Improved oscillation criteria for second order quasilinear dynamic equations of noncanonical type

Abstract

The present discussion is to study the following second order nonlinear delay dynamic equation of the form: [r(θ)(WΔ(θ))α]Δ+P(θ)Wβ(η(θ))=0,θ∈T0=[θ0,∞)∩T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}{}[r(\ heta )(\\mathcal {W}^{\\Delta }(\ heta ))^{\\alpha }]^{\\Delta } +\\mathcal {P}(\ heta )\\mathcal {W}^{\\beta }(\\eta (\ heta ))=0,\\;\ heta \\in \\mathbb {T}_{0}=[\ heta _{0},\\infty )\\cap \\mathbb {T} \\end{aligned}$$\\end{document}under the assumption ∫θ0θr-1/α(s)Δs<∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{\ heta _{0}}^{\ heta }r^{-1/\\alpha }(s)\\Delta s<\\infty . \\end{aligned}$$\\end{document}We divide the research into two halves, α>β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\beta $$\\end{document} and α<β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha <\\beta $$\\end{document}, and look for some lim sup\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\limsup $$\\end{document} type conditions that cause all solutions to oscillate. In addition, we extend the Philos-type oscillation criteria. To illustrate the analytical findings, two examples are provided.