We study xed point theory for semi-strongly asymptotically nonexpansive and strongly asymptotically nonexpansive mappings. We consider these mappings for renormings of l 1 and c0, and show that l 1 cannot be equivalently renormed to have the xed point property for semi-strongly asymptotically nonexpansive mappings, while c0 cannot be equivalently renormed to have the xed point property for strongly asymptotically nonexpansive mappings. Next and more importantly, we show reexivity is equivalent

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... o the xed point property for ane semi-strongly asymptotically nonexpansive mappings in Banach lattices. Finally, we give an application of our results in Lorentz-Marcinkiewicz spaces l 0 w,∞ , and some examples of these new types of mappings associated with a large class of c0-summing basic sequences in c0. Keywords: nonexpansive mapping, reexive Banach space, xed point property, weak xed point property, closed bounded convex set, asymptotically isometric c 0 -summing basic sequence, Lorentz-Marcinkiewicz spaces, semi-strongly asymptotically nonexpansive mapping, strongly asymptotically nonexpansive mapping. 2000 AMS Classication: 47H09, 46B42, 46B45, 46B10.