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.