A quantitative version of Helly’s selection principle in Banach spaces and its applications

Abstract

We present a novel generalization, in Banach spaces, of the celebrated Helly’s principle selection. Specifically, our main result is a quantitative version of such principle selection. Our main tool is the so called Degree of Nondensifiability, which measures (in the specified sense) the distance of a given convex subset of a Banach space to the class of its Peano Continua. As application of our results, we analyze the solvability of certain Volterra integral equations.

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References

  1. 1.

    Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measure of Noncompactness and Condensing Operators. Birkhäuser Verlag, Basel (1992)

    Google Scholar 

  2. 2.

    Appell, J.: Measures of noncompactness, condensing operators and fixed points: an application-oriented survey. Fixed Point Theory 6(2), 157–229 (2005)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ayerbe Toledano, J.M., Domínguez Benavides, T., López Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser Verlag, Basel (1997)

    Google Scholar 

  4. 4.

    Balcerak, S., Belov, S.A., Chistyakov, V.V.: On Helly’s principle for metric semigroup valued BV mappings of two real variables. Bull. Austr. Math. Soc. 66(2), 245–257 (2002)

    MATH  Google Scholar 

  5. 5.

    Belov, S.A., Chistyakov, V.V.: A selection principle for mappings of bounded variation. J. Math. Anal. Appl. 249(2), 351–366 (2000)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Carothers, N.L.: Real Analysis. Cambridge University Press, New York (2000)

    Google Scholar 

  7. 7.

    Cherruault, Y., Mora, G.: Optimisation Globale. Théorie des Courbes \(\alpha \)-denses. Económica, Paris (2005)

    Google Scholar 

  8. 8.

    Chistyakov, V.V.: The optimal form of selection principles for functions of a real variable. J. Math. Anal. Appl. 310(2), 609–625 (2005)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Chistyakov, V.V., Tretyachenko, Y.V.: Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles. J. Math. Anal. Appl. 369(1), 82–93 (2010)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Duchoň, M., Debiéve, C.: Functions with bounded variation in locally convex space. Tatra Mt. Math. Publ. 49, 89–98 (2011)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Filipóv, R., Mrozek, N., Recław, I., Szuca, P.: I-selection principles for sequences of functions. J. Math. Anal. Appl. 396(2), 680–688 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Fraňková, D.: Regulated functions. Math. Bohem. 116(1), 20–59 (1991)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    García, G.: A quantitative version of the Arzelá-Ascoli theorem based on the degree of nondensifiability and applications. Appl. Gen. Topol. 20(1), 265–279 (2019)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    García, G.: Existence of solutions for infinite systems of ordinary differential equations by densifiability techniques. Filomat 32(10), 3419–3428 (2018)

    MathSciNet  Google Scholar 

  15. 15.

    García, G.: Solvability of initial value problems with fractional order differential equations in Banach spaces by \(\alpha \)-dense curves. Fract. Calc. Appl. Anal. 20(3), 646–661 (2017)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    García, G., Mora, G.: A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral equations. J. Math. Anal. Appl. 472(1), 1220–1235 (2019)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    García, G., Mora, G.: The degree of convex nondensifiability in Banach spaces. J. Convex Anal. 22(3), 871–888 (2015)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Springer-Verlag, New York (1996)

    Google Scholar 

  19. 19.

    Heinz, H.P.: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. 7(12), 1351–1371 (1983)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mora, G.: The Peano curves as limit of \(\alpha \)-dense curves. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 99(1), 23–28 (2005)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Mora, G., Cherruault, Y.: Characterization and generation of \(\alpha \)-dense curves. Comput. Math. Appl. 33(9), 83–91 (1997)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Mora, G., Mira, J.A.: Alpha-dense curves in infinite dimensional spaces. Inter. J. Pure App. Math. 5(4), 257–266 (2003)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Mora, G., Redtwitz, D.A.: Densifiable metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105(1), 71–83 (2011)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Porter, J.E.: Helly’s selection principle for functions of bounded \(p\)-variation. Rocky Mountain J. Math. 35(2), 675–679 (2005)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Sagan, H.: Space-filling Curves. Springer-Verlag, New York (1994)

    Google Scholar 

  26. 26.

    Schwabik, S., Guoju, Y.: Topics in Banach Space Integration, Series in Real Analysis, vol. 10. Word Scientifc, Singapore (2005)

    Google Scholar 

  27. 27.

    Szufla, S.: On the Volterra integral equation with weakly singular kernel. Math. Bohem. 131(3), 225–231 (2006)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Tretyachenko, Y.V., Chistyakov, V.V.: Selection principle for pointwise bounded sequences of functions. Math. Notes 84(3–4), 396–406 (2008)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Väth, M.: Volterra and integral equations of vector functions. Pure Appl. Math. 224, New York (2000)

  30. 30.

    Willard, S.: General Topology. Dover Pubs. Inc., New York (1970)

    Google Scholar 

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Correspondence to G. García.

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Communicated by Constantin Niculescu.

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García, G. A quantitative version of Helly’s selection principle in Banach spaces and its applications. Ann. Funct. Anal. (2020). https://doi.org/10.1007/s43034-020-00083-9

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Keywords

  • Helly’s selection principle
  • Degree of Nondensifiability
  • \(\alpha \)-dense curves
  • Bochner integrals
  • Volterra integral equations

Mathematics Subject Classification

  • 26A45
  • 46B50
  • 46G12
  • 45D05
  • 40A10