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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 2π-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, and p is a continuous and 2π-periodic function. Also, the limits lim x - + F(x),F(x)= 0 x f(u)du, and lim x - + g(x) exist and are finite. By using some previous ideas of related works, the authors define two functions Σ 1 and Σ 2 which involve the quantities a,b, F( - + ), g( - + ) and function p. Then, they prove the existence of 2π-periodic solutions under some additional restrictions on the zeros of Σ 1 and Σ 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:
34C25Periodic solutions of ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)