Nonemptiness and boundedness of solution sets for vector variational inequalities via topological method. (English) Zbl 1322.49011
Summary: In this paper, some characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are studied in finite and infinite dimensional spaces, respectively. By using a new proof method which is different from the one used in [X. X. Huang et al., J. Optim. Theory Appl. 162, No. 2, 548–558 (2014; Zbl 1315.90055)], a sufficient and necessary condition for the nonemptiness and boundedness of solution sets is established. Based on this result, some new characterizations of nonemptiness and boundedness of solution sets for vector variational inequalities are proved. Compared with the known results in [loc. cit.], the key assumption that \(K_\infty\cap (F(K))^{w\circ}_C=\{0\}\) is not required in finite dimensional spaces. Furthermore, the corresponding result of [loc. cit.] is extended to the case of infinite dimensional spaces. Some examples are also given to illustrate the main results.
MSC:
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H05 | Monotone operators and generalizations |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
vector variational inequalities; \(C\)-pseudomonotone mapping; boundedness; connectedness; recession coneCitations:
Zbl 1315.90055References:
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