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:
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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.

[2]

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

[3]

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

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[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.

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.

[2]

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

[3]

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

[4]

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

[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.

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[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.

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