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MATHEMATICA BOHEMICA, Vol. 130, No. 4, pp. 409-425 (2005)
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Singular Dirichlet problem for ordinary differential equations with $\unusedphi$-Laplacian

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Vladimir Polasek, Irena Rachunkova

* Vladimir Polasek*, * Irena Rachunkova*, Department of Mathematics, Palacky University, Tomkova 40, 779 00 Olomouc, Czech Republic, e-mail: ` polasek.vlad@seznam.cz, rachunko@inf.upol.cz`

**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

**Full text of the article:**

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