On the open problems connected to the results of Lazer and Solimini. Issue 1 (30th January 2014)

• Record Type: Journal Article
• Authors: Hakl, Robert; Zamora, Manuel
• Is Part Of: Proceedings. Volume 144:Issue 1(2014)
• Page Start: 109
• Page End: 118
• Publication Date: 2014-01-30
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210512001862
• Languages: English
• ISSNs: 0308-2105

Abstract

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 p th 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.