August  2015, 8(4): 773-792. doi: 10.3934/dcdss.2015.8.773

Hysteresis operators in metric spaces

1. 

Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi, 24, I 10129 Torino, Italy

Received  January 2014 Revised  July 2014 Published  October 2014

Motivated by the sweeping processes, we develop an abstract theory of continuous hysteresis operators acting between rectifiable curves with values in metric spaces. In particular we study the continuity properties of such operators and how they can be extended from the space of Lipschitz continuous functions to the space of rectifiable curves. Applications to the sweeping processes and to the vector play operator are shown.
Citation: Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773
References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis,, Third edition, (2006).   Google Scholar

[2]

L. Ambrosio, Metric space valued functions of bounded variation,, Ann. Sc. Norm. Sup. Pisa, 17 (1990), 439.   Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford Mathematical Monographs, (2000).   Google Scholar

[4]

L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces,, Oxford University Press, (2004).   Google Scholar

[5]

H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,, North-Holland Mathematical Studies, (1973).   Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[7]

N. Dinculeanu, Vector Measures,, International Series of Monographs in Pure and Applied Mathematics, (1967).   Google Scholar

[8]

J. L. Doob, Measure Theory,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-0877-8.  Google Scholar

[9]

N. Dunford and J. Schwartz, Linear Operators, Part 1,, Wiley Interscience, (1958).   Google Scholar

[10]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (1969).   Google Scholar

[11]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis,, Springer-Verlag, (1989).  doi: 10.1007/978-3-642-61302-9.  Google Scholar

[12]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations,, Gakuto International Series Mathematical Sciences and Applications, (1996).   Google Scholar

[13]

P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators,, in Nonlinear Differential Equations (Chvalatice, (1998), 47.   Google Scholar

[14]

P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159.   Google Scholar

[15]

P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[16]

P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes,, J. Convex Anal., 21 (2014), 121.   Google Scholar

[17]

P. Krejčí and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces,, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 637.  doi: 10.3934/dcdsb.2011.15.637.  Google Scholar

[18]

S. Lang, Real and Functional Analysis - Third Edition,, Graduate Text in Mathematics, (1993).  doi: 10.1007/978-1-4612-0897-6.  Google Scholar

[19]

A. Mielke, Evolution in rate-independent systems,, in Evolutionary Equations, (2005), 461.   Google Scholar

[20]

J. J. Moreau, Rafle par un convexe variable, II,, in Travaux du Sminaire d'Analyse Convexe, (1972).   Google Scholar

[21]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[22]

V. Recupero, On locally isotone rate independent operators,, Appl. Math. Letters, 20 (2007), 1156.  doi: 10.1016/j.aml.2006.10.006.  Google Scholar

[23]

V. Recupero, $\BV$-extension of rate independent operators,, Math. Nachr., 282 (2009), 86.  doi: 10.1002/mana.200610723.  Google Scholar

[24]

V. Recupero, The play operator on the rectifiable curves in a Hilbert space,, Math. Methods Appl. Sci., 31 (2008), 1283.  doi: 10.1002/mma.968.  Google Scholar

[25]

V. Recupero, On a class of scalar variational inequalities with measure data,, Appl. Anal., 88 (2009), 1739.  doi: 10.1080/00036810903397446.  Google Scholar

[26]

V. Recupero, Sobolev and strict continuity of general hysteresis operators,, Math. Methods Appl. Sci., 32 (2009), 2003.  doi: 10.1002/mma.1124.  Google Scholar

[27]

V. Recupero, $\BV$ solutions of rate independent variational inequalities,, Ann. Sc. Norm. Super. Pisa Cl. Sc. (5), 10 (2011), 269.   Google Scholar

[28]

V. Recupero, Extending vector hysteresis operators,, J. Phys.: Conf. Ser., 268 (2011).  doi: 10.1088/1742-6596/268/1/012024.  Google Scholar

[29]

V. Recupero, A continuity method for sweeping processes,, J. Differ. Equations, 251 (2011), 2125.  doi: 10.1016/j.jde.2011.06.018.  Google Scholar

[30]

A. Visintin, Differential Models of Hysteresis,, Applied Mathematical Sciences, (1994).  doi: 10.1007/978-3-662-11557-2.  Google Scholar

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis,, Third edition, (2006).   Google Scholar

[2]

L. Ambrosio, Metric space valued functions of bounded variation,, Ann. Sc. Norm. Sup. Pisa, 17 (1990), 439.   Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford Mathematical Monographs, (2000).   Google Scholar

[4]

L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces,, Oxford University Press, (2004).   Google Scholar

[5]

H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,, North-Holland Mathematical Studies, (1973).   Google Scholar

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[7]

N. Dinculeanu, Vector Measures,, International Series of Monographs in Pure and Applied Mathematics, (1967).   Google Scholar

[8]

J. L. Doob, Measure Theory,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4612-0877-8.  Google Scholar

[9]

N. Dunford and J. Schwartz, Linear Operators, Part 1,, Wiley Interscience, (1958).   Google Scholar

[10]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (1969).   Google Scholar

[11]

M. A. Krasnosel'skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis,, Springer-Verlag, (1989).  doi: 10.1007/978-3-642-61302-9.  Google Scholar

[12]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations,, Gakuto International Series Mathematical Sciences and Applications, (1996).   Google Scholar

[13]

P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators,, in Nonlinear Differential Equations (Chvalatice, (1998), 47.   Google Scholar

[14]

P. Krejčí and P. Laurençot, Generalized variational inequalities,, J. Convex Anal., 9 (2002), 159.   Google Scholar

[15]

P. Krejčí and M. Liero, Rate independent Kurzweil processes,, Appl. Math., 54 (2009), 117.  doi: 10.1007/s10492-009-0009-5.  Google Scholar

[16]

P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes,, J. Convex Anal., 21 (2014), 121.   Google Scholar

[17]

P. Krejčí and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces,, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 637.  doi: 10.3934/dcdsb.2011.15.637.  Google Scholar

[18]

S. Lang, Real and Functional Analysis - Third Edition,, Graduate Text in Mathematics, (1993).  doi: 10.1007/978-1-4612-0897-6.  Google Scholar

[19]

A. Mielke, Evolution in rate-independent systems,, in Evolutionary Equations, (2005), 461.   Google Scholar

[20]

J. J. Moreau, Rafle par un convexe variable, II,, in Travaux du Sminaire d'Analyse Convexe, (1972).   Google Scholar

[21]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[22]

V. Recupero, On locally isotone rate independent operators,, Appl. Math. Letters, 20 (2007), 1156.  doi: 10.1016/j.aml.2006.10.006.  Google Scholar

[23]

V. Recupero, $\BV$-extension of rate independent operators,, Math. Nachr., 282 (2009), 86.  doi: 10.1002/mana.200610723.  Google Scholar

[24]

V. Recupero, The play operator on the rectifiable curves in a Hilbert space,, Math. Methods Appl. Sci., 31 (2008), 1283.  doi: 10.1002/mma.968.  Google Scholar

[25]

V. Recupero, On a class of scalar variational inequalities with measure data,, Appl. Anal., 88 (2009), 1739.  doi: 10.1080/00036810903397446.  Google Scholar

[26]

V. Recupero, Sobolev and strict continuity of general hysteresis operators,, Math. Methods Appl. Sci., 32 (2009), 2003.  doi: 10.1002/mma.1124.  Google Scholar

[27]

V. Recupero, $\BV$ solutions of rate independent variational inequalities,, Ann. Sc. Norm. Super. Pisa Cl. Sc. (5), 10 (2011), 269.   Google Scholar

[28]

V. Recupero, Extending vector hysteresis operators,, J. Phys.: Conf. Ser., 268 (2011).  doi: 10.1088/1742-6596/268/1/012024.  Google Scholar

[29]

V. Recupero, A continuity method for sweeping processes,, J. Differ. Equations, 251 (2011), 2125.  doi: 10.1016/j.jde.2011.06.018.  Google Scholar

[30]

A. Visintin, Differential Models of Hysteresis,, Applied Mathematical Sciences, (1994).  doi: 10.1007/978-3-662-11557-2.  Google Scholar

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