  We consider structured optimisation problems defined in terms of the sum of a
smooth and convex function, and a proper, l.s.c., convex (typically non-smooth)
one in reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due
to their intrinsic space-variant properties, such spaces can be naturally used
as solution space and combined with space-variant functionals for the solution
of ill-posed inverse problems. For this purpose, we propose and analyse two
instances (primal and dual) of proximal gradient algorithms in
$L_{p(\cdot)}(\Omega)$, where the proximal step, rather than depending on the
natural (non-separable) $L_{p(\cdot)}(\Omega)$ norm, is defined in terms of its
modular function, which, thanks to its separability, allows for the efficient
computation of algorithmic iterates. Convergence in function values is proved
for both algorithms, with convergence rates depending on problem/space
smoothness. To show the effectiveness of the proposed modelling, some numerical
tests highlighting the flexibility of the space $L_{p(\cdot)}(\Omega)$ are
shown for exemplar deconvolution and mixed noise removal problems. Finally, a
numerical verification on the convergence speed and computational costs of both
algorithms in comparison with analogous ones defined in standard
$L_{p}(\Omega)$ spaces is presented.
