Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 27, pp. 1-12.
Title: Singular periodic problem for nonlinear ordinary
differential equations with $\phi$-Laplacian
Authors: Vladimir Polasek (Palacky Univ., Czech Republic)
Irena Rachunkova (Palacky Univ., Czech Republic)
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 Carath\'eodory 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.