Viewing nonoscillatory second order linear differential equations from the angle of Riccati equations

Abstract

We build an existence theory for nonoscillatory second order differential equations of the form (A) $(p(t)x^{\prime }{)}^{\prime }=q(t)x,$ $p(t)$ and $q(t)$ being positive continuous functions on $[a,\infty )$, in which a crucial role is played by a pair of the Riccati differential equations (R1) $u^{\prime }=q(t)-u^2/p(t)$, (R2) $v^{\prime }=1/p(t)-q(t)v^2$, associated with (A). An essential part of the theory is the construction of a pair of linearly independent nonoscillatory solutions $x_1(t)$ and $x_2(t)$ of (A) enjoying explicit exponential-integral representations in terms of solutions $u_1(t)$ and $u_2(t)$ of (R1) or in terms of solutions $v_1(t)$ and $v_2(t)$ of (R2).