OSCILLATORY AND ASYMPTOTIC BEHMIOUR OF A CLASS OF NONLINEAR DIFFERENTIALEQUATIONS OF THIRD ORDER

This paper deals with oscillatory/nonoscillatory behaviour of solutions of third order nonlinear differential equations of the formy"'+a(t)y"+b(t)y'+c(t)y=0 (1)andy"'+a(t)y"+b(t)y'+c(t)f(y)=0,(2)where a,b,c ∈ C([σ,∞),R) such that a(t) does not change sign, b(t) ≤ 0, c(t) > 0,f∈C(R, R) such that (f(y)/y) ≥ β > 0 for y ≠ 0 and γ > 0 is a quotient of odd integers.It has been shown, under certain conditions on coefficient functions, that a solution of (1) and (2) which Las a zero is oscillatory and the nonoscillatory solutions of these equations tend to zero as t → ∞. The motivation for this work came from the observation that the equationu"'+ay"+by'+cy=0,(3)where a, b, c are constants such that b ≤ 0, c > 0, has an oscillatory solution if(2a³)/27-ab/3+c-2/3γ3(a²/3-b)^3/2>0and only ifand all nonoscillatory solutions of (3) tend to zero if and only if the equation has an oscillatory solution.