Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
843166 | Nonlinear Analysis: Theory, Methods & Applications | 2008 | 6 Pages |
Abstract
In this work, we prove among other results that if S is a right amenable semigroup and Ï={Ts:sâS} is a (quasi-)nonexpansive semigroup on a closed, convex subset C in a strictly convex reflexive Banach space E such that the set F(Ï) of common fixed points of Ï is nonempty, then there exists a (quasi-)nonexpansive retraction P from C onto F(Ï) such that PTt=TtP=P for each tâS and every closed convex Ï-invariant subset of C is also P-invariant. Moreover, if the mappings are also affine then Tμ [G. Rode, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl. 85 (1982) 172-178. [12]] is a quasi-contractive affine retraction from C onto F(Ï), such that TμTt=TtTμ=Tμ for each tâS, and Tμxâco¯{Ttx:tâS} for each xâC; and if R is an arbitrary retraction from C onto F(Ï) such that Rxâco¯{Ttx:tâS} for each xâC, then R=Tμ. It is shown that if the Tt's are F(Ï)-quasi-contractive then the results hold without the strict convexity condition on E.
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Authors
Shahram Saeidi,