Two oscillation theorems are
provided for certain nonlinear nonautonomous third order differential equations. Both
results involve integral conditions and are of the form that any solution which has
one zero is oscillatory. Theorem. Let p(t), q(t) be continuous and let p′(t) be
nonpositive. Define Q(t) =∫0tq(s)ds. If A + Bt −∫t0tQ(s)ds < 0 for t sufficiently
large, any A and B, if γ is positive and the quotient of two odd integers, then
any continuable solution of y′′′+ p(t)y′ + q(t)yγ= 0 which has a zero is
oscillatory.
Theorem. Let p(t) and q(t) be continuous and nonnegative and let f(y)∕y ≧ α > 0
for some α. If αq(t) − p′(t) is positive and if ∫∞t(αq(t) − p′)(t)dt = ∞ then
any continuable solution of y′′′+ p(t)y′ + q(t)f(y) = 0 which has a zero is
oscillatory.
