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In this paper, we study the maximum number of limit cycles for the piecewisesmooth system of differential equations $\dot{x}=y, \ \dot{y}=-x-\varepsilon\cdot (f(x)\cdot y +{\rm sgn}(y)\cdot g(x))$. Using the averaging method, wewere able to generalize a previous result for Li\'enard systems. In ourgeneralization, we consider $g$ as a polynomial of degree $m$. We conclude thatfor sufficiently small values of $|\epsilon|$, the number$\left[\frac{n}{2}\right]+\left[\frac{m}{2}\right]+1$ serves as a lower boundfor the maximum number of limit cycles in this system, which bifurcates fromthe periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$.Furthermore, we demonstrate that it is indeed possible to achieve such a numberof limit cycles.

Sharp estimates for the number of limit cycles in discontinuous generalized Li\'enard equations