The super fixed point property for asymptotically nonexpansive mappings

Abstract

We show that the super fixed point property for nonexpansive mappings and for asymptotically nonexpansive mappings in the intermediate sense are equivalent. As a consequence, we obtain fixed point theorems for asymptotically nonexpansive mappings in uniformly nonsquare and uniformly noncreasy Banach spaces. The results are generalized for commuting families of asymptotically nonexpansive mappings

The Fixed Point Property in Direct Sums and Modulus

“…Theorem 2.4 in [30] shows that X has the super fixed point property for nonexpansive mappings if and only if X has the super fixed point property for asymptotically nonexpansive mappings in the intermediate sense. Since the direct sum (X 1 ⊕ • • • ⊕ X r ) ψ of uniformly nonsquare spaces is stable under passing to the Banach space ultrapowers, it follows from the properties of ultrapowers and Corollary 3.6 (Theorem 3.7) that (X 1 ⊕ • • • ⊕ X r ) ψ with a strictly monotone norm (X 1 ⊕ ψ X 2 with any monotone norm) has the super fixed property for nonexpansive mappings and, consequently, for asymptotically nonexpansive mappings in the intermediate sense.…”

“…Theorem 2.4 in [32] shows that X has the super fixed point property for nonexpansive mappings if and only if X has the super fixed point property for asymptotically nonexpansive mappings in the intermediate sense. Since the direct sum (X 1 ⊕...⊕X r ) ψ of uniformly nonsquare spaces is stable under passing to the Banach space ultrapowers, it follows from the properties of ultrapowers and Corollary 3.6 (resp.…”