Electron. J. Diff. Eqns., Vol. 2006(2006), No. 27, pp. 1-12.

Singular periodic problem for nonlinear ordinary differential equations with $\phi$-Laplacian

Vladimir Polasek, Irena Rachunkova

Abstract:
We investigate the singular periodic boundary-value problem with $\phi$-Laplacian,
$$\displaylines{ (\phi (u'))' = f(t, u, u'), \cr u(0) = u(T),\quad u'(0) = u'(T), }$$
where $\phi$ is an increasing homeomorphism, $\phi(\mathbb{R} )=\mathbb{R}$, $\phi(0)=0$. We assume that $f$ satisfies the Caratheodory conditions on each set $[a, b]\times \mathbb{R}^{2}$, $[a, b]\subset (0, T)$ and $f$ does not satisfy the Caratheodory conditions on $[0, T]\times \mathbb{R}^{2}$, which means that $f$ has time singularities at $t=0$, $t=T$.
We provide sufficient conditions for the existence of solutions to the above problem belonging to $C^{1}[0, T]$. We also find conditions which guarantee the existence of a sign-changing solution to the problem.

Submitted January 5, 2006. Published March 9, 2006.
Math Subject Classifications: 34B16, 34C25.
Key Words: Singular periodic problem; $\phi$-Laplacian; smooth sign-changing solutions; lower and upper functions.

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Vladimír Polásek
Department of Mathematics, Palacky University
Tomkova 40, 779 00 Olomouc, Czech Republic
email: polasek.vlad@seznam.cz
Irena Rachunková
Department of Mathematics, Palacky University
Tomkova 40, 779 00 Olomouc, Czech Republic
email: rachunko@inf.upol.cz

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