This paper further studies global asymptotic stability of the zero solusion of Liènard's equation x+f(x)x+g(x) = 0 (1) We obtain the following result: Theorem. The zero solution of equation (1) is globally as...This paper further studies global asymptotic stability of the zero solusion of Liènard's equation x+f(x)x+g(x) = 0 (1) We obtain the following result: Theorem. The zero solution of equation (1) is globally asymptoticly stable if f(x) and g(x) satisfy one of the following conditions. Condition 1: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) integral from n-0 to x g(x)dx→+∞, as |x|→+∞, Condition 2: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) F(x) and F(-x)(x>0) are all infinity,where F(x)= integral from n=0 to x f(x)dx. Compared with [1—3], this result further weakens condition on f(x), thus, it has more extensive working field.展开更多
文摘This paper further studies global asymptotic stability of the zero solusion of Liènard's equation x+f(x)x+g(x) = 0 (1) We obtain the following result: Theorem. The zero solution of equation (1) is globally asymptoticly stable if f(x) and g(x) satisfy one of the following conditions. Condition 1: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) integral from n-0 to x g(x)dx→+∞, as |x|→+∞, Condition 2: 1) xg(x)>0, for all x≠0; 2) x integral from n=0 to x f(x)dx≥0 and on any interval of x, integral from n=0 to x f(x)dx0; 3) F(x) and F(-x)(x>0) are all infinity,where F(x)= integral from n=0 to x f(x)dx. Compared with [1—3], this result further weakens condition on f(x), thus, it has more extensive working field.
