Fixed point for mappings of asymptotically nonexpansive type in Lebesgue spaces with variable exponents

Assume that (Ω, Σ, μ) is a σ-finite measure space and pΩ→ [1,∞] a variable exponent. In the case of a purely atomic measure, we prove that the w-FPP for mappings of asymptotically nonexpansive type in the Nakano space ᵖ⁽ᵏ⁾, where $p(k)$ is a sequence in [1,∞], is equivalent to several geometric properties of the space, as weak normal structure, the w-FPP for nonexpansive mappings and the impossibility of containing isometrically L¹([0,1]). In the case of an arbitrary σ-finite measure, we prove that this characterization also holds for pointwise eventually nonexpansive mappings. To determine if the w-FPP for nonexpansive mappings and for mappings of asymptotically nonexpansive type are equivalent is a long standing open question {Ki3}. According to our results, this is the case, at least, for pointwise eventually nonexpansive mappings in Lebesgue spaces with variable exponents.