Projection schemes and the fixed point property in nonreflexive Banach spaces

We introduce the notion of a projection scheme on a Banach space and develop an abstract framework for studying the fixed point property (FPP) for nonexpansive mappings. Within this framework, we obtain a general criterion ensuring the FPP for a broad class of nonreflexive Banach spaces. In particular, every uniformly convex Banach space admits an isometric embedding into a nonreflexive Banach space having the FPP. This criterion unifies and extends several previously known fixed point results for nonreflexive Banach spaces with Schauder bases. As an application of our general result, we characterize the FPP for variable Lebesgue spaces. More precisely, we prove that if $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ is a $\sigma$-finite measure space and $p:\mathrm{\Omega }\to [1,\mathrm{\infty }]$ is a measurable function, then $L^{p(\cdot )}(\mathrm{\Omega })$ has the FPP if and only if it does not contain an isometric copy of ${\mathrm{\ell }}_1$. Consequently, as in the classical Lebesgue space framework, the presence of an isometric copy of ${\mathrm{\ell }}_1$ completely characterizes the failure of the FPP in variable Lebesgue spaces. This result yields a class of nonreflexive Banach spaces with the fixed point property that contains isometric copies of infinitely many $L^p$-spaces, $1<p<\mathrm{\infty }$. Although nonreflexive sequence spaces with the FPP are known, to the best of our knowledge, these variable Lebesgue spaces provide the first examples of nonreflexive Banach function lattices with the FPP.