Renormalized Solutions for the Non-local Equations in Fractional Musielak–Sobolev Spaces

We consider the non-local equations with non-negative L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-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.