Swaminathan, Srinivasa

Thompson, Anthony C.

English

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

Commentationes Mathematicae Universitatis CarolinaeSrinivasa Swaminathan; Anthony C. ThompsonSizes of sets and some fixed point theoremsCommentationes Mathematicae Universitatis Carolinae, Vol. 12 (1971), No. 3, 537--549Persistent URL: http://dml.cz/dmlcz/105364Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1971Institute 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 12,3 (1971) SIZES OF SETS AND SOME FIXED POINT THEOREMS S. SWAMINATHAN and A.C. THOMPSON, Halifax** 1. If x and AL, are distinct elements of a metric spa-ce (Xtd ) the distance d(xffy) is a measure of the "size" of the setfx,/!^! and a contraction mapping can be viewed as one which reduces, in a uniform way, the size of all two-element subsets. More generally, the diameter d (A) * *uft> id(xfty) : xf/y» * At is a mea-sure of the size of an arbitrary bounded subset A of X , A further notion of the size of an arbitrary boun-ded subset A of X was introduced by Kuratowski CHJ as the infimum of those positive numbers 6 such that A can be covered by a finite number of subsets of X of diameter less than £ , The closely related notion of the infimum of those €, such that A can be covered by a finite £, -net has been considered by Sadovskii tl2j for Banach spaces. Sadovskii and earlier Darbo £53 -who used Kuratowski's definition - proved fixed point theorems for mappings which reduce the size of bounded subsets of the spaces considered by them. «) Research supported in part by. the National Research Council under research grants A-5615 and A-4066. AMS, Primary 47H10 Ref.2. 7.978.53 537 The purpose of this note is to present the fixed point theorem of Browder-Gohde-Kirk 13, 6, 1C1 in a similar light and to give a proof of the Darbo-Sadovskii theo-rem using Bourbaki's fixed point theorem instead of Zorn's Lemma. 2. A type of uniform etructuree Let X be a topological space. Let 3 be a family of subsets of X such that (i) X € © , (ii) 3 € 5i implies S e 3 ,(iii) the intersection of members of any subfamily of © belongs to tB . The family of all closed subsets of a topological space and the family of all closed convex subsets of a linear topological space are simple examples of such fa-milies. Definition 2.1. A & -uniform structure on X is a family U of subsets of X x X such that (i) A £ U for every U m 11; (A - <(x,x)t xe X}) and X x X e 11 , (ii) Every U € It is symmetric. (iii) For every x e X and U e UL , the set U ^ m » C < x I x X ) n U is closed and belongs to fa * (iv) For every x * X , (\i\lM t U e 1l\ m {*% . We write 11^ to denote the collection { U^ t It m 11 1 . Remarks: (a) We do not require all the properties of Hausdorff uniformity in the usual sense. (b) The only connection between the topology and the 538 uniform structure is that specified in (iii). This al-lows, for example, the possibility of X being a nor-med linear space with the weak topology, Ji the fami-ly of all closed convex sets and the S -uniform structure being such that %x is the family of all closed balls with positive radius centered at x • (c) "Symmetric" in (ii) means that for all x , <y. e U , if ty € U^ , then x e VL„ . (d) Conditions (iii) and (iv) together imply some fur-ther conditions on the family di , for example; ix I e e Sb for every x c X . 3. Notions of the "size" of sets-Let X be a topological space with a & -uniform structure % , For a subset A of X we consider the following families of subsets of U * Definition 3»1. (i) B ( A ) - f l i « t t : for every x € A, A £ U* 1 j ( i i ) X ( A ) ° -C U 6 UL i there exists x e A with A S ttK\ 5 (iii) ft. (A) •* iU € % t there exists a finite subset i F of X with A & U <% s x e T } ) } (iv) fl»(A) ss ill e 1L i there exists a finite subset F of A with A ft IU Uj i x « P! J . These sets are non-empty since X X X belongs to each one of them. The set J is called the 11 -diameter of A \ X is called the ft-A -radius of A and depends on the - 539 -"shape** of A since points x in A are needed for the "centers" of the measuring sets U^ j Q.̂ . are measures of the "total boundedness" (or precompactness) of A relative to 41 , It should be noted that a mea-sure of non-compactness defined by means of " % -nets" was introduced in £73 and [81, and also, independently, in [121. Observe that when X is a normed linear space and 1iy is the family of all closed balls of positive ra-dius centered at x , we can identify each 11 # 4t with a ball of poaitive radius centered at the origin. Then, for bounded subsets A of X , dCA) m 4M,f{tt3(Qft,) « 3(A)t is the diameter of A , *CA)» imSit i B ( 0 , e)€jUA)} is the"radius" of A f and $4<A) * Smf its 3(0, e>) € fl^CA)! U-4,1) is such that for all e > q^i CA ) there exists a finite e -net for A (with elements in X and in A respectively). We further remark that in this context there is a semi-linear structure available for bounded subsets of X . A,*A2-» i*4 ^x^t x^cA^ J, AAm { A x / x € A H A & 0) and that with this structure d, ft and ^ are "semi-norms" in the sense that &l\ + A1)^kdCAH)^d(Az)f d(XA)m%d(A) (A2J) and similarly, for jc and g^ • Moreover, d and <%* are monotonic, i.e., if A * S* A * then d CA. ) .6 & d(A^) • Also the g,£ satisfy the inequalities - 540 -%iCA) ^ (LiCA) ^ 2ii CA) for an A fi X . Definition 3*2. The set A is said to be % -totally bounded if fl1 CA) m <)L . Definition 3*3* The set A has % -normal struc-ture if X ( J ) ' 3 j ( ' 3 ) for every B such that 3 c 5i f B m A and B is not a singleton. Remarks: (i) We use c (or 3 ) to mean strict inclusion throughout. (ii) The set A is a singleton if and only if J> CA) m m % , (iii) The definition of normal structure comes directly from the definition of Brodskii and Milman [2 J for a bounded convex subset of a normed linear space, namely, A has normal structure if and only if ft CO ^ dCC) for every non-trivial convex subset C of A . Next we are concerned with fixed point theorems for map-pings which satisfy inequalities with respect to the above measures of size. In what follows we shall usually suppress explicit reference to % and speak of normal structure, diameter, etc. 4. The Browder-Qo'hde-Kirk fixed point theorem* Let X be a topological space with a j9 -uniform structure* Let f be a function v&ich maps X into it-self. Definition 4.1. (i) f is non-exoansive if 541 B C f C A ) ) 2 D C A ) for every two-element set A - <x9ty 1 fif X . (ii) f is normalizing if K CcH f CB )) => T> (3) for every non-trivial subset B of SB . Here ch A (with respect to S ) denotes the smallest closed member B of 33 which contains A (and is the intersection of all such B ). We observe also that RCefe A ) 2 K ( A ) and is, in general, larger since there may be suitable "centers" in ch> A which are not in A * Theorem 1. Let X be a compact topological space with a Ji -uniform structure. Let f be a mapping of X into itself, which is both normalizing and non-ex-pansive. Then f has a fixed point. Proof: (similar to that in £103) Consider the set P of all non-empty closed subsets A of X which are in Si and which are invariant under f , The set (P is non-empty since X belongs to it. Also it is inductive-ly ordered by inclusion since any chain in (P has the finite intersection property and hence has a lower bound (the intersection of all members of the chain) which is non-empty since X is compact and belongs to J3 since % is stable for such intersections. Hence, by Zorn's Lemma, P has minimal elements. If A 0 is such a mini-mal element it is a fixed point of the mapping q. defi-ned by 9^CA) m ch, i£CA)) . Suppose, if possible, that A 0 is not a singleton, i.e. J>CAQ) + 11 . Then, - 542 -since £ is normalizing, R(^(A 0)) » &(A 0) ̂  D(A 0) . Let U' e 4£ be chosen so that U1 £ P ( A 0 ) but IT • H (A0> and let A^ m f x e A 0 t A0 £ Ux i . Then (i) A,, is non-empty since W e R(A0) and so there exists an x such that A 0 £ \JL*X \ Cii) A. £ A0 by definition; (iii) A. 4» A 0 since U' ^ P C A 0 ) and so not every x in A 0 belongs to \ * (iv) A1 - A 0n i U ^ j A^ e AQ \ for, if x e A , then /y, e U'^ for all /y. c A0 and hence by the symmetry of Mf , x e U* for all ̂  in Ap , i.e., •X e A 0 A -C U' * ty c A 0 J . Conversely, if x is in the intersection then x c U* for all ny e A0 and so ny e e U ^ for all ny e A 0 , i.e., A p £ U^ * Now since A- and U' are all members of & f (iv) implies that AA e 3b , Moreover, since A 0 e (P and 1 * since W is closed for each ty , A^ is closed. We prove, finally, that A,- is invariant under f . Let X e A then A 0 £ VL9X , i.e., /^fill^ for all q, e € A 0 and hence also, K C VL* so that \V e eVio(,nj,i for all X e A. , /$> ̂  A 0 # Now f is non-expansive so that U. feP{f(*),£ Cty) i for all x € e A^ ,ty e AQ , i.e., U ^ } 2 £ (A0 ) , But then, since U' is a closed member of Si , U'fCx) 2 2 cfo £(A0) * ^fA 0 ) m A 0 . Thus f (x) e A^ and A^ is invariant under f and hence is in & # But (ii) and (iii) contradict the minimality of A 0 . Thus A 0 con-sists of a single point i a, J . Hence ch> i£ Cap)i • m ia,\ .This implies that £ (a,0) m a, because, for - 543 -any Jj e X , <% 1 S eh it* J * <3 c © t Ar0 * 3 and B i s closed J & n <U^ '. U, € % 1 ** < fir0 I . The purpose of the next lemma and proposition i s to establ ish a connection between normalizing mappings and se t s with normal structure. Lemma 4 . 2 . If X i s a topological space with 3B -uniform structure and i f A i s a subset of X , then ( i ) JD(c#i, A ) • P ( A ) and ( i i ) i f f t X ~» X i s non-expansive, then $(£ (A)) 3 3(A) • Proof: ( i ) Since A S e-H A i t i s c lear that i f 11 € $(&4v A) then U 6 3) (A) . Suppose conversely that It € 2 (A ) . Let * € A . Then A C tty and so , since tt^ 6 33 and i s c losed, c4* A ff U*x * Thus, i f /^ € cfo A , ^ e tt^ and hence x c tt^_ for every tf € A , i . e . , A S tt^ and so t i t A C IL. for eve-ry ty € ch A f i . e . , tt«P (c#t A ) * ( i i ) Let U f i J X A ) , then U e 3 M . x , ^ } for every pair x9fu,eA . Since f i s non-expansive, th i s means IL e D i i ( * ) , f (/JJJ ? for every pair x ^ ^ A , i . e . , UL e J H x ^ / v ? for every pair w,nr e i(A) , i . e . , w € U^ for every w , /tr e £ (A) , Thus f (A) £ VL^ for eve-ry AT € l ( A ) and so tt e D ( * ( A ) ) . Remark: Part (ii) of the lemma means that the res-triction to pairs in the definition of non-expansive map-pings is unnecessary. Proposition 4.3. If X has normal structure, then •very non-expansive mapping of X into itself is norma-lizing. - 544 • Proof: For every non-trivial subset 3 of A for which 3 e 33 , it is true that H (c*t £ (B)) 3 oDCcJh, £ C3)) , by the definition of normal struc-ture! and by Lemma 4.2, it follows that KCch £ (B)) o T)C£C3)) 2 DC3) . 5. The Darbo-Sadovakii fixed point theorem* As in Section 4, let X be a topological space with a $ -uniform structure 1t and let £ be a map-ping of X into itaelf. In this section, however, we shall assume more: namely that % is a base for a uni-formity on X and that the topology on X is the uni-form topology generated by % . The definition (3.2 abo-ve) of % -totally bounded sets then coincides with the more usual definition (see for example t93» p.198). We shall also assume throughout this section that the map-ping f is continuous. For the main theorem we make use of Tychonoff 's fixed point theorem and, therefore, re-quire a linear structure and a specific type of Si -uni-formity; for this reason this section is somewhat diffe-rent in character from the preceding ones. Definition 5.1. The mapping I is condensing on JC if ft,(f(A)) D fl,(Jl) for every A in X which is not totally bounded. The following lemma is clear. Lemma 5.2. If £ is condensing on X and if A m\ £ X and i (A) m A then A is totally bounded. - 545 -Lemma 5.3« I-? -* i s condensing on X then every orbit Otx) * ii^tx) I m* « 4, ft, 3, ... f is to-tally bounded. Proof: Since itOtx)) & Otx) we have Q titOlx)) & 6L(0(*)) . On the other hand, sin-i " ce 0 (X) m i (Otx )) U <x I , if F is a finite U -net for i(0(x)) , then P U < ^ f is a finite % -net for 0(x) . Hence A^tOtx)) » fi,, Cf C0C.X )); and we must have 0(«x) totally bounded. Now suppose that X is complete with respect to the uniformity % , It is well known (C9]p.l98) that if A is totally bounded and complete then A is compact. It is, therefore, a corollary to Lemma 5-3, that if X is complete and if f is condensing on X then K «• Otx) is a compact subset of X for all x in X . Moreover, since we are assuming that £ is continuous, £tK) £ K . Thus, there exist non-empty compact subsets of X which are invariant under £ , The purpose of the rest of this section is to show that when iS is the family of convex sets in a locally convex linear topological space X , then there are compact elements of 43 which are inva-riant under £ . The argument is divided into two lemmas. Lemma 5.4. Let #J be the family of convex sets and % a uniformity for the topology of a locally convex linear topological space X generated by closed convex neighborhoods of 0 , then fi<| Celt A ) m ft C A) for each A in X , - 546 -This lemma is proved in the same way as Lemma 1 of £12J. Lemma 5*5. With the hypotheses of the preceding lemma, and if f is a condensing mapping on some ele-ment A of H which is complete then there is a compact element of 55 invariant under f 0 Proof: Let K denote the closure of an orbit 0 (x ) . Let (p be the set of all closed non-empty elements C « Si such that {(C) fi C and C fl K 4» 4* 0 and consider (P partially ordered by inclusion. Then P is non-empty since, A C 4* • Consider the map-ping f i P —-> P defined by 3- CC ) » eft C£ ( C )) -Then, since C is closed and convex (in Si ) and f C O e c. C , we have 9, C C ) £. C • Moreover every chain in (P has a lower bound L « D(C: C in the chain } . ( L fi K is non-empty since, for each C in the chain, C fl K is a non-empty compact set and these sets have the finite intersection property, and, clearly, L has the other defining properties of IP •) By the Bourbaki fixed point theorem (see, for example, [1] p.41) the mapping q, has a fixed point in (P • i.e., <fr(C0) m CQ . Thus cto, i ( C0 ) m C0 . Now ft^Ce-h, f CC0)) m a^Cf CC0)) 2 ^ CC^) m 0, Cc*t, i(C0 )) with the inclusion strict, by the condensing property of f f unless CQ is totally bounded. Thus CQ is to-tally bounded. Since CQ is also a closed subset of the complete set A , Cfl is compact. Theorem 2 (Darbo-Sadovskii). Let A be a comple-te, convex, bounded subset of a locally convex linear 547 topological space E and let £ be continuous and condensing on A » Then f has a fixed point in A # Proof. By Lemma 5*5 there is a compact convex sub-set (L of A invariant under £ . The result now fol-lows from Tychonoff's fixed point theorem applied to f and the set CQ . Remark: Lemma 5.5 can be obtained from Propositions 1 and 4 of t4l» and Theorem 2 is contained in Theorem 3 of [41. However the above proof does not use Zorn's Lemma. We thank the referee for his helpful comments on this pa-per. R e f e r e n c e s tl] C. BEROE: Topological space, McMillan,New York,1963. t2] M.S. HRODSKII and D.P. MILMAN: On the center of a convex set, Dokl.Akad.Nauk SSSR(N.S.)59 (1948),837-840. [31 F.E. BROWDER: Nonexpansive nonlinear operators in a Banach space, Proc.Nat.Acad.Sci.U.S.A. 54(1965),1041-1044. t4] J. DANES: Generalized concentrative mappings and their fixed points, Comment.Math.Univ.Caro-linae 11(1970),115-136. (51 G. DARBO: Punti uniti trasformazioni a codomino non-compatto, Rend.Sem.Mat.Univ.Padova 24(1955), 84-92. C6] D. GOHDE: Zum Prinzip der kontraktiven Abbildung, Math.Nachrichten 30(1965),251-258. - 548 -[7] L.S. GOLDSTEIN, I.C. GOHBERG, and A.S. MARKUS: On some properties of linear boonded opera-tors in connection with their q-norms (Russian). UC.Zap.KiSinev.un-ta 29(1957), 29-36. [8] L.S. GOLDSTEIN and A.S. MARKUS: On a measure of non-compactness of bounded sets and line-ar operators, Issled.po Algebre i Matem. Analizu,KiSinev (Russian), 1965. [9] J.L. KELLEY: General Topology, Van Nostrand, 1955. [10] W.A. KIRK: A fixed point theorem for mappings which do not increase distance, Amer.Math.Month-ly 72(1965),1004-1006. [11] K. KURATOWSKI: Sur les espaces complets, Fundamenta Mathematicae 15(1930), 300-309. [12] B.N. SADOVSKII: A fixed point principle, Functional Anal.and Appl.l(1967),151-153 . Dalhousie University Halifax, Nova Scotia (Oblatum 26.10.1970) - 549 

Sizes of sets and some fixed point theorems

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Sizes of sets and some fixed point theorems

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