Abstract
It is a well known theorem for Sturmian boundary value problems Lx=r Rx=0 that the pair (L, R=0) is inverse monotone (i. e. Lx ≧0, Rx=0 ⇒ x≧0) if there exists a weak majorizing element, i. e. a function z≧0 satisfying Lz≧0, Rz=0. We show that this criterion carries over to ordinary boundary value problems of arbitrary order if in addition there exists an inverse monotone pair ‘larger” than (L, R) in a certain sense. This follows from a variant of Schröder's theorem [10] combined with a result on strict monotonicity of nonnegative Green's functions. However, it will also be shown that the additional condition can only be dispensed with, if the boundary value problem is at most of the second order. Furthermore an analogous result holds, for elliptic boundary value problems in arbitrary dimensions.
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Beyn, WJ. Schwach Majorisierende Elemente und die Besonderen Monotonie-Eigenschaften von Randwertaufgaben Zweiter Ordnung. Manuscripta Math 28, 317–336 (1979). https://doi.org/10.1007/BF01954612
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DOI: https://doi.org/10.1007/BF01954612