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Linear Functional-Differential Equations in a Banach Algebra*

Published online by Cambridge University Press:  20 November 2018

W. J. Fitzpatrick  and
L. J. Grimm
W. J. Fitzpatrick
Affiliation:
Department of Mathematics, College of Arts & Sciences, University of Missouri-Rolla, Rolla, Missouri 65401
L. J. Grimm
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, California 90007
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The theory of analytic differential systems in Banach algebras has been investigated by E. Hille and others, see for instance Chapter 6 in [4].

In this paper we show how a projection method used by W. A. Harris, Jr., Y. Sibuya, and L. Weinberg [3] can be applied to study a class of functional differential equations in this setting. The method, based on functional analysis, had been used extensively by L. Cesari [1] in similar forms for boundary value problems, and by J. K. Hale, S. Bancroft, and D. Sweet [2]. We also obtain as corollaries several results for ordinary differential equations in Banach algebras which were proved in a different way by Hille.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 4 , 01 December 1978 , pp. 435 - 439
Copyright
Copyright © Canadian Mathematical Society 1978

Footnotes

*

Research supported by University of Missouri Faculty Research Grant and by NSF Grant MCS 76-08229.

References

1. Cesari, L., Functional analysis and an alternative method. Michigan Math. J. 11, (1964), 385-414.Google Scholar
2. Hale, J. K., Bancroft, S., and Sweet, D., Alternative problems for nonlinear functional equations. J. Differential Equations 4, (1968), 40-56.Google Scholar
3. Harris, W. A. Jr., Sibuya, Y., and Weinberg, L., Holomorphic solutions of linear differential systems at singular points, Arch. Rat. Mech. Anal. 35, (1969), 245-248.Google Scholar
4. Hille, E., Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass., 1969.Google Scholar