Abstract
In this paper, we study the third-order functional dynamic equation {r2(t)ϕα2([r1(t)ϕα1(xΔ(t))]Δ)}Δ+q(t)ϕα(x(g(t)))=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl\\{ r_{2}(t)\\phi_{\\alpha_{2}} \\bigl( \\bigl[ r_{1}(t) \\phi _{\\alpha _{1}} \\bigl( x^{\\Delta}(t) \\bigr) \\bigr] ^{\\Delta} \\bigr) \\bigr\\} ^{\\Delta}+q(t)\\phi_{\\alpha} \\bigl( x\\bigl(g(t)\\bigr) \\bigr) =0, $$\\end{document} on an upper-unbounded time scale mathbb{T}. We will extend the so-called Hille and Nehari type criteria to third-order dynamic equations on time scales. This work extends and improves some known results in the literature on third-order nonlinear dynamic equations and the results are established for a time scale mathbb{T} without assuming certain restrictive conditions on mathbb{T}. Some examples are given to illustrate the main results.
Highlights
We are concerned with the oscillatory behavior of the third-order half-linear functional dynamic equation r (t)φα r (t)φα x (t)+ q(t)φα x g(t) = ( . )on an upper-unbounded time scale T, where φα(u) := |u|α– u, α, α, α := α α > ; ri, i =, are positive rd-continuous functions on T such that, for t ∈ T, ∞ ri αi (t)t = ∞; t q is a positive rd-continuous function on T; and g : T → T is a rd-continuous function such that limt→∞ g(t) = ∞
We will assume that the reader is familiar with the basic facts of time scales and time scale notation, for an excellent introduction to the calculus on time scales, see Bohner and Peterson [, ]
We state some oscillation results for differential equations that will be related to our oscillation results for ( . ) on time scales and explain the important contributions of this paper
Summary
We are concerned with the oscillatory behavior of the third-order half-linear functional dynamic equation r (t)φα r (t)φα x (t). Peterson and Saker [ ] established Hille and Nehari oscillation criteria for the third-order dynamic equation x (t) + q(t)x(t) = , where q is a positive real-valued rd-continuous function on T, we list the main results of [ ] as follows. ). Wang and Xu in [ ] considered the third-order dynamic equation r (t) r (t)x (t) α + q(t)x(t) = , where α ≥ is a quotient of odd positive integers and with the condition. Consider the nonlinear third-order advanced dynamic equation tα – φα φα x (t) ηα + ltα+ φα x g(t) where η positive constant and l lim inft→∞(. Proof Without loss of generality, assume that x g(t) > , x[ ](t) > , x[ ](t) > , x[ ](t) < on [t , ∞)T Integrating both sides of the dynamic equation
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