Existence of fixed points for mappings on a ball of a reflexive and separable Banach space

Let X be a reflexive and separable Banach space, Br denotes the closed ball of radius r in X and f : Br \to X a continuous mapping in the weak topology of X satisfying the following boundary condition \vert fx \vert < \vert fx - x \vert + \vert x \vert for all x ϵ \partial B_r. This paper contains some theorems of existence of a fixed point under the above boundary condition. Moreover a connection with Leray Schauder nonlinear alternative.