NORMAL EIGENVALUES IN EVOLUTIONARY PROCESS

Title & Authors
NORMAL EIGENVALUES IN EVOLUTIONARY PROCESS
Kim, Dohan; Miyazaki, Rinko; Naito, Toshiki; Shin, Jong Son;

Abstract
Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a $\small{C_0}$-semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on $\small{P_{\tau}(X)}$ associated with the $\small{{\tau}}$-periodic evolutionary process on a Banach space X, where $\small{P_{\tau}(X)}$ stands for the space of all $\small{{\tau}}$-periodic continuous functions mapping $\small{{\mathbb{R}}}$ to X.
Keywords
$\small{C_0}$-semigroup;evolution semigroup;monodromy operator;normal eigenvalue;order of pole;ascent;
Language
English
Cited by
References
1.
F. E. Browder, On the spectral theory of elliptic differential operators I, Math. Ann. 142 (1961), 22-130.

2.
K.-J. Engel and R. Nagel, One-Parameter Semigroups of Linear Evolution Equations, Springer, 1999.

3.
J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer, 1993.

4.
E. Hille and R. S. Phillip, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R. I., 1957.

5.
R. Miyazaki, D. Kim, T. Naito, and J. S. Shin, Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems, J. Differential Equations 257 (2014), no. 11, 4214-4247.

6.
R. Miyazaki, D. Kim, T. Naito, and J. S. Shin, Generalized eigenspaces of generators of evolution semigroups, to appear in J. Math. Anal. Appl..

7.
R. Miyazaki, D. Kim, T. Naito, and J. S. Shin, Solutions of higher order inhomogeneous periodic evolutionary process, in preparation.

8.
T. Naito and N. V. Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Differential Equations 152 (1999), no. 2, 358-376.

9.
T. Naito and J. Shin, On solution semigroups of functional differential equations, RIMS Kokyuuroku 940 (1996), 161-175.

10.
J. S. Shin and T. Naito, Representations of solutions, translation formulae and as-ymptotic behavior in discrete linear systems and periodic continuous linear systems, Hiroshima Math. J. 44 (2014), no. 1, 75-126.

11.
J. S. Shin, T. Naito, and N. V. Minh, On stability of solutions in linear autonomous functional differential equations, Funkcial. Ekvac. 43 (2000), no. 2, 323-337.

12.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.

13.
A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley-Sons. Inc., 1980.

14.
C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 394-418.

15.
G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Pure and Appl. Math. Vol.89, Dekker, 1985.