Mehta, Ghanshyam

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeGhanshyam B. MehtaFixed points, equilibria and maximal elements in linear topological spacesCommentationes Mathematicae Universitatis Carolinae, Vol. 28 (1987), No. 2, 377--385Persistent URL: http://dml.cz/dmlcz/106548Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1987Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE 28,2(1987) FIXED POINTS, EQUILIBRIA AND MAXIMAL ELEMENTS IN LINEAR TOPOLOGICAL SPACES x) Ghanshyam MEHTA Abstract: In this paper we have proved some generalizati-ons of the fixed-point theorems of Browder and Tarafdar in line-ar topological spaces. These results are used to prove some ge-neral theorems on the existence of maximal elements and equili-bria in linear topological spaces. Key words: Fixed points, maximal element, equilibrium. Classification: 47H05, 90A14 1. Introduction. Browder [1968 Theorem 1] proved that a nonempty convex-valued multifunction with open inverse images defined on a compact, convex subset of a Hausdorff linear topo-logical space, has a fixed-point. This theorem of Browder was generalized by Tarafdar [1977] who replaced the assumption that the multifunction has open inverse images by a weaker condition. Tarafdar used this generalized theorem to prove the existence of a solution to a nonlinear variational inequality. Tarafdar s theorem was also used in Mehta and Tarafdar [1985], Tarafdar and Mehta [1984] to prove generalized versions of the Gale-Ni-kaido-Oebreu theorem in mathematical economics. The object of this paper is to prove some generalizations of the Browder and Tarafdar theorems and to give some applica-tions. In section 2, a generalization of the Browder and Taraf-dar fixed-point theorems is proved in a locally convex linear topological space. In section 3, a different approach is employ-ed and a generalization of the Browder and Tarafdar theorems is proved, in a linear topological space using a recent theorem of x) This work was done while visiting the University of Bonn. I should like to thank Professor Hildebrand for his hospitality and encouragement. Financial support from the Sonderforschungs-bereich is gratefully acknowledged. - 377 -Fan [1984]. Finally, in section 4, these generalizations are used to prove some general theorems on the existence of maximal elem-ents and the existence of equilibria of qualitative games or ab-stract economies. We shall use the following notation.If K is a subset of a linear topological space, then co K denotes the convex hull of K, cl K denotes the topological closure of K and int K denotes the topological interior of K. 2. A fixed-point theorem in a locally convex space. The following theorem has been proved by Browder [1968, p. 285 3. Theorem 2.1. Let K be a compact, convex subset of a Haus-dorff linear topological space. Let T be a multifunction on K into 2 such that (a) for each xe K, T(x) is a non-empty convex subset of K; (b) for each x e K, T~ (x)= { y e K:x e T(y)} is open in K. Then there is a point x £ K such that x e T(x ). o o o Remark. Instead of condition (b) in the above theorem one can make the following weaker assumption: ( b ) for each x e K, there is a y in K such that x 6int LT"1(y)3. Tarafdar [1977] has proved the existence of a fixed point of T under the weaker condition (b'). The question arises if the assumption in the Browder and Ta-rafdar theorems, that the domain of the multifunction is compact, can be significantly weakened. We show in this section that this can be done in locally convex linear topological spaces. More specifically, we prove that a multifunction defined on a para-compact, convex subset X of a locally convex linear topological space with values in a compact subset 0 of X, and which satisfi-es the other conditions, has a fixed-point. The method of proof we employ in this section is similar to that of Browder and consists of two steps. First, a partition-of-unity argument is used to get a continuous selection for the gi-ven multifunction. Secondly, a fixed-point theorem is applied to get a fixed-point for the continuous selection. The fixed-point theorem we use is a generalization due to Himmelberg [ 19723 of the classical fixed-point theorem of Fan. - 378 -Theorem 2.2. Let X be a paracompact, convex, nonempty sub-set of a Hausdorff locally convex linear topological space. Let D be that a compact subset of X and T:X —=*• 2 a multifunction such (i) for all x in X, T(x) is convex and nonempty; (ii) for each x in X, there exists a y in D such that x & int T (y). Then there exists a point x eD such that x e T(x ). ^ o o o Proof. Let 0 =int T (y) for y in D. By assumption (ii), 0= {0 :yt 0} is an open cover of X. Since X is paracompact, the-re exists an open locally finite refinement V= *[V :ae A£ of 0 where A is an index set [Kelley, 1955, p. 156J. Since every pa-racompact space is normal [Kelley, 1955, p. 159J, there exists »a family of continuous functions •£ f :aeA? on X with non-negative real values, such that for each xeX, JE . f„(x) = l and each f„va-CL e A a a nishes outside V [Kelley, 1955, p. 171]. Now for each aeA, there exists a y in D such that V £0 g a a Va -1 £• T (yg) since V refines 0. Define a function f on X as fol-lows: f (x)= 2LA f ( x )y . Since V is locally finite, each point x has a neighbourhood which intersects only finitely many sets of the family -t V :ae.AJ. Consequently, only finitely many functions have a non-zero value for each x. Thus f is a continuous functi-on. If fa(x)=0, xeVag.O &T"1(ya) so that y a&T(x). Hence, by ya assumption (i), f(x) e T(x) for all x & X, since f(x) is a convex combination of the points y in T(x). This proves the existence of a continuous selection f for the multifunction T on X. Now f is a continuous single-valued function on a convex subset X of a separated locally convex topological vector space with values in a compact subset D of X. It follows that f is an upper semi-continuous multifunction on X with closed and convex values. Hence, by a theorem of Himmelberg £1972, Theorem 23 f has a fixed-point x . Consequently, x 0= f ( x 0 )e K x 0 ) and I n a s a fix-ed point. q.e.d. Remark 1. The above theorem shows that in a locally convex - 379 -linear topological space, it is possible to generalize the Brow-der and Tarafdar theorems, to multifunctions defined on paracom-pact, convex subsets. Remark 2. If we assume that for yeD, T~ (y) is open in X, then assumption (ii) is automatically satisfied since T(x) is non-empty for each xeX. Hence, the fixed-point theorem of Yann-elis and Prabhakar [19833 is a special case of the above theorem. 3. A generalization of the Browder and Tarafdar theorems. The object of this section is to generalize the Browder and Ta-rafdar theorems in arbitrary linear topological spaces and to prove them in a different way. The proofs of Browder's theorem [1968] and the previous fixed-point theorem were based on a par-tition of unity. The approach used in the next theorem does not rely on this method. Instead, we prove the existence of a fixed-point by using a recent generalization of the classic Knaster-Kuratowski-Mazurkiewicz theorem due to Fan [19843. Theorem 3.1. Let K be a convex subset of a Hausdorff line-ar topological space E and T:K —-v 2 a multifunction such that (i) for each xe.K, T(x) is nonempty and convex; (ii) for each yeK, T~ (y)= i x e K:y e. T(x) I contains an op-en subset 0 of K; (iii) LHO :y fc K]=K; (iv) there exists a nonempty subset K of K such that . r\, itO ]CJ is compact and K is contained in a compact con-A 6 KQ X 0 vex subset of E. Then T has a fixed point. Proof. Suppose that T has no fixed point. Then x + T (x) for all x. This implies that x^O for all x. cx For xeK, define F(x)= [ 0 "j . Since 0 is open in K for each x e K, it follows that F(x) is a relatively closed subset of K for each x e. K . By assumption (iv), r v F(x ) is compact for some K con-* e i\0 o o tained in a compact, convex subset. Let ix,,x«,...,x } be a finite subset of X. We want to show that the convex hull S of i x,,x0,...,x } is contained in ft 1 z n .U^ F(x.). We argue by contradiction. Suppose there is x eS such - 380 -that x 4 - ^ F(x.). Hence, for each i = l,...,n, xiCO ] c which implies that for each i, x e 0 . Now, by assumption (ii),Q £ xi i -1 -l £T (x.), so that x e T (x.) for all i. This implies that x..e e T(x) for all i. By assumption (i), T(x) is a convex set for all x which implies that x e T(x) and this contradicts our suppo-sition that T has no fixed points. We conclude, therefore, that the convex hull of any finite subset {x.,x9,...,x } is contain-ed in the corresponding union -U/JF(x.). Thus all the conditions of Theorem 4 of Fan [19843 are sa-tisfied and we conclude that f^ F(x)=j=- 0. Let x^ be a point in K £ K o this intersection. Then x e F(x)= CO 3 C for all x in K, and this contradicts assumption (iii). The contradiction proves the exisr tence of a fixed point for T. q.e.d. Remark. If K is compact, as in Browder [19683 or Tarafdar C19773 condition (iv) is automatically satisfied so that these theorems are a special case of the above theorem. 4. Applications. Let K be a subset of some linear topolo-gical space E. With each binary relation P on K one can associa-te te a multifunction T:K—>- 2 in the following way: yeT(x) if and i/ only if (x,y)e P. Conversely, if T:K —> • 2 is a multifunction, then a binary relation P is defined on K by the condition that (x,y)eP if and only if y<s.T(x). Hence, we have the following: Definition. A point x„ is said to be a maximal element of 0  the multifunction T:K—-> 2* if T(x ) = 0. We now prove the following theorems on the existence of ma-ximal elements of a multifunction T. Theorem 4.1. Let X be a non-empty, paracompact, and convex subset of a separated locally convex linear topological space and D a compact, convex subset of X. Let T:X —> 2 be a multifunction such that (i) for each xeX, x £ co T(x); (ii) for each x e X, there exists a y in D such that xe int Q (y), where Q(x)=co T(x) for x in X. - 381 -Then there is a maximal element, i.e. a point x such that T(x )= -0. Proof. If there is no maximal element, T(x), and therefo-re Q(x), is nonempty for each x e X, so that Q:X —>- 2 is non-empty, convex-valued multifunction. Assumption (ii) of Theorem 4.1 implies that assumption (ii) of Theorem 2.2 is satisfied. By Theorem 2.2, we conclude that there exists a point x such that ' o x e T(x ) £ Q(x ). This is a contradiction since assumption (i) implies that x<£Q(x) for all x e X . Hence, T has a maximal elem-ent. Remark. Assumption (ii) of Theorem 4.1 is weaker than the assumption used by Yannelis and Prabhakar that for all y e D, T (y) is open in X. The reason for this is the following. If T (y) is open, then Q (y) is also open, where Q(y)=co T(y) [1983, p. 239J . Hence, Theorem 4.1 is a generalization of the result of Yannelis and Prabhakar [1983, p. 240]. Theorem 4.2. Let K be a convex subset of a Hausdorff line-ar topological space E and T:K — > 2 a multifunction such that: (i) for each xeK, T(x) is convex; (ii) for each xeK, x^T(x); (iii) for each y e K , T" (y) contains an open subset 0 of (iv) IH0 :y£K?=K; (v) there exists a nonempty subset K of K such that X Q K "-[0 1 C l S compact and K is contained in a compact convex subset of E. Then T has a maximal element. Proof. Follows immediately from Theorem 3.1. For a recent general theorem on the existence of maximal e-lements see HadzAc [1986, Proposition]. We turn now to the problem of the existence of equilibria of qualitative games or abstract economies. Let X. be a nonempty set for is I, where I is an index set. An abstract economy or qualitative game E is defined by a family of ordered triples (X.,A.,P.) where A.: TT. TX . — > X. and P.: X X X X J f e X J X X : TT. TX, — > X. are multifunctions. An equilibrium for E is an jel j l —a  - 382 -K; x* G ^j^jX. such that for each i: (i) x*e cl A.(x*), (ii) P.(x*)n Ai(x*)= 0 • We now prove the following generalization of the main theo-rem of Yannelis and Prabhakar £1983.1. Theorem 4.3. Let E=(X.,A.,P.) be an abstract economy satis-fying for each i in some countable index set I. (i) X. is nonempty, compact, convex, metrizable subset of a locally convex linear topological space; (ii) A.(x) is convex and nonempty for, all xeX; (iii) the correspondence A..-X —*• 2 defined by A.(x) = = cl A.(x) for all x£ X is upper semicontinuous. (iv) the set U.= ixeX: $-(x)*0? is open in X; (v) for each x e U., there exists y- in X. such that x € int $ 7 (y-; )» where <J>.(x)=A.(x)r.co P,(x) for all x e X ; (vi) x.^'co P.(x) for all xeX. then E has an equilibrium. X. Proof. Consider the correspondence ^&. :U.—> 2 . Assump-tions (ii) and (iv) imply that <£. is nonempty and convex valued for each xcU.. Assumption (v) implies that assumption (ii) of Theorem 2.2, is satisfied. Hence, as the proof of Theorem 2.2, shows, there exists a continuous function f.:U.—> X. such that f.(x) e $.;(x) for all xeU., since U. is paracompact as a subset of the metrizable space TT._TX.. The rest of the proof is based on an idea used by Gale and Mas-Colell L1975, p. 103 and is carried out as in Flam E1979J or Yannelis and Prabhakar 11983!. For completeness we give it here. Oefine the correspondence F.:X-^>2 by F.(x)= 4f.(xH if x eU t, Fi(x)=A.(x) if x 4 U r V F. is easily seen to be upper semi-continuous. Define F:X-->2 by F(x)= 7T. -rF. (x). It is easily verified that F is an upper se-mi-continuous multifunction with non-empty, closed and convex values. Hence, by Himmelberg's theorem E1972J, F has a fixed point x* . For each i, x*$ U. since if x*e U-, then x^ = f.(x*)e - 383 -€. $ . ( x * ) £ c o P . ( x * ) , a contradiction to assumption (vi). Hence, x*4-U. and it is easily checked that x* is an equilibrium point of the economy E. . 7 q . e. d . Remark. Yannelis and Prabhakar L1983.1 assume that A. and -1 -1 P. have open inverse images, i.e. A. (y) and P. (y) are open for each y. It is easily verified that these assumptions imply that conditions (iv) and (v) of Theorem 4.3 are satisfied. Consequen-tly Theorem 4.3 is a generalization of the main theorem of Yan-nelis and Prabhakar [1983, Theorem 6.1J. R e f e r e n c e s [1] BROWDER F.: The fixed-point theory of multi-valued mappings in topological vector spaces, Mathematische Annalen 177(1968), 283-301. [2] FAN K.: Some properties of convex sets related to fixed point theorems, Mathematische Annalen 266(1984), 519-537. [3] FLAM S.: Abstract economies and games, Soochow 3ournal of Mathematics 5(1979), 155-162. [4] GALE D. and MAS-COLELL A.: An equilibrium existence theorem for a general model without ordered preferences, 3ou 3ournal of Mathematical Economics 2(1975), 9-16. [5) GRANAS A.: BEN-EL-MECHAIEKH and DEGUIRE P.: Fixed points and coincidences for setvalued maps of type $> and §**, Comptes Rendus A c a d . S c , Paris, October 1982, pp. 381-384. [6] HAD^ld 0.: A coincidence theorem in topological vector spa-ces, Bulletin of the Australian Mathematical Society 33(1986), 373-382. [7] HIMMELBERG C.3.: Fixed points for compact multifunctions, 3ournal of Mathematical Analysis and Applications 38(1972), 205-207. [8] KELLEY 3.: General Topology, Van Nostrand, Princeton, 1955. [9] MEHTA G. and TARAFDAR E.: Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Taraf-dar, 1985, 3ournal of Economic Theory (to appear). [10] TARAFDAR E.: On nonlinear variational inequalities, Procee-dings of the American Mathematical Society 67(1977), 95-98. [Ill TARAFDAR E. and MEHTA 6.: On the existence of quasi-equilib-rium in a competitive economy, International 3ournal of Science and Engineering 1(1984), 1-12. [12] YANNELIS N. and PRABHAKAR N.: Existence of maximal elements and equilibria in linear topological spaces, 3ournal of Mathematical Economics 12(1983), 233-245. - 384 -Departments of Economics and Mathematics University of Queensland St. Lucia, Q. 4067 Australia (Oblatum 31.3. 1987) 385 

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Fixed points, equilibria and maximal elements in linear topological spaces

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Fixed points, equilibria and maximal elements in linear topological spaces

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