Integral averaging techniques for the oscillation and nonoscillation of solutions of second order ordinary differential equations (Q1894991)

The second-order ordinary differential equations \(\ddot x+ a(t) f(x)= 0\) are considered under some complementary conditions. The author introduces the integral averaging \(A_p(t)\) in a special manner. Moreover, the definitions of strictly superlinear, strictly sublinear and linear equation play here an important role. It is shown that if the equation is strictly superlinear, sublinear or linear and there is a nonoscillatory solution \(x(t)\) of the above equation, then either a finite \(\lim A_p(t)\) (as \(t\to \infty\)) exists or \(\lim A_q(t)= -\infty\) (as \(t\to \infty\)) for some \(q\) described in detail. Corollaries devoted to oscillation of all continuable solutions are of considerable interest.