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Existence result for a singular nonlinear boundary value problem at resonance. (English) Zbl 1134.34013
Summary: We study the existence of solutions of the second-order boundary value problem $$\align &u''(t)+ \pi^2u(t)+ a(t)g(u(t))= h(t) \quad \text{a.e. }t\in (0,1),\\ &u'\in AC_{\text{loc}}(0,1),\\ &u(0)=u(1)=0, \endalign$$ where $g:\Bbb R\to\Bbb R$ is continuous, $a,h\in\{z\in L_{\text{loc}}^1(0,1)\mid \int_0^1 t|z(t)|\,dt< \infty\}$. The proof of the main result is based upon the Lyapunov-Schmidt procedure and the connectivity properties of the solution set of parametrized families of compact vector fields.

34B16Singular nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
Full Text: DOI
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