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 and 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, Springer, Berlin - Heidelberg, 2006.

[2]

L. Ambrosio, Metric space valued functions of bounded variation, Ann. Sc. Norm. Sup. Pisa, 17 (1990), 439-478.

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000.

[4]

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

[5]

H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert, North-Holland Mathematical Studies, Vol. 5, North-Holland Publishing Company, Amsterdam, 1973.

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, Vol. 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[7]

N. Dinculeanu, Vector Measures, International Series of Monographs in Pure and Applied Mathematics, Vol. 95, Pergamon Press, Berlin, 1967.

[8]

J. L. Doob, Measure Theory, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0877-8.

[9]

N. Dunford and J. Schwartz, Linear Operators, Part 1, Wiley Interscience, New York, 1958.

[10]

H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg, 1969.

[11]

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

[12]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto International Series Mathematical Sciences and Applications, Vol. 8, Gakkōtosho, Tokyo, 1996.

[13]

P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in Nonlinear Differential Equations (Chvalatice, 1998), Chapman & Hall/CRC Res. Notes Math., 404, Chapman & Hall/CRC, Boca Raton, FL, 1999, 47-110.

[14]

P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.

[15]

P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.

[16]

P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21 (2014), 121-146.

[17]

P. Krejčí and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 637-650. doi: 10.3934/dcdsb.2011.15.637.

[18]

S. Lang, Real and Functional Analysis - Third Edition, Graduate Text in Mathematics, Vol. 142, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[19]

A. Mielke, Evolution in rate-independent systems, in Evolutionary Equations, Vol. II (eds. C. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 461-559.

[20]

J. J. Moreau, Rafle par un convexe variable, II, in Travaux du Sminaire d'Analyse Convexe, Vol. II, Exp. No. 3, Secretariat des Math., Publ. No. 122, U.É.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier, 1972, 36 pp.

[21]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.

[22]

V. Recupero, On locally isotone rate independent operators, Appl. Math. Letters, 20 (2007), 1156-1160. doi: 10.1016/j.aml.2006.10.006.

[23]

V. Recupero, $\BV$-extension of rate independent operators, Math. Nachr., 282 (2009), 86-98. doi: 10.1002/mana.200610723.

[24]

V. Recupero, The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31 (2008), 1283-1295. doi: 10.1002/mma.968.

[25]

V. Recupero, On a class of scalar variational inequalities with measure data, Appl. Anal., 88 (2009), 1739-1753. doi: 10.1080/00036810903397446.

[26]

V. Recupero, Sobolev and strict continuity of general hysteresis operators, Math. Methods Appl. Sci., 32 (2009), 2003-2018. doi: 10.1002/mma.1124.

[27]

V. Recupero, $\BV$ solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sc. (5), 10 (2011), 269-315.

[28]

V. Recupero, Extending vector hysteresis operators, J. Phys.: Conf. Ser., 268 (2011), 0120124. doi: 10.1088/1742-6596/268/1/012024.

[29]

V. Recupero, A continuity method for sweeping processes, J. Differ. Equations, 251 (2011), 2125-2142. doi: 10.1016/j.jde.2011.06.018.

[30]

A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences, Vol. 111, Springer-Verlag, Berlin Heidelberg, 1994. doi: 10.1007/978-3-662-11557-2.

show all references

References:
[1]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, Third edition, Springer, Berlin - Heidelberg, 2006.

[2]

L. Ambrosio, Metric space valued functions of bounded variation, Ann. Sc. Norm. Sup. Pisa, 17 (1990), 439-478.

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000.

[4]

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

[5]

H. Brezis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert, North-Holland Mathematical Studies, Vol. 5, North-Holland Publishing Company, Amsterdam, 1973.

[6]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, Vol. 121, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4048-8.

[7]

N. Dinculeanu, Vector Measures, International Series of Monographs in Pure and Applied Mathematics, Vol. 95, Pergamon Press, Berlin, 1967.

[8]

J. L. Doob, Measure Theory, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0877-8.

[9]

N. Dunford and J. Schwartz, Linear Operators, Part 1, Wiley Interscience, New York, 1958.

[10]

H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg, 1969.

[11]

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

[12]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Gakuto International Series Mathematical Sciences and Applications, Vol. 8, Gakkōtosho, Tokyo, 1996.

[13]

P. Krejčí, Evolution variational inequalities and multidimensional hysteresis operators, in Nonlinear Differential Equations (Chvalatice, 1998), Chapman & Hall/CRC Res. Notes Math., 404, Chapman & Hall/CRC, Boca Raton, FL, 1999, 47-110.

[14]

P. Krejčí and P. Laurençot, Generalized variational inequalities, J. Convex Anal., 9 (2002), 159-183.

[15]

P. Krejčí and M. Liero, Rate independent Kurzweil processes, Appl. Math., 54 (2009), 117-145. doi: 10.1007/s10492-009-0009-5.

[16]

P. Krejčí and V. Recupero, Comparing BV solutions of rate independent processes, J. Convex Anal., 21 (2014), 121-146.

[17]

P. Krejčí and T. Roche, Lipschitz continuous data dependence of sweeping processes in BV spaces, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 637-650. doi: 10.3934/dcdsb.2011.15.637.

[18]

S. Lang, Real and Functional Analysis - Third Edition, Graduate Text in Mathematics, Vol. 142, Springer Verlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[19]

A. Mielke, Evolution in rate-independent systems, in Evolutionary Equations, Vol. II (eds. C. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005, 461-559.

[20]

J. J. Moreau, Rafle par un convexe variable, II, in Travaux du Sminaire d'Analyse Convexe, Vol. II, Exp. No. 3, Secretariat des Math., Publ. No. 122, U.É.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier, 1972, 36 pp.

[21]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.

[22]

V. Recupero, On locally isotone rate independent operators, Appl. Math. Letters, 20 (2007), 1156-1160. doi: 10.1016/j.aml.2006.10.006.

[23]

V. Recupero, $\BV$-extension of rate independent operators, Math. Nachr., 282 (2009), 86-98. doi: 10.1002/mana.200610723.

[24]

V. Recupero, The play operator on the rectifiable curves in a Hilbert space, Math. Methods Appl. Sci., 31 (2008), 1283-1295. doi: 10.1002/mma.968.

[25]

V. Recupero, On a class of scalar variational inequalities with measure data, Appl. Anal., 88 (2009), 1739-1753. doi: 10.1080/00036810903397446.

[26]

V. Recupero, Sobolev and strict continuity of general hysteresis operators, Math. Methods Appl. Sci., 32 (2009), 2003-2018. doi: 10.1002/mma.1124.

[27]

V. Recupero, $\BV$ solutions of rate independent variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sc. (5), 10 (2011), 269-315.

[28]

V. Recupero, Extending vector hysteresis operators, J. Phys.: Conf. Ser., 268 (2011), 0120124. doi: 10.1088/1742-6596/268/1/012024.

[29]

V. Recupero, A continuity method for sweeping processes, J. Differ. Equations, 251 (2011), 2125-2142. doi: 10.1016/j.jde.2011.06.018.

[30]

A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences, Vol. 111, Springer-Verlag, Berlin Heidelberg, 1994. doi: 10.1007/978-3-662-11557-2.

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