Fixed point theorems in reflexive Banach spaces

In this paper fixed point theorems are established first for mappings T T , mapping a closed bounded convex subset K K of a reflexive Banach space into itself and satisfying \[ | | T x − T y | | ≦ 1 2 { | | x − T x | | + | | y − T y | | , x , y ∈ K , ||Tx - Ty|| 1/2\{ ||x - Tx|| + ||y - Ty||, x,y ∈ K, \] and then an analogous result is obtained for nonexpansive mappings giving rise to a question regarding the unification of these theorems.