zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
49K40Sensitivity, stability, well-posedness of optimal solutions
49J40Variational methods including variational inequalities
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 · doi:10.1023/A:1017531323904
[2]Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)
[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)
[4]Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982) · Zbl 0502.90080 · doi:10.1287/moor.7.2.211
[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)
[6]Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)
[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)
[10]Lignola, M.B.: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 128, 119–138 (2006) · Zbl 1093.49005 · doi:10.1007/s10957-005-7561-2
[11]Lignola, M.B., Morgan, J.: Semicontinuity and episemicontinuity: equivalence and applications. Boll. Unione Mat. Ital. 8B, 1–6 (1994)
[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 · doi:10.1023/A:1008370910807
[13]Lignola, M.B., Morgan, J.: Vector quasivariational inequalities: approximate solutions and well-posedness. J. Convex Anal. 13, 373–384 (2006)
[14]Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969) · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[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)
[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 · doi:10.1007/s10957-006-9077-9
[18]Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)
[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)
[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 · doi:10.1007/s10255-004-0185-8
[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
[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 · doi:10.1007/s10898-005-5509-6