EXTENSIONS OF BEST APPROXIMATION AND COINCIDENCE THEOREMS

Let X be a Hausdorff compact space, E a topological vector space on which E" separates points, F X 2E an upper semicontinuous multifunction with compact acyclic values, and g: X E a continuous function such that g(X) is convex and g-1 (y) is acyclic for each y e g(X). Then either (1) there exists an x0 E X such that gxo . Fxo or (2) there exist an (x0, z0) on the graph of F and a continuous seminorm p on E such that O < p(gzo zo) <_ p(yzo) for all y g(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.


INTRODUCTION
One of the most interesting extensions of Ky Fan's best approxi,rnation theorems [1] was due to Prolla [2] for two functions.Subsequently, a number of its generalizations or variations followed, and some applications to coincidence theory were also given.See [3][4][5][6][7][8].
Moreover, there have also appeared some generalizations of such results for two maps and two different space settings for example [3,18].
Usually, those results are obtained for single-valued maps or convex-valued upper semicontin- uous multifunctions.However, more recently, the author [11,13,19] showed that some of such best approximation and fixed point theorems can be extended for a large "admissible" class of multifunctions.
In the present paper, we obtain best approximation and coincidence theorems for such large class of multifunctions and two different space settings.Our new results are general enough to subsume more than fifty known results of other authors as particular cases.
S. ARK 2. PRELIMINARIES A multifunction or set-valued map (simply, map) F X 2 Y is a function with nonempty set-values Fx C Y for each x E X.The set (x, y) y Fx} is called either the graph of F or, simply, F. So (x, y) E F if and only if y Fx.For topological spaces X and Y, a map F X 2 Y is upper semzcontinuous (u.s.c.) if, for if Y is compact Hausdorff and Fx is closed for each x E X, then F is u.s.c, if and only if the graph of F is closed in X x Y.A nonempty topological space is acyclic if all of its reduced (ech homology groups over rationals vanish.
A convex space C is a nonempty convex set with any topology that induces the Euclidean topology on the convex hulls of its finite subsets.Such convex hulls are called polytopes.
Given a class X of maps, X(X, Y) denotes the set of all maps F X 2 " belonging to and )c the set of all finite composites of maps in X.A class 9.l of maps is one satisfying the following: (i) 9.1 contains the class C of (single-valued) continuous functions; (ii) each F E 92c is u.s.c, and compact-valued; and (iii) for any polytope P, each f E 92(P, P) has a fixed point.
Examples of 9.1 are C, the Kakutani maps K (with convex values in convex spaces), the acyclic maps V (with acyclic values), the approachable maps A in topological vector spaces [20][21][22], admissible maps in the sense of G6rniewicz [23], permissible maps in Dzedzej [24], and others.
Moreover, we define F 92(X,Y) == for any a-compact subset K of X, there is a F 92(K,Y) such that Fx C Fx for each x K.
F 92'(X,Y) for any compact subset K of X, there is a F 92c(K,Y) such that Fx C Fx for each x K.
A class 92 is said to be admissible.Note that 92 C 92 C 92 C 92. Examples of 92 are due to Lassonde [25] and V due to Park et al. [26].Note that IK includes classes K, R, and "IF in [25].The approximable maps recently due to Ben-E1-Mechaiekh and Itzik [27] belong to Therefore, any compact-valued u.s.c, map F X 2E, where E is a locally convex t.v.s, and X C E, belongs to 92 if its values are all convex, contractible, decomposable, or oo-proximally connected.See [27].
Let E (E, v) be a topological vector space, E" its topologiqal dual, and S(E) S(E, 7-) the family of all continuous seminorms on (E, v).Let w denote the weak topology of E. We say that E" separates points of E if for each x E with x 0, there exists a E" such that b(x) 0; that is, if x 0, then p(x) > 0 for some p E S(E, w) C S(E, T) by taking p(x) [(x)[   for all x E E.
The following is due to the author [19,II]: Let X be a compact convex subset of a topological vector space E on which E" separates points.Then any F 92(X,X) has a fixed point.
Let C be a nonempty subset of a Hausdorff topological vector space E and p S(E).For each y E E, define dp(y, C) inf{p(y-x) x E C} and the set of best approximations to y E E from C by Qp(y) {x 6 C'p(y-x) dp(y, C)}.The multifunction Qp thus defined is called the metric projection onto C if Q(y) @ for each y 6 E. It is well-known that if C is compact convex, then the metric projection Q" E 2 c belongs to IK(E, C).
In (E, T), let Bd, Int, and-denote the boundary, interior, and closure, respectively, with respect to T.
For a topological space X, a real function f X ]R is lower semicontinuous (1.s.c.) if {xX'fx>r} is open for eachrR.
The following is well known: LEMMA 2.2.Let X and Y be topological spaces, h" X x Y ]R 1.s.c. and F" X 2 " a compact-valued u.s.c, multifunction.Then x inf{h(x, y) "y Fx} is 1.s.c. on X.

MAIN RESULTS
From Theorem 2.1, we obtain the following generalization of many best approximation and fixed point theorems: THEOREM 3.1.Let X be a Hausdorff compact convex space, E (E, r) a topological vector space on which E" separates points, F 6 9/:(X, (E, w)), and g C(X, (E, w)) such that g(X) is convex.Suppose that either (I) g-' (y) is convex for y 6 g(X) and ]K(g(X),X) C (g(X),X); or (II) g-'(y) is acyclic for y 6 g(X) and V(g(X),X) C PI(g(X),X).
PROOF.Since X is compact, we may assume that F 6 9/c(X, (E, w)).Since the graph of g-1 g(X) 2 x is closed in g(X) x X and X is compact Hausdorff, we know that g-1 is u.s.c, and g-'(y) is closed for each y 6 g(X).Therefore, either (I) g-6 ]K(g(X),X) if g-' is convex-valued; or (II) g-* e V(g(X), X) if g-* is acyclic-valued.Let p 6 S(E, w).Consider the composite of multifunctions g-1 g(X) X F_. (E, w) g(X).
Since g(X) is a weakly compact convex subset of a Hausdorff topological vector space (E, w), the metric projection Q; belongs to IK((E, w),g(X)).Hence, in any case we have QFg-' e fa:(g(X),g(X)), which has a fixed point yo g(X) by Theorem 2.1.Then yo (QpF)Xo for some x0 g-'(Y0); and hence gxo Qpzo for some zo Fxo, which is equivalent to p(gxo Zo) <_ p(y-zo) for all y g(X).
Considering (g(X),E) instead of (X, Y) in Lemma 2.2, put h(9, z) (-) for x e X, z E. Then z g(gx, Fx) is 1.s.c.Therefore, F[p] is closed in X.Moreover, for a finite subset {P,,P2, ,P,} of S(E, w), we have p E=, P, S(E, w) and F[p] FIE=, p,] c 5, F[p,].Therefore, {F[p]'p S(E,w)} is a family of closed subsets of X with the finite intersection property.Since X is compact, there exists a u {F[p]'p S(E,w)}.Now we claim that 9u Fu.
Suppose that gu Fu.Then the origin 0 does not belong to the compact set K gu Fu of (E, w).Let z K. Since E" separates points of (E, w), there exists a 4 E" (E, w)" such that (z) -0.By putting pz (x) 14(x)l for x E, we know that pz S(E, w) and p, (z) > 0.
Since p, is continuous on K, there exists an open neighborhood U, of z in K such that p, (y) > 0 for every y Uz.Let {Uz,,..., Uz } be a finite subcover of the cover {U, }zero of g and let p P*, S(E, w) Since PIK is continuous it attains its infimum on K. Since the infimum can not be zero, we have d,(gu, Fu) > 0. This contradicts u {F[p]'p S(E, w)} #-@.This completes our proof.
2. If F' P.I:(X, (E, w)), where F' is defined by F'z 2gx Fx for z X, then the inward set in the conclusion (2) of Theorem 3.1 can be replaced by the corresponding outward set.
PARTICULAR FORMS 3.1.Theorem generalizes, unifies, and improves many of well- known best approximation and fixed point theorems.We list some major particular forms in the chronological order.
1. Fan [1, Theorem 1]: Let E be locally convex, X a nonempty compact convex subset of E, F f C(X,E), and g x the identity map.
2. Fan [1, Theorem 2]: E is a normed vector space in the above.
3. Halpern [32, Theorem 20]: X is a subset of a Banach space E, g lx, and F 6 V(X, E) under some restrictions.
5. Fitzpatrick and Petryshyn [33, Theorem 3(i)]: X is a subset of a strictly convex Banach space E, g 1x, and F E V(X, E).
6. Reich [34, Theorems and 2]: X is a subset of a locally convex Hausdorff topological vector space, g lx, and F ]K(X, E). 7. Prolla [2,Theorem]: X is a subset of a normed vector space E, F f C(X, E), and g C(X, X) is an almost affine surjection.
8. Sehgal and Singh [15, Theorem 1 and Corollary 1], Sehgal et al. [16, Corollary 1]: X is a nonempty weakly compact convex subset of a real locally convex Hausdorff topological vector space (E,T), F= f C((X, w), (E, T)) [that is, strongly continuous], and g--lx.9. Ha [18, Theorem 3]: X is a compact convex subset of a Hausdorff topological vector space, E a locally convex Hausdorff topological vector space, and F f, g E C(X, E) such that g(X) and g-l(y) are convex for y e g(X).
10. Ha [35, Theorem 3]: X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space E, g x, and F ]K(X, E). 11. Park [6, Theorem 2.1]: X is a subset of a normed vector space E, and F f, g C(X,E) such that g(X) X and g is almost affine.
13. Carbone [4, Theorem 1]: X is a subset of a normed vector space E, g C(X,X) is an almost quasiconvex surjection, and f 6 C(X, E).
(iv) It is well known that (iv) (iii) [37].(v) If Fx C g(X), then for each z Fx, we can choose A 0 in (iii).(vi) Clearly.we have (vi) == (v).REMARKS 3.2.1.If F' P.t[(X, (E, w)), where F' is defined by F'x 2gx Fx for x X, then the inward set in Theorem 3.2 can be replaced by the corresponding outward set.

If g
Ix and P/ is replaced by K in Theorem 3.2(I), then the boundary conditions in Theorem 3.2 can be replaced by more general ones.See [19].
PARTICULAR FORMS 3.2.We list major particular forms of Theorem 3.2.