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State Dependent Sweeping Process in p-Uniformly Smooth and q-Uniformly Convex Banach Spaces

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Abstract

We prove the existence of solutions for the state depending convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces via new recent results on proximal analysis in reflexive smooth Banach spaces.

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References

  1. Alber, Y., Ryazantseva, I.: Nonlinear Ill-posed Problems of Monotone Type. Springer, Dordrecht (2006)

    MATH  Google Scholar 

  2. Alber, Y.: Generalized projection operators in banach spaces: properties and applications. Funct. Different. Equations 1(1), 1–21 (1994)

    Google Scholar 

  3. Alber, Y.: Metric and generalized projection operators in Banach spaces: properties and appklications. In: Kartsatos, A. (ed.) Theory and Applications of Nonlinear Operators of Monotonic and Accretive Type, pp. 15–50. Marcel Dekker, New York (1996)

    Google Scholar 

  4. Al-Senan, B.: Applications of Nonsmooth Analysis to Nonconvex Variational Inequalities and Nonconvex differential Inclusions. Master thesis, Department of Mathematics, College of Sciences, King Saud University, Saud Arabia (2005)

  5. Al-Yusof, R.: Sweeping Processes and Proximal Analysis in Reflexive Banach spaces. Ph.D. thesis, Department of Mathematics, College of Sciences, King Saud University, Saud Arabia (2008)

  6. Bounkhel, M.: Existence results for convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces (2011, to appear)

  7. Bounkhel, M.: Existence results for second order convex sweeping processes in p-uniformly smooth and q-uniformly convex Banach spaces. In: Proceedings of Applied Mathematics. ICMSAO 2011—International Conference on Modeling, Simulation and Applied Optimization 2011, Kuala Lumpur, Malaysia (2011, accepted for publication)

  8. Bounkhel, M., Al-yusof, R.: First and second order convex sweeping processes in reflexive smooth banach spaces. Set-Valued Var. Anal. 18(2), 151–182 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bounkhel, M., Al-yusof, R.: Proximal analysis in reflexive smooth Banach spaces. Nonlinear Anal.: Theory, Methods and Applications 73(7), 1921–1939 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Bounkhel, M., Thibault, L.: On various notions of regularity of sets. Nonlinear Anal.: Theory, Methods and Applications 48(2), 223–246 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Castaing, C., Ibrahim, A.G., Yarou, M.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10(1), 1–20 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Castaing, C., Saadoune, M.: Dunford-Pettis types theorem and Convergences in Set-Valued integration. J. Nonlinear Convex Anal. 1(1), 37–71 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Castaing, C., and Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math., vol. 580. Springer, Berlin and New York (1977)

    MATH  Google Scholar 

  14. Chemetov, N., Monteiro Marques, M.D.P.: Non-convex quasi-variational differential inclusions. Set-Valued Anal. 15(3), 209–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chemetov, N., Monteiro Marques, M.D.P., Stefanelli, U.: Ordered non-convex quasi-variational sweeping processes. J. Convex Anal. 15(2), 201–214 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  17. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer (1998)

  18. Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow, UK (1993)

  19. Deimling, K.: Nonlinear functional analysis. Springer, Berlin (1985) xiv+450 pp. ISBN: 3-540-13928-1

  20. Diestel, J.: Geometry of Banach Spaces, Selected Topics. Lecture Notes in Mathematics, vol. 485. Springer (1975)

  21. Edmonds, J.F.: Problemes d’évolution associés a des ensembles prox-reguliers, Inclusions et integration de sous-différentiels. These de Doctorat, Université Montpellier II, Juin (2004)

  22. Figiel, T.: On the moduli of convexity and smoothness. Stud. Math. 56, 121–155 (1976)

    MathSciNet  MATH  Google Scholar 

  23. Kunze, M., Monteiro Marques, M.D.P.: On parabolic quasi-variational inequalities and state-dependent sweeping processes. Topol. Methods Nonlinear Anal. 12(1), 179–191 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Li, J.: The generalized projection operator on reflexive Banach spaces and its applications. J. Math. Anal. Appl. 306, 55–71 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26, 347–374 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. Penot, J.P., Ratsimahalo, R.: Characterizations of metric projections in Banach spaces and applications. Abstr. Appl. 3(1–2), 85–103 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers (2000) iv+276 pp. ISBN: 4-946552-04-9

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

  29. Vainberg, M.M.: Variational Methods and Method of Monotone Operators in the Theory of Nonlinear Equations. Wiley, New York (1973)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

    Messaoud Bounkhel

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

    Charles Castaing

Authors
  1. Messaoud Bounkhel
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  2. Charles Castaing
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Corresponding author

Correspondence to Messaoud Bounkhel.

Additional information

The first author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No. RGP-VPP-024.

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Bounkhel, M., Castaing, C. State Dependent Sweeping Process in p-Uniformly Smooth and q-Uniformly Convex Banach Spaces. Set-Valued Anal 20, 187–201 (2012). https://doi.org/10.1007/s11228-011-0186-8

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  • DOI: https://doi.org/10.1007/s11228-011-0186-8

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