Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

AbstractLetU=(U(t, s))t≥s≥O be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsGO,GX andIX on certain spaces ofX-valued continuous functions connected with the integral equation $$u(t) = U(t,s)u(s) + \int_s^t {U(t,\xi )f(\xi )d\xi }$$ , and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofGO,GX andIX, respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.

[1]  田辺 広城,et al.  Equations of evolution , 1979 .

[2]  George R. Sell,et al.  Dichotomies for linear evolutionary equations in Banach spaces , 1994 .

[3]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[4]  N. Minh.  On the proof of characterizations of the exponential dichotomy , 1999 .

[5]  R. Datko,et al.  Uniform Asymptotic Stability of Evolutionary Processes in a Banach Space , 1972 .

[6]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

[7]  Yuri Latushkin,et al.  Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators , 1995 .

[8]  V. V. Zhikov,et al.  Almost Periodic Functions and Differential Equations , 1983 .

[9]  Roland Schnaubelt,et al.  The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions , 1996 .

[10]  J. J. Schaffer,et al.  Linear differential equations and function spaces , 1966 .

[11]  M. A. Kaashoek,et al.  Invertibility and dichotomy of differential operators on a half-line , 1993 .

[12]  Kenneth J. Palmer,et al.  Exponential dichotomies and Fredholm operators , 1988 .

[13]  S. Montgomery-Smith,et al.  Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces , 1993 .

[14]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[15]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[16]  Israel Gohberg,et al.  Dichotomy of systems and invertibility of linear ordinary differential operators , 1992 .

[17]  J. Neerven,et al.  Characterization of Exponential Stability of a Semigroup of Operators in Terms of Its Action by Convolution on Vector-Valued Function Spaces overR+ , 1996 .

[18]  Semigroups and stability of nonautonomous differential equations in Banach spaces , 1994 .

[19]  C. Batty,et al.  Stability of individual elements under one-parameter semigroups , 1990 .

[20]  Weinian Zhang.  The Fredholm Alternative and Exponential Dichotomies for Parabolic Equations , 1995 .

[21]  M. Kreĭn,et al.  Stability of Solutions of Differential Equations in Banach Spaces , 1974 .

[22]  René T. Rau,et al.  Hyperbolic evolution semigroups on vector valued function spaces , 1994 .

[23]  T. Randolph,et al.  Exponential Dichotomy and Mild Solutions of Nonautonomous Equations in Banach Spaces , 1998 .

[24]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[25]  Auf der Morgenstelle.  Exponential Bounds and Hyperbolicity of Evolution Families , 1996 .

[26]  Yuri Latushkin,et al.  EVOLUTIONARY SEMIGROUPS AND DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON LOCALLY COMPACT SPACES WITH BANACH FIBERS , 1996 .