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In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following formwhere, Ω is an open bounded subset inwith Lipschitz boundary, andis a Carathéodory function with suitable growth conditions.

On a class of nonlocal problems in new fractional Musielak-Sobolev spaces