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On the existence of harmonic solutions of Liénard systems. (English) Zbl 0735.34033

The author considers Liénard systems of the form (*) \(\ddot x+(d/dt)\text{grad}F(x)+\text{grad}G(x)=p(t)\), \(F\in C^ 2(R^ n,R)\), \(G\in C^ 1(R^ n,R)\), \(p\in C^ 0(R,R^ n)\) and \(p(t+T)=p(t)\) for a constant \(T>0\), and proves two theorems for the existence of harmonic solutions, in which the conditions are somewhat weaker than those assumed in other papers quoted by the author. Denoting by \((\cdot,\cdot)\) the inner product in \(R^ n\) and assuming \(| x|=\sqrt{(x,x)}\) for \(x\in R^ n\), Theorem 2 is:
Suppose that \(\int^ T_ 0 p(s)ds=0\), and that there is a nonsingular matrix A and a constant \(R_ 0>0\) such that \((Ax,\text{grad}G(x))>0\) for \(x\in R^ n\) with \(| x|>R_ 0\), \[ \lim_{| x| \to \infty}|(\text{grad}F(x),\text{grad}G(x))|/|\text{grad}G(x)|>M=\max_{[0,T]}| P(t)|. \] Then system (*) has at least one harmonic solution.
Reviewer: S.Nocilla (Torino)

MSC:

34C25 Periodic solutions to ordinary differential equations
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References:

[1] Ding, Tongren, An existence theorem for harmonic solutions of periodically perturbed systems of Duffing’s type (1987), preprint · Zbl 0757.34032
[2] Ding, Weiyue, On the existence of periodic solutions for Liénard systems, Acta math. Sinica, 25, 626-632 (1982) · Zbl 0514.34035
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