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

Burton, T. A.; Furumochi, Tetsuo

Funkc. Ekvacioj, Ser. Int. 44, No. 1, 73-82 (2001).

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