Common Fixed Point Theorems for Almost Weakly Periodic Nonexpansive Mappings
Kok-Keong Tan · Proceedings of the American Mathematical Society · 1972
The notions of normal structure, (convex) diminishing orbital diameters, regular orbital diameters (r.o.d.) have been generalized into a Hausdorff locally convex space $(X,\tau )$ whose topology $\tau$ is generated by a family $\mathcal {P}$ of seminorms. Theorem 1. Let $K \subseteq X$ be nonempty weakly compact convex with normal structure w.r.t. $\mathcal {P}$ and $\mathcal {F}$ be a (not necessarily finite nor commuting) family of almost weakly periodic nonexpansive mappings w.r.t. $\mathcal {P}$ on K. Then $\mathcal {F}$ has a common fixed point. Theorem 2. Let $K \subseteq X$ be nonempty weakly compact convex and $\mathcal {F}$ be a semigroup with identity of almost weakly periodic nonexpansive mappings w.r.t. $\mathcal {P}$ on K. If $\mathcal {F}$ has r.o.d. w.r.t. $\mathcal {P}$, then $\mathcal {F}$ has a common fixed point. Corollary. If $K \subseteq X$ is nonempty weakly compact convex and $\mathcal {F} = \{ {f_1}, \cdots ,{f_n}\}$ is a finite commuting family of pointwise periodic nonexpansive mappings w.r.t. $\mathcal {P}$ on K, then $\mathcal {F}$ has a common fixed point.