Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations - Taher S. Hassan; Mnaouer Kachout; Bassant M. El-Matary; Loredana Florentina Iambor; Ismoil Odinaev; Akbar Ali (Mathematics)
In this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments <semantics><mfenced separators="" open="{" close="}">α2(η)ϕδ2<mfenced separators="" open="(" close=")"><mfenced separators="" open="[" close="]">α1<mfenced open="(" close=")">η</mfenced>ϕδ1<mfenced separators="" open="(" close=")">uΔ(η)</mfenced></mfenced>Δ</mfenced></mfenced>Δ+p(η)ϕδ<mfenced separators="" open="(" close=")">u(g(η))</mfenced>=0,</semantics> on an arbitrary unbounded-above time scale <semantics>T</semantics>, where <semantics>η∈[η0,∞)T:=[η0,∞)∩T</semantics>, <semantics>η0≥0</semantics>, <semantics>η0∈T</semantics> and <semantics>ϕζ(w):=<mfenced open="|" close="|">w</mfenced>ζ<mspace width="3.33333pt"></mspace>sgnw</semantics>, <semantics>ζ>0</semantics>. 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.