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On the open problems connected to the results of Lazer and Solimini

Published online by Cambridge University Press:  30 January 2014

Robert Hakl  and
Manuel Zamora
Robert Hakl
Affiliation:
Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, 616 62 Brno, Czech Republic (hakl@ipm.cz)
Manuel Zamora
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva, 18071 Granada, Spain (mzamora@ugr.es)
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Abstract

A well-known theorem proved by Lazer and Solimini claims that the singular equation

has a periodic solution if and only if the mean value of the continuous external force is positive. In this paper, we show that this result cannot be extended to the case when h is an integrable function, unless additional assumptions are introduced. In addition, for each p ≥ 1 and h-integrable function in the pth power, we give a sharp condition guaranteeing the existence of periodic solutions to the above-mentioned equation, showing that there is a close relation between p and the order of the singularity λ.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 1 , February 2014 , pp. 109 - 118
Copyright
Copyright © Royal Society of Edinburgh 2014 

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