On cyclic α(·)-monotone multifunctions

S. Rolewicz

Studia Mathematica (2000)

  • Volume: 141, Issue: 3, page 263-272
  • ISSN: 0039-3223

Abstract

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Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let Γ : X → 2 Φ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), Γ ( x ) = ∂ Φ - α f | x .

How to cite

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Rolewicz, S.. "On cyclic α(·)-monotone multifunctions." Studia Mathematica 141.3 (2000): 263-272. <http://eudml.org/doc/216784>.

@article{Rolewicz2000,
abstract = {Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let $Γ: X → 2^\{Φ\}$ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), $Γ(x)=∂^\{-α\}_\{Φ\}f|_\{x\}$.},
author = {Rolewicz, S.},
journal = {Studia Mathematica},
keywords = {Fréchet Φ-differentiability; cyclic α(·)-monotone multi- function; weak -subdifferential},
language = {eng},
number = {3},
pages = {263-272},
title = {On cyclic α(·)-monotone multifunctions},
url = {http://eudml.org/doc/216784},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Rolewicz, S.
TI - On cyclic α(·)-monotone multifunctions
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 3
SP - 263
EP - 272
AB - Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let $Γ: X → 2^{Φ}$ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), $Γ(x)=∂^{-α}_{Φ}f|_{x}$.
LA - eng
KW - Fréchet Φ-differentiability; cyclic α(·)-monotone multi- function; weak -subdifferential
UR - http://eudml.org/doc/216784
ER -

References

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  1. R. Correa, A. Jofré and L. Thibault (1994), Subdifferential monotonicity as characterization of convex functions, Numer. Funct. Anal. Optim. 15, 531-535. Zbl0807.49015
  2. A. Jourani (1996), Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions, Control Cybernet. 25, 721-737. Zbl0862.49018
  3. D. Pallaschke and S. Rolewicz (1997), Foundations of Mathematical Optimization, Math. Appl. 388, Kluwer, Dordrecht. Zbl0887.49001
  4. R. T. Rockafellar (1970), On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33, 209-216. Zbl0199.47101
  5. R. T. Rockafellar (1980), Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32, 257-280. Zbl0447.49009
  6. S. Rolewicz (1999), On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions, Studia Math. 133, 29-37. Zbl0920.47047

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