In this thesis we are interested in fixed point properties of representations of semi-topological semigroups of non-expansive mappings on weak and weak* compact convex sets in Banach or dual spaces. More particularly, we study the following problems : Problem 1 : Let F be any commuting family of non-expansive mappings on a non-empty weakly compact convex subset of a Banach space such that for each f ∈ F there is an x whose f -orbit has a cluster point (in the norm topology). Does F possess a

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... mon fixed point ? Problem 2 : What amenability properties of a semi-topological semigroup do ensure the existence of a common fixed point for any jointly weakly continuous non-expansive representation on a non-empty weakly compact convex subset of a Banach space ? Problem 3 : Does any left amenable semi-topological semigroup S possess the following fixed point property : (F * ) : Whenever S defines a weak* jointly continuous non-expansive representation on a non-void weak* compact convex set in the dual of a Banach space E, there is a common fixed point for S ? Problem 4 : Is there a fixed point proof of the existence of a left Haar measure for locally compact groups ? Our approach is essentially based on the use of the axiom of choice through Zorn's lemma, amenability techniques and the concept of an ii