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Analytic contractive vector fields in Fréchet spaces

Published online by Cambridge University Press:  17 April 2009

Duong Minh Duc
Duong Minh Duc
Affiliation:
Faculty of Mathematics, University of Hochiminh City, Box 923, Hochiminh City, S.R. Vietnam.
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Abstract

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Using degree theory we establish the inverse mapping theorem and a Sard-Smale theorem for analytic contractive vector fields in Fréchet spaces. Applying these theorems we obtain Cronin's theorem relating topological degree to the number of zeros of analytic contractive vector fields.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 27 , Issue 2 , April 1983 , pp. 165 - 183
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Browder, Felix E. and Gupta, Chaitan P., “Topological degree and nonlinear mappings of analytic type in Banach spaces”, J. Math. Anal. Appl. 26 (1969), 390–402.CrossRefGoogle Scholar
[2]Cronin, Jane, “Analytic functional mappings”, Ann. of Math. (2) 58 (1953), 175–181.CrossRefGoogle Scholar
[3]Duc, Duong Minh, “Differential calculus in Fréchet spaces”, Bull. Austral. Math. Soc. 24 (1981), 93–122.CrossRefGoogle Scholar
[4]Duc, Duong Minh, “Measure of noncompactness and spectral theory”, Math. Nachr. (to appear).Google Scholar
[5]Красноселъский, M.A., Забрейко, P.P. [Krasnosel'skiĭ, M.A., Zabreĭko, P.P.], Геомеmрuческuе меmо∂ы нелuнеŭноƨо ɑнɑлuзɑ [Geometric methods of nonlinear analysis] (Izdat. “Nauka”, Moscow, 1975).Google Scholar
[6]Sadovskii, B.N., “Limit-compact and condensing operators”, Russian Math. Surveys 27 (1972), no. 1, 85–155.CrossRefGoogle Scholar
[7]Sard, Arthur, “The measure of the critical values of differentiate maps”, Bull. Amer. Math. Soc. 48 (1942), 883–890.CrossRefGoogle Scholar
[8]Schwartz, J., “Compact analytical mappings of B-spaces and a theorem of Jane Cronin”, Comm. Pure Appl. Math. 16 (1963), 253–260.CrossRefGoogle Scholar
[9]Smale, S., “An infinite dimensional version of Sard's theorem”, Amer. J. Math. 87 (1965), 861–866.CrossRefGoogle Scholar
[10]Stuart, C.A. and Toland, J.F., “The fixed point index of a linear k-set contraction”, J. London Math. Soc. (2) 6 (1972/3), 317–320.Google Scholar