Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-30T17:09:20.273Z Has data issue: false hasContentIssue false

Mean value theorems and a Taylor theorem for vector valued functions

Published online by Cambridge University Press:  17 April 2009

Rudolf Výborný
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, Queensland 4067, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Aziz, A.K. and Diaz, J.B., “On a neam-value theorem of the differential calculus of vector-valued functions, and uniqueness theorems for ordinary differential equations in a linear-normed space”, Contrib. Differential Equations 1 (1963), 251269.Google Scholar
[2]Aziz, A.K., Diaz, J.B. and Mlak, W., “On a mean value theorem for vector-valued functions, with applications to uniqueness theorems for ‘right-hand-derivative’ equations”, J. Math. Anal. Appl. 16 (1966), 302307.CrossRefGoogle Scholar
[3]Darboux, G., “Sur les développments en série des fonctions d'une seule variable”, Liouville J. (3) 11 (1876), 291312.Google Scholar
[4]Diaz, J.B. and Výborný, R., “On mean value theorems for strongly continuous vector valued functions”, Contrib. Differential Equations 3 (1964), 107118.Google Scholar
[5]Diaz, Joaquin B. and Výborný, R., “A fractional mean value theorem, and a Taylor theorem, for strongly continuous vector valued functions”, Czechoslovak Math. J. 15 (90) (1965), 299304.CrossRefGoogle Scholar
[6]Diaz, J.B. and Výborný, R., “Generalized mean value theorems of the differential calculus”, J. Austral. Math. Soc. Ser. A 20 (1975), 290300.Google Scholar
[7]Dieudonné, J., Foundations of modern analysis (Pure and Applied Mathematics, 10. Academic Press, New York and London, 1960).Google Scholar
[8]Eberlein, W.F., “The mean value theorem for vector valued functions”, preprint.Google Scholar
[9]Flett, T.M., “Mean value theorems for vector-valued functions”, Tôhoku Math. J. 24 (1972), 141151.CrossRefGoogle Scholar
[10]Flett, T.M., “Some historical notes and speculations concerning the mean value theorems of the differential calculus”, Bull. Inst. Math. Appl. 10 (1974), no. 3, 6672.Google Scholar
[11]Garnir, H.G., “Solovay's axiom and functional analysis”, Functional analysis and its applications, 189204 (Proc. Internat. Conf., Madras, 1973. Lecture Notes in Mathematics, 399. Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[12]Hermite, Ch., Cours de la Faculté des Sciences de Paris sur les intégrales définies, la théorie des fonctions d'une variable imaginaire, et les fonctions elliptiques, 4 éd., entièrement refondue (Rédigé par M. Andoyer. Hermann, Paris, 1891).Google Scholar
[13]McLeod, Robert M., “Mean value theorems for vector valued functions”, Proc. Edinburgh Math. Soc. (2) 14 (19641965), 197209.CrossRefGoogle Scholar
[14]Nashed, M.Z., “Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis”, Nonlinear functional analysis and applications, 103309 (Proc. Advanced Seminar, Mathematics Research Center, University of Wisconsin, Madison, 1970. Publication No. 26 of the Mathematics Research Center, The University of Wisconsin. Academic Press, New York, London, 1971).CrossRefGoogle Scholar
[15]Szarski, Jacek, Differential inequalities (Monografie Matematiyczne, 43. PWN – Polish Scientific Publishers, Warszawa, 1965).Google Scholar
[16]Wazewski, T., “Une généralisation des théorèmes sur les accroissements finis au cas des espaces de Banach et application à la généralisation du théorème de l'Hôpital”, Ann. Soc. Polon. Math. 24 (1951), no. 2, 132147 (1954).Google Scholar
[17]Yamamuro, Sadayuki, Differential calculus in topological linear spaces Lecture Notes in Mathematics, 374. Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar