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Perturbed Evolution Problems with Continuous Bounded Variation in Time and Applications

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Abstract

This paper is devoted to the study of evolution problems of the form \(-\frac {du}{dr}(t) \in A(t)u(t) + f(t, u(t))\) in a new setting, where, for each t, A(t) : D(A(t)) → 2H is a maximal monotone operator in a Hilbert space H and the mapping t↦A(t) has continuous bounded or Lipschitz variation on [0, T], in the sense of Vladimirov’s pseudo-distance. The measure dr gives an upper bound of that variation. The perturbation f is separately integrable on [0, T] and separately Lipschitz on H. Several versions and new applications are presented.

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References

  1. Adly, S., Haddad, T., Thibault, L.: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Program 148(1-2, Ser. B), 5–47 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Attouch, H., Cabot, A., Redont, P.: The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and Penalty Approximations. Advances in Mathematical Sciences and Applications, Gakkotosho, Tokyo, vol. 12 no. 1, pp. 273–306 (2002)

  3. Azzam-Laouir, D., Izza, S., Thibault, L.: Mixed semicontinuous perturbation of nonconvex state-dependent sweeping process. Set Valued Var. Anal. 22, 271–283 (2014)

    MATH  Google Scholar 

  4. Azzam-Laouir, D., Makhlouf, M., Thibault, L.: On perturbed sweeping process. Appl. Analysis. 95(2), 303–322 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff Int. Publ., Leyden (1976)

    Book  MATH  Google Scholar 

  6. Benabdellah, H., Castaing, C.: BV Solutions of multivalued differential equations on closed moving sets in Banach spaces. Banach center publications, vol. 32. Institute of Mathematics, Polish academy of Sciences, Warszawa (1995)

    MATH  Google Scholar 

  7. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland (1979)

  8. Castaing, C.: Topologie de la convergence uniforme sur les parties uniformément intégrables de \({L_{E}^{1}}\) et théorèmes de compacité faible dans certains espaces du type Köthe-Orlicz. Travaux Sém. Anal. Convexe 10(1), 27 (1980). exp. no. 5

    Google Scholar 

  9. Castaing, C., Duc Ha, T.X., Valadier, M.: Evolution equations governed by the sweeping process. Set-Valued Anal. 1, 109–139 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castaing, C., Marcellin, S.: Evolution inclusions with pln functions and application to viscosity and control. JNCA 8(2), 227–255 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Castaing, C., Monteiro Marques, M.D.P.: BV periodic solutions of an evolution problem associated with continuous convex sets. Set-Valued Anal. 3, 381–399 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Castaing, C., Marques, M.M., Raynaud de Fitte, P.: Some problems in optimal control governed by the sweeping process. J. Nonlinear Convex Analysis 15(5), 1043–1070 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Castaing, C., Marques, M.M., Raynaud de Fitte, P.: A Skorohod problem governed by a closed convex moving set. J. Convex Analysis 23(2), 387–423 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Castaing, C., Raynaud de Fitte, P., Salvadori, A.: Some variational convergence results with application to evolution inclusions. Adv. Math. Econ 8, 33–73 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Castaing, C., Raynaud de Fitte, P., Valadier, M.: Young measures on topological spaces with applications in control theory and probability theory. Kluwer Academic Publishers, Dordrecht (2004)

    MATH  Google Scholar 

  16. Castaing, C., Salvadori, A., Thibault, L.: Functional evolution equations governed by nonconvex sweeeping process. J. Nonlinear Convex Analysis 2(2), 217–241 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Castaing, C., Le Xuan, T., Raynaud de Fitte, P., Salvadori, A.: Some problems in second order evolution inclusions with boundary condition: a variational approach, to appear in Adv. Math. Econ. 21 (2017)

  18. Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions, lectures notes in mathematics, p. 580. Springer-Verlag, Berlin (1977)

    Book  MATH  Google Scholar 

  19. Colombo, G., Goncharov, V.V.: The sweeping processes without convexity. Set- Valued Anal. 7, 357–374 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Edmond, J.F., Thibault, L.: Relaxation and optimal control problem involving a perturbed sweeping process. Math. Program, Ser. B 104, 347–373 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grothendieck, A.: Espaces Vectoriels Topologiques Mat. São Paulo, São Paulo, 3rd edn. Publ. Soc. (1964)

  22. Kenmochi, N.: Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Educ. Chiba Univ. 30 (1981)

  23. Florescu, L.C., Godet-Thobie, C.: Young measures and compactness in measure spaces. De Gruyter, Berlin (2012)

    Book  MATH  Google Scholar 

  24. Kunze, M., Monteiro Marques, M.D.P.: BV Solutions to evolution problems with time-dependent domains. Set-Valued Anal. 5, 57–72 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moreau, J.J., Valadier, M.: A chain rule involving vector functions of bounded variations. J. Funct. Anal. 74(2), 333–345 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Monteiro Marques, M.D.P.: Perturbations convexes semi-continues supérieurement de problèmes d’évolution dans les espaces de Hilbert, Séminaire d’Analyse Convexe, Montpellier, vol. 14. exposé n. 2. (1984)

  27. Monteiro Marques, M.D.P.: Differential inclusions nonsmooth mechanical problems, shocks and dry friction. Progress in Nonlinear Differential Equations and Their Applications, Birkhauser vol. 9 (1993)

  28. Paoli, L.: An existence result for non-smooth vibro-impact problem. J. Differ. Equ. 211(2), 247–281 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Saidi, S., Thibault, L., Yarou, M.: Relaxation of optimal control problems involving time dependent subdifferential operators. Numer. Funct. Anal. Optim. 34(10), 1156–1186 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schatzman, M.: Problèmes unilatéraux d’ évolution du second ordre en temps. Thèse de Doctorat d’ Etat es Sciences Mathématiques, Université Pierre et Marie Curie, Paris 6 (1979)

  31. Thibault, L.: Propriétés des sous-différentiels de fonctions localement Lipschitziennes définies sur un espace de Banach séparable. Applications. Thèse, Université Montpellier II (1976)

  32. Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193, 1–26 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

  34. Valadier, M.: Lipschitz approximations of the sweeping process (or Moreau) process. J. Differ. Equ. 88(2), 248–264 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vladimirov, A.A.: Nonstationary dissipative evolution equations in a Hilbert space. Nonlinear Anal. 17, 499–518 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vrabie, I.L.: Compactness methods for nonlinear evolution equations, Pitman Monographs and Surveys in Pure and Applied mathematics, Longman Scientific and Technical, vol. 32. Wiley, New York (1987)

    Google Scholar 

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Acknowledgments

M. D. P. Monteiro Marques was partially supported by Fundação para a Ciência e a Tecnologia, grant UID/MAT/04561/2013.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Pures et Appliquées, FSEI, Université Mohammed Seddik Benyahia-Jijel, Jijel, Algeria

    Dalila Azzam-Laouir

  2. Département de Mathématiques, Université Montpellier II, 34095, Case courrier 051, Montpellier, Cedex 5, France

    Charles Castaing

  3. CMAF-CIO, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016, Lisboa, Portugal

    M. D. P. Monteiro Marques

Authors
  1. Dalila Azzam-Laouir
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  2. Charles Castaing
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  3. M. D. P. Monteiro Marques
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Correspondence to M. D. P. Monteiro Marques.

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Azzam-Laouir, D., Castaing, C. & Monteiro Marques, M.D.P. Perturbed Evolution Problems with Continuous Bounded Variation in Time and Applications. Set-Valued Var. Anal 26, 693–728 (2018). https://doi.org/10.1007/s11228-017-0432-9

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  • DOI: https://doi.org/10.1007/s11228-017-0432-9

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