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Differential equations are often classified according to oscillatomj/nonoscillatory properties of their solutions as equations having property A or property B. The aim of the paper is to state an equivalence theorem between property A and property B for third order differential equations. Some applications, to linear as well as to nonlinear equations, are given too. Particularly, we give integral criteria ensuring property A or B for nonlinear equations. Our only assumption on nonlinearity is its superlinearity in neighbourhood of infinity, hence our results apply also to Emden-Fowler type equations.

The characteristic properties of the principal solution for half-linear differential equation ðaðtÞFðx 0 ÞÞ 0 þ bðtÞFðxÞ ¼ 0;where the functions a; b are positive and continuous for tX0 and FðuÞ ¼ juj pÀ2 u; p41; are investigated. In the linear case it is well-known that the principal solution is the ''smallest one'' in a neighbourhood of infinity; we show that this property continues to hold in the half-linear case. In addition, it is proved that the principal solutions can be fully characterized by means of two different integral criteria, which reduce to that one well known in the linear case. r

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