Fractional eigenvalues in Orlicz spaces with no ${\mathrm{\Delta }}_2$ condition

We study the eigenvalue problem for the $g-$Laplacian operator in fractional order Orlicz-Sobolev spaces, where $g=G^{\mathrm{\prime }}$ and neither $G$ nor its conjugated function satisfy the ${\mathrm{\Delta }}_2$ condition. Our main result is the existence of a nontrivial solution to such a problem; this is achieved by first showing that the corresponding minimization problem has a solution and then applying a generalized Lagrange multiplier theorem to get the existence of an eigenvalue. Further, we prove closedness of the spectrum and some properties of the eigenvalues and, as an application, we show existence for a class of nonlinear eigenvalue problems.