Applicable Analysis and Discrete Mathematics, 2 (2008), 1–30. Available electronically at http://pefmath.etf.bg.ac.yu OSCILLATION THEOREMS FOR CERTAIN HIGHER ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS Ravi P. Agarwal, Said R. Grace, Patricia J. Y. Wong Some new oscillation theorems for higher-order nonlinear functional differen- tial equations of the form d n dt n  a(t)  d n x(t) dt n  α  + q(t)f ( x ( g(t) )) =0, α> 0, are established. 1. INTRODUCTION This paper is concerned with the oscillatory behavior of the higher-order nonlinear functional differential equation (1.1) L 2n x(t)+ q(t)f ( x ( g(t) )) =0, where the differential operator, L 2n , is defined recursively by (1.1) ′              L 0 x = x, L i x = d dt L i−1 x, i =1, 2,...,n − 1, L j x = d j−n dt j−n  a  d dt L n−1 x  α  = d j−n dt j−n L n x, j = n, n +1,..., 2n. Clearly L i x = d dt L i−1 x, i =1, 2,...,n − 1,n +1,..., 2n, 2000 Mathematics Subject Classification. 34C10. Keywords and Phrases. Functional differential equations, oscillation, nonoscillation, comparison. 1