Global Kneser solutions to nonlinear equations with indefinite weight

The paper deals with the nonlinear differential equation [ igl(a(t)Phi(x^{prime})igr)^{prime}+b(t)F(x)=0, quad tin [1,infty) ] in the case when the weight $b$ has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions $x$ such that $x(t)>0, x^{prime}(t)<0$ on the whole closed interval $[1,infty),$ is considered. Moreover, conditions assuring that these solutions tend to zero as $t ightarrowinfty$ are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight }$b$ {small is a periodic function or it is unbounded from below.