Semi-strongly Asymptotically Non-Expansive Mappings and Their Applications on Fixed Point Theory

Chris Lennard
2017 Hacettepe Journal of Mathematics and Statistics  
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
more » ... vity is equivalent to 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.
doi:10.15672/hjms.20174620775 fatcat:dgy6na7ctjc6bes62om36lkp4m