Abstract
Assume that $(\Omega, \Sigma, \mu)$ is a $\sigma$-finite measure space and $p\colon\Omega\to [1,\infty]$ 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 $\ell^{p(k)}$, where $p(k)$ is a sequence in $[1,\infty]$, 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 $L^1([0,1])$. In the case of an arbitrary $\sigma$-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 [19]. According to our results, this is the case, at least, for pointwise eventually nonexpansive mappings in Lebesgue spaces with variable exponents.
Citation
Tomas Domínguez Benavides. "Fixed point for mappings of asymptotically nonexpansive type in Lebesgue spaces with variable exponents." Topol. Methods Nonlinear Anal. 63 (1) 23 - 38, 2024. https://doi.org/10.12775/TMNA.2023.044
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