Asymptotic integrations of nonoscillatory second order differential equations

Abstract:

The linear differential equation (1) $(r(t)x’)’ + (f(t) + q(t))x= 0$ is viewed as a perturbation of the equation (2) $(r(t)y’)’ + (f(t)y = 0$, where $r > 0$, $f$ and $q$ are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and ${y_1}$, ${y_2}$ are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution ${x_1}$ of (1). Under some mild assumptions, we characterize that (1) has a principal solution ${x_1}$ satisfying ${x_1}= {y_1}(1 + o(1))$. Sufficient (sometimes necessary) conditions under which the nonprincipal solution ${x_2}$ of (1) behaves, in three different degrees, like ${y_2}$ as $t \to \infty$ are also established.