Preprint no. 134
Mathematical Institute of the Academy of Sciences of the Czech Republic

Existence of Nonnegative and Nonpositive Solutions
for Second Order Periodic Boundary Value Problems

Irena Rachunkova, Milan Tvrdy, Ivo Vrkoc

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/;
Ivo Vrkoc, Mathematical Institute, Acad. Sci.of the Czech Republic, Zitna 25, 115 67 Praha, Czech Republic, e-mail: vrkoc@matsrv.math.cas.cz

Summary: In the paper we are interested in nonnegative and nonpositive solutions of the boundary value problem

u''=f(t,u), u(0)=u(1), u'(0)=u'(1),

where f fulfils the Caratheodory conditions on [0.1] x R. We generalize the results reached by M. N. Nkashama, J. Santanilla and L. Sanchez and present estimates for solutions. Besides, we apply our existence theorems to periodic boundary value problems for nonlinear Duffing equations whose right-hand sides have a repulsive or attractive singularity at the origin. We extend or generalize existence results by A. C. Lazer and S. Solimini and other authors. Moreover, we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity, in constrast to the previous papers on this subject. Our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples.

Keywords: Second order nonlinear ordinary differential equation, periodic solution, lower and upper functions, differential inequalities, nonnegative solution, nonpositive solution, attractive and repulsive singularity, Duffing equation.

AMS Subject Classification: 34 B 15, 34 C 25

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