On the homotopy index for infinite-dimensional semiflows

Rybakowski, Krzysztof P.

doi:10.1090/S0002-9947-1982-0637695-7

On the homotopy index for infinite-dimensional semiflows
HTML articles powered by AMS MathViewer

by Krzysztof P. Rybakowski PDF

Trans. Amer. Math. Soc. 269 (1982), 351-382 Request permission

Abstract:

In this paper we consider semiflows whose solution operator is eventually a conditional $\alpha$-contraction. Such semiflows include solutions of retarded and neutral functional differential equations, of parabolic and certain other classes of partial differential equations. We prove existence of (nonsmooth) isolating blocks and index pairs for such semiflows, via the construction of special Lyapunov functionals. We show that index pairs enjoy all the properties needed to define the notion of a homotopy index, thus generalizing earlier results of Conley [2]. Finally, using a result of Mañé [9], we prove that, under additional smoothness assumptions on the semiflow, the homotopy index is essentially a finite-dimensional concept. This gives a formal justification of the applicability of Ważewski’s Principle to infinite-dimensional problems. Several examples illustrate the theory.

References

Richard C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330–352. MR 336763, DOI 10.1016/0022-0396(72)90036-8
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61. MR 279830, DOI 10.1090/S0002-9947-1971-0279830-1
Klaus Deimling, Nichtlineare Gleichungen und Abbildungsgrade, Hochschultext [University Textbooks], Springer-Verlag, Berlin-New York, 1974 (German). MR 0500322
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721

Geometric theory of semilinear parabolic equations

John Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations 22 (1976), no. 2, 331–348. MR 423399, DOI 10.1016/0022-0396(76)90032-2
Ricardo Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 230–242. MR 654892
John Montgomery, Cohomology of isolated invariant sets under perturbation, J. Differential Equations 13 (1973), 257–299. MR 334173, DOI 10.1016/0022-0396(73)90018-1
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
F. Wesley Wilson Jr., Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413–428. MR 251747, DOI 10.1090/S0002-9947-1969-0251747-9
F. Wesley Wilson Jr. and James A. Yorke, Lyapunov functions and isolating blocks, J. Differential Equations 13 (1973), 106–123. MR 385251, DOI 10.1016/0022-0396(73)90034-X
N. P. Bhatia and O. Hájek, Local semi-dynamical systems, Lecture Notes in Mathematics, Vol. 90, Springer-Verlag, Berlin-New York, 1969. MR 0251328
Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
J. Donald Monk, Introduction to set theory, McGraw-Hill Book Co., New York-London-Sydney, 1969. MR 0286668