Let X be a Banach space, B a closed ball centered at the origin in X, and T: B → X a pseudo-contractive mapping (i.e., (λ − 1) x − y (λI − T)(x) − (λI − T) (y) for all x, y ε B and λ > 1). It is shown here that the antipodal boundary condition: T(x) = −T(−x) for all x ε δB assures existence of a fixed point of T in B provided that the ball B has the fixed point property with respect to non-expansive self-mappings. Also included are some fixed point theorems which involve the Leray-Schauder condition.