Infinity weak solutions for a nonlocal Fractional problem with different boundary conditions

In this paper, we investigate two nonlinear equations of fractional type under different external conditions. The first part of this study aims to prove the existence of an infinite number of solutions for nonlocal elliptic problems with non-homogeneous Neumann boundary conditions. The proof is guaranteed by exploiting the correct oscillatory behavior of non-smooth terms. The second section of the paper examines a class of nonlocal elliptic problems in which non-smooth components exhibit a mixed effect of concave and convex nonlinearity at Dirichlet boundary conditions. The nonlinearities do not satisfy Ambrosetti-Rabinowitz and monotonicity conditions. Our framework is a Fractional Orlicz-Sobolev space. To establish the main result, we apply variational approaches paired with Ekeland's variational principle.