  In this paper the weak fixed point property ($w$-FPP) and the fixed point
property (FPP) in Variable Lebesgue Spaces are studied. Given
$(\Omega,\Sigma,\mu)$ a $\sigma$-finite measure and $p(\cdot)$ a variable
exponent function, the $w$-FPP is completely characterized for the variable
Lebesgue space $L^{p(\cdot)}(\Omega)$ in terms of the exponent function
$p(\cdot)$ and the absence of an isometric copy of $L_1[0,1]$. In particular,
every reflexive $L^{p(\cdot)}(\Omega)$ 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 $L^p$-spaces. In
connection with the FPP, we prove that Maurey's result for $L^1$-spaces can be
extended to the larger class of variable $L^{p(\cdot)}(\Omega)$ spaces with
order continuous norm, that is, every reflexive subspace of
$L^{p(\cdot)}(\Omega)$ has the FPP. Never\-theless, Maurey's converse does not
longer hold in the variable setting, since some nonreflexive subspaces of
$L^{p(\cdot)}(\Omega)$ satisfying the FPP can be found. As a consequence, we
discover that several nonreflexive Nakano sequence spaces $\ell^{p_n}$ 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 asympto\-tically isometric copies of $\ell_1$ in
$L^{p(\cdot)}(\Omega)$ is also analyzed.
