×

Well-posedness for mixed quasivariational-like inequalities. (English) Zbl 1157.49033

In this paper, mixed quasivariational like inequalities are studied where the underlying map is multivalued. For various concepts of well-posedness (for example \(L\)-well-posedness) necessary and sufficient conditions for well-posedness are established.

MSC:

49K40 Sensitivity, stability, well-posedness
49J40 Variational inequalities
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ansari, Q.H., Yao, J.C.: Iterative schemes for solving mixed variational-like inequalities. J. Optim. Theory Appl. 108, 527–541 (2001) · Zbl 0999.49008
[2] Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984) · Zbl 0551.49007
[3] Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980) · Zbl 0441.47056
[4] Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982) · Zbl 0502.90080
[5] Del Prete, I., Lignola, M.B., Morgan, J.: New concepts of well-posedness for optimization problems with variational inequality constraints. J. Inequal. Pure Appl. Math. 4, 26–43 (2003) · Zbl 1029.49024
[6] Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993) · Zbl 0797.49001
[7] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research. Springer, Berlin (2003). Vols. 32 I and 32 II
[8] Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I. Kluwer Academic, Dordrecht (1997)
[9] Lechicki, A.: On bounded and subcontinuous multifunctions. Pac. J. Math. 75, 191–197 (1978) · Zbl 0387.54008
[10] Lignola, M.B.: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 128, 119–138 (2006) · Zbl 1093.49005
[11] Lignola, M.B., Morgan, J.: Semicontinuity and episemicontinuity: equivalence and applications. Boll. Unione Mat. Ital. 8B, 1–6 (1994) · Zbl 0799.49013
[12] Lignola, M.B., Morgan, J.: Well-posedness for optimization problems with constraints defined by a variational inequality having a unique solution. J. Glob. Optim. 16, 57–67 (2000) · Zbl 0960.90079
[13] Lignola, M.B., Morgan, J.: Vector quasivariational inequalities: approximate solutions and well-posedness. J. Convex Anal. 13, 373–384 (2006) · Zbl 1105.49016
[14] Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969) · Zbl 0192.49101
[15] Mosco, U.: Implicit variational problems and quasivariational inequalities. In: Summer School, Nonlinear Operators and the Calculus of Variations, Bruxelles, Belgium, 1975. Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)
[16] Rockafellar, T.: Convex Analysis. Princeton University Press, Princeton (1970) · Zbl 0193.18401
[17] Schaible, S., Yao, J.C., Zeng, L.C.: Iterative method for set-valued mixed quasivariational inequalities in a Banach space. J. Optim. Theory Appl. 129, 425–436 (2006) · Zbl 1123.49006
[18] Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958) · Zbl 0081.11502
[19] Tykhonov, A.N.: On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6, 631–634 (1966)
[20] Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (eds.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic Press, New York (1971) · Zbl 0281.47043
[21] Zeng, L.C.: Iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities. Acta Math. Appl. Sinica 20, 477–486 (2004). English Series · Zbl 1049.49013
[22] Zeng, L.C.: Perturbed proximal point algorithm for generalized nonlinear set-valued mixed quasi-variational inclusions. Acta Math. Sinica 47, 11–18 (2004). Chinese Series · Zbl 1167.49304
[23] Zeng, L.C., Yao, J.C.: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. J. Glob. Optim. 36, 483–496 (2006) · Zbl 1115.49005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.