# Resonance and Multiplicity in Periodic Boundary Value Problems with Singularity

## Irena Rachunkova, Milan Tvrdy, Ivo Vrkoc

Irena Rachunkova, Department of Math., Palacky University, 779 00 OLOMOUC, Tomkova 40, Czech Republic, e-mail: rachunko@risc.upol.cz;
Milan Tvrdy, Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail: tvrdy@math.cas.cz, http://www.math.cas.cz/~tvrdy/;
Ivo Vrkoc, Mathematical Institute, Acad. Sci.of the Czech Republic, 115 67 PRAHA 1, Zitna 25, Czech Republic, e-mail: vrkoc@matsrv.math.cas.cz

Summary: The paper deals with the boundary value problem $u''+k u=g(u)+e(t), u(0)=u(2\pi), u'(0)=u'(2\pi),$ where $k\in R,$ $g: (0,\infty)\to\R$ is ontinuous, $e\in\L[0,2\pi]$ and $\lim_{x\to 0+}\int_x^1 g(s) ds=\infty.$ In particular, the existence and multiplicity results are obtained using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

Keywords:   Second order nonlinear ordinary differential equation,  periodic problem, lower and upper functions.

AMS Subject Classification: 34B15, 34C25

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