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

  • H. S. Hassan (a1) and N. G. Lloyd (a1)

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).

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1Coddington, E. A. and Levinson, N.. Theory of ordinary differential equations (New York: McGraw-Hill, 1955).
2Gunning, R. C.. Lectures on complex analytic varieties: the local parametrization theorem (Princeton Mathematical Notes, 1970).
3Hervé, M.. Several complex variables: local theory (TATA Institute Research Studies in Mathematics, Oxford University Press, 1963).
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), 667700.
5Lloyd, N. G.. On analytic differential equations. Proc. London Math. Soc. 30 (1975), 430444.
6Lloyd, N. G.. On a class of differential equations of Riccati type. J. London Math. Soc. 10 (1975), 110.
7Lloyd, N. G.. Differential equations on complex manifolds. J. London Math. Soc. 13 (1976), 258262.
8Nachbin, L.. Holomorphic functions, domains of holomorphy and local properties, (Amsterdam: North-Holland, 1970).
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10Whitney, H.. Complex analytic varieties (London: Addison Wesley, 1972).

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