Fixed point properties and reflexivity in variable Lebesgue spaces

In this paper the weak fixed point property (w-FPP) and (FPP) in Variable Lebesgue Spaces are studied. Given (Ω,Σ,μ) a σ-finite measure p(⋅) variable exponent function, w-FPP is completely characterized for space Lp(⋅)(Ω) terms of function absence an isometric copy L1[0,1]. particular, every reflexive has FPP our results bring to light existence some nonreflexive spaces satisfying w-FPP, sharp contrast with classic Lp-spaces. connection FPP, we prove that Maurey's result L1-spaces can be extended larger class order continuous norm, is, subspace FPP. Nevertheless, converse does not longer hold setting, since subspaces found. As consequence, discover several Nakano sequence ℓpn do have endowed Luxemburg norm. far as authors concerned, family gives rise first known Banach enjoying without requiring any renorming procedure. The failure asymptotically copies ℓ1 also analyzed.