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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
\begin{aligned} &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, \end{aligned}
where $$g:\mathbb R\to\mathbb R$$ 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.

##### MSC:
 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:
##### References:
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