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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.
⟨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.
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
199-212
Opis fizyczny
Daty
wydano
2007
otrzymano
2005-08-01
Twórcy
autor
- Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, Republic of China
autor
- Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan, Republic of China
Bibliografia
- [1] J.P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, 1984.
- [2] D. Chan and J.S. Pang, The generalized quasi-variational inequality problem, Math. Operations Research 7 (1982), 211-222.
- [3] L.J. Chu and C.Y. Lin, Variational inequalities in noncompact nonconvex regions, Disc. Math. Differential Inclusions, Control and Optimization 23 (2003), 5-19.
- [4] P. Cubiotti, Finite-dimensional quasi-variational inequalities associated with discontinuous functions, J. Optimization Theory and Applications 72 (1992), 577-582.
- [5] P. Cubiotti, An existence theorem for generalized quasi-variational inequalities, Set-Valued Analysis 1 (1993), 81-87.
- [6] P. Cubiotti, An application of quasivariational inequalities to linear control systems, J. Optim. Theory Appl. 89 (1) (1996), 101-113.
- [7] P. Cubiotti, Generalized quasi-variational inequalities without continuities, J. Optim. Theory Appl. 92 (3) (1997), 477-495.
- [8] P. Cubiotti, Generalized quasi-variational inequalities in infinite-dimensional normed spaces, J. Optim. Theory Appl. 92 (3) (1997), 457-475.
- [9] E. Klein and A.C. Thompson, Theorem of Correspondences, Wiley, New York, 1984.
- [10] H. Kneser, Sur un théoreme fondamantal de la théorie des jeux, Comptes Rendus de l'Academie des Sciences, Paris 234 (1952), 2418-2420.
- [11] M.L. Lunsford, Generalized variational and quasivariational inequalities with discontinuous operators, J. Math. Anal. Appl. 214 (1997), 245-263.
- [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).
- [13] R. Saigal, Extension of the generalized complemetarity problem, Math. Operations Research 1 (3) (1976), 260-266.
- [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.
- [15] J.C. Yao and J.S. Guo, Variational and generalized variational inequalities with discontinuous mappings, J. Math. Anal. Appl. 182 (1994), 371-392.
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1083
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