OSCILLATION OF SECOND-ORDER NONLINEAR NEUTRAL DYNAMIC EQUATIONS WITH MIXED ARGUMENTS ON TIME SCALES
H. A. AGWA *
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt.
AHMED M. M. KHODIER
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt.
HEBA M. ARAFA
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we use Riccati transformation technique to establish some new oscillation criteria for the second-order nonlinear neutral dynamic equation with mixed arguments
(r(t)[(x(t) +p1(t)x(η1(t)) +p2(t)x(η2(t)))∆]γ)∆+f(t,x(τ1(t))) +g(t,x(τ2(t))) = 0,
on a time scale T. Our results improve and extend some results of Tao Ji et al. [1]. Also, our results unify the oscillation of second order nonlinear neutral di erential and di erence equations. Finally, we give some examples to illustrate our results.
Keywords: Oscillation, neutral dynamic equations, time scales, generalized Riccati technique