Common Fixed Point Theorems for Almost Weakly Periodic Nonexpansive Mappings

The notions of normal structure, (convex) diminishing orbital diameters, regular orbital diameters (r.o.d.) have been generalized into a Hausdorff locally convex space (X,τ ) whose topology τ is generated by a family P of seminorms. Theorem 1. Let K ⊆ X be nonempty weakly compact convex with normal structure w.r.t. P and F be a (not necessarily finite nor commuting) family of almost weakly periodic nonexpansive mappings w.r.t. P on K. Then F has a common fixed point. Theorem 2. Let K ⊆ X be nonempty weakly compact convex and F be a semigroup with identity of almost weakly periodic nonexpansive mappings w.r.t. P on K. If F has r.o.d. w.r.t. P, then F has a common fixed point. Corollary. If K ⊆ X is nonempty weakly compact convex and F = \ f₁, ⋯ ,fₙ\ is a finite commuting family of pointwise periodic nonexpansive mappings w.r.t. P on K, then F has a common fixed point.