Positive solutions of nonlinear differential equations with prescribed decay of the first derivative

An existence and uniqueness result for bounded, positive solutions x(t) of the equation u′′+f(t,u,u′)=0, t⩾t0⩾0, is established by means of the Banach contraction principle. For such a solution it is shown that α(t)⩽x′(t)⩽β(t), t⩾t0, where α, β are given nonnegative, continuous functions which are integrable over [t0,+∞). The result complements others known in the literature.