Properties of solutions of a class of third-order linear differential equations

Abstract

where r, qi for i --~ 1, 2, are real valued continuous functions on I : [a, oo), with r ~0. Following the classification of solutions of third order linear equations due to Hanan [6], Jones in [8], considered a second order linear equation and related its s tudy to a third order equation. In Section 2 of this paper we shall obtain conditions under which the equation (1.1) satisfies Hanan's classification. Also we shall present disconjugacy criteria for (1.1). In Section 3, we shall discuss oscillation results for (1.1). Motivation for this s tudy stems basically from the works of [3], [7], [8]. We notice that the equation (1.1) is more general than those studied in [1], [4], [5], [6], [8], [9]. A nontrivial solution of (1.1) is said to be oscillatory on I if it has infinitely many zeros on I; otherwise, nonoseillatory. The differential equation (1.1) is said to be oscillatory or nonoscillatory, respectively, depending on the existence or nonexistence of an oscillatory solution. Moreover, the equation (1.1) is called disconjugate on I if no nontrivial solution has more than two zeros in I. Furthermore, the equation (1.1) is said to be C 1 (or C2) on I, if for each c E (a, ~ ) and nontrivial solution y of (1.1) satisfying the conditions y(c) -~ y'(c) : O, we have y(t) ~= 0 for t E [a, c) (or t ~ (c, o~)). I t follows from [2] that if (1.1) is C1 and C~ on I, then it is disconjugate on I.