Singular Dirichlet problem for ordinary differential equations with $\unusedphi$-Laplacian

Abstract: We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with $\phi$-Laplacian
\ogather(\phi(u'))' = f(t, u, u'),
u(0) = A, u(T) = B,
where $\phi$ is an increasing homeomorphism, $\phi(\R)=\R$, $\phi(0)=0$, $f$ satisfies the Carath{é}odory conditions on each set $[a, b]\times\R^{2}$ 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 $[0, T]$.

Keywords: singular Dirichlet problem, $\phi$-Laplacian, existence of smooth solution, lower and upper functions

Classification (MSC2000): 34B16, 34B15

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