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In this work, we establish some abstract results on the perspective of thefractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Rieszproperty with respect to the modular function, $(S_{+})$-property, Brezis-Liebtype Lemma to the modular function and monotonicity results. Moreover, we applythe theory developed to study the existence of solutions to the following classof nonlocal problems  \begin{equation*}  \left\{  \begin{array}{ll}  (-\Delta)_{\Phi_{x,y}}^s u = f(x,u),& \mbox{in }\Omega,  u=0,& \mbox{on }\mathbb{R}^N\setminus \Omega,  \end{array}  \right.  \end{equation*}  where $N\geq 2$, $\Omega\subset \mathbb{R}^N$ is a bounded domain withLipschitz boundary $\partial \Omega$ and $f:\Omega \times \mathbb{R}\rightarrow \mathbb{R}$ is a Carath\'{e}odory function not necessarilysatisfying the Ambrosetti-Rabinowitz condition. Such class of problems enablesthe presence of many particular operators, for instance, the fractionaloperator with variable exponent, double-phase and double-phase with variableexponent operators, anisotropic fractional $p$-Laplacian, among others.

On Fractional Musielak-Sobolev spaces and applications to nonlocal problems