invariant manifolds for delay differential equations with state space in $C^{1}((-∞, 0], \mathbb{R}^{n})$, Electron. J. Qual. Theory Differ. Equ., 2016, Paper no. 85, 29 pp. Google Scholar

show all references

References:
[1]

E. R. Aragão-CostaT. CaraballoA. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, J. Nonlinearity, 24 (2011), 2099-2117.  doi: 10.2307/2152750.  Google Scholar

[2]

E. R. Aragão-Costa and A. P. Silva, On the generation of groups of bounded linear operators on Fréchet spaces, preprint. Google Scholar

[3]

L. BarreiraD. Dragičević and C. Valls, A Perron-type theorem for nonautonomous difference equations, Nonlinearity, 26 (2013), 855-870.  doi: 10.2307/2152750.  Google Scholar

[4]

L. BarreiraD. Dragičević and C. Valls, Nonuniform spectrum on Banach spaces, Advances in Mathematics, 321 (2017), 547-591.  doi: 10.2307/2152750.  Google Scholar

[5]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences 182 Springer, New York, 2013. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[6]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., Springer-Verlag, Berlin, 1978. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[7]

J. L. Daleckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Trans. Math. Monographs, vol. 43, Amer. Math. Soc., Providence, 1974. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[8]

G. B. Folland, Real Analysis, Modern Techniques and Their Aplications, 2$^{nd}$ edition, Wiley Intercience, 1999. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[10]

J. L. Massera and J. J. Schfer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[11]

M. MeganA. L. Sasu and B. Sasu, Theorems of Perron type for uniform exponential stability of linear skew-product semiflows, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 23-43.  doi: 10.2307/2152750.  Google Scholar

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1963. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[13]

O. Perron, Die stabilitätsfrage bei differentialgleichungen, Math. Z., 32 (1930), 703-728.   Google Scholar

[14]

Preda Petre and Mureȿan Raluca, Uniform exponential stability for evolution families on the half-line, J. Nonlinear Sci. Appl., 6 (2013), 68-73.  doi: 10.2307/2152750.  Google Scholar

[15]

W. Rudin, Functional Analysis, 2$^{nd}$ edition, MacGraw-Hill, New York, 1991. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[16]

A. L. Sasu, Exponential instability and complete admissibility for semigroups in Banach spaces, Rend. Sem. Mat. Univ. Politec. Torino, 63 (2005), 141-151.   Google Scholar

[17]

L. Ta, Die stabilitätsfrage bei differenzengleichungen, Acta Math., 63 (1934), 99-141.   Google Scholar

[18]

F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York 1967. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[19]

F. Treves, Study of a model in the theory of complexes of pseudodifferential operators, Ann. of Math. (2) 104 (1976), 269–324. Google Scholar

[20]

H. O. Walther, Semiflows for differential equations with locally bounded delay on solution manifolds in the space $C^{1}((-∞, 0], \mathbb{R}^{n})$, Topol. Methods Nonlinear Anal., 48 (2016), 507-537.   Google Scholar

[21]

H. O. Walther, Local invariant manifolds for delay differential equations with state space in $C^{1}((-∞, 0], \mathbb{R}^{n})$, Electron. J. Qual. Theory Differ. Equ., 2016, Paper no. 85, 29 pp. Google Scholar

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