×

Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems. (English) Zbl 1063.90062

The paper derives Fan-Knaster-Kuratowski-Mazurkiewicz type theorems and geometric properties of convex spaces, applying these to obtain coincidence theorems and fixed point theorems in convex spaces, with which they obtain existence results for a solution of generalized vector equilibrium problems. Some details follow.
Let \(X\) be a convex space, i.e. a non-empty convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets. Let \(Y\) be a Hausdorff topological space. If \( S,T:X\rightarrow 2^{Y}\)are maps such that \(T(coN)\sqsubseteq S(N)\), for each \(N\in \langle X\rangle \) (where \(coN\) denotes the convex hull of \(N\) and \( \langle X\rangle \) denotes the class of all finite subsets of \(X\) ) then \(S\) is said to be {generalized KKM mapping w.r.t. }\({T}\). The map \(T\) is said to have the {KKM property} if \(S\) is a generalized KKM map w.r.t. \(T\) such that the family \(\{\overline{S(X)}:x\in X\}\) has the finite intersection property, where \(\overline{S(X)}\) denotes the closure of \(S\left( X\right) \). Denote by \(KKM(X,Y)\) the family of all maps having the KKM property. A map \(G:X\rightarrow 2^{Y}\) is {transfer closed} if for any \(x\in X\) and \(y\notin P(x)\), there exists \(\widetilde{x}\in X\) such that \(y\notin P(\widetilde{x})\) and it is {transfer open} if for any \(x\in X\) and \(y\in P(x)\), there exists \(\widetilde{x}\in X\) such that \( y\in intP(\widetilde{x}).\)
The following theorem is proved: let \(X\) be a convex space, \(Y\) a topological space and \(T\in KKM(X,Y)\) be compact. Assume that \( G:X\rightarrow 2^{Y}\) is transfer closed and, for any \(N\in \langle X\rangle ,T(coN)\sqsubseteq G(N).\) Then, \(\overline{T(X)}\cap [ \bigcap \{G(x):x\in X\}]\neq \varnothing .\) This leads to results concerning geometric properties of convex spaces and to coincidence theorems such as the following. Let \(Y\) be a convex space, \(X\) a topological space and \( T\in KKM(Y,X)\) be compact. Let \(P:X\rightarrow 2^{Y}\) be a map such that for all \(x\in X\) , \(P(x)\) is convex and \(X=\bigcup \{intP^{-1}(y):y\in Y\}.\) Then there exists \(\left( \widetilde{x},\widetilde{y}\right) \in X\times Y\) such that \(\widetilde{x}\in T(\widetilde{y})\) and \(\widetilde{y}\in P( \widetilde{x}).\)
Let \(K\) be a non-empty convex subset of a topological vector space \(X\). Let \(Z\) be a a topological vector space. Let \(C:K\rightarrow 2^{Z}\) be a map such that, for each \(x\in K\), \(C(x)\) is a closed and convex cone with \( intC(x)\neq \emptyset \). Let \(F:K\times K\rightarrow 2^{Z}\backslash \{\emptyset \}\), then the {generalized vector equilibrium problem} ( {GVEP}) is to find \(\widetilde{x}\in K\) such that \(F(\widetilde{x} ,y)\nsubseteqq -int C(\widetilde{x})\), for all \(y\in K\). Theorems establishing the existence of a solution to GVEPs are obtained as applications of coincidence theorems.

MSC:

90C47 Minimax problems in mathematical programming
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
91B50 General equilibrium theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] FAN, K., A Generalization of Tychonoff’s Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305-310, 1961. · Zbl 0093.36701 · doi:10.1007/BF01353421
[2] HA, C.W., Minimax and Fixed-Point Theorems, Mathematische Annalen, Vol. 248, pp. 73-77, 1980. · Zbl 0413.47042 · doi:10.1007/BF01349255
[3] DING, X. P., KIM, W. K., and TAN, K.K., A New Minimax Inequality on H-Space with Applications, Bulletin of the Australian Mathematical Society, Vol. 41, pp. 457-473, 1990. · Zbl 0697.49008 · doi:10.1017/S0004972700018347
[4] PARK, S., Foundations of the KKM Theory via Coincidence of Composite of Upper Semicontinuous Maps, Journal of the Korean Mathematical Society, Vol. 31, pp. 493-519, 1994. · Zbl 0829.49002
[5] PARK, S., BAE, J. S., and KANG, H.K., Geometric Properties, Minimax Inequalities, and Fixed-Point Theorems on Convex Spaces, Proceedings of the American Mathematical Society, Vol. 121, pp. 429-439, 1994. · Zbl 0806.47054 · doi:10.1090/S0002-9939-1994-1231303-6
[6] YUAN, G.X.Z., KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, NY, 1999. · Zbl 0936.47034
[7] TIAN, G.Q., Generalizations of the KKM Theorem and the Ky Fan Minimax Inequality with Applications to Maximal Elements, Price Equilibrium, and Complementarity, Journal of Mathematical Analysis and Applications, Vol. 170, pp. 457-471, 1992. · Zbl 0767.49007 · doi:10.1016/0022-247X(92)90030-H
[8] LIN, L.J., Applications of a Fixed-Point Theorem in G-Convex Space, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 46, pp. 601-608, 2001. · Zbl 1001.47041 · doi:10.1016/S0362-546X(99)00456-3
[9] WU, X., and SHEN, S., A Further Generalization of Yannelis-Prabhakar’s Continuous Selection Theorem and Its Applications, Journal of Mathematical Analysis and Applications, Vol. 197, pp. 61-74, 1996. · Zbl 0852.54019 · doi:10.1006/jmaa.1996.0007
[10] LASSONDE, M., On the Use of KKM Multifunctions in Fixed-Point Theory and Related Topics, Journal of Mathematical Analysis and Applications, Vol. 97, pp. 151-201, 1983. · Zbl 0527.47037 · doi:10.1016/0022-247X(83)90244-5
[11] CHANG, T. H., and YEN, C. L., KKM Properties and Fixed-Point Theorems, Journal of Mathematical Analysis and Applications, Vol. 203, pp. 224-235, 1996. · Zbl 0883.47067 · doi:10.1006/jmaa.1996.0376
[12] LIN, L.J., A KKM Type Theorem and Its Applications, Bulletin of the Australian Mathematical Society, Vol. 59, pp. 481-493, 1999. · Zbl 0955.47037 · doi:10.1017/S0004972700033189
[13] AUBIN, J. P., and CELLINA, A., Differential Inclusions, Springer Verlag, Berlin, Germany, 1994. · Zbl 0538.34007
[14] CHANG, S. S., LEE, B. S., WU, X., CHO, Y. J., and LEE, G.M., On the Generalized Quasivariational Inequality Problems, Journal of Mathematical Analysis and Applications, Vol. 203, pp. 686-711, 1996. · Zbl 0867.49008 · doi:10.1006/jmaa.1996.0406
[15] KOMIYA, H., Coincidence Theorem and Saddle-Point Theorem, Proceedings of the American Mathematical Society, Vol. 96, pp. 599-602, 1986. · Zbl 0657.47055 · doi:10.1090/S0002-9939-1986-0826487-0
[16] ANSARI, Q. H., WONG, N. C., and YAO, J.C., The Existence of Nonlinear Inequalities, Applied Mathematics Letters, Vol. 12, pp. 89-92, 1999. · Zbl 0940.49010 · doi:10.1016/S0893-9659(99)00062-2
[17] BIANCHI, M., and SCHAIBLE, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31-43, 1996. · Zbl 0903.49006 · doi:10.1007/BF02192244
[18] BLUM, E., and OETTLI, W., From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Students, Vol. 63, pp. 123-146, 1994. · Zbl 0888.49007
[19] CHADLI, O., CHBANI, Z., and RIAHI, H., Equilibrium Problems and Noncoercive Variational Inequalities, Optimization, Vol. 49, pp. 1-12, 1999. · Zbl 1022.49013
[20] CHADLI, O., CHBANI, Z., and RIAHI, H., Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 105, pp. 299-323, 2000. · Zbl 0966.91049 · doi:10.1023/A:1004657817758
[21] KONNOV, I. V., and SCHAIBLE, S., Duality for Equilibrium Problems under Generalized Monotonicity, Journal of Optimization Theory and Applications, Vol. 104, pp. 395-408, 2000. · Zbl 1016.90066 · doi:10.1023/A:1004665830923
[22] TARAFDAR, E., and YUAN, G.X.Z., Generalized Variational Inequalities and Its Applications, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 30, pp. 4171-4181, 1997. · Zbl 0912.49004 · doi:10.1016/S0362-546X(96)00142-3
[23] ANSARI, Q.H., Vector Equilibrium Problems and Vector Variational Inequalities, Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 1-16, 2000. · Zbl 0992.49012
[24] BIANCHI, M., HADJISAVVAS, N., and SCHAIBLE, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527-542, 1997. · Zbl 0878.49007 · doi:10.1023/A:1022603406244
[25] GIANNESSI, F., Editor, Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, Holland, 2000. · Zbl 0952.00009
[26] HADJISAVVAS, N., and SCHAIBLE, S., From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297-309, 1998. · Zbl 0903.90141 · doi:10.1023/A:1022666014055
[27] LEE, G. M., KIM, D. S., and LEE, B. S., On Noncooperative Vector Equilibrium, Indian Journal of Pure and Applied Mathematics, Vol. 27, pp. 735-739, 1996. · Zbl 0858.90141
[28] LIN, L. J., and YU, Z.T., Fixed-Point Theorems and Equilibrium Problems, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 43, pp. 987-999, 2001. · Zbl 0989.47051 · doi:10.1016/S0362-546X(99)00202-3
[29] OETTLI, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213-221, 1997. · Zbl 0914.90235
[30] TAN, N. X., and TINH, P.N., On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Applications, Vol. 19, pp. 141-156, 1998. · Zbl 0896.90161
[31] ANSARI, Q. H., and YAO, J.C., An Existence Result for the Generalized Vector Equilibrium Problems, Applied Mathematics Letters, Vol. 12, pp. 53-56, 1999. · Zbl 1014.49008 · doi:10.1016/S0893-9659(99)00121-4
[32] ANSARI, Q.H., KONNOV, I. V., and YAO, J.C., On Generalized Vector Equilibrium Problems, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 47, pp. 543-554, 2001. · Zbl 1042.90642 · doi:10.1016/S0362-546X(01)00199-7
[33] KONNOV, I. V., and YAO, J.C., Existence of Solutions for Generalized Vector Equilibrium Problems, Journal of Mathematical Analysis and Applications, Vol. 233, pp. 328-335, 1999. · Zbl 0933.49004 · doi:10.1006/jmaa.1999.6312
[34] LIN, L. J., YU, Z. T., and KASSAY, G., Existence of Equilibria for Monotone Multivalued Mappings and Its Applications to Vectorial Equilibria, Journal of Optimization Theorey and Applications, Vol. 114, pp. 189-208, 2002. · Zbl 1023.49014 · doi:10.1023/A:1015420322818
[35] OETTLI, W., and SCHLÄGER, D., Existence of Equilibria for Monotone Multivalued Mappings, Mathematical Methods of Operations Research, Vol. 48, pp. 219-228, 1998. · Zbl 0930.90077 · doi:10.1007/s001860050024
[36] ANSARI, Q. H., and YAO, J.C., Generalized Vector Equilibrium Problems, Journal of Statistics and Management Systems (to appear). · Zbl 1079.90594
[37] ANSARI, Q.H., OETTLI, W., and SCHLÄGER, D., A Generalization of Vectorial Equilibria, Mathematical Methods of Operation Research, Vol. 46, pp. 147-152, 1997. · Zbl 0889.90155 · doi:10.1007/BF01217687
[38] OETTLI, W., and SCHLÄGER, Generalized Vectorial Equilibria and Generalized Monotonicity, Functional Analysis with Current Applications in Science, Technology, and Industry, Edited by M. Brokate and A.H. Siddiqi, Pitman Research Notes in Mathematics, Longman, Essex, England, Vol. 373, pp. 145-154, 1998.
[39] SONG, W., Vector Equilibrium Problems with Set-Valued Mappings, Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 403-422, 2000. · Zbl 0993.49011
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.