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

Abstract

Abstract 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 ) x μ ( t ) − 1 x γ ( t ) = e ( t ) , $$ x''(t)+f(x(t))x'(t)+\varphi (t)x^{\mu}(t)-\frac{1}{x^{\gamma}(t)}=e(t), $$ where f : ( 0 , + ∞ ) → R $f:(0,+\infty )\rightarrow R$ is continuous, which may have a singularity at the origin, the sign of φ ( t ) $\varphi (t)$ , 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 , + ∞ ) $\mu \in [0,+\infty )$ .

Keywords

• Liénard equation
• Periodic solutions
• Singularity
• Continuation theorem