Oscillation Criteria of Even Order Delay Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

Abstract

We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: $(p(t){\left|x^{{\mathrm{\Delta }}^{n-1}}(t)\right|}^{\alpha -1}{x}^{{\mathrm{\Delta }}^{n-1}}(t){)}^{\mathrm{\Delta }}+f(t,x(\delta (t)))\hspace{0.17em}+\hspace{0.17em}{\int }_a^{\sigma (b)}k(t,s){\left|x(g(t,s))\right|}^{\theta (s)}\text{s}\text{g}\text{n}(x(g(t,s)))\mathrm{\Delta }\xi (s)=0,$ where $t\in [t_0,\mathrm{\infty }{)}_{𝕋}:=[t_0,\mathrm{\infty })\cap 𝕋$, $𝕋$ a time scale which is unbounded above, $n⩾2$ is even, $\left|f(t,u)\left|⩾q(t)\right|u^{\alpha }\right|$, $\alpha >0$ is a constant, and $\theta :[a,b{]}_{{𝕋}_1}\to ℝ$ is a strictly increasing right-dense continuous function; $p,q:[t_0,\mathrm{\infty }{)}_{𝕋}\to ℝ$, $k:[{\mathrm{t}}_0,\mathrm{\infty }{)}_{𝕋}\times [a,b{]}_{{𝕋}_1}\to ℝ$, $\delta :[t_0,\mathrm{\infty }{)}_{𝕋}\to [t_0,\mathrm{\infty }{)}_{𝕋}$, and $g:[t_0,\mathrm{\infty }{)}_{𝕋}\times [a,b{]}_{{𝕋}_1}\to [t_0,\mathrm{\infty }{)}_{𝕋}$ are right-dense continuous functions; $\xi :[a,b{]}_{{𝕋}_1}\to ℝ$ is strictly increasing. Our results extend and supplement some known results in the literature.