Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case

• Main Authors: Zuzana Došlá, Petr Hasil, Serena Matucci, Michal Veselý
• Format: Article
• Language: English
• Published: SpringerOpen 2019-07-01
• Series: Journal of Inequalities and Applications
• Subjects: Half-linear equations; Linear equations; Euler type equations; Oscillation theory; Oscillation criterion; Non-oscillation criterion
• Online Access: http://link.springer.com/article/10.1186/s13660-019-2137-0
• ISSN: 1029-242X

Abstract This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.