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A note on stability for parametric equilibrium problems. (English) Zbl 1112.90082
The authors study the stability of perturbed equilibrium problems in vector metric spaces where the function $f$ and the set $K$ are perturbed by the parameters ${\epsilon},\eta$. They study the stability of the solutions, providing some results in the peculiar framework of generalized monotone functions, first in the particular case where $K$ is fixed, then under both data perturbation. It is well known that equilibrium problems include variational inequalities as special case. The references are not up to date. Equilibrium problems considered in this paper were introduced and studied by {\it E. Blum} and {\it W. Oettli} [ Math. Stud. 63, 123--145 (1994; Zbl 0888.49007)] and {\it M. Aslam Noor} and {\it W. Oettli} [Mathematiche 49, 313--331 (1994; Zbl 0839.90124)]. For senstivity analysis of variational inclusions, see, for example, {\it M. A. Noor} [Comput. Math. Appl. 44, 1175--1181 (2002; Zbl 1034.49007)] and references therein.

90C31Sensitivity, stability, parametric optimization
49K40Sensitivity, stability, well-posedness of optimal solutions
90C47Minimax problems
47N10Applications of operator theory in optimization, convex analysis, programming, economics
Full Text: DOI
[1] M. Ait Mansour, Stability and sensitivity for equilibria and quasi-equilibria: application to variational and hemivariational inequalities, Ph.D. Thesis, Cadi Ayad University, 2002.
[2] Aliprantis, C. D.; Border, K. C.: Infinite dimensional analysis. (1999) · Zbl 0938.46001
[3] Aubin, J. -P.; Cellina, A.: Differential inclusions. (1984) · Zbl 0538.34007
[4] Bank, B.; Guddat, J.; Klatte, B.; Kummer, B.; Tammer, K.: Non-linear parametric optimization. (1983) · Zbl 0502.49002
[5] Berge, C.: Espaces topologiques. (1959) · Zbl 0088.14703
[6] Bianchi, M.; Pini, R.: A note on equilibrium problems for PQM bifunctions. J. global optim. 20, 67-76 (2001) · Zbl 0985.90090
[7] Hogan, W. W.: Point-to-set maps in mathematical programming. SIAM rev. 15, 591-603 (1973) · Zbl 0256.90042
[8] Luc, D. T.: Theory of vector optimization. (1989) · Zbl 0688.90051
[9] Yen, N. D.: Hölder continuity of solutions to a parametric variational inequality. Appl. math. Optim. 31, 245-255 (1995) · Zbl 0821.49011