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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

Citations:

Zbl 0595.00009