A new result on the existence of periodic solutions for Rayleigh equation with a singularity

Abstract

Abstract In this paper, we study the existence of periodic solutions for Rayleigh equation with a singularity of repulsive type x ″ ( t ) + f ( x ′ ( t ) ) + φ ( t ) x ( t ) − 1 x α ( t ) = p ( t ) , $$x''(t)+f\bigl(x'(t)\bigr)+\varphi (t)x(t)-\frac{1}{x^{\alpha }(t)}=p(t), $$ where α ⩾ 1 $\alpha \geqslant 1$ is a constant, and φ and p are T-periodic functions. The proof of the main result relies on a known continuation theorem of coincidence degree theory. The interesting point is that the sign of the function φ ( t ) $\varphi (t)$ is allowed to change for t ∈ [ 0 , T ] $t\in [0,T]$ .

Keywords

• second order differential equation
• continuation theorem
• singularity
• periodic solution