Eigenvalue type problem in $s(\mathrm{.},\mathrm{.})$-fractional Musielak-Sobolev spaces

In this paper, first we introduce the $s(\mathrm{.},\mathrm{.})$-fractional Musielak-Sobolev spaces $W^{s(x,y)}{L}_{{\Phi }_{x,y}}(\mathrm{\Omega })$. Next, by means of Ekeland’s variational principal, we show that there exists ${\lambda }_{\ast }>0$ such that any $\lambda \in (0,{\lambda }_{\ast })$ is an eigenvalue for the following problem $$({\mathcal{P}}_a)\{\begin{array}{ll}{(-\mathrm{\Delta })}_{a_{(x,\mathrm{.})}}^{s(x,\mathrm{.})}u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q(x)-2}u& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0& \mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^N$ with $C^{0,1}$-regularity and bounded boundary.