Skip to main content
SpringerLink
Log in

Integral Functionals on Nonseparable Banach Spaces With Applications

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results. Then, we provide an interchange formula between integration and infimum, which enables us to get explicit formulas for the conjugate and Clarke subdifferential of integral functionals. Applications to expected functionals from stochastic programming, optimality conditions for a calculus of variation problem and sweeping processes are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artstein, Z., Prikry, K.: Carathéodory selections and the Scorza Dragoni property. J. Math. Anal. Appl. 127(2), 540–547 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Arvanitakis, A.D.: A simultaneous selection theorem. Fund. Math. 219(1), 1–14 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Bogachev, V.I.: Measure Theory, vol. I. Springer, Berlin (2007)

    MATH  Google Scholar 

  4. Borwein, J., Zhu, Q.: Techniques of Variational Analysis, Volume 20 of CMS Books in Mathematics/Ouvrages Mathematics SMC. Springer, New York (2005)

  5. Bourbaki, N.: General Topology. Chapters 5–10 Elements of Mathematics. Springer, Berlin (1998)

    MATH  Google Scholar 

  6. Brogliato, B.: Nonsmooth Mechanics, Communications and Control Engineering Series, 3rd edn. Springer, Cham (2016)

    MATH  Google Scholar 

  7. Cascales, B., Kadets, V., Rodríguez, J.: Measurability and selections of multi-functions in Banach spaces. J. Convex Anal. 17(1), 229–240 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Castaing, C.: Une nouvelle extension du théorème de Dragoni-Scorza. C. R. Acad. Sci. Paris Sér. A-B 271, A396–A398 (1970)

    MATH  Google Scholar 

  9. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Maths. Springer, Berlin (1977)

    MATH  Google Scholar 

  10. Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. Graduate Texts in Math, vol. 178. Springer, New York (1998)

    Google Scholar 

  11. Correa, R., Hantoute, A., Pérez-Aros, P.: Characterizations of the subdifferential of convex integral functions under qualification conditions. J. Funct. Anal. 277(1), 227–254 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Correa, R., Hantoute, A., Pérez-Aros, P.: Subdifferential calculus rules for possibly nonconvex integral functions. SIAM J. Control Optim. 58(1), 462–484 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Correa, R., Hantoute, A., Pérez-Aros, P.: Qualification conditions-free characterizations of the \(\varepsilon \)-subdifferential of convex integral functions. Appl. Math. Optim. 83(3), 1709–1737 (2021)

    MathSciNet  MATH  Google Scholar 

  14. Deimling, K.: Multivalued differential equations. de Gruyter Series in Nonlinear Analysis and Applications, vol. 1. Walter de Gruyter & Co., Berlin (1992)

    Google Scholar 

  15. Diestel, J., Uhl, J.J., Jr.: Vector Measures. Mathematical Surveys, No. 15. American Mathematical Society, Providence, RI (1977)

  16. Dugundji, J.: Topology. Allyn and Bacon, Inc., Boston, MA (1978)

    MATH  Google Scholar 

  17. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, Revised Textbooks in Mathematics. CRC Press, Boca Raton, FL (2015)

    MATH  Google Scholar 

  18. Folland, G.B.: Real Analysis, 2nd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  19. Himmelberg, C.J., Van Vleck, F.S.: An extension of Brunovský’s Scorza Dragoni type theorem for unbounded set-valued functions. Math. Slovaca 26(1), 47–52 (1976)

    MathSciNet  MATH  Google Scholar 

  20. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, Volume 419 of Mathematics and its Applications, vol. I. Kluwer Academic Publishers, Dordrecht (1997)

    Google Scholar 

  21. Krejci, P., Roche, T.: Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete Contin. Dyn. Syst. Ser. B 15(3), 637–650 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Krejčí, P., Liero, M.: Rate independent Kurzweil processes. Appl. Math. 54(2), 117–145 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Kucia, A.: Scorza-Dragoni type theorems. Fund. Math. 138(3), 197–203 (1991)

    MathSciNet  MATH  Google Scholar 

  24. Levin, V.L.: Convex integral functionals and the theory of lifting. Russ. Math. Surv. 30(2), 119 (1975)

    MathSciNet  MATH  Google Scholar 

  25. Łojasiewicz, S.: Some theorems of Scorza-Dragoni type for multifunctions with application to the problem of existence of solutions for differential multivalued equations. In: Mathematical Control Theory, Volume 14 of Banach Center Publications, pp. 625–643. PWN, Warsaw (1985)

  26. Michael, E.: A selection theorem. Proc. Am. Math. Soc. 17, 1404–1406 (1966)

    MathSciNet  MATH  Google Scholar 

  27. Mordukhovich, B., Pérez-Aros, P.: Generalized Leibniz rules and Lipschitzian stability for expected-integral mappings. SIAM J. Optim. 31(4), 3212–3246 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Mordukhovich, B.S., Pérez-Aros, P.: Generalized sequential differential calculus for expected-integral functionals. Set-Valued Var. Anal. 29(3), 621–644 (2021)

    MathSciNet  MATH  Google Scholar 

  29. Mordukhovich, B.S., Sagara, N.: Subdifferentials of nonconvex integral functionals in Banach spaces with applications to stochastic dynamic programming. J. Convex Anal. 25(2), 643–673 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3–4), 329–349 (1999)

    MathSciNet  MATH  Google Scholar 

  31. Moreau, J.J.: Rafle par un convexe variable. I. In: Travaux du Séminaire d’Analyse Convexe, Vol. I, Exp. No. 15, pp. 1–43. U.É.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier (1971)

  32. Moreau, J.J.: Rafle par un convexe variable. II. In Travaux du Séminaire d’Analyse Convexe, Vol. II, Exp. No. 3, pp. 1–36. U.É.R. de Math., Univ. Sci. Tech. Languedoc, Montpellier (1972)

  33. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. J. Differ. Equ. 26(3), 347–374 (1977)

    MathSciNet  MATH  Google Scholar 

  34. Papageorgiou, N.S., Kyritsi-Yiallourou, S.T.H.: Handbook of Applied Analysis, vol. 19. Springer, New York (2009)

    MATH  Google Scholar 

  35. Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)

    Google Scholar 

  36. Recupero, V.: BV continuous sweeping processes. J. Differ. Equ. 259(8), 4253–4272 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Rockafellar, R.T.: Integrals which are convex functionals. Pac. J. Math. 24, 525–539 (1968)

    MathSciNet  MATH  Google Scholar 

  38. Rockafellar, R.T.: Integrals which are convex functionals. II. Pac. J. Math. 39, 439–469 (1971)

    MathSciNet  MATH  Google Scholar 

  39. Rockafellar, R. T.: Conjugate duality and optimization. In: CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA (1974)

  40. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren Math Wiss, vol. 317. Springer, Berlin (1998)

    Google Scholar 

  41. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., Columbus (1987)

    MATH  Google Scholar 

  42. Rzeżuchowski, T.: Scorza-Dragoni type theorem for upper semicontinuous multivalued functions. Bull. Acad. Pol. Sci. Sér. Sci. Math. 28(1–2), 61–66 (1981)

    MathSciNet  MATH  Google Scholar 

  43. Shirali, S., Vasudeva, H.L.: Metric Spaces. Springer, London (2006)

    MATH  Google Scholar 

  44. Stewart, D.: Dynamics with Inequalities. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011)

    MATH  Google Scholar 

  45. Thibault, L.: Moreau sweeping process with bounded truncated retraction. J. Convex Anal. 23(4), 1051–1098 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Valov, V.: On a theorem of Arvanitakis. Mathematika 59(1), 250–256 (2013)

    MathSciNet  MATH  Google Scholar 

  47. Zălinescu, C.: Convex Aalysis in General Vector Spaces. World Scientific Publishing Co. Inc, River Edge, NJ (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile

    Juan Guillermo Garrido

  2. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Rancagua, Chile

    Juan Guillermo Garrido, Pedro Pérez-Aros & Emilio Vilches

Authors
  1. Juan Guillermo Garrido
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Pérez-Aros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilio Vilches
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Guillermo Garrido.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

J.G. Garrido and E. Vilches were supported by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence from ANID-Chile. P. Pérez-Aros and E. Vilches were supported by ANID-Chile under grants Fondecyt regular N\(^{\circ }\) 1200283 and Fondecyt regular N\(^{\circ }\) 1220886.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Garrido, J.G., Pérez-Aros, P. & Vilches, E. Integral Functionals on Nonseparable Banach Spaces With Applications. Appl Math Optim 87, 29 (2023). https://doi.org/10.1007/s00245-022-09942-4

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-022-09942-4

Keywords

Search

Navigation