Abstract
LetU=(U(t, s)) t≥s≥O be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG O,G X andI X 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 ofG O,G X andI X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.
Similar content being viewed by others
References
[AuM] Aulbach B., Nguyen Van Minh,Semigroups and exponential stability of nonautonomous linear differential equations on the half-line, (R.P. Agrawal Ed.), Dynamical Systems and Aplications, World Scientific, Singapore 1995, pp. 45–61.
[BaV] Batty C., Vũ Quoc Phóng,Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), 805–818.
[BeG] Ben-Artzi A., Gohberg I.,Dichotomies of systems and invertibility of linear ordinary differential operators, Oper. Theory Adv. Appl. 56 (1992), 90–119.
[BGK] Ben-Artzi A., Gohberg I., Kaashoek M.A.,Invertibility and dichotomy of differential operators on the half-line, J. Dyn. Differ. Equations 5 (1993), 1–36.
[Bus] Buşe C.,On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces, preprint.
[Cop] Coppel W.A., “Dichotomies in Stability Theory”, Springer-Verlag, Berlin Heidelberg, New York, 1978.
[DaK] Daleckii Ju. L., Krein M.G., “Stability of Solutions of Differential Equations in Banach Spaces”, Amer. Math. Soc., Providence RI, 1974.
[Dat] Datko R.,Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal. 3 (1972), 428–445.
[Hen] Henry D., “Geometric Theory of Semilinear Parabolic Equations”, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
[Kat] Kato, T., “Perturbation Theory for Linear Operators”, Springer-Verlag, Berlin, Heidelberg, New York, 1966.
[LaM] Latushkin Y., Montgomery-Smith S.,Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal. 127 (1995), 173–197.
[LMR1] Latushkin Y., Montgomery-Smith S., Randolph T.,Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibers, J. Diff. Eq. 125 (1996), 73–116.
[LMR2] Latushkin Y., Montgomery-Smith S., Randolph T.,Evolution semigroups and robust stability of evolution operators on Banach spaces, preprint.
[LaR] Latushkin Y., Randolph T.,Dichotomy of differential equations on Banach spaces and an algebra of weighted composition operators, Integral Equations Oper. Theory 23 (1995), 472–500.
[LRS] Latushkin Y., Randolph T., Schnaubelt R.,Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, to appear in J. Dynamics Diff. Equations.
[LeZ] Levitan B.M., Zhikov V.V., “Almost Periodic Functions and Differential Equations”, Cambridge Univ. Press, 1982.
[MaS] Massera J.J., Schäffer J.J., “Linear Differential Equations and Function Spaces”, Academic Press, New York, 1966.
[Mi1] Nguyen Van Minh,Semigroups and stability of nonautonomous differential equations in Banach spaces, Trans. Amer. Math. Soc. 345 (1994), 223–242.
[Mi2] Nguyen Van Minh,On the proof of characterizations of the exponential dichotomy, preprint.
[Nee] van Neerven, J.M.A.M.,Characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector-valued function spaces over ℝ+, J. Diff. Eq. 124 (1996), 324–342.
[Nic] Nickel G.,On evolution semigroups and well-posedness of non-autonomous Cauchy problems, PhD thesis, Tübingen, 1996.
[Pal] Palmer K.J.,Exponential dichotomy and Fredholm operators, Proc. Amer. Math. Soc. 104 (1988), 149–156.
[Paz] Pazy A., “Semigroups of Linear Operators and Applications to Partial Differential Equations”, Springer-Verlag, Berlin, Heidelberg, New York, 1983.
[RRS] Räbiger F., Rhandi A., Schnaubelt R., Voigt J.,Non-autonomous Miyadera perturbations, preprint.
[RS1] Räbiger F., Schnaubelt R.,The spectral mapping theorem for evolution semigroups on spaces of vector valued functions, Semigroup Forum 48 (1996), 225–239.
[RS2] Räbiger F., Schnaubelt R.,Absorption evolution families with applications to non-autonomous diffusion processes, Tübinger Berichte zur Funktionalanalysis 5 (1995/96), 335–354.
[Rau] Rau R.,Hyperbolic evolution semigroups onvector valued function spaces, Semigroup Forum 48 (1994), 107–118.
[SaS] Sacker R., Sell G.,Dichotomies for linear evolutionary equations in Banach spaces, J. Diff. Eq. 113 (1994), 17–67.
[Sch] Schnaubelt R.,Exponential bounds and hyperbolicity of evolution families, PhD thesis, Tübingen, 1996.
[Tan] Tanabe H., “Equations of Evolution”, Pitman, London, 1979.
[Vũ] Vũ Quôc Phóng,On the exponential stability and dichotomy of C o -semigroups, preprint.
[Zha] Zhang W.,The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl. 191 (1995), 180–201.
[Zhi] Zhikov V.V.,On the theory of the admissibility of pairs of function spaces, Soviet Math. Dokl. 13 (1972), 1108–1111.
Author information
Authors and Affiliations
Additional information
This work was done while the first author was visiting the Department of Mathematics of the University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in Tübingen for their kind hospitality and constant encouragement.
Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Nguyen Van Minh, Räbiger, F. & Schnaubelt, R. Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integr equ oper theory 32, 332–353 (1998). https://doi.org/10.1007/BF01203774
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01203774