The paper studies the nonlinear operator equation
on ordered Banach spaces, where
an increasing sub-homogeneous operator, and
a homogeneous operator. By using the properties of cones and a fixed point theorem for increasing general
-concave operators, some new results on the existence and uniqueness of positive solutions are obtained. Applications are made to two classes of nonlinear problems; they include fourth-order two-point boundary value problems for elastic beam equations and elliptic boundary value problems for Lane-Emden-Fowler equations.