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.

Show me the PDF file (269K), TEX file, and other files for this article.

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

Return to the EJDE web page