Applications of Schauder’s fixed point theorem to singular differential equations. (English) Zbl 1128.34027

Chu, Jifeng; Torres, Pedro J.

Bull. Lond. Math. Soc. 39, No. 4, 653-660 (2007).

The paper studies the existence of positive periodic solutions to the second-order singular differential equation \[ x''+a(t)x=f(t,x)+e(t), \] where \(a(t)\) and \(e(t)\) are continuous and \(1\)-periodic and \(f(t,x)\) is \(1\)-periodic in \(t\). The interest is focused on the case when \(f(t,x)\) is singular at \(x=0\). The proof relies on Schauder’s fixed point theorem. It is pointed out that in some situation weak singularities can help to create periodic solutions.

Reviewer: Ivan Ginchev (Varese)

MSC:

34C25	Periodic solutions to ordinary differential equations
47H10	Fixed-point theorems

Keywords:

singular differential equations; periodic solutions; Schauder fixed point theorem