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

• Main Authors: Lijuan Chen, Shiping Lu
• Format: Article
• Language: English
• Published: SpringerOpen 2017-04-01
• Series: Advances in Difference Equations
• Subjects: Rayleigh equation; topological degree; singularity; periodic solution
• Online Access: http://link.springer.com/article/10.1186/s13662-017-1136-z

Abstract In this paper, the problem of the existence of periodic solutions is studied for the second-order differential equations with a singularity of repulsive type, x ″ ( t ) + f ( x ′ ( t ) ) + φ ( t ) x ( t ) − 1 x r ( t ) = h ( t ) , $$ x''(t)+f\bigl(x'(t)\bigr)+\varphi(t)x(t)- \frac{1}{x^{r}(t)}=h(t), $$ where φ and h are T-periodic functions. By using topological degree theory, a new result on the existence of positive periodic solutions is obtained. The interesting thing is that the sign of the function φ ( t ) $\varphi(t)$ is allowed to be changed for t ∈ [ 0 , T ] $t\in[0,T]$ .