我們首先推廣了許多早期有關於一致點的結果。事實上，我們藉由Gorniewicz固定點定理得到一個在G-space上的一致點定理，以及它們的一些推論。這些是很強的結果，因為在G-space上不需要有線性或凸性結構。接著我們藉由Nikaido的一致點定理，導出一些關鍵性結果。西元1959年Nikaido建立了在緊緻Hausdorff拓樸空間上一個很著名的一致點定理，並推廣了Gale的一些有關經濟平衡點（economic equilibrium）存在性與競賽問題（game problems）的結果。的確，我們簡化並改良了某些一般的擬變分不等式（quasi-variational inequalities）與關於非循環（acyclic）多值函數的 Lefschetz-type的固定點定理，以及一些相關定理。在函數不必是單調（monotonicity）且約束集合不必是賦距空間之下，我們的結果統整了許多有關古典優選問題的定理。順道一提，就某種意義下我們也證明了Nikaido一致點定理是這些結果的推論。

The paper focuses on two important folds in nonlinear optimization; namely, coincidence theory and variational inequalities. We first extend and generalize many earlier results about coincidences. Indeed, by using Gorniewicz fixed point theorem, we obtain a coincidence theorem on G-space and some corollaries from it. Those are very strong results, because in G-space there is not necessarily any linear and convex structure. Secondly, we shall deduce several generalized key results based on a very powerful result from Nikaido. In 1959, Nikaido established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. Indeed, we shall simplify and reformulate a few generalized variational inequalities, quasi-variational inequalities, and a Lefschetz-type fixed point theorem on acyclic multifunctions, as well as some related theorems. Beyond the realm of monotonicity nor
metrizability, the results derived here generalize and unify various earlier ones from classic optimization. In the sequel, we shall deduce that Nikaido's coincidence theorem becomes a consequence of our result.

item {[1]} {

f R.A. Al`{o} & H.L. Shapiro}. {it
``Normal Topological Spaces"/}, Cambridge university press,
London, 1974.
item {[2]} {

f J.-P. Aubin & Cellina}. {it ``Differential
Inclusion"/}, Springer-Verlag, Berlin, Heidelberg, 1994.
item {[3]} {

f J.-P. Aubin & I. Ekeland}. {it Applied nonlinear analysis/},
John Wiley & Sons, New York, 1984.
item {[4]} {

f J.-P. Aubin & H. Frankowska}. {it
``Set-Valued Analysis" /}, Birkh"{a}user, Berlin, 1990.
item {[5]} {

f C. Baiocchi & A. Capelo}. {it
Variational and quasi-variational inequalities, Applications to
free-boundary problems/}, John Wiley and Sons Ltd, New York,
1984.
item {[6]} {

f E.G. Begle}. {it Locally
connected spaces and generalized manifolds/}, Amer. Math. J. {

f
64} (0942), 553-574.
item {[7]} {

f E.G. Begle}. {it The vietoris mapping
theorem for bicompact space/}, Ann. of Math. {

f 51} (1950), 534-543.
item {[8]} {

f E.G. Begle}. {it A fixed point
theorem/}, Ann. of Math. {

f 51} (1950), 544-550.
item {[9]} {

f F.E. Browder}. {it Coincidence theorems,
minimax theorems, and variational inequalities/}, Contemp. Math.
{

f 26} (1984), 67-80.
item {[10]} {

f F.E. Browder}. {it The fixed point
theory of multivalued mappings in topological vector spaces/},
Math. Ann. {

f 177} (1968), 283-301
item {[11]} {

f S.Y. Chang}. {it On the nash equilibrium /},
Soochow J. Math. {

f 16} (1990), 241-248.
item {[12]} {

f D. Chan & J.S. Pang}.
{it The generalized quasi-variational inequality problem /},
Math. Oper. Res. {

f 7} (1982), 211-222.
item {[13]} {

f L.J. Chu}. {it Unified approaches to nonlinear
optimization/}, Optimization {

f 46} (1999), 25-60.
item {[14]} {

f L.J. Chu}. {it On Fan's minimax inequality/},
J. Math. Anal. Appl. {

f 201} (1996), 103-113.
item {[15]} {

f G. Debreu}. {it Mathematical
economics/}, Cambridge Univ. Press, New York 1983.
item {[16]} {

f X.P. Ding & E. Tarafdar}. {it Some
coincidence theoremsand applications},
Bull. Austral. Math. Soc. {

f 50 } (1994), 73-80.
item {[17]} {

f K. Fan}. {it A minimax
inequality and applications, Inequalities/} {

f 3}, (O.
Shisha ed.), Academic Press, New York 1972, 103-113.
item {[18]} {

f K. Fan}. {it A generalization of
Tychnoff's fixed point theorem /}, Math. Ann. {

f 142} (1961), 305-310.
item {[19]} {

f K. Fan}. {it Fixed-point and minimax
theorems in locally convex topological linear spaces /}, Proc.
Nat. Acad. Sci. USA {

f 38} (1952), 121-126.
item {[20]} {

f D. Gale}. {it The law of supply and demand/},
Math. Scand. {

f 3} (1955), 155-169.
item {[21]} {

f F. Giannessi}. {it Theorems of alternative,
quadratic problems and Complementary problems/}, Variational
Inequalities and Complementary Problems (Cottle, R.W., Giannessi, F,
& Lions, J.L., Eds), Wiley, New York 1980, 151-186.
item {[22]} {

f I.L. Glicksberg}. {it A
further generalization of the Kakutani fixed point theorem, with
application to Nash equilibrium points/}, Proc. Amer. Math.
Soc. {

f 3} (1952), 170-174.
item {[23]} {

f L. Gorniewicz}. {it A Lefschetz-type fixed point theorem}, Fund.Math.
{

f 88} (1975), 103-115.
item {[24]} {

f A. Granas & F.-C. Liu}.
{it Coincidences for set-valued maps and minimax inequalities/},
J. Math. Pures et Appl. {

f 65} (1986), 119-148.
item {[25]} {

f C.W. Ha}. {it Minimax and fixed point theorems /},
Math. Ann. {

f 248} (1980), 73-77.
item {[26]} {

f C.W. Ha}. {it Extensions of two fixed point
theorems of Ky Fan /}, Math. Z. {

f 190} (1985), 13-16.
item {[27]} {

f B. Knaster, C. Kuratowski, &
S. Mazurkiewicz}. {it Ein beweis des fixpunktsatzes
fur $n$-dimensionale simplexe/}, Fund. Math. {

f 14} (1929), 132-137.
item {[28]} {

f H. Komiya}. {it Coincidences
theorem and saddle point theorem/}, Proc. Amer. Math. Soc. {

f
14} (1986), 599-602.
item {[29]} {

f S. Kum}
. {it A generalization of generalized
quasi-variational inequalities/}, J. Math. Anal. Appl. {

f 182} (1994), 158-164.
item {[30]} {

f L.J. Lin & S. Park}. {it On some generalized
quasi-equilibrium problems/}, to appear in J. Math. Anal. Appl.,
1999.
item {[31]} {

f W.S. Massey}. {it ``Singular Homology Theory"
/}, Springer-Verlag, New York 1980.
item {[32]} {

f G. Mehta & S. Sessa}. {it Coincidence
theorems and maximal elements in topological vector spaces /},
Math. Japan. {

f 47} (1992), 839-845.
item {[33]} {

f J. Nagata}. {it ``Modern General
Topology" /}, North-Holland, New York 1985.
item {[34]} {

f J.F. Nash}. {it Equilibrium
points in $n$-person games/}, Proc. Nat. Acad. Sci. U.S.A.
{

f 36} (1950), 48-49.
item {[35]} {

f H. Nikaid^o}. {it
Coincidence and some systems of inequalities/}, J. Math. Soc.,
Japan {

f 11} (1959), 354-373.
item {[36]} {

f S. Park}. {it Generalizations of Ky Fan's
matching theorems and applications /},
J. Math. Anal. Appl. {

f 141} (1989), 164-176.
item {[37]} {

f S. Park}. {it Coincidences of composites of
admissible u.s.c. maps and applications/}, C.R. Math. Acad. Sci. Canada
{

f 15} (1993), 125-130.
item {[38]} {

f S. Park & H. Kim}. {it Coincidence theorems for admissible
multifunctions on generalized convex space}, J. Math. Anal. Appl.
{

f 197} (1996), 173-186.
item {[39]} {

f S. Sessa}. {it Some remarks and
applications of an extensions of a lemma of Ky Fan /}, Math.
Univ. Carolin. {

f 29} (1988), 567-575.
item {[40]} {

f M.-H. Shih & K.-K. Tan}. {it Generalized
quasi-variational inequalities in locally convex topological vector spaces
/}, J. Math. Anal. Appl. {

f 108} (1985), 333-343.
item {[41} {

f N. Shioji}. {it A further generalization of the
Knaster-Kuratowski-Mazurkiewicz theorem/}, Proc. Amer. Math. Soc. {

f III@
(1)} (1991), 187-195.
item {[42]} {

f S. Simons}. {it Two-function minimax theorems and variational inequalities for
functions on compact and noncompact sets, with some comments on fixed-point theorems
/}, Proc. Symp. Pure Math. {

f 45} (1986), Pt. 2, 377-392.
item {[43]} {

f S. Simons}. {it Cyclical coincidences
of multivalued maps /}, J. Math. Soc. Japan {

f 38} (1986), 515-525.
item {[44]} {

f E. Tarafdar}. {it A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-
Mazurkiewicz theorem}, J. Math. Anal. Appl. {

f 128} (1987),
475-479.
item {[45]} {

f E. Tarafdar}. {it Fixed point
theorems in H-space and equilibrium point of an abstract
economies /}, J. Austral. Math. Soc. Ser. A {

f 53} (1992), 252-260.
item {[46]} {

f E. Tarafdar}. {it On nonlinear
variational inequalities /}, Proc. Amer. Math. Soc. {

f 67} (1977),
95-98.
item {[47]} {

f X. Wu & S. Shen}. {it A futher generalization of
Yannelis-Prabhakar's continuous selection theorem and its applications
/}, J. Math. Anal. Appl. {

f 197} (1996), 61-74.
item {[48]} {

f X.Q. Yang & G.Y. Chen}. {it A class of nonconvex functions
and pre-variational inequalities /}, J. Math. Anal. Appl. {

f 169} (1992), 359-373.
item {[49]} {

f J.-C. Yao}. {it On the generalized
variational inequality /}, J. Math. Anal. Appl. {

f 174} (1993), 550-555.
item {[50]} {

f Z.T. Yu & L.J. Lin}. {it
Continuous selection and fixed point theorems /}, 1999, to appear.