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The characteristic properties of the principal solution for half-linear differential equation (a(t)Φ(x′))′+b(t)Φ(x)=0, where the functions a,b are positive and continuous for t⩾0 and Φ(u)=|u|p−2u, p>1, 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.

are positive continuous function on the interval [0, + co). By solution we always mean a continuously differentiable function such that r(t) x’(t) has a continuous derivative satisfying (E,). When the func- tion r is continuous but does not have a continuous derivative, (E,) can be interpreted as the first order system for the vector (x, y) = (x, rx’). 381

We give necessary and sufficient conditions for the solutions of the differential equation ( p ( t ) x ′( t ))′ = q ( t ) x ( t ) to be bounded together with their first derivatives. We also study the asymptotic behavior of the solutions.