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

We study fixed point theory for semi-strongly asymptotically nonexpansive and strongly asymptotically nonexpansive mappings. We consider these mappings for renormings of $l^1$ and $c_0$, and show that $l^1$ cannot be equivalently renormed to have the fixed point property for semi-strongly asymptotically nonexpansive mappings, while $c_0$ cannot be equivalently renormed to have the fixed point property for strongly asymptotically nonexpansive mappings Next and more importantly, we show reflexivity is equivalent to the fixed point property for affine semi-strongly asymptotically nonexpansive mappings in Banach lattices. Finally, we give an application of our results in Lorentz-Marcinkiewicz spaces $l_{w,\infty}^0$, and some examples of these new types of mappings associated with a large class of $c_0$-summing basic sequences in $c_0$.