Oscillatory and asymptotic behaviour of a class of nonlinear differential equations of third order

This paper deals with oscillatory/nonoscillatory behaviour of solutions of third order nonlinear differential equations of the form y''' + a(t)y " + b(t)y' + c(t)y(gamma) = 0 (1) and y''' + a(t)y " + b(t)y' + c(t)f(y) = 0, (2) where a,b,c is an element of C([sigma,infinity),R) such that a(t) does not change sign, b(t) less than or equal to 0, c(t) > 0, f is an element of C(R, R) such that (f(y)/y) greater than or equal to beta > 0 for y not equal 0 and gamma > 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 has a zero is oscillatory and the nonoscillatory solutions of these equations tend to zero as t --> infinity. The motivation for this work came from the observation that the equation y''' + ay " + by' + cy = 0, (3) where a,b,c are constants such that b less than or equal to 0; c > 0, has an oscillatory solution if and only if 2a(3)/27 - ab/3 + c - 2/3 gamma 3(a(2)/3 - b)(3/2) > 0 and all nonoscillstory solutions of (3) tend to zero if and only if the equation has an oscillatory solution.