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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

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