# On cyclic α(·)-monotone multifunctions

Studia Mathematica (2000)

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

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topRolewicz, 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

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

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