Abstract
In this paper, we consider the following second order neutral dynamic equations with deviating arguments on time scales: \t\t\t(r(t)(zΔ(t))α)Δ+q(t)f(y(m(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}$$\\bigl(r(t) \\bigl(z^{\\Delta}(t)\\bigr)^{\\alpha}\\bigr)^{\\Delta}+q(t)f \\bigl(y\\bigl(m(t)\\bigr)\\bigr)=0, $$\\end{document} where z(t)=y(t)+p(t)y(tau(t)), m(t)leq t or m(t)geq t, and tau(t)leq t. Some new oscillatory criteria are obtained by means of the inequality technique and a Riccati transformation. Our results extend and improve many well-known results for oscillation of second order dynamic equations. Some examples are given to illustrate the main results.
Highlights
The study of dynamic equations on time scales which goes back to its founder Hilger [1] as an area of mathematics that has received a lot of attention
Sui and Han Advances in Difference Equations (2018) 2018:337 of the most interesting problems is the study of the oscillation of solutions of dynamic equation with deviating arguments
Saker [5] studied the oscillation of second order nonlinear neutral delay dynamic equation r(t) y(t) + p(t)y(t – τ ) α + f t, y(t – δ) = 0, under the condition ∞ r–1/α(t) t = ∞
Summary
The study of dynamic equations on time scales which goes back to its founder Hilger [1] as an area of mathematics that has received a lot of attention. We consider the following second order neutral dynamic equations with deviating arguments on time scales: (r(t)(z (t))α) + q(t)f (y(m(t))) = 0, where z(t) = y(t) + p(t)y(τ (t)), m(t) ≤ t or m(t) ≥ t, and τ (t) ≤ t. Some new oscillatory criteria are obtained by means of the inequality technique and a Riccati transformation.
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