₁ spreading models and the FPP for Ces\`aro mean nonexpansive maps

Let K be a nonempty set in a Banach space X. A mapping T K→ K is called { cm-nonexpansive} if for any sequence (xᵢ)ᵢ₌₁ⁿ and y in K, one has \|(1/n) ∑ᵢ₌₁ⁿ Txᵢ -Ty\|≤ \|(1/n)∑ᵢ₌₁ⁿ xᵢ - y\|. As a subsclass of nonexpansive maps, the FPP for such maps is well-established in a great variety of spaces. The main result of this paper is a fixed point result relating cm-nonexpansiveness, ₁ spreading models and Schauder bases with not-so-large basis constants. As a result, we deduce that every Banach space with weak Banach-Saks property has the fixed point property for cm-nonexpansive maps.