Quasilinear and quadratic singularly perturbed periodic boundary value problem. (English) Zbl 1048.34094

Vrábel’, Róbert

Arch. Math., Brno 36, No. 1, 1-7 (2000).

The author deals with existence and asymptotic behaviour of solutions of the periodic boundary value problem with a small parameter \[ \varepsilon y''=F(t,y,y'), \quad y(a)=y(b),\;y'(a)=y'(b), \tag{*} \] in the particular cases when \(F(t,y,y')=f(t,y)y'+g(t,y)\) or \(F(t,y,y')\!=\!f(t,y)y'{}^2+g(t,y)\). Using the method of upper and lower solutions, solutions of (*) are compared with solutions of the equation \(F(t,u,u')=0\) (which corresponds to the case \(\varepsilon =0\) in (*)).

Reviewer: Ondřej Došlý (Brno)

MSC:

34E15	Singular perturbations for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

singularly perturbed periodic boundary value problem