A sharp oscillation result for second-order half-linear noncanonical delay differential equations

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2020; no. 46; pp. 1 - 14
• Main Authors: Džurina, Jozef, Jadlovská, Irena
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2020
• Subjects: delay; half-linear; oscillation; second-order differential equation
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2020.1.46

In the paper, new single-condition criteria for the oscillation of all solutions to a second-order half-linear delay differential equation in noncanonical form are obtained, relaxing a traditionally posed assumption that the delay function is nondecreasing. The oscillation constant is best possible in the sense that the strict inequality cannot be replaced by the nonstrict one without affecting the validity of the theorem. This sharp result is new even in the linear case and, to the best of our knowledge, improves all the existing results reporting in the literature so far. The advantage of our approach is the simplicity of the proof, only based on sequentially improved monotonicities of a positive solution.