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Hiriart-Urruty–Phelps-Like Formula for the Subdifferential of Integral Sums

Abstract

We provide subdifferential calculus rules for continuous sums parametrized in measurable spaces that use the approximate subdifferentials of the data functions. As in Hiriart-Urruty and Phelps (J. Funct. Anal. 118: 154–166, 1993) given for finite sums, the resulting formulas hold without any conditions of continuity type on the involved functions. All this analysis is done in the setting of locally convex Suslin spaces.

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Acknowledgements

We are very grateful to an anonymous referee for his/her suggestions and comments.

Funding

This work is partially supported by CONICYT grant Fondecyt 1151003, Conicyt-Redes no. 150040, and Mathamsud 17-MATH-06.

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Correspondence to A. Hantoute.

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Dedicated to Prof. Michel Théra for his 70th birthday.

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Hantoute, A., Jourani, A. Hiriart-Urruty–Phelps-Like Formula for the Subdifferential of Integral Sums. Vietnam J. Math. 46, 391–405 (2018). https://doi.org/10.1007/s10013-018-0291-1

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  • DOI: https://doi.org/10.1007/s10013-018-0291-1

Keywords

  • Integral sums
  • Convex normal integrands
  • ε-subdifferential
  • Subdifferential calculus

Mathematics Subject Classification (2010)

  • 26B05
  • 26J25
  • 49H05