Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations

Abstract

In this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments α2(η)ϕδ2α1ηϕδ1uΔ(η)ΔΔ+p(η)ϕδu(g(η))=0, on an arbitrary unbounded-above time scale T, where η∈[η0,∞)T:=[η0,∞)∩T, η0≥0, η0∈T and ϕζ(w):=wζsgnw, ζ>0. Using the integral mean approach and the known Riccati transform methodology, several improved Hille-type and Ohriska-type oscillation criteria have been derived that do not require some restrictive assumptions in the relevant results. Illustrative examples and conclusions show that these criteria are sharp for all third-order dynamic equations compared to the previous results in the literature.

Keywords

• oscillation criteria
• Hille-type
• Ohriska-type
• differential equations
• dynamic equations
• time scales