Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

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In this paper, we study the limit cycles for m-piecewise discontinuous polynomial Lienard differential systems of degree n with m/2 straight lines passing through the origin whose slopes are $$\tan (\alpha + 2j\pi /m)$$ for $$j = 0, 1, \ldots , m/2 -1$$ , and prove that for any positive even number m, if $$\sin ( m\alpha /2)\ne 0$$ , then there always exists such a system possessing at least $$\left[ \frac{1}{2}(n-\frac{m-2}{2}) \right] $$ limit cycles. This result verifies a conjecture proposed by Llibre and Teixerira (Z Angew Math Phys 66:51–66, 2015).