A boundary value problem on a half-line for differential equations with indefinite weight

The boundary value problem on the half-line for the second order differential equation with general -Laplacian (a(t) P(x′))′= b(t)F(x), t ≥ 0, x(0) = c > 0, 0 < lim x(t) < ∞, x(t) > 0, lim x′(t) = 0, is considered, where a, b are continuous functions on [0,∞), a is positive and b can change its sign. The cases of regular variation, slow variation, and rapid variation of the inverse function of the generalized p-laplacian P are considered. Some applications of the main results complete the paper.