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Nonlinear periodic boundary value problems with sign-changing Green’s function. (English) Zbl 1220.34038

The author considers the existence and nonexistence of positive solutions for the nonlinear boundary value problem with periodic boundary conditions
\[ u'' + a(t) u = \lambda b(t) f(u), \text{ a.e. } t \in [0, T], \]
\[ u(0) = u(T), \qquad u'(0) = u'(T). \]
Here, \(a(t), b(t) \geq 0\) for a.e. \(t \in [0, T]\) and are positive on a set of positive measure. The author assumes that the associated linear problem has a unique solution and that the Green function changes sign. Under the assumption that there exists an \(\varepsilon > 0\) such that
\[ \int_0^T \! [G^+(t, s) - (1 + \varepsilon ) G^-(t, s)] b(s) \, ds > 0, \quad t \in [0,T], \]
the author shows that there exists a \(\lambda_0 > 0\) such that for all \(\lambda \in (0, \lambda_0)\), the problem has a positive solution. The author also establishes conditions under which the problem has no positive solutions for \(\lambda\) sufficiently small as well as conditions for which the problem has no positive solution for \(\lambda\) large enough.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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