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

In this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments α 2 ( η ) ϕ δ 2 α 1 η ϕ δ 1 u Δ ( η ) Δ Δ + p ( η ) ϕ δ u ( g ( η ) ) = 0 , on an arbitrary unbounded-above time scale T , where η ∈ [ η 0 , ∞ ) T : = [ η 0 , ∞ ) ∩ T , η 0 ≥ 0 , η 0 ∈ T and ϕ ζ ( w ) : = w ζ sgn w , 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.