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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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