On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions

S. Rolewicz

Studia Mathematica (1999)

  • Volume: 133, Issue: 1, page 29-37
  • ISSN: 0039-3223

Abstract

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Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction Γ : X → 2 Φ to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.

How to cite

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Rolewicz, S.. "On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions." Studia Mathematica 133.1 (1999): 29-37. <http://eudml.org/doc/216603>.

@article{Rolewicz1999,
abstract = {Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction $Γ: X → 2^Φ$ to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.},
author = {Rolewicz, S.},
journal = {Studia Mathematica},
keywords = {Fréchet Φ-differentiability; γ-paraconvex functions; α(·)-monotone multifunctions; metric space; -monotone multifunction; weakly angle-small set; -paraconvex function; Fréchet differentiable},
language = {eng},
number = {1},
pages = {29-37},
title = {On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions},
url = {http://eudml.org/doc/216603},
volume = {133},
year = {1999},
}

TY - JOUR
AU - Rolewicz, S.
TI - On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions
JO - Studia Mathematica
PY - 1999
VL - 133
IS - 1
SP - 29
EP - 37
AB - Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction $Γ: X → 2^Φ$ to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.
LA - eng
KW - Fréchet Φ-differentiability; γ-paraconvex functions; α(·)-monotone multifunctions; metric space; -monotone multifunction; weakly angle-small set; -paraconvex function; Fréchet differentiable
UR - http://eudml.org/doc/216603
ER -

References

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  1. [1] E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31-47. Zbl0162.17501
  2. [2] A. Jourani, Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions, Control Cybernet. 25 (1996), 721-737. Zbl0862.49018
  3. [3] S. Mazur, Über konvexe Mengen in linearen normierten Räumen, Studia Math. 4 (1933), 70-84. Zbl59.1074.01
  4. [4] D. Pallaschke and S. Rolewicz, Foundations of Mathematical Optimization, Math. Appl. 388, Kluwer, Dordrecht, 1997. Zbl0887.49001
  5. [5] D. Preiss and L. Zajíček, Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions, Suppl. Rend. Circ. Mat. Palermo (2) 3 (1984), 219-223. Zbl0547.46026
  6. [6] S. Rolewicz, On paraconvex multifunctions, in: Third Symposium on Operation Research (Mannheim, 1978), Operations Res. Verfahren 31, Hain, Königstein/Ts., 1979, 539-546. Zbl0403.49021
  7. [7] S. Rolewicz, On γ-paraconvex multifunctions, Math. Japon. 24 (1979), 293-300. Zbl0434.54009
  8. [8] S. Rolewicz, Generalization of Asplund inequalities on Lipschitz functions, Arch. Math. (Basel) 61 (1993), 484-488. Zbl0791.49019
  9. [9] S. Rolewicz, On an extension of Mazur's theorem on Lipschitz functions, ibid. 63 (1994), 535-540. Zbl0813.49018
  10. [10] S. Rolewicz, On subdifferentials on non-convex sets, in: Different Aspects of Differentiablity, D. Przeworska-Rolewicz (ed.), Dissertationes Math. 340 (1995), 301-308. Zbl0957.49012
  11. [11] S. Rolewicz, Convexity versus linearity, in: Transform Methods and Special Functions 94, P. Rusev, I. Dimovski and V. Kiryakova (eds.), Science Culture Technology Publ., Singapore, 1995, 253-263. Zbl0928.49018
  12. [12] S. Rolewicz, On Φ-differentiability of functions over metric spaces, Topol. Methods Nonlinear Anal. 5 (1995), 229-236. Zbl0894.46030
  13. [13] S. Rolewicz, On approximations of functions on metric spaces, Acta Univ. Lodz. Folia Math. 8 (1996), 99-108. Zbl0881.46032
  14. [14] I. Singer, Abstract Convex Analysis, Wiley, 1997. 

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