SINGULAR STRONGLY INCREASING SOLUTIONS OF HIGHER-ORDER QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS

In the present paper, we consider quasilinear ordinary differential equations of the form D(alpha(n), alpha(n-1), ..., alpha(1))x = p(t)vertical bar x vertical bar(beta) sgn x, t >= a, (1.1) where D(alpha(n), alpha(n-1), ..., alpha(1)) is the nth-order iterated differential operator such that D(alpha(n), alpha(n-1), ..., alpha(1))x equivalent to D(alpha(n))D(alpha(n-1))...D(alpha(1))x and, in general, D(alpha) is the first-order differential operator defined by D(alpha)x = (d/dt)(vertical bar x vertical bar(alpha) sgn x) for alpha > 0. For the case where alpha(1)alpha(2) ... alpha(n) < beta, we present a new sufficient condition for all strongly increasing solutions of (1.1) to be singular. If alpha(1) = alpha(2) = ... = alpha(n) = 1, then one of the main results, Corollary 3.2, gives an extension of the well-known theorem of Kiguradze and Chanturia [2].