Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type

Abstract

Abstract In this paper, we study the periodic problem for the Liénard equation with an indefinite singularity of attractive type x″+f(x)x′+φ(t)x+r(t)xμ=0, $$ x''+f(x)x'+\varphi (t)x+\frac{r(t)}{x^{\mu }}=0, $$ where f:(0,+∞)→R $f:(0,+\infty )\rightarrow R$ is continuous and may have singularities at zero, r, φ:R→R $\varphi : R\rightarrow R$ are T-periodic functions, and μ is a positive constant. Using the method of upper and lower functions, we obtain some new results on the existence of positive periodic solutions to the equation.

Keywords

• Boundary value problem
• Periodic solution
• Singularity
• Upper and lower functions