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.


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


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