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We study the eigenvalue problem for the $g-$Laplacian operator in fractionalorder Orlicz-Sobolev spaces, where $g=G'$ and neither $G$ nor its conjugatedfunction satisfy the $\Delta_2$ condition. Our main result is the existence ofa nontrivial solution to such a problem; this is achieved by first showing thatthe corresponding minimization problem has a solution and then applying ageneralized Lagrange multiplier theorem to get the existence of an eigenvalue.Further, we prove closedness of the spectrum and some properties of theeigenvalues and, as an application, we show existence for a class of nonlineareigenvalue problems.

Fractional eigenvalues in Orlicz spaces with no $\Delta_2$ condition