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Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach
Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators
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 |
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.
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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