Fixed point properties of semigroups of non-expansive mappings |
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Authors: | Anthony To-Ming Lau Yong Zhang |
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Institution: | a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada b Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Canada |
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Abstract: | In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976. |
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Keywords: | Fixed point property Non-expansive mapping Weakly compact convex set Weakly almost periodic Reversible semigroup Invariant mean Bicyclic semigroup |
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