On discontinuous quasi-variational inequalities

Liang-Ju Chu; Ching-Yang Lin

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2007)

  • Volume: 27, Issue: 2, page 199-212
  • ISSN: 1509-9407

Abstract

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In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and T : X → 2 X * and S : C → 2 D are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian’s condition, we give a partial affirmative answer to an unbounded quasi-variational inequality problem.

How to cite

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Liang-Ju Chu, and Ching-Yang Lin. "On discontinuous quasi-variational inequalities." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 27.2 (2007): 199-212. <http://eudml.org/doc/271142>.

@article{Liang2007,
abstract = {In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and $T:X → 2^\{X*\}$ and $S:C → 2^D$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian’s condition, we give a partial affirmative answer to an unbounded quasi-variational inequality problem.},
author = {Liang-Ju Chu, Ching-Yang Lin},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {variational inequality; quasi-variatioal inequality; Ricceri's conjecture; Karamardian condition; Hausdorff continuous multifunction; Kneser's minimax inequality},
language = {eng},
number = {2},
pages = {199-212},
title = {On discontinuous quasi-variational inequalities},
url = {http://eudml.org/doc/271142},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Liang-Ju Chu
AU - Ching-Yang Lin
TI - On discontinuous quasi-variational inequalities
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2007
VL - 27
IS - 2
SP - 199
EP - 212
AB - In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and $T:X → 2^{X*}$ and $S:C → 2^D$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian’s condition, we give a partial affirmative answer to an unbounded quasi-variational inequality problem.
LA - eng
KW - variational inequality; quasi-variatioal inequality; Ricceri's conjecture; Karamardian condition; Hausdorff continuous multifunction; Kneser's minimax inequality
UR - http://eudml.org/doc/271142
ER -

References

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  11. [11] M.L. Lunsford, Generalized variational and quasivariational inequalities with discontinuous operators, J. Math. Anal. Appl. 214 (1997), 245-263. Zbl0945.49003
  12. [12] B. Ricceri, Basic existence theorem for generalized variational and quasi-variational inequalities, Variational Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A. Maugeri, Plenum Press, New York, 1995 (251-255). Zbl0847.49011
  13. [13] R. Saigal, Extension of the generalized complemetarity problem, Math. Operations Research 1 (3) (1976), 260-266. Zbl0363.90091
  14. [14] M.H. Shih and K.K. Tan, Generalized quasi-variational inequaloties in locally convex topological vector spaces, J. Math. Anal. Appl. 108 (1985), 333-343. Zbl0656.49003
  15. [15] J.C. Yao and J.S. Guo, Variational and generalized variational inequalities with discontinuous mappings, J. Math. Anal. Appl. 182 (1994), 371-392. Zbl0809.49005

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