Oscillation theorems and asymptotic behaviour of certain third-order neutral differential equations with distributed deviating arguments

The purpose of this paper is to study the oscillation criteria for a class of third-order neutral differential equations with distributed deviating arguments [b(t)((a(t)(z'(t))(alpha 1))')(alpha 2)]' + integral(d)(c) q(t, xi)f(x(sigma(t, xi)))d xi = 0, t >= t(0) where z(t) = x(t) +integral(n)(m) p(t, xi)x(tau(t, xi))d xi and alpha(i) are ratios of positive odd integers, i = 1, 2. By using a generalised Riccati transformation and an integral averaging technique, we establish some new theorems, which ensure that all solutions of this equation oscillate or converge to zero. Some examples are given to illustrate our main results.