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Optimal conditions for maximum and antimaximum principles of the periodic solution problem. (English) Zbl 1200.34001
The aim of this paper is to give several optimal criteria for the existence of maximum or antimaximum principles for the second order operator ${L}_{q}x={x}^{\text{'}\text{'}}+qx$, with periodic conditions, where $q$ is a given periodic, integrable potential. This problem can be fully characterized in terms of periodic and antiperiodic eigenvalues. Alternatively, the maximum or antimaximum principle is closely related with the associated Green’s function and its sign. The author presents an illuminating and unifying review of this topic.
##### MSC:
 34-02 Research monographs (ordinary differential equations) 34C25 Periodic solutions of ODE 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 34B27 Green functions
##### References:
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