A new result on the existence of periodic solutions for Liénard equations with a singularity of repulsive type

摘要

In this paper, the problem of the existence of a periodic solution is studied for the second order differential equation with a singularity of repulsive type x ″ ( t ) + f ( x ( t ) ) x ′ ( t ) − g ( x ( t ) ) + φ ( t ) x ( t ) = h ( t ) , $$x''(t)+figl(x(t)igr)x'(t)-gigl(x(t) igr)+arphi(t)x(t)=h(t), $$ where g ( x ) $g(x)$ is singular at x = 0 $x=0$ , φ and h are T-periodic functions. By using the continuation theorem of Manásevich and Mawhin, a new result on the existence of positive periodic solution is obtained. It is interesting that the sign of the function φ ( t ) $arphi(t)$ is allowed to change for t ∈ [ 0 , T ] $tin[0,T]$ .