Existence results of non-local integro-differential problem with singularity under a new fractional Musielak–Sobolev space*

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Abstract In this article, we construct a new fractional Musielak-Sobolev space designed to address the non-locality associated with the corresponding integro-differential operators. The completeness, reflexivity, and uniform convexity of this new space, along with an associated embedding theorem are established. As a practical application, we investigate the solvability of a class of non-local problems with singular nonlinearities under the new fractional Musielak-Sobolev space. Based on appropriate assumptions, we derive an existence result for two weak solutions utilizing the fibering method in form of the Nehari manifold. Furthermore, we provide two concrete operator examples within the framework of the fractional Musielak-Sobolev space as examples.