Statements analogous to the Ky Fan approximation theorem are established for \(1\)-set-contractive, semi-contractive, and LANE mappings of the intersection of a closed ball, annulus, and a sphere with a closed cone, \(P\), in a real Banach space. As an application, some fixed point theorems for the continuous mappings \(f : \{x\in P: \left\|x\right\|\leq R\} \to P\), where \(R>0\), are proved.