Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, x''(t)+f(x(t))x'(t)+φ (t)xμ(t)-1xγ(t)=e(t), x ″ ( t ) + f ( x ( t ) ) x ′ ( t ) + φ ( t ) x μ ( t ) − 1 x γ ( t ) = e ( t ) , where f:(0,+∞ )→ R f : ( 0 , + ∞ ) → R is continuous, which may have a singularity at the origin, the sign of φ (t) φ ( t ) , $e(t)$ e ( t ) is allowed to change, and μ , γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T -periodic solution when μ ∈ [0,+∞ ) μ ∈ [ 0 , + ∞ ) .