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On the integrable selections of certain multifunctions

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Abstract

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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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