## 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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### References:

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