zbMATH — the first resource for mathematics

Existence and uniqueness results for a class of rate-independent hysteresis problems. (English) Zbl 1121.34052
Summary: In this paper, we address the problem of existence, approximation, and uniqueness of solutions to an abstract doubly nonlinear equation, modeling a rate-independent process with hysteretic behavior. Models of this kind arise in, e.g. plasticity, solid phase transformations, and several other problems in non smooth mechanics. Existence of solutions is proved via passage to the limit in a time-discretization scheme, whereas uniqueness results are obtained by means of convex analysis techniques.

34C55 Hysteresis for ordinary differential equations
47J40 Equations with nonlinear hysteresis operators
49J40 Variational inequalities
74N30 Problems involving hysteresis in solids
34G25 Evolution inclusions
Full Text: DOI
[1] Aubin J.-P., Applied Nonlinear Analysis (1984)
[2] DOI: 10.1137/0322035 · Zbl 0549.49005 · doi:10.1137/0322035
[3] Brokate M., J. Convex Anal. 11 pp 111–
[4] Brezis H., North-Holland Mathematics Studies, in: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (1973)
[5] DOI: 10.1007/BF03167565 · Zbl 0757.34051 · doi:10.1007/BF03167565
[6] DOI: 10.1080/03605309908820706 · Zbl 0707.34053 · doi:10.1080/03605309908820706
[7] Francfort G., J. Reine Angew. Math. 595 pp 55–
[8] Han W., Interdisciplinary Applied Mathematics, in: Plasticity (Mathematical Theory and Numerical Analysis) (1999)
[9] Kunze M., J. Convex Anal. 4 pp 165–
[10] Kunze M., Topol. Meth. Nonlinear Anal. 12 pp 179– · Zbl 0923.34018 · doi:10.12775/TMNA.1998.036
[11] Krejčí P., Chapman & Hall/CRC Res. Notes Math., in: Nonlinear Differential Equations (Chvalatice 1998) (1999)
[12] A. Mielke, Handbook of Differential Equations, Evolutionary Equations 2, eds. C. Dafermos and E. Feireisl (Elsevier, 2005) pp. 461–559. · doi:10.1016/S1874-5717(06)80009-5
[13] DOI: 10.1007/s00526-004-0267-8 · Zbl 1161.74387 · doi:10.1007/s00526-004-0267-8
[14] DOI: 10.1007/978-3-0348-7614-8 · doi:10.1007/978-3-0348-7614-8
[15] Moreau J.-J., C. R. Acad. Sci. Paris Sér. A-B 276 pp A791–
[16] DOI: 10.1016/0022-0396(77)90085-7 · Zbl 0356.34067 · doi:10.1016/0022-0396(77)90085-7
[17] Mielke A., Nonlinear Diff. Eqns. Appl. (NoDEA) 11 pp 151–
[18] DOI: 10.1007/s002050200194 · Zbl 1012.74054 · doi:10.1007/s002050200194
[19] DOI: 10.1515/9781400873173 · Zbl 0932.90001 · doi:10.1515/9781400873173
[20] DOI: 10.1051/cocv:2006013 · Zbl 1116.34048 · doi:10.1051/cocv:2006013
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.