Geodesic
Singular periodic problem for nonlinear ordinary differential equations with \(\phi\)-Laplacian
Electronic journal of differential equations, Tome 2006 (2006)
We investigate the singular periodic boundary-value problem with

$\displaylines{ (\phi (u'))' = f(t, u, u'), \cr u(0) = u(T),\quad u'(0) = u'(T), }$

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.
Classification : 34B16, 34C25
Keywords: singular periodic problem, $\phi$-Laplacian, smooth sign-changing solutions, lower and upper functions 
@article{EJDE_2006__2006__a79,
     author = {Pol\'a\v{s}ek,  Vladim{\'\i}r and Rach\r{u}nkov\'a,  Irena},
     title = {Singular periodic problem for nonlinear ordinary differential equations with {\(\phi\)-Laplacian}},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1098.34014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a79/}
}
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Polášek,  Vladimír; Rachůnková,  Irena. Singular periodic problem for nonlinear ordinary differential equations with \(\phi\)-Laplacian. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a79/