  In this paper, we consider the existence of solutions of the following
nonhomogeneous fractional $p(x,.)$-Laplacian Dirichlet problem:
\begin{equation*} \left\{\begin{aligned} \Big(-\Delta_{p(x,.)}\Big)^s u
(x)&=f(x, u) &\text { in }& \Omega, u &=g &\text { in }& \mathbb{R}^N
\setminus\Omega, \end{aligned}\right. \end{equation*} where
$\Omega\subset\mathbb{R}^N$ is a smooth bounded domain,
$\Big(-\Delta_{p(x,.)}\Big)^s$ is the fractional $p(x,.)$-Laplacian, $f$ is a
Carath\'eodory function with suitable growth condition and $g$ is a given
boundary data. The proof of our main existence results relies on the study of
the fractional $p(x, \cdot)$-Poisson equation with a nonhomogeneous Dirichlet
boundary condition and the theory of fractional Sobolev spaces with variable
exponents, together with Schauder's fixed point theorem.
