Asymptotic behaviour of \(T\)-periodic solutions of singularly perturbed second-order differential equation. (English) Zbl 0863.35026

Vrábeľ, Róbert

Math. Bohem. 121, No. 1, 73-76 (1996).

The singular problem \(\mu y''=f(t,y)\) is considered when \(f\in C^1(\mathbb{R}^2)\) is \(T\)-periodic in \(t\) and \(\mu >0\) is small. Conditions are shown under which the problem has a unique \(T\)-periodic solution defined on \(\mathbb{R}\) for sufficiently small \(\mu >0\) which converges uniformly to the solution of the reduced problem \(f(t,u)=0\) for \(\mu \to 0\).

Reviewer: Š.Schwabik (Praha)

MSC:

34E15	Singular perturbations for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

singularly perturbed equation; periodic solution