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 $$M=[-1,1]^\mathbb {N}$$ M = [ - 1 , 1 ] N has the fixed point property for d-nonexpansive mappings where $$d(\cdot ,\cdot )$$ d ( · , · ) is a metric verifying certain restrictions. Applications to some Nakano-type norms are also given.

We prove that modular spaces L ρ have the uniform Kadec-Klee property w.r.t. the convergence ρ-a.e. when they are endowed with the Luxemburg norm. We also prove that these spaces have the uniform Opial condition w.r.t. the convergence ρ-a.e. for both the Luxemburg norm and the Amemiya norm. Some assumptions over the modular ρ need to be assumed. The above geometric properties will enable us to obtain some fixed point results in modular spaces for different kind of mappings.  2004 Elsevier Inc. All rights reserved.In this paper we consider modular function spaces L ρ , where ρ is a convex additive modular satisfying the ∆ 2 -type condition. Our main goal is to prove that these spaces satisfy certain geometric properties which let us deduce some fixed point results for different classes of mappings. In order to do that we endow L ρ with either the Luxemburg norm or the Amemiya norm. The convergence ρ-a.e. will be essential in our purpose.The organization of this paper is the following: We begin by introducing the definitions and notations which will be used later. In Section 2 we prove some technical results which are our main tools throughout the paper. In Section 3 we prove that the modular space L ρ ✩

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