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Semigroup of nonexpansive mappings on a Hilbert space. (English) Zbl 0656.47049

A result of Opial states the following: “Let X be a uniformly convex Banach space with a weakly continuous duality mapping and let C be a closed convex subset of X. If T:C$\to C$ is a nonexpansive mapping having at least one fixed point and if ${lim}_{n\to \infty }\parallel {T}^{n}x-{T}^{n-1}x\parallel =0$, then ${T}^{n}y$ converges weakly to a fixed point of T.”

The present author generalizes this result to right (or left) reversible semitopological semigroups of nonexpansive mappings.

Other existence theorems on common fixed points are proved under certain assumptions of existence of an invariant mean and relative compactness or boundedness on the orbits.

MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 47H20 Semigroups of nonlinear operators