Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications

Abderrahim Jourani

Studia Mathematica (1996)

  • Volume: 117, Issue: 2, page 123-136
  • ISSN: 0039-3223

Abstract

top
We extend the open mapping theorem and inversion theorem of Robinson for convex multivalued mappings to γ-paraconvex multivalued mappings. Some questions posed by Rolewicz are also investigated. Our results are applied to obtain a generalization of the Farkas lemma for γ-paraconvex multivalued mappings.

How to cite

top

Jourani, Abderrahim. "Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications." Studia Mathematica 117.2 (1996): 123-136. <http://eudml.org/doc/216247>.

@article{Jourani1996,
abstract = {We extend the open mapping theorem and inversion theorem of Robinson for convex multivalued mappings to γ-paraconvex multivalued mappings. Some questions posed by Rolewicz are also investigated. Our results are applied to obtain a generalization of the Farkas lemma for γ-paraconvex multivalued mappings.},
author = {Jourani, Abderrahim},
journal = {Studia Mathematica},
keywords = {-paraconvex multivalued mapping; open mapping theorem; inversion theorem; Lipschitz property; Farkas lemma},
language = {eng},
number = {2},
pages = {123-136},
title = {Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications},
url = {http://eudml.org/doc/216247},
volume = {117},
year = {1996},
}

TY - JOUR
AU - Jourani, Abderrahim
TI - Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications
JO - Studia Mathematica
PY - 1996
VL - 117
IS - 2
SP - 123
EP - 136
AB - We extend the open mapping theorem and inversion theorem of Robinson for convex multivalued mappings to γ-paraconvex multivalued mappings. Some questions posed by Rolewicz are also investigated. Our results are applied to obtain a generalization of the Farkas lemma for γ-paraconvex multivalued mappings.
LA - eng
KW - -paraconvex multivalued mapping; open mapping theorem; inversion theorem; Lipschitz property; Farkas lemma
UR - http://eudml.org/doc/216247
ER -

References

top
  1. [1] J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), 441-459. Zbl0654.49004
  2. [2] F. H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), 165-174. Zbl0404.90100
  3. [3] R. Cominetti, Metric regularity, tangent sets and second-order optimality conditions, Appl. Math. Optim. 21 (1990), 265-288. Zbl0692.49018
  4. [4] S. Dolecki, Open relation theorem without closedness assumption, Proc. Amer. Math. Soc. 109 (1990), 1019-1024. Zbl0721.46003
  5. [5] B. M. Glover, V. Jeyakumar and W. Oettli, A Farkas lamma for difference sublinear systems and quasi-differentiable programming, Math. Programming 63 (1994), 109-125. Zbl0804.49015
  6. [6] J. Gwinner, Results of Farkas type, Numer. Funct. Anal. Optim. 9 (1987), 471-520. Zbl0598.49017
  7. [7] V. Jeyakumar, A general Farkas lemma and characterization of optimality for a nonsmooth program involving convex processes, J. Optim. Theory Appl. 55 (1987), 449-461. Zbl0616.90072
  8. [8] A. Jourani, Intersection formulae and the marginal function in Banach spaces, J. Math. Anal. Appl. 192 (1995), 867-891. Zbl0831.46044
  9. [9] A. Jourani, Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions, 1995, submitted. Zbl0862.49018
  10. [10] A. Jourani and L. Thibault, The use of metric graphical regularity in approximate subdifferential calculus rule in finite dimensions, Optimization 21 (1990), 509-519. Zbl0721.49018
  11. [11] J.-P. Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), 629-643. Zbl0687.54015
  12. [12] S. M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), 130-143. Zbl0418.52005
  13. [13] R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), 665-698. Zbl0593.49013
  14. [14] S. Rolewicz, On paraconvex multifunctions, Oper. Res. Verfahren 31 (1979), 539-546. Zbl0403.49021
  15. [15] S. Rolewicz, On γ-paraconvex multifunctions, Math. Japon. 24 (1979), 293-300. Zbl0434.54009
  16. [16] S. Rolewicz, On conditions warranting Φ 2 -subdifferentiability, Math. Programming Study 14 (1981), 215-224. Zbl0444.90106
  17. [17] K. Shimizu, E. Aiyoshi and R. Katayama, Generalized Farkas's theorem and optimization of infinitely constrained problems, J. Optim. Theory Appl. 40 (1983), 451-462. Zbl0494.90062
  18. [18] C. Swartz, A general Farkas lemma, ibid. 46 (1985), 237-244. Zbl0543.49010
  19. [19] C. Ursescu, Multifunctions with convex closed graph, Czechoslovak Math. J. 7 (1975), 438-441. 
  20. [20] D. E. Ward and J. M. Borwein, Nonsmooth calculus in finite dimensions, SIAM J. Control Optim. 25 (1987), 1312-1340. Zbl0633.46043
  21. [21] C. Zalinescu, On Gwinner's paper "Results of Farkas type", Numer. Funct. Anal. Optim. 10 (1989), 401-414. Zbl0679.49024

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.