Principal and nonprincipal solutions of impulsive differential equations with applications (Q972162)

The authors consider the scalar impulsive equation \[ (r(t)z')'+ q(t)z= 0,\quad r(t)> 0,\tag{1} \] \[ \Delta r(t)z'+ q_iz= 0,\quad t=\theta_i,\quad t\geq t_0.\tag{2} \] The solutions \(u(t)\) and \(v(t)\) of (1), (2) are called principal and nonprincipal if \[ \lim_{t\to\infty} {u(t)\over v(t)}= 0,\;\int^\infty_0{dt\over ru^2}= \infty,\quad \int^\infty_a {dt\over ru^2}<\infty, \] \[ {r(t)v'(t)\over v(t)}> {r(t)u'(t)\over u(t)} \] for \(t\) sufficiently large. The authors find a connection between the existence of a positive solution on \([a,\infty)\) of (1), (2) and the existence of principal and nonprincipal solutions of (1), (2).