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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 E. Blum and W. Oettli [ Math. Stud. 63, 123–145 (1994; Zbl 0888.49007)] and M. Aslam Noor and W. Oettli [Mathematiche 49, 313–331 (1994; Zbl 0839.90124)]. For senstivity analysis of variational inclusions, see, for example, M. A. Noor [Comput. Math. Appl. 44, 1175–1181 (2002; Zbl 1034.49007)] and references therein.

##### MSC:
 90C31 Sensitivity, stability, parametric optimization 49K40 Sensitivity, stability, well-posedness 90C47 Minimax problems in mathematical programming 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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##### References:
 [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), Springer Berlin · Zbl 0938.46001 [3] Aubin, J.-P.; Cellina, A., Differential inclusions, (1984), Springer Berlin [4] Bank, B.; Guddat, J.; Klatte, B.; Kummer, B.; Tammer, K., Non-linear parametric optimization, (1983), Birkhäuser Basel, Boston · Zbl 0502.49002 [5] Berge, C., Espaces topologiques, (1959), Dunod Paris · 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), Springer Berlin [9] Yen, N.D., Hölder continuity of solutions to a parametric variational inequality, Appl. math. optim., 31, 245-255, (1995) · Zbl 0821.49011
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