Estimates for the Number of Limit Cycles in Discontinuous Generalized Liénard Systems with Bivariate Perturbation

We study the number of limit cycles of the piecewise smooth differential system $$\dot{x}=y, \quad \dot{y}=-x-\varepsilon\bigl(f(x)y+\operatorname{sgn}(x)g(x,y)\bigr),$$ where $f(x)$ is a polynomial of degree $n\geq 1$, $g(x,y)$ is a bivariate polynomial of degree $m\geq 2$, and $\varepsilon$ is a s