On a theorem of Picard

Gesztesy, F.; Sticka, W.

doi:10.1090/S0002-9939-98-04668-1

On a theorem of Picard
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by F. Gesztesy and W. Sticka PDF

Proc. Amer. Math. Soc. 126 (1998), 1089-1099 Request permission

Abstract:

We extend Picard’s theorem on the existence of elliptic solutions of the second kind of linear homogeneous ${n}^{\mathrm {th}}$-order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order $n\times n$ systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determined. Connections with completely integrable systems are mentioned.

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