Abstract
In this note, we extend and improve the corresponding result of Takahashi7. Fixed point theorem for amenable semi group of non-expansive mappings.
Highlights
Let K be a subset of a Banach space E
Let S be a semi-topological semigroup, i.e. S is a semigroup with a Hausdorff topology such that for each a ∈ S, the mappings s 1 sa and s as from S into S are continuous
S is called left amenable if LUC (S) has a left invariant mean (LIM)
Summary
Let K be a subset of a Banach space E. In [1] DeMarr proved the following theorem: Theorem 1.1: For any non-empty compact convex subset K of a Banach space E, each commuting family of non-expansive selfmappings on K has a common fixed point in K. Takahashi [6] proved a generalization of DeMarr’s fixed point theorem as follows: Theorem 1.2: Let K be a non-empty compact convex subset of a Banach space E and S be an amenable discrete semigroup which acts on K separately continuous and non-expansive. It is well-known that every left amenable discrete semigroup is left reversible [4], so Mitchell [7] proved the following theorem: Theorem 1.3: Let K be a non-empty compact convex subset of a Banach space E and S be a left reversible discrete semigroup which acts on K separately continuous and non- expansive. Our theorem is new and is not a result of any previous work
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