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Singular Dirichlet problem for ordinary differential equations with \(\phi \)-Laplacian. (English) Zbl 1114.34017
Summary: We provide sufficient conditions for the solvability of a singular Dirichlet boundary value problem with \(\phi \)-Laplacian \[ (\phi (u'))' = f(t, u, u'),\quad u(0) = A, \;u(T) = B, \] where \(\phi \) is an increasing homeomorphism, \(\phi (\mathbb R )=\mathbb R \), \(\phi (0)=0\), \(f\) satisfies the Carathéodory conditions on each set \([a, b]\times \mathbb R ^{2}\) with \([a, b]\subset (0, T)\) and \(f\) is not integrable on \([0, T]\) for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on \([0, T]\).

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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