Proceedings of the American Mathematical Society

Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces

Author(s): Shui-Nee Chow; Hugo Leiva
Journal: Proc. Amer. Math. Soc. 124 (1996), 1071-1081.
MSC (1991): Primary 34G10; Secondary 35B40
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Abstract: In this paper we introduce a concept of exponential dichotomy for linear skew-product semiflows (LSPS) in infinite dimensional Banach spaces, which is an extension of the classical concept of exponential dichotomy for time dependent linear differential equations in Banach spaces. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães in recent years is stronger than this one, but they are equivalent under suitable conditions. Using this concept we where able to find a formula for all the bounded negative continuations. After that, we characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also, this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS.

1.: S. N. Chow and H. Leiva, Dynamical spectrum for time dependent linear systems in banach spaces, Japan J. Indust. Appl. Math. 11 (1994), 379-415. MR 95i:34106
2.: S. N. Chow and H. Leiva, Dynamical spectrum for skew-product flow in banach spaces, Boundary Problems for Functional Differential Equations, World Sci. Publ., Singapore, 1995, pp 85-105.
3.: S. N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in banach spaces, J. Differential Equations 120 (1995), 429--477. CMP 95:17
4.: W. A. Coppel, Dichotomies in stability theory, Lect. Notes in Math, vol. 629, Springer-Verlag, New York, 1978. MR 58:1332
5.: J. L. Daleckii and M. G. Krein, Stability of solutions of differential equations in Banach space, Transl. Math. Monographs, vol. 43, Amer. Math. Soc., Providence, RI, 1974. MR 50:5126
6.: J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys and Monographs, vol. 25, Amer. Soc., Providence, R.I., 1988. MR 89g:58059
7.: D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, New York, 1981. MR 83j:35084
8.: N. Levinson, The asymptotic behavior of system of linear differential equations, Amer. J. Math. vol. 68, pp. 1--6, 1946. MR 7:381f
9.: X. B. Lin , Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations 63 (1986), 227--254. MR 87j:34138
10.: X. B. Lin, Exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator-prey model, J. Differential Equations 108 (1994), 36--63. MR 95c:35139
11.: L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, in Dynamics Of Infinite Dimensional Systems, NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 37, Springer Verlag, New York, 1987, pp. 161--168. CMP 20:06
12.: J. L. Massera and J. J. Schäffer, Linear differential equations and function spaces , Academic Press, New York, 1966. MR 35:3197
13.: K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, vol. 55, pp. 225--256, 1984. MR 86d:58088
14.: O. Perron, Die stabilit[??]atsfrage bei differentialgleichungen, Math. Z vol. 32, pp. 703--728, 1930.
15.: R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splitting for linear differential systems I, II, III J. Differential Equations. 15 (1974), 429--458, 22 (1976), 478--496, 497--525. MR 49:6209
16.: R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 (1994), 17--67. CMP 95:01

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34G10, 35B40

Shui-Nee Chow
Affiliation: CDSNS Georgia Tech, Atlanta, Georgia 30332
Email: chow@math.gatech.edu

Hugo Leiva
Affiliation: CDSNS Georgia Tech, Atlanta, Georgia 30332 and ULA-Venezuela
Email: leiva@math.gatech.edu

DOI: 10.1090/S0002-9939-96-03433-8
PII: S 0002-9939(96)03433-8
Keywords: Skew-product semiflow, exponential dichotomy, stable and unstable manifolds
Received by editor(s): April 14, 1994
Additional Notes: This research was partially supported by NSF grant DMS-9306265.
Communicated by: Hal L. Smith
Copyright of article: Copyright 1996, American Mathematical Society

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