Let C be a convex subset of a Banach space X and let T be a mapping from C into C. Fix \(\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)\) a multi-index in \({\mathbb {R}}^n\) such that \(\alpha _i\ge 0\) (\(1\le i\le n\)), \(\sum _{i=1}^n\alpha _i=1\), \(\alpha _1,\alpha _n>0\), and consider the mapping \(T_\alpha :C\rightarrow C\) given by \(T_\alpha =\sum _{i=1}^n \alpha _i T^i\). Every fixed point of T is a fixed point for \(T_\alpha \) but the converse does not hold in general. In this paper we study necessary and sufficient conditions to assure the existence of fixed points for T in terms of the existence of fixed points of \(T_\alpha \) and the behaviour of the T-orbits of the points in the domain of T. As a consequence, we prove that the fixed point property for nonexpansive mappings and the fixed point property for mean nonexpansive mappings are equivalent conditions when the involved mappings are affine. Some extensions for more general classes of mappings are also achieved.

令C为Banach空间X的凸子集，令T为从C到C的映射。修复\（{\ mathbb {R}} ^ n \）中的多索引\（\ alpha =（\ alpha _1，\ alpha _2，\ ldots，\ alpha _n）\）使得\（\ alpha _i \ ge 0 \）（\（1 \ le i \ le n \）），\（\ sum _ {i = 1} ^ n \ alpha _i = 1 \），\（\ alpha _1，\ alpha _n> 0 \ ），并考虑由\（T_ \ alpha = \ sum _ {i = 1} ^ n \ alpha _i T ^ i \）给出的映射\（T_ \ alpha：C \ rightarrow C \）。T的每个固定点都是\（T_ \ alpha \）的固定点但相反的说法并不普遍。在本文中，我们研究的充分必要条件，以确保固定点的存在Ť中的固定点的存在方面\（T_ \阿尔法\）和行为Ť中的域的点的-orbits Ť。结果，我们证明了当涉及的映射是仿射时，非扩展映射的定点属性和平均非扩展映射的定点属性是等效条件。还实现了对更通用的映射类的一些扩展。