## Periodic solutions of partial differential equations with hysteresis.(English)Zbl 0614.35004

Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 221-226 (1986).
[For the entire collection see Zbl 0595.00009.]
The author summarizes some theorems and sketches part of the proofs concerning existence of periodic solutions to the problems $u_{tt}- u_{xx}\pm F(u)=H(t,x),\quad u(t,0)=u(t,\pi)=0,$
$u_ t-(F(u_ x))_ x=H(t,x),\quad u(t,0)=u(t,1)=0,$
$u_{tt}-(F(u_ x))_ x=H(t,x),\quad u_ x(t,0)=u_ x(t,\pi)=0.$ Here F denotes Ishlinski hysteresis operator and H is a given time-periodic function. He uses Galerkin approximation and time discretization; the third problem is first transformed into an auxiliary problem.
Reviewer: M.Brokate

### MSC:

 35B10 Periodic solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35L65 Hyperbolic conservation laws 47A15 Invariant subspaces of linear operators 35L70 Second-order nonlinear hyperbolic equations 35A35 Theoretical approximation in context of PDEs

Zbl 0595.00009