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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 (*) $\ddot x+(d/dt)\text{grad}F(x)+\text{grad}G(x)=p(t)$, $F\in C\sp 2(R\sp n,R)$, $G\in C\sp 1(R\sp n,R)$, $p\in C\sp 0(R,R\sp n)$ 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 $R\sp n$ and assuming $\vert x\vert=\sqrt{(x,x)}$ for $x\in R\sp n$, Theorem 2 is: Suppose that $\int\sp T\sb 0 p(s)ds=0$, and that there is a nonsingular matrix A and a constant $R\sb 0>0$ such that $(Ax,\text{grad}G(x))>0$ for $x\in R\sp n$ with $\vert x\vert>R\sb 0$, $$\lim\sb{\vert x\vert \to \infty}\vert(\text{grad}F(x),\text{grad}G(x))\vert/\vert\text{grad}G(x)\vert>M=\max\sb{[0,T]}\vert P(t)\vert.$$ Then system (*) has at least one harmonic solution.

34C25Periodic solutions of ODE
Full Text: DOI
[1] Ding, Tongren: An existence theorem for harmonic solutions of periodically perturbed systems of Duffing’s type. (1987)
[2] Ding, Weiyue: On the existence of periodic solutions for Liénard systems. Acta math. Sinica 25, 626-632 (1982) · Zbl 0514.34035
[3] Gaines, R. E.; Mawhin, J. L.: Coincidence degree, and non-linear differential equations. (1977) · Zbl 0339.47031
[4] Ge Weigao, On harmonic solutions of n-dimensional Liénard systems, Chin. Annls Math. (to appear). · Zbl 0718.34049
[5] Liu, Bin: Existence theorems of harmonic solutions of periodically perturbed systems of Duffing’s type. (1989)
[6] Reissig, R.: Contractive mappings and periodically perturbed nonconservative systems. Atti. accad. Naz. lincei rend. Cl. sci. Fis. mat. Natur. 58, 696-702 (1975) · Zbl 0344.34033
[7] Rouche, N.; Mawhin, J. L.: Equations differentielles ordinaires. 2 (1973) · Zbl 0289.34001