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March  2020, 9(1): 219-254. doi: 10.3934/eect.2020004

Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

Dalila Azzam-Laouir 1,, , Warda Belhoula 1, , Charles Castaing 2, and  M. D. P. Monteiro Marques 3,

1. 

Laboratoire LAOTI, FSEI, Université Mohammed Seddik Benyahia de Jijel, Algérie
2. 

IMAG, Université Montpellier, CNRS, Montpellier Cedex 5, France
3. 

CMAFcIO, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Portugal

* Corresponding author: Dalila Azzam-Laouir

Received  December 2018 Revised  May 2019 Published  March 2020 Early access  August 2019

In this paper, we study the existence of solutions for evolution problems of the form $ -\frac{du}{dr}(t) \in A(t)u(t) + F(t, u(t))+f(t, u(t)) $, where, for each $ t $, $ A(t) : D(A(t)) \to 2 ^H $ is a maximal monotone operator in a Hilbert space $ H $ with continuous, Lipschitz or absolutely continuous variation in time. The perturbation $ f $ is separately integrable on $ [0, T] $ and separately Lipschitz on $ H $, while $ F $ is scalarly measurable and separately scalarly upper semicontinuous on $ H $, with convex and weakly compact values. Several new applications are provided.

Keywords: Absolutely continuous, bounded variation, Lipschitz mapping, maximal monotone operators, pseudo-distance, perturbations.
Mathematics Subject Classification: Primary: 34A60, 28C20; Secondary: 28A25.
Citation: Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations and Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004
References:
[1]

S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program, 148 (2014), 5-47.  doi: 10.1007/s10107-014-0754-4.

[2]

H. Attouch, A. Cabot and M.-O. Czarnecki, Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc., 370 (2018), 755-790.  doi: 10.1090/tran/6965.

[3]

H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, Nonlinear Anal., 2 (1978), 329-353.  doi: 10.1016/0362-546X(78)90021-4.

[4]

D. Azzam-Laouir, W. Belhoula, C. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Ttheory Appl., 21 (2019), Art. 40, 32 pp. https://doi.org/10.1007/s11784-019-0666-2. doi: 10.1007/s11784-019-0666-2.

[5]

D. Azzam-Laouir, C. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with continuous bounded variation in time and applications, Set-Valued Var. Anal., 26 (2018), 693-728.  doi: 10.1007/s11228-017-0432-9.

[6]

D. Azzam-Laouir, S. Izza and L. Thibault, Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process, Set Valued Var. Anal., 22 (2014), 271-283.  doi: 10.1007/s11228-013-0248-1.

[7]

D. Azzam-Laouir, A. Makhlouf and L. Thibault, On perturbed sweeping process, Appl. Analysis, 95 (2016), 303-322.  doi: 10.1080/00036811.2014.1002482.

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Int. Publ. Leyden, 1976,352 pp.

[9]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5.

[10]

H. Benabdellah and A. Faik, Perturbations convexes et non convexes des équations d'évolution, Port. Math., 53 (1996), 187-208. 

[11]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North Holland, 1973.

[12]

B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control, Springer, 3rd edition, 2016. doi: 10.1007/978-3-319-28664-8.

[13]

C. Castaing, T. X. Dúc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal., 1 (1993), 109-139.  doi: 10.1007/BF01027688.

[14]

C. Castaing and M. D. P. Monteiro Marques, Evolution problems associated with nonconvex closed moving sets with bounded variation, Portugal. Math., 53 (1996), 73-87. 

[15]

C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, A Skorohod problem governed by a closed convex moving set, Journal of Convex Analysis, 23 (2016), 387-423. 

[16]

C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, Second-order evolution problems with time-dependent maximal monotone operator and applications, Adv. Math. Econ., 22 (2018), 25-77. 

[17]

C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/1-4020-1964-5.

[18]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580. Springer-Verlag, Berlin, 1977.

[19]

B. Cornet, Contributions à la Théorie Mathématique des Mécanismes Dynamiques D'allocation des Ressources Thèse, Université de Paris IX, 1981.

[20]

J. Cortes, Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability, Control Systems Magazine, 28 (2008), 36-73.  doi: 10.1109/MCS.2008.919306.

[21]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program, Ser. B, 104 (2005), 347–373. doi: 10.1007/s10107-005-0619-y.

[22]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differential Equations, 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.

[23]

A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Mathematics, 841. Springer-Verlag, 1981.

[24]

M. Kunze and M. D. P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal., 5 (1997), 57-72.  doi: 10.1023/A:1008621327851.

[25]

M. D. P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d'évolution dans les espaces de Hilbert, Séminaire d'Analyse Convexe, Montpellier, 14 (1984), 23 pp.

[26]

M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, Shocks and Dry Friction, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-7614-8.

[27]

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.

[28]

F. Nacry, Perturbed BV sweeping process involving prox-regular sets, Journal of Nonlinear and Convex Analysis, 18 (2017), 1619-1651. 

[29]

J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Prog., Ser. A, 113 (2008), 345–424. doi: 10.1007/s10107-006-0052-x.

[30]

A. Tanwani, B. Brogliato and C. Prieur, Well-posedness and output regulation for implicit time-varying evolution variational inequalities, SIAM J. Control Optim., 56 (2018), 751-781.  doi: 10.1137/16M1083657.

[31]

L. Thibault, Propriétés des Sous-Différentiels de Fonctions Localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier Ⅱ, 1976, 32 pp.

[32]

M. Valadier, Quelques résultats de base concernant le processus de la rafle, Sém. Anal. Convexe, Montpellier, 3 (1988), 30 pp.

[33]

A. A. Vladimirov, Nonstationary dissipative evolution equation in Hilbert space, Nonlinear Anal., 17 (1991), 499-518.  doi: 10.1016/0362-546X(91)90061-5.

[34]

I. I. Vrabie, Compactness Methods for Nonlinear Evolution, Pitman Monographs and Surveys in Pure and Applied Mathematics, 32. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.

show all references

References:
[1]

S. Adly, T. Haddad and L. Thibault, Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities, Math. Program, 148 (2014), 5-47.  doi: 10.1007/s10107-014-0754-4.

[2]

H. Attouch, A. Cabot and M.-O. Czarnecki, Asymptotic behavior of nonautonomous monotone and subgradient evolution equations, Trans. Amer. Math. Soc., 370 (2018), 755-790.  doi: 10.1090/tran/6965.

[3]

H. Attouch and A. Damlamian, Strong solutions for parabolic variational inequalities, Nonlinear Anal., 2 (1978), 329-353.  doi: 10.1016/0362-546X(78)90021-4.

[4]

D. Azzam-Laouir, W. Belhoula, C. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with absolutely continuous variation in time and applications, J. Fixed Point Ttheory Appl., 21 (2019), Art. 40, 32 pp. https://doi.org/10.1007/s11784-019-0666-2. doi: 10.1007/s11784-019-0666-2.

[5]

D. Azzam-Laouir, C. Castaing and M. D. P. Monteiro Marques, Perturbed evolution problems with continuous bounded variation in time and applications, Set-Valued Var. Anal., 26 (2018), 693-728.  doi: 10.1007/s11228-017-0432-9.

[6]

D. Azzam-Laouir, S. Izza and L. Thibault, Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process, Set Valued Var. Anal., 22 (2014), 271-283.  doi: 10.1007/s11228-013-0248-1.

[7]

D. Azzam-Laouir, A. Makhlouf and L. Thibault, On perturbed sweeping process, Appl. Analysis, 95 (2016), 303-322.  doi: 10.1080/00036811.2014.1002482.

[8]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Int. Publ. Leyden, 1976,352 pp.

[9]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5.

[10]

H. Benabdellah and A. Faik, Perturbations convexes et non convexes des équations d'évolution, Port. Math., 53 (1996), 187-208. 

[11]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North Holland, 1973.

[12]

B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control, Springer, 3rd edition, 2016. doi: 10.1007/978-3-319-28664-8.

[13]

C. Castaing, T. X. Dúc Ha and M. Valadier, Evolution equations governed by the sweeping process, Set-Valued Anal., 1 (1993), 109-139.  doi: 10.1007/BF01027688.

[14]

C. Castaing and M. D. P. Monteiro Marques, Evolution problems associated with nonconvex closed moving sets with bounded variation, Portugal. Math., 53 (1996), 73-87. 

[15]

C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, A Skorohod problem governed by a closed convex moving set, Journal of Convex Analysis, 23 (2016), 387-423. 

[16]

C. Castaing, M. D. P. Monteiro Marques and P. Raynaud de Fitte, Second-order evolution problems with time-dependent maximal monotone operator and applications, Adv. Math. Econ., 22 (2018), 25-77. 

[17]

C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces with Applications in Control Theory and Probability Theory, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/1-4020-1964-5.

[18]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580. Springer-Verlag, Berlin, 1977.

[19]

B. Cornet, Contributions à la Théorie Mathématique des Mécanismes Dynamiques D'allocation des Ressources Thèse, Université de Paris IX, 1981.

[20]

J. Cortes, Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability, Control Systems Magazine, 28 (2008), 36-73.  doi: 10.1109/MCS.2008.919306.

[21]

J. F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program, Ser. B, 104 (2005), 347–373. doi: 10.1007/s10107-005-0619-y.

[22]

J. F. Edmond and L. Thibault, BV solutions of nonconvex sweeping process differential inclusion with perturbation, J. Differential Equations, 226 (2006), 135-179.  doi: 10.1016/j.jde.2005.12.005.

[23]

A. Haraux, Nonlinear Evolution Equations-Global Behavior of Solutions, Lecture Notes in Mathematics, 841. Springer-Verlag, 1981.

[24]

M. Kunze and M. D. P. Monteiro Marques, BV solutions to evolution problems with time-dependent domains, Set-Valued Anal., 5 (1997), 57-72.  doi: 10.1023/A:1008621327851.

[25]

M. D. P. Monteiro Marques, Perturbations convexes semi-continues supérieurement de problèmes d'évolution dans les espaces de Hilbert, Séminaire d'Analyse Convexe, Montpellier, 14 (1984), 23 pp.

[26]

M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, Shocks and Dry Friction, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-7614-8.

[27]

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.

[28]

F. Nacry, Perturbed BV sweeping process involving prox-regular sets, Journal of Nonlinear and Convex Analysis, 18 (2017), 1619-1651. 

[29]

J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Prog., Ser. A, 113 (2008), 345–424. doi: 10.1007/s10107-006-0052-x.

[30]

A. Tanwani, B. Brogliato and C. Prieur, Well-posedness and output regulation for implicit time-varying evolution variational inequalities, SIAM J. Control Optim., 56 (2018), 751-781.  doi: 10.1137/16M1083657.

[31]

L. Thibault, Propriétés des Sous-Différentiels de Fonctions Localement Lipschitziennes définies sur un espace de Banach séparable. Applications, Thèse, Université Montpellier Ⅱ, 1976, 32 pp.

[32]

M. Valadier, Quelques résultats de base concernant le processus de la rafle, Sém. Anal. Convexe, Montpellier, 3 (1988), 30 pp.

[33]

A. A. Vladimirov, Nonstationary dissipative evolution equation in Hilbert space, Nonlinear Anal., 17 (1991), 499-518.  doi: 10.1016/0362-546X(91)90061-5.

[34]

I. I. Vrabie, Compactness Methods for Nonlinear Evolution, Pitman Monographs and Surveys in Pure and Applied Mathematics, 32. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987.

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