The purpose of this work is two-fold. On the one side, we focus on the space of real convergent sequences c where we study non-weakly compact sets with the fixed point property. Our approach brings a positive answer to a recent question raised by Gallagher et al. in (J Math Anal Appl 431(1):471–481, 2015). On the other side, we introduce a new metric structure closely related to the notion of relative uniform normal structure, for which we show that it implies the fixed point property under adequate conditions. This will provide some stability fixed point results in the context of hyperconvex metric spaces. As a particular case, we will prove that the set \(M=[-1,1]^\mathbb {N}\) has the fixed point property for d-nonexpansive mappings where \(d(\cdot ,\cdot )\) is a metric verifying certain restrictions. Applications to some Nakano-type norms are also given.

这项工作的目的有两个。一方面，我们关注实收敛序列c的空间，在那里我们研究具有不动点性质的非弱紧集。我们的方法为 Gallagher 等人最近提出的一个问题带来了积极的答案。在 (J Math Anal Appl 431(1):471–481, 2015)。另一方面，我们引入了一种与相对统一正规结构的概念密切相关的新度量结构，为此我们表明它暗示了在适当条件下的不动点属性。这将在超凸度量空间的上下文中提供一些稳定性不动点结果。作为一个特例，我们将证明集合\(M=[-1,1]^\mathbb {N}\)具有d -nonexpansive 映射的不动点性质，其中\(d(\cdot ,\cdot )\)是验证某些限制的指标。还给出了一些 Nakano 型规范的应用。