## Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance.(English)Zbl 1044.34002

The purpose of this work is to present some new results on the existence of $$2\pi-$$periodic solutions of the second-order differential Liénard equation with asymmetric nonlinearities $$x''+f(x)x'+ax^{+}-bx^{-}+g(x) = p(t).$$ Here, $$a,b$$ are positive constants satisfying $$a^{-1/2}+b^{-1/2} = 2/n, \;n \in {\mathbb{N}}$$, and $$p$$ is a continuous and $$2\pi$$-periodic function. Also, the limits $$\lim_{x\rightarrow^{+}_{-} \infty}\;F(x), \;F(x) = \int_{0}^{x}f(u)\;du,$$ and $$\lim_{x\rightarrow^{+}_{-} \infty}\;g(x)$$ exist and are finite. By using some previous ideas of related works, the authors define two functions $$\Sigma_{1}$$ and $$\Sigma_{2}$$ which involve the quantities $$a,b,$$ $$F(^{+}_{-}\infty),$$ $$g(^{+}_{-}\infty)$$ and function $$p$$. Then, they prove the existence of $$2\pi$$-periodic solutions under some additional restrictions on the zeros of $$\Sigma_{1}$$ and $$\Sigma_{2}.$$
On the other hand, new nonresonant conditions are discussed if $$F$$ is unbounded and oscillatory and $$g$$ is sublinear. In this case, phase-plane analysis methods are used in the proofs.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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