Preprint no. 135
Mathematical Institute of the Academy of Sciences of the Czech Republic

On the Construction of Nonconstant Lower and Upper Functions to Second Order Nonlinear Periodic Boundary Value Problems

Irena Rachunkova, Milan Tvrdy

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

Summary: In this paper we present effective conditions ensuring the existence of lower and upper functions for the periodic boundary value problem  u''=f(t,u), u(0)=u(2 Pi), u'(0)=u'(2 Pi).   They are constructed as solutions of some related generalized linear problems and they need not be constant.  As applications, two new results  concerning singular periodic boundary value problems  for nonlinear Duffing equations of both attractive  and repulsive type are delivered.

Keywords: Second order nonlinear ordinary differential equation, periodic solution, singular problem, lower and upper functions, generalized linear differential equation, attractive and repulsive singularity, Duffing equation.

AMS Subject Classification: 34 B 15, 34 C 25

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