這論文的要旨有兩層:
(1)我們利用一般化的KKM mapping 概念,首先得到廣義的 G´orniewicz固定點定理;
(2)藉著應用我們的固定點定理,建立幾個廣義的向量值擬變分不等式(GVQVI)的存在性定理.我們在既不是凸的亦不是緊緻的,而僅僅只是在 nearly convex 的區域上推論出幾個結果.

We introduce a new class of nonconvex sets, which are named nearly convex set, and then extend several existence results on nonconvex optimization problems. In fact, the purpose of this paper is two fold: (1) we first obtain a rather general version of the G´orniewicz fixed point theorem by using the concept of generalized KKM mappings; (2) we establish some existence theorems for generalized vector quasi-variational inequality problems by applying our fixed point theorem. We derive several results here neither convex nor compact on constraint

{1} Q. H. Ansari & J. C. Yao (2000). On nondifferentiable and nonconvex vector optimization problems, J. Optim. Theory Appl. 106, 487-500.
{2} J.-P. Aubin & I. Ekeland (1984). Applied Nonlinear Analysis, John Wiley & Sons, New York.
{3} E. G. Begle (1942). Locally connected spaces and generalized manifolds, Amer. Math. J. 64, 553-574.
(4) E. G. Begle (1950). The Vietoris mapping theorem for bicompact space, Ann. Math. 51, 534-543
(5) F. E. Browder (1984). Coincidence theorems, minimax theorems and variational inequalities, Comtemporary Math. 26, 67-80.
(6) S. S. Chang, H. B. Thompson & G. H. Z. Yuan (1999). The existence theorems of
solutions for generalized vector-valued variational-like inequalities, Comput. Math. Appl. 37, 1-9.
(7) T. H. Chang & C. L. Yen (1996). KKM properties and fixed point theorem, J. Math. Anal. Appl. 203, 224-235.
(8) G. Y. Chen (1992). Existence of solution for a vector variational inequality : an extension of the Hartman-Stampacchia theorem, J. Optim. Theory Appl. 74(3), 445-456.
(9) L. J. Chu (1999). Unified approaches to nonlinear optimization, Optimization 46, 25-60.
(10) K. Fan (1952). Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S.A. 38, 121-126.
(11) I. L. Glicksberg (1952). A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3, 170-174.
(12) L. G’orniewicz (1975). A Lefschetz-type fixed point theorem, Fund. Math., 88, 103-115.
(13) H. Halkin (1967). Finite convexity in infinite-dimensional spaces, Proc. of the Colloquium on Convexity, Copenhagen (1965),W. Fenchel (ed.), Copenhagen, 126-131.
(14) V. Klee (1960). Leray-Schauder theory without local convexity, Math. Ann. 141, 286-297.
(15) M. Lassonde (1990). Fixed points for Kakutani factorizable multifunctions,
J. Math. Anal. Appl. 152, 46-60.
(16) L. J. Lin & Z. T. Yu (1999). Fixed points theorems of KKM-type maps,Nonlinear Analysis 38, 265-275.
(17) D. T. Luc (1989). Theory of vector optimization, Lecture Notes in Economics and Math. Systems 319, Springer, Berlin.
(18) W. S. Massey (1980). Singular homology theory, Springer-Verlag, New York.
(19) E. Michael (1956). Continuous selections I, Ann. Math. 63, 361-382.
(20) G. J. Minty (1961). On the maximal domain of a “monotone” function, Michigan Math. J. 8, 135-137.
(21) D. T. Nhan (1991). On coincidence theorems for set-valued mapping and variational inequalities, Acat Math. Vietnamica 16(1), 61-67.
(22) S. Park (1993). Coincidences of composites of admissible u.s.c. maps and applications, C. R. Math. Acad. Sci. Canada 15, 125-130.
(23) S. Park (1994). Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps, J. Korean Math. Soc. 31, 493-519.
(24) R. T. Rockafellar (1970). On the virtual convexity of the domain and range of a nonlinear maximal monotone operator, Math. Ann. 185, 81-90.
(25) R. T. Rockafellar (1970). Convex Analysis, Princeton, New Jersey, Princeton University Press.
(26) J. C. Yao & I. V. Konnov (1997). On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206, 42-58.