NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS TO A SECOND-ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH IMPULSES

In this work, we obtain necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system { (r(t)(z '(t))gamma)' + Sigma(m)(i=1) q(i)(t)x(alpha i)(sigma(i)(t)) = 0, t >= t(0), t not equal lambda k, Delta(r(lambda(k))(z '(lambda(k)))gamma) + Sigma(m)(i=1) h(i)(lambda(k))x(alpha i)(sigma(i)(lambda(k))) = 0, k = 1,2,3, ..., where z(t) = x(t) + p(t)x(tau(t)). Under the assumption integral(infinity)(0) r(eta)(-1/gamma) d eta = infinity, we consider two cases when gamma > alpha(i) and 7 < alpha(i). Our main tool is Lebesgue's Dominated Convergence theorem. Examples are given to illustrate our main results and we state an open problem.