Generalized vector variational inequality and fuzzy extension. (English) Zbl 0804.49004

Lee, Gue Myung; Kim, Do Sang; Lee, Byung Soo; Cho, Sung Jin

Appl. Math. Lett. 6, No. 6, 47-51 (1993).

A generalized vector variational inequality is considered. The definitions of \(C\)-pseudomonotonicity and \(V\)-hemicontinuity are given. An existence result for the generalized vector variational inequality is obtained. The aim of this paper is to obtain the fuzzy extension of a result of Chen and Yang.

Reviewer: R.Stavre (Bucureşti)

MSC:

49J20	Existence theories for optimal control problems involving partial differential equations
49J27	Existence theories for problems in abstract spaces

Keywords:

fuzzy extension; vector variational inequality; \(C\)-pseudomonotonicity; \(V\)-hemicontinuity

References:

[1]	Giannessi, F., Theorems of alternative, quadratic programs and complementarity problems, (), 151-186 · Zbl 0484.90081
[2]	Chen, G.Y.; Cheng, G.M., Vector variational inequality and vector optimization, (), 408-416
[3]	Chen, G.Y.; Craven, B.D., Approximate dual and approximate vector variational inequality for multiobjective optimization, J. austral. math. soc., 47, 418-423, (1989), (Series A) · Zbl 0693.90089
[4]	Chen, G.Y.; Craven, B.D., A vector variational inequality and optimization over an efficient set, Zeitscrift fur operations research, 3, 1-12, (1990) · Zbl 0693.90091
[5]	Chen, G.Y.; Yang, X.Q., The vector complementarity problem and its equivalence with the weak minimal element in ordered sets, J. math. anal. appl., 153, 136-158, (1990)
[6]	Chen, G.Y., Existence of solutions for a vector variational inequality: an extension ofthe hartman-Stampacchia theorem, J. optim. th. appl., 74, 3, 445-456, (1992) · Zbl 0795.49010
[7]	Chang, S.S.; Zhu, Y.G., On variational inequalities for fuzzy mappings, Fuzzy sets and systems, 32, 359-367, (1989) · Zbl 0677.47037
[8]	Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. math. anal. appl., 97, 151-201, (1983) · Zbl 0527.47037
[9]	Shin, M.H.; Tan, K.K., Generalized quasi-variational inequalities in locally convex topological spaces, J. math. anal. appl., 108, 333-343, (1985)
[10]	Takahashi, W., Nonlinear variational inequalities and fixed point theorems, J. math. soc. japen, 28, 168-181, (1976) · Zbl 0314.47032
[11]	Yen, C.L., A minimax inequality and its applications to variational inequalities, Pacific J. math., 97, 142-150, (1981)
[12]	B.S. Lee, G.M. Lee, S.J. Cho and D.S. Kim, A variational inequality for fuzzy mappings, Proceedings of Fifth International Fuzzy Systems Association World Congress (to appear).
[13]	Kim, W.K.; Tan, K.K., A variational inequality in non-compact sets and its applications, Bull. austral. math. soc., 46, 139-248, (1992)
[14]	Hartman, G.J.; Stampacchia, G., On some nonliner elliptic differential functional equations, Acta math., 115, 271-310, (1966) · Zbl 0142.38102
[15]	Cottle, R.W.; Yao, J.C., Pseudo-monotone complementarity problems in Hilbert space, J. optim. th. appl., 75, 2, 281-295, (1992) · Zbl 0795.90071
[16]	Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. ann., 142, 305-310, (1961) · Zbl 0093.36701
[17]	Weiss, M.D., Fixed points, separation and induced topologies for fuzzy sets, J. math. anal. appl., 50, 142-150, (1975) · Zbl 0297.54004