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On the integrable selections of certain multifunctions

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The aim of this paper is to establish a result of which the following is a particular case: If F is a nonempty closed-valued measurable multifunction, from a nonatomic σ-finite measure space (T, F, μ) into a separable real Banach space E, such that

$$d(0,F( \cdot )) \in L^1 (T) and \mathop {\lim }\limits_{\lambda \to + \infty } \frac{{d(\lambda x,F(t))}}{\lambda } = 0$$

for almost every tT and for every xE, then each closed hyperplane of L 1(T,E) contains a selection of F. Also, some consequences are indicated.

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  1. 1.

    AubinJ.-P. and EkelandI.: Applied Nonlinear Analysis, Wiley, New York, 1984.

  2. 2.

    BeerG.: Topologies on Closed and Closed Convex Sets, Kluwer Acad. Publ., Dordrecht, 1993.

  3. 3.

    Ionescu TulceaA. and Ionescu TulceaC.: Topics in the Theory of Lifting, Springer-Verlag, Berlin, 1969.

  4. 4.

    RicceriB.: Sur l'approximation des sélections mesurables, C.R. Acad. Sci. Paris, Série I 295 (1982), 527–530.

  5. 5.

    RicceriB.: Some topological mini-max theorems via an alternative principle for multifunctions, Arch. Math. 60 (1990), 367–377.

  6. 6.

    RicceriB.: A variational property of integral functionals on L p-speces of vector-valued functions, C.R. Acad. Sci. Paris, Série 1 318 (1994), 337–342.

  7. 7.

    Ricceri, B.: A variational property of integral functionals and related conjectures, Banach Center Publ., to appear.

  8. 8.

    Saint RaymondJ.: Connexité des sous-niveaux des fonctionnelles intégrales, Rend. Circ. Mat. Palermo 44 (1995), 162–168.

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Ricceri, B. On the integrable selections of certain multifunctions. Set-Valued Anal 4, 91–99 (1996). https://doi.org/10.1007/BF00419375

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Mathematics Subject Classifications (1991)

  • 28B20
  • 49J99

Key words

  • measurable multifunctions
  • integrable selections
  • integral functionals
  • Lipschitzian functionals
  • closed hyperplanes
  • Aumann integral
  • Wijsman convergence
  • Ekeland's variational principle