Domínguez Benavides, TomásJapón Pineda, María de los Ángeles2025-07-212025-07-212021-03-15Domínguez Benavides, T. y Japón Pineda, M.d.l.Á. (2021). Fixed point properties and reflexivity in variable Lebesgue spaces. Journal of Functional Analysis, 280 (6), 108896-º. https://doi.org/https://doi.org/10.1016/j.jfa.2020.108896.0022-12361096-0783https://hdl.handle.net/11441/175464In this paper the weak fixed point property (w-FPP) and the fixed point property (FPP) in Variable Lebesgue Spaces are studied. Given (Ω,Σ,μ) a σ-finite measure and p(⋅) a variable exponent function, the w-FPP is completely characterized for the variable Lebesgue space Lp(⋅)(Ω) in terms of the exponent function p(⋅) and the absence of an isometric copy of L1[0,1]. In particular, every reflexive Lp(⋅)(Ω) has the FPP and our results bring to light the existence of some nonreflexive variable Lebesgue spaces satisfying the w-FPP, in sharp contrast with the classic Lebesgue Lp-spaces. In connection with the FPP, we prove that Maurey's result for L1-spaces can be extended to the larger class of variable Lp(⋅)(Ω) spaces with order continuous norm, that is, every reflexive subspace of Lp(⋅)(Ω) has the FPP. Nevertheless, Maurey's converse does not longer hold in the variable setting, since some nonreflexive subspaces of Lp(⋅)(Ω) satisfying the FPP can be found. As a consequence, we discover that several nonreflexive Nakano sequence spaces ℓpn do have the FPP endowed with the Luxemburg norm. As far as the authors are concerned, this family of sequence spaces gives rise to the first known nonreflexive classic Banach spaces enjoying the FPP without requiring of any renorming procedure. The failure of asymptotically isometric copies of ℓ1 in Lp(⋅)(Ω) is also analyzed.application/pdf22 p.engAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Variable Lebesgue spacesFixed point propertyNonexpansive mappingsBanach function latticesinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.1016/j.jfa.2020.108896