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On the sign of the Green’s function associated to Hill’s equation with an indefinite potential. (English) Zbl 1161.34014

Summary: We give an \(L^p\)-criterion for the positiveness of the Green’s function of the periodic boundary value problem:
\[ x''+a(t)x= 0, \quad x(0)= x(T), \quad x'(0)= x'(T) \]
with an indefinite potential \(a(t)\). Moreover, we prove that such Green’s function is negative provided \(a(t)\) belongs to the image of a suitable periodic Riccati type operator.

MSC:

34B27 Green’s functions for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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