Hysteresis in singular perturbation problems with nonuniqueness in limit equation. (English) Zbl 0793.34040

Lovicar, V.; Straškraba, I.; Krejčí, P.

Visintin, A. (ed.), Models of hysteresis. Proceedings of a workshop, held in Trento, Italy in September 1991. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 286, 91-101 (1993).

We investigate a problem (1) \(u_ \varepsilon(a)=\zeta_ \varepsilon\), \(\varepsilon u_ \varepsilon'(t)+F(u_ \varepsilon(t))=z_ \varepsilon(t)\) for a.e. \(t \in (a,b)\), where \(\varepsilon \geq 0\) is a parameter, \((a,b)\) is a bounded interval in \(R\), \(\zeta_ \varepsilon \in R\) and \(z_ \varepsilon \in L^ \infty (a,b)\). The main goal is the study of the convergence (or compactness) of the solutions \(u_ \varepsilon\) to (1) as \(\varepsilon \to 0+\) provided that the \(z_ \varepsilon\)’s are convergent (or compact) in an appropriate sense.
For the entire collection see [Zbl 0785.00016].

MSC:

34E15

Singular perturbations for ordinary differential equations

Keywords:

hysteresis; singular perturbation problems; nonuniqueness in limit equation; convergence; compactness