Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces

Journal of Dynamics and Differential Equations

Volumn 10, Issue 3, 1998, Pages 489-510

  Latushkin, Y ^a   Randolph, T ^b   Schnaubelt, R ^c  

^a UNIVERSITY OF MISSOURI   (United States)

^b MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY   (United States)

^c UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Central manifolds; Exponential dichotomy; Mild solutions

Indexed keywords

EID: 54649083842     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1023/A:1022609414870     Document Type: Article

References (48)

1. Ben-Artzi, A., and Gohberg, I. (1992). Dichotomy of systems and invertibility of linear ordinary differential operators. Oper. Theory Adv. Appl. 56, 90-119.
2. Chicone, C., and Swanson, R. (1981). Spectral theory for linearization of dynamical systems. J. Diff. Eq. 40, 155-167.
3. Chow, S.-N., and Leiva, H. (1994). Dynamical spectrum for time dependent linear systems in Banach spaces. Japan J. Ind. Appl. Math. 11, 379-415.
4. Chow, S.-N., and Leiva, H. (1995). Existence and roughness of the exponential dichotomy for skew product semiflow in Banach space. J. Diff. Eq. 120, 429-477.
5. Curtain, R., and Pritchard, A. J. (1978). Infinite Dimensional Linear System Theory. Lecture Notes in Control and Information Sciences, Vol. 8, Springer-Verlag, Berlin, Heidelberg, New York.
6. Daleckij, J., and Krein, M. (1974). Stability of Solutions of Differential Equations in Banach Space. Am. Math. Soc., Providence, RI.
7. p regularity for abstract differential equations. In Functional Analysis and Related Topics, Lecture Notes Math., No. 1540, Springer-Verlag, Berlin, pp. 25-38.
8. Henry, D. (1981). Geometric Theory of Nonlinear Parabolic Equations, Lecture Notes Math., No. 840, Springer-Verlag, Berlin.
9. Howland, J. S. (1974). Stationary scattering theory for time-dependent hamiltonians. Math. Ann. 207, 315-335.
10. Johnson, R. (1980). Analyticity of spectral subbundles. J. Diff. Eqs. 35, 366-387.
11. Johnson, R., Palmer, K., and Sell, G. (1987). Ergodic properties of linear dynamical systems. SIAM J. Math. Anal. 18, 1-33.
12. Kurbatov, V. G. (1990). Lyneinye differentsial'no-rasnostnye uravneniya (Linear differential-difference equations), Voronez University, Voronez.
13. Latushkin, Y., and Montgomery-Smith, S. (1994). Lyapunov theorems for Banach spaces. Bull. Am. Math. Soc. (N.S.) 31(1), 44-49.
14. Latushkin, Y., and Montgomery-Smith, S. (1995). Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Fund. Anal. 127, 173-197.
15. Latushkin, Y., and Randolph, T. (1995). Dichotomy of differential equations on Banach spaces and an algebra of weighted composition operators, Integr. Eq. Oper. Theory 23, 472-500.
16. Latushkin, Y., and Stepin, A. M. (1992). Weighted composition operators and linear extensions of dynamical systems. Russ. Math. Surv. 46(2), 95-165.
17. Latushkin, Y., Montgomery-Smith, S., and Randolph, T. (1996). Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibers. J. Diff. Eq. 125, 73-116.
18. Levitan, B. M., and Zhikov, V. V. (1982). Almost Periodic Functions and Differential Equations, Cambridge University Press.
19. Massera, J., and Schaeffer, J. (1966). Linear Differential Equations and Function Spaces, Academic Press, New York.
20. Mather, J. (1968). Characterization of Anosov diffeomorphisms. Indag. Math. 30, 479-483.
21. Nagel, R. (Ed.) (1984). One Parameters Semigroups of Positive Operators, Lecture Notes Math., No. 1184, Springer-Verlag, Berlin.
22. Nagel, R. (1995). Semigroup methods for non-autonomous Cauchy problems. In G. Ferreyra, G. Ruiz Goldstein, and F. Neubrander (Eds.), Lecture Notes Pure Appl. Math. 168, 301-316.
23. Neidhardt, H. (1981). On abstract linear evolution equations I. Math. Nachr. 103, 283-293.
24. Palmer, K. (1988). Exponential dichotomy and Fredholm operators. Proc. Am. Math. Soc. 104, 149-156.
25. Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York/Berlin.
26. Perron, O. (1930). Die Stabilitatsfrage bei Differentialgleichungen. Math. Z. 32(5), 703-728.
27. 0-semigroups. Trans. Am. Math. Soc. 284(2), 847-857.
28. Räbiger, F., and Schnaubelt, R. (1994). A spectral characterization of exponentially dichotomic and hyperbolic evolution families. Tübinger Berichte Funktionalanal. 3, 204-221.
29. Räbiger, F., and Schnaubelt, R. (1996). The spectral mapping theorem for evolution semi-groups on spaces of vector-valued functions. Semigroups Forum 52, 225-239.
30. Rau, R. (1994a). Hyperbolic evolutionary semigroups on vector-valued function spaces. Semigroup Forum 48, 107-118.
31. Rau, R. (1994b). Hyperbolic evolution groups and dichotomic of evolution families. J. Diff. Eq. 6(2), 335-350.
32. Rodrigues, M. M., and Ruas-Filho, J. G. (1995). Evolution equations: Dichotomies and the Fredholm alternative for bounded solutions. J. Diff. Eq. 119, 263-283.
33. Sacker, R., and Sell, G. (1974, 1976). Existence of dichotomies and invariant splitting for linear differential systems, I, II, III. J. Diff. Eq. 15, 22, 429-458, 478-522.
34. Sacker, R., and Sell, G. (1978). A spectral theory for linear differential systems, J. Diff. Eq. 27, 320-358.
35. Sacker, R., and Sell, G. (1994). Dichotomies for linear evolutionary equations in Banach spaces. J. Diff. Eq. 113, 17-67.
36. Selgrade, J. (1975). Isolated invariant sets for flows on vector bundles. Trans. Amer. Math. Soc. 203, 359-390.
37. Sell, G. (1995). References on dynamical systems. IMA Preprint 1300.
38. Shen, W., and Yi, Y. (1995). Dynamics and almost periodic scalar equations. J. Diff. Eq. 122, 114-136.
39. Shen, W., and Yi, Y. (1995). Asymptotic almost periodicity of scalar parabolic equations with almost periodic time dependence. J. Diff. Eq. 122, 373-397.
40. Shen, W., and Yi, Y. (1995). On minimal sets of scalar parabolic equations with skew product structures. Trans. Am. Math. Soc. 347, 4413-4431.
41. Shen, W., and Yi, Y. (1996). Ergodicity of minimal sets of scalar parabolic equations. J. Dynam. Diff. Eq. 8, 299-323.
42. Shen, W., and Yi, Y. Almost automorphic and almost periodic dynamics in skew product semiflows, Mem. Am. Math. Soc. (in press).
43. Vanderbauwhede, A., and Iooss, G. (1992). Center manifold theory in infinite dimensions. Dynam. Rep. 1 (new. ser.), 125-163.
44. Van Minh, N. (1994). Semigroups and stability of nonautonomous differential equations in Banach spaces. Trans. Am. Math. Soc. 345, 223-241.
45. van Neerven, J. M. A. M. (1996). The Asymptotic Behavior of Semigroups of Linear Operators, Operator Theory Adv. Appl. 88, Birkhauser.
46. Yi, Y. Almost automorphy and almost periodicity. In Almost Automorphic and Almost Periodic Dynamics in Skew Product Semiflows, Mem. Am. Math. Soc. (in press).
47. Yi, Y. (1993). A generalized integral manifold theorem. J. Diff. Eq. 102, 153-187.
48. Zhang, W. (1995). The Fredholm alternative and exponential dichotomies for parabolic equations. J. Math. Anal. Appl. 191, 180-201.