Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T09:02:58.422Z Has data issue: false hasContentIssue false
  • English
  • Français

On the structure of the set of solutions of certain holomorphic two-point boundary value problems

Published online by Cambridge University Press:  14 November 2011

H. S. Hassan  and
N. G. Lloyd
H. S. Hassan
Affiliation:
Department of Pure Mathematics, The University College of Wales, Aberystwyth
N. G. Lloyd
Affiliation:
Department of Pure Mathematics, The University College of Wales, Aberystwyth
Get access Rights & Permissions [Opens in a new window]

Synopsis

Suppose that f: ℝ×ℂN→ℂN is holomorphic in z and continuous in t, and that Φ: ℂN×ℂN→ℂN is holomorphic. Boundary value problems of the form

are considered. The particular interest is in the structure and topological properties of the set of solutions. The paper is motivated by the corresponding properties of the set of periodic solutions of ż = f(t, z) when f is periodic in t. Consideration of this complex equation gives information about the periodic solutions of the real equation ẋ = f(t, x).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 3-4 , 1981 , pp. 235 - 247
Copyright
Copyright © Royal Society of Edinburgh 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: McGraw-Hill, 1955).Google Scholar
2Gunning, R. C.. Lectures on complex analytic varieties: the local parametrization theorem (Princeton Mathematical Notes, 1970).Google Scholar
3Hervé, M.. Several complex variables: local theory (TATA Institute Research Studies in Mathematics, Oxford University Press, 1963).Google Scholar
4Lloyd, N.G.. The number of periodic solutions of the equation ż = z N + p 1(t z N−1 + … + P N(t). Proc. London Math. Soc. 27 (1973), 667–700.CrossRefGoogle Scholar
5Lloyd, N. G.. On analytic differential equations. Proc. London Math. Soc. 30 (1975), 430–444.CrossRefGoogle Scholar
6Lloyd, N. G.. On a class of differential equations of Riccati type. J. London Math. Soc. 10 (1975), 1–10.CrossRefGoogle Scholar
7Lloyd, N. G.. Differential equations on complex manifolds. J. London Math. Soc. 13 (1976), 258–262.CrossRefGoogle Scholar
8Nachbin, L.. Holomorphic functions, domains of holomorphy and local properties, (Amsterdam: North-Holland, 1970).Google Scholar
9Narasimhan, R.. Several complex variables (University of Chicago Press, 1971).Google Scholar
10Whitney, H.. Complex analytic varieties (London: Addison Wesley, 1972).Google Scholar