We consider the non-local equations with non-negative \(L^1\) -data in the fractional Musielak–Sobolev spaces. Utilizing approximation and energy methods, we establish the existence and uniqueness of non-negative renormalized solutions for such problems. The operators discussed in this work include the fractional Orlicz operators with variable exponents, the fractional double-phase operators with variable exponents, and the anisotropic fractional p-Laplacian operators, among others.
