A fixed point theorem for eventually nonexpansive semigroups of mappings

Let K be a subset of a Banach space X. A semigroup $\text{F}$ = {?_α ∥ α ∞ A} of Lipschitz mappings of K into itself is called eventually nonexpansive if the family of corresponding Lipschitz constants $\text{{k}{}_{\mathrm{\alpha }}\text{¦ α ? A}}$ satisfies the following condition: for every ? > 0, there is a γ?A such that k_β < 1 + ? whenever ?_β ? ?_γ$\text{F}$ = {?_gg?_α ¦ ?_α ? $\text{F}$}. It is shown that if K is a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space, and if $\text{F}$:K → K is an eventually nonexpansive, commutative, linearly ordered semigroup of mappings, then $\text{F}$ has a common fixed point. This result generalizes a fixed point theorem by Goebel and Kirk.