臺灣博碩士論文加值系統

令 T 是一個從拓樸向量空間 X 的子集映至 X* 的多值函數。此文章主要是討論 T 滿足 (S)_+ 條件和 T 滿足 (S)_+^1 條件之間的關係。為了整合單值與多值函數之 (S)_+ 條件的定義，我們將 [9] 中對於單值函數定義的 (S)_+^w 條件延拓至多值函數的情形。上列這些條件對於映至 L(X,Z) 之函數亦可定義，其中 Z 是賦序拓樸向量空間。利用這些條件，我們推導了一些廣義向量變分不等式與廣義變分不等式的存在性結果。

Let T be a multivalued mapping from a nonempty subset of a topological vector space into its topological dual. In this paper, we discuss the relationship between the multivalued mapping T satisfying the (S)_+ condition and T satisfying the (S)_+^1 condition. To unify the (S)_+ condition for single-valued and multivalued mappings, we introduce the weak (S)_+ condition for single-valued mappings defined in [9] to multivalued mappings. The above
conditions extend naturally to mappings into L(X,Z), where Z is an ordered Hausdorff topological vector space. We also derive some existence results for generalized vector
variational inequalities and generalized variational inequalities associated with mappings which satisfy the (S)_+, (S)_+^1 or weak (S)_+ condition.

1. Introduction 2
2. Preliminary 4
3. The (S)_+ and (S)_+^1 Conditions 7
4. Generalized Vector Variational Inequalities 19
5. Appendix 27

[1] C.D. Aliprantis and K.C. Border, Infinite Dimensional
Analysis, Springer-Verlag, Berlin, 1999.

[2] Q.H. Ansari, X.Q. Yang, and J.C. Yao, Existence and Duality of Implicit Vector Variational Problems, Numerical Functional Analysis and Optimization, Vol.22, pp.815-829, 2001.

[3] C. Berge, Topological Spaces, includung a treat of multivalued Functions, Vector Spaces and Convexity, Oliver and Boyd Ltd., 1963.

[4] F.E. Browder, Existence Theorems for Nonlinear Partial Differential Equations, Proceedings of Symposia in Pure Mathematics, Vol.16, pp.1-60, 1970.

[5] O. Chadli, Y. Chiang, and S. Huang, Topological pseudomonotonicity and vector equilibrium problems, Journal of Mathematical Analysis and Applications, Vol.270, pp.435-450, 2002.

[6] O. Chadli, N.C. Wong, and J.C. Yao, Equilibrium Problems with Applications to Eigenvalue Problems,
Journal of Optimization Theory and Applications, Vol.117, No.2, pp.245-266, 2003.

[7] Y. Chiang, Vector Superior and Inferior, Taiwanese Journal of Mathematics, Vol.8, No.3, pp.477-487, 2004.

[8] Y. Chiang, The (S)_+^1 Condition for Generalized Vector Variational Inequalities, Journal of Optimization Theory and Applications, Vol.124, No.3, pp.581-594, 2005.

[9] Y. Chiang, The (S)_+-Condition for Vector Equilibrium Problems, Taiwanese Journal of Mathematics, Vol.10, No.1, pp.31-43, 2006.

[10] Y. Chiang, Generalized Vector Variational Inequalities Assosiated with Nonmonotone Multivalued Mappings, preprint.

[11] Y. Chiang and J.C. Yao, Vector Variational Inequalities and the (S)_+ Condition, Journal of Optimization Theory and Applications, Vol.123, No.2, pp.271-290, 2004.

[12] J.B. Conway, A Course in Functional Analysis, 2nd Edition, Springer-Verlag, New York, 1990.

[13] P. Cubiotti, On the Discontinuous Infinite-Dimensional Generalized Quasivariational Inequality Problem, Journal of Optimization Theory and Applications, Vol. 115, pp. 97-111, 2002.

[14] P. Cubiotti, Generalized Quasivariational Inequalities in Infinite-Dimensional Normed Spaces, Journal
of Optimization Theory and Applications, V0l. 92, pp. 457-475, 1997.

[15] P. Cubiotti, Generalized Quasivariational Inequalities without Continuities, Journal of Optimization
Theory and Applications, V0l. 92, pp. 477-495. 1997.

[16] P. Cubiotti and J.C. Yao, Multivalued (S)_+^1 Operators and Generalized Variational Inequalities, Computers and Mathematics with Applications, Vol.29, No.12, pp.49-56, 1995.

[17] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984.

[18] J.S. Guo and J.C. Yao, Variational Inequalities with Nonmonotone Operators, Journal of Optimization Theory and Applications, Vol.80, No.1, pp.63-74, 1994.

[19] G. Isac and M.S. Gowda, Operators of Class (S)_+^1, Altman''s Condition and the Complementarity Problem,
Journal of the Faculty of Science, University of Tokyo, Section IA, Mathematics Vol. 40, pp.1-16, 1993.

[20] L.V. Kantorovich and G.P. Akilov, Functional Analysis, 2nd Edition, pergamon, 1982.

[21] E. Klein and A.C. Thompson, Theory of Correspondences, John Wiley and Sons, New York, 1984.

[22] G. Kothe, Topological Vector Spaces I, Springer-Verlag, Berlin, Heidelberg, 1983.

[23] J.R. Munkres, Topology, 2nd Edition, Prentice Hall, Inc., NJ, 2000.

[24] L. Narici and E. Beckenstein, Topological Vector Spaces, Marcel Dekker, Inc., 1985.

[25] W. Rudin, Functional Analysis, 2nd Edition, McGraw-Hill, 1991.

[26] H.H. Schaefer and M.P. Wolff, Topological Vector Spaces, 2nd Edition, Springer-Verlag, New York, 1999.

[27] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol II/A, Linear Monotone
Operators, Springer-Verlag, New York, 1990.