Krejčí, Pavel A monotonicity method for solving hyperbolic problems with hysteresis. (English) Zbl 0668.35065 Apl. Mat. 33, No. 3, 197-203 (1988). A method of Minty-Browder type is used for proving the existence and uniqueness of a weak \(\omega\)-periodic solution to the model equation for vibrating processes in elasto-plastic solids or in ferromagnetics \(u_{tt}-div(F(\text{grad} u))=g\) in a bounded domain \(\Omega \subset R^ N\), \(u=0\) on \(\partial \Omega\), where g is given \(\omega\)-periodic function and F is the Ishlinskij hysteresis operator. Its hyperbolicity is confirmed by the finite speed of propagation. The proof is based on sharp estimates of hysteresis energy losses. Reviewer: P.Krejčí Cited in 4 Documents MSC: 35L70 Second-order nonlinear hyperbolic equations 35B10 Periodic solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 74H99 Dynamical problems in solid mechanics 74H45 Vibrations in dynamical problems in solid mechanics Keywords:quasilinear; method of Minty-Browder type; existence; uniqueness; weak \(\omega\)-periodic solution; vibrating processes; elasto-plastic solids; ferromagnetics; hysteresis operator; finite speed of propagation; sharp estimates; hysteresis energy losses PDFBibTeX XMLCite \textit{P. Krejčí}, Apl. Mat. 33, No. 3, 197--203 (1988; Zbl 0668.35065) Full Text: EuDML References: [1] S. Fučík A. Kufner: Nonlinear differential equations. (Czech). SNTL, Praha, 1978. [2] P. Krejčí: Hysteresis and periodic solutions to semilinear and quasilinear wave equations. Math. Z. 193 (1986), 247-264. · Zbl 0658.35065 · doi:10.1007/BF01174335 [3] P. Krejčí: On Ishlinskii model for non-perfectly elastic bodies. Apl. mat. 33 (1988), No. 2, 133-144. · Zbl 0653.73013 [4] A. Kufner O. John S. Fučík: Function spaces. Academia, Praha, 1977. [5] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris, 1969. · Zbl 0189.40603 [6] О. В. Бесов, В П. Ильин С. М. Никольский: Интегральные представления функций и теоремы вложения. Hauka, Москва, 1975. · Zbl 1231.90252 · doi:10.1287/mnsc.21.10.1113 [7] А. Ю. Ишлинский: Некоторые применения статистики к описанию законов деформирования тел. Изв. АН СССР, OTH, 1944, Ho 9, 583-590. · Zbl 0149.19102 · doi:10.1007/BF02768548 [8] М. А. Красносельский А. В. Покровский: Системы с гистерезисом. Hauka, Москва, 1983. · Zbl 1229.47001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.