In this paper, we consider the existence of solutions of the following nonhomogeneous fractional \( \varvec{p(x,.)} \) -Laplacian Dirichlet problem: \(\begin{aligned} \left\{ \begin{aligned} \Big (-\varvec{\Delta }_{\varvec{p(x,.)}}\Big )^{\varvec{s}} \varvec{u(x)}&=\varvec{f(x, u)}&\text{ in }&\varvec{\Omega ,} \\ \varvec{u}&=\varvec{g}&\text{ in }&\mathbb {R}^{\varvec{N}} \setminus \varvec{\Omega ,} \end{aligned}\right. \end{aligned}\) where \( \varvec{\Omega }\subset \mathbb {R}^{\varvec{N}} \) is a smooth bounded domain, \( \Big (-\varvec{\Delta }_{\varvec{p(x,.)}}\Big )^{\varvec{s}} \) is the fractional \( \varvec{p(x,.)} \) -Laplacian, \( \varvec{f} \) is a Carathéodory function with suitable growth condition and \( \varvec{g} \) is a given boundary data. The proof of our main existence results relies on the study of the fractional \( \varvec{p(x, \cdot )} \) -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.
