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Periodic solutions of Liénard equations with resonant isochronous potentials

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  • In this paper, we study the existence and multiplicity of periodic solutions of Liénard equations $$ x''+f(x)x'+V'(x)+g(x)=p(t), $$ where $V$ is a $2\pi/n$-isochronous potential. When $F(F(x)=\int_0^xf(s)ds)$ and $g$ are bounded, we provide new sufficient conditions to ensure the existence of periodic solutions of this equations. Moreover, we prove the multiplicity of periodic solutions of the given equations under certain bounded conditions by using topological degree method. When $F, g$ satisfy a certain class of unbounded conditions, we also give sufficient conditions to ensure the existence of $2\pi$-periodic solutions of the given equations.
    Mathematics Subject Classification: Primary: 34C25; Secondary: 34B15.

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