NECESSARY AND SUFFICIENT CONDITIONS FOR ROBUST OSCILLATION

Consider the n th-order difference equation x(p + n) + Sigma(k = 1)(n) a(k)x(n - k + p) = 0 and the differential equation x((n)) + Sigma(k = 1)(n) a(k)x((n - k)) (t - tau) = 0 (tau greater than or equal to 0), where a(k) is an element of [<(a)under bar (k)>, <(a)over bar (k)>]. I n this paper we establish the concept that (1) and (2) are robustly os cillatory (i.e. (1) or (2) is oscillatory for all a(k) is an element o f [<(a)under bar (k)>, <(a)over bar (k)>], k = 1,..., n). Necessary an d sufficient conditions are given by the oscillation of special extrem e point equations. Moreover, we obtain explicit necessary and sufficie nt conditions for the oscillation of the extreme equations so that ans wers to two problems of Ladas are obtained.