## A note on stability by Schauder’s theorem.(English)Zbl 1158.34329

From the introduction: We consider an equation
$x''+2f(t)x'+x+g(t,x)=0,\quad t\in \mathbb R^+,\tag{1}$
with a prototype being $$f(t)=1/(t+1)$$ and $$g(t,x) =x^2/(t+1)$$. The linear part is asymptotically stable, but not uniformly asymptotically stable; and this makes it difficult to obtain asymptotic stability for a perturbed system.
The question we propose to answer here is: “How can we effectively use fixed point theory to prove that the zero solution of (1) is asymptotically stable?”

### MSC:

 34D20 Stability of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 47H10 Fixed-point theorems