Oscillation Criteria for Functional Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integral

Abstract

We present new oscillation criteria for the second order nonlinear dynamic equation ${[r(t){\varphi }_{\gamma }(x^{\mathrm{\Delta }}(t))]}^{\mathrm{\Delta }}+q_{\mathrm{0}}(t){\varphi }_{\gamma }(x(g_{\mathrm{0}}(t)))+{\int }_a^b\mathrm{‍}q(t,s){\varphi }_{\alpha (s)}(x(g(t,s)))\mathrm{\Delta }\zeta (s)=\mathrm{0}$ under mild assumptions. Our results generalize and improve some known results for oscillation of second order nonlinear dynamic equations. Several examples are worked out to illustrate the main results.