Remarks on multivalued nonexpansive mappings
Abstract
Convergence of fixed point sets of multivalued nonexpansive mappings is studied under both the Mosco and Hausdorff senses. A characterization for *- nonexpansive multivalued mappings is given. Also a counterexample is constructed to show a negative answer to a question raised by A. Canbtti, G. Marino and P. Pibtramala.
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SOOCHOW JOURNAL OF MATHEMATICS Volume 21, No. 1, pp. 107-115, January 1995 REMARKS ON MULTIVALUED NONEXPANSIVE MAPPINGS BY GENAROL OPEZ ACEDO AND HONG-KUN XU Abstract. Convergence of xed p ointsetsof multivalued nonexpansive mappings is studied under b oth the Mosco and Hausdor senses. A characterization for *-nonexpansivemultivalued mappings is given. Also acounterexample is constructed to show a negativeanswer to a question raised by A. Canbtti, G. Marino and P. Pibtramala. Let H b e a Hilb ert space, C a b ounded closed convex subset of H and T : C ! C is a (single-valued) nonexpansive mapping (i.e., k Tx ; Ty kk x ; y k , x y 2 C ). Then for eachxed x 0 2 C and 2 0 1), the mapping T : C ! C dened by T x =(1 ; ) x 0 + T x x 2 C (1) is a contraction on C . Hence, Banach's Contraction Principle yields a unique x 2 C such that T x = x namely, x =(1 ; ) x 0 + T x : (2) An elegant result in the xed p oint theory of (single-valued) nonexpansive mappings is Browder's theorem 1] which states that the approximating curve x dened by (2) converges strongly as ! 1 to a xed p ointof T . This result Received February 22, 1994 revised April 27, 1994. AMS Sub ject Classication. Primary 47H09, 47H10. KeyWords. Multivalued nonexpansive mapping, xed p oint, Hausdor metric, Mosco convergence, multivalued *-nonexpansive mapping. Research partially supp orted by DGICYT under pro ject PB 90-903 and the Junta de Andalucia under pro ject 1241. 107
108 GENAROL OPEZ ACEDO AND HONG-KUN XU was extended by Reich 11] to a framework of a uniformly smo oth Banach space. For recent progress along the line, the reader is referred to 12], 7], 14]. Now we turn to the multivalued case. For a metric space ( X d ), we use CB ( X ) to denote the family of all nonempty closed bounded subsets of X , K ( X ) the family of all nonempty compact subsets of X ,and H the Hausdor metric on CB ( X ) induced by the metric d of X that is, for A , B 2 CB ( X ), H ( A B ) = max sup a 2 A d ( a B ) sup b 2 B d ( b A ) where d ( x E ) = inf f d ( x y ): y 2 E g is the distance from a point x 2 X to a subset E X . Now recall that a multivalued mapping T : X ! CB ( X ) is said to b e nonexpansive if H ( TxTy ) d ( x y ) x y 2 X: Recall also that a sequence f A n g in CB ( X ) is said to converge to an elemen t A 2 CB ( X ) under the Mosco sense if lim sup n !1 A n = lim inf n !1 A n = A where lim sup n !1 A n = f x 2 X : there are subsequences f n k g and f x n k g with x n k 2 A n k such that x n k ! x g and lim inf n !1 A n = f x 2 X : there exists x n 2 A n for each n such that x n ! x g . It is not hard to see that if H ( A n A ) ! 0 ( A n A 2 CB ( X )), then A n ! A under the sense of Mosco. Assume now H and C areasaboveand T : C ! K ( C ) is nonexpansive. For eachxed x 0 2 C and 2 0 1), we dene the mapping T : C ! K ( C ) by the same formula (1) ab ove. Then T is a multivalued contraction and hence has a (nonunique, in general) xed p oint x 2 C (see8]) i.e., x 2 (1 ; ) x 0 + T x : (3) Let y 2 Tx be suchthat x =(1 ; ) x 0 + y : (4)
REMARKS ON MULTIVALUED NONEXPANSIVE MAPPINGS 109 A natrual question nowgives rise to whether Browder's theorem can be extended to the multivalued case. The following simple example presents a negativeanswer. Example 1 .10] Let C =0 1] 0 1] b e the square in the real plane and T : C ! K ( C ) b e dened by T ( a b ) = the triangle with vertices (0 0) ( a 0) (0 b ) ( a b ) 2 C: Then it is easy to see that for any( a i b i ) 2 C , i =1 2 H ( T ( a 1 b 1 ) T ( a 2 b 2 )) = max fj a 1 ; a 2 j j b 1 ; b 2 jg k ( a 1 b 1 ) ; ( a 2 b 2 ) k showing that T is nonexpansive. It is also easy to see that the xed p ointset of T is F ( T ) = f ( a 0) : 0 a 1 gf (0 b ) : 0 b 1 g . Let x 0 =(1 0). Then the map T dened by (1) has the xed p ointset F ( T )= f ( a 0) : 1 ; a 1 g : Let x = ( ( 1 n 0) if =1 ; 1 n (1 0) otherwise. Then f x g satises (3) but is not convergent. The same example also shows that the net f F ( T ) g of xed p oint sets of the T 's do es not converge as ! 1to the xed point set F ( T )of T under either the Hausdor metric or the Mosco sense. However, this will be so if we put some restrictions on the xed p oint set F ( T )of T . First recall that a Banach space X is said to satisfy Opial's property 9] if for any sequence f x n g in X , the condition that f x n g converges weakly to x implies that lim sup n !1 k x n ; x k < lim sup n !1 k x n ; y k 8 y 2 X y 6 = x: Spaces saisfying this prop erty include all Hilb ert spaces and ` p for 1 <p< 1 . Also it is known 3] that any separable Banach space can be equivalently renormed so that it p ossesses Opial's prop erty.
110 GENAROL OPEZ ACEDO AND HONG-KUN XU Theorem 1. Let C be a nonempty closed bounded convex subset of a Banach space X satisfying Opial's property and T : C ! K ( C ) be a nonexpansive mapping such that F ( T ) = f z g . Then for any x 0 2 C ,the net f F ( T ) g of xed point sets of the T 's weakly converges as ! 1 to the xed point set F ( T ) of T under the Mosco sense, i.e., w ; lim sup ! 1 F ( T )= w ; lim inf ! 1 F ( T )= F ( T ) : Pro of. It is sucient to show that (i) F ( T ) w ; lim sup ! 1 F ( T )and (ii) w ; lim inf ! 1 F ( T ) F ( T ). To show (i), we assume that x 2 w ; lim sup ! 1 F ( T ), which means that there exist a sequence n 2 0 1) converging to 1and a sequence f x n g such that x n 2 F ( T n )and x n ! x weakly. Let y n 2 Tx n be such that x n =(1 ; n ) x 0 + n y n . Cho ose z n 2 Tx satisfying k y n ; z n k H ( Tx n Tx ) k x n ; x k : (5) Since Tx is compact, we mayassume that z n ! z 1 2 Tx strongly. Noting that k x n ; y n k! 0, we obtain by (5) that lim sup k x n ; z 1 k lim sup k x n ; x k : (6) Since x n ! x weakly, it follows from (6) and Opial's prop erty that x = z 1 and x 2 Tx . This concludes the pro of of (i). Next we show (ii). For each 2 0 1), cho ose any x 2 F ( T ) and y 2 Tx satisfying (4). Then by the same pro of as ab ove, we see that every weak cluster point of f x g is a xed pointof T . But, by assumption, F ( T )= f z g . Hence f x g converges weakly as ! 1to z . If the unique xed p oint z of T is such that Tz = f z g , then we have the following strong convergence result. Theorem 2. Let C be a nonempty closedc onvex subset of a Hilbert space H and T : C ! K ( C ) be a nonexpansive mapping with a unique xed point z . Suppose in addition that Tz = f z g . Then H ( F ( T ) F ( T )) ! 0 as ! 1 .
REMARKS ON MULTIVALUED NONEXPANSIVE MAPPINGS 111 Pro of. First we observe that f F ( T ) g is uniformly b ounded. In fact, given any x 2 F ( T ), wehave some y 2 Tx such that x =(1 ; ) x 0 + y . However, k y ; z k = d ( y Tz ) H ( Tx Tz ) k x ; z k : Hence k x ; z k k y ; z k +(1 ; ) k x 0 ; z k k x ; z k +(1 ; ) k x 0 ; z k : This implies that k x ; z kk x 0 ; z k and f x g is uniformly b ounded. Now cho ose x 2 F ( T )such that H ( F ( T ) F ( T )) = sup x 2 F ( T ) k x ; z k < k x ; z k +1 ; : We shall show that k x ; z k ! 0 as ! 1. Indeed, we have y 2 Tx satisfying (4). Since k y ; z k = d ( y Tz ) H ( Tx Tz ) k x ; z k , we obtain x ; (1 ; ) x 0 ; z k x ; z k that is, x ; x 0 +( x 0 ; z ) 2 k ( x ; x 0 )+( x 0 ; z ) k 2 which leads to k ; x 0 k 2 2 1+ h x ; x 0 z ; x 0 ik x ; x 0 kk z ; x 0 k : Therefore, k x ; x 0 kk z ; x 0 k : (7) From the pro of of theorem 1, we knowthat x ! z weakly as ! 1. It then easily follows from (7) that lim sup ! 1 k x kk z k . On the other hand, due to the lower weak continutiy of the norm of H , we have lim inf ! 1 k x kk z k . Therefore, wehav elim ! 1 k x k = k z k and k x ; z k 2 = k x k 2 ; 2 h x z i + k z k 2 ! 0 as ! 1 : This completes the pro of of the theorem.
112 GENAROL OPEZ ACEDO AND HONG-KUN XU Corollary 1. 10] Let the assumptions of theorem 2 be satised. Then w ; lim sup ; 1 F ( T )= kk; lim inf ! 1 F ( T )= F ( T ) : Remark 1. The example ab ove shows that the conclusions of theorems 1 and 2 are not valid if the xed p ointset F ( T )of T is not a singleton. However, it is an op en question whether the restriction Tz = f z g in Theorem 2 can b e removed. We also do not know if Theorem 2 is valid outside a Hilb ert space. Next welet( X d ) b e a metric space. Amultivalued map f : X ! K ( X )is said to b e *-nonexpansive 4] if for all x y 2 X and u x 2 f ( x ) with d ( x u x )= inf f d ( x z ) : z 2 f ( x ) g , there exists u y 2 f ( y )with d ( y u y ) = inf f d ( y w ) : w 2 f ( y ) g such that d ( u x u y ) d ( x y ) : It is obvious that this notion is identical with the notion of nonexpansiveness for singlevalued mappings. But they are dierent for multivalued mappings (see 13]). Wenowgiveacharacterization of multivalued *-nonexpansivemap- pings. Denote by P f the map x 7! f u x 2 f ( x ) : d ( x u x )=inf f d ( u x ) : u 2 f ( x ) gg . Note that P f ( x ) is nonemptyfor f ( x ) is compact. Theorem 3. A multivalued map f : X ! K ( X ) is *-nonexpansive if and only if the associated map P f : X ! K ( X ) is nonexpansive. Pro of. First assume that f is *-nonexpansive. Given any x y 2 X and u x 2 P f ( x ). By deniition, there is u x 2 f ( y )such that d ( u x u y ) d ( x y ). It follows that sup u x ! P f ( x ) d ( u x P f ( y )) sup u x ! P f ( x ) d ( u x u y ) d ( x y ) : The same argument shows that sup u y ! P f ( y ) d ( u y P f ( x )) d ( x y ) : Hence H ( P f ( x ) P f ( y )) d ( x y )
REMARKS ON MULTIVALUED NONEXPANSIVE MAPPINGS 113 and P f is nonexpansive. Conversely,we assume that P f is nonexpansive. Then given any x y 2 X and u x 2 f ( x )with d ( x u x ) = inf f d ( x z ) : z ! f ( x ) g (i.e., u x 2 P f ( x )). By compactness, we can cho ose u y 2 P f ( y ) such that d ( u x u y )= d ( u x P f ( y )). Hence d ( u x u y ) H ( P f ( x ) P f ( y )) d ( x y ) and f is *-nonexpansive. Remark 2. Theorem 3 indicates that the xed p oint theory of multivalued nonexpansive mappings applies to multivalued *-nonexpansive mappings in particular, we have the following results whose nonexpansivecounterparts were proved in 5] and 6], resp ectively. Corollary 2 Let X be a Banach space satisfying Opial's property, C a nonempty weakly compact convex subset of X ,and T : C ! K ( C ) a *-nonexpansive mapping. Then T has a xed point. Corollary 3. Let X be a uniformly convex Banach space, C a nonempty closed bounded convex subset of X , and T : C ! K ( C ) a *-nonexpansive mapping. Then T has a xed point. Corollaries 2 and 3 improve up on the corresp onding results of 4] and 13]. We conclude the pap er with a counterexample that presents anegative answer to a question raised by A .Canbtti, G. Marino and P. Pibtramala 2]. Supp ose that H is a Hilb ert space and K is a nonempty closed convex subset of H . We denote by KC ( K ) the family of all nonempty compact convex subsets of K ,andby d ( A B ) the distance b etween two subsets A B H , i.e., d ( A B )=inf fk x ; y k : x 2 A y 2 B g . With each mapping T : K !KC ( H ) one can asso ciate amultivalued mapping ^ T : K !KC ( H ) dened as follows: ^ Tx := f y 2 Tx : d ( y K )= d ( TxK ) g : The question raised by A . Canbtti, G. Marino and P. Pibtramala (see 2, Remark 1, p. 207]) is whether the nonexpansiveness of T implies that ^ T is nonexpansive. The following example shows that the answer is negative.
114 GENAROL OPEZ ACEDO AND HONG-KUN XU Example 2. Let OABC b e the unit square 0 1] 0 1] in the plane H . Let D and E b e the midp oints of the segments AB and OC , resp ectively.Let K be the triangle 4 AD F . To eachpoint P 2 K ,let P 0 be the symmetric pointof P with resp ect to the diagonal segment AC .Let P 0 1 b e the pro jection of P 0 onto the segment OA . Nowwe dene a map T : K !KC ( H )by setting (see the gure b elow) T ( P ) := The segment CP 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... ...................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .......................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O D 0 1 P 0 1 A E F D C B P 0 P G It is then easy to see that T is a nonexpansive mappipng with the unique xed p oint A . Wealsohave the following facts: (i) T ( A )= AC and hence d ( T ( A ) K )=0 (ii) ^ T ( A ) is the segment AF (iii) T ( D )= CD 0 1 , where D 0 1 is the midp ointof OA (iv) ^ T ( D ) = f G g , where G is the nearest point pro jection of F onto the segment CD 0 1 . Hence H ( ^ T ( A ) ^ T ( D )) = sup f d ( G M ): M 2 AF g =The length of the segment GA > The length of the segment AD = d ( A D ) showing that ^ T is not nonexpansive.
REMARKS ON MULTIVALUED NONEXPANSIVE MAPPINGS 115 Acknowledgement Part of this work was done while the second author was visiting Universidad de Sevilla. He would like to thank Universidad de Sevilla for nancial supp ort and Departamento de Analisis Matematico for hospitality. He aloso wishes to thank Professor K. K. Tan for providing him with a copyofthepaper 2]. Both authors are grateful to the anonymous referee for careful reading and go o d sugestions for the manuscript. References 1] F. E. Browder, Convergenceofapproximantstofixedpointsofnonexpansivenonlinearmappings inBanachspaces , Arch. Rational Mech. Anal., 24(1967), 82-90. 2] A. Canbtti, G. Marino and P. Pibtramala, Fan'sbestapproximationandfixedpointtheorems formultivaluednonexpansivemappingsinHilbertspace s , Indian J. Math., 33(1991), 203-220. 3] D. van Dulst, Equivalentnormsandthefixedpointpropertyfornonexpansivemappings ,J. London Math. So c., 25(1982), 139-144. 4] T. Husain and A. Latif, Fixedpointsofmultivaluednonexpansivemaps , Math. Jap onica, 33(1988), 385-391. 5] E. Lami Dozo, MultivaluednonexpansivemappingsandOpial'scondition , Pro c. Amer. Math. So c., 38(1973),286-292. 6] T. C. Lim, Afixedpointtheoremformultivaluednonexpansivemappingsinaunif ormlyconvexBanachspace , Bull. Amer. Math. So c., 80(1974), 1123-1126. 7] G. Marino and G. Trometta, Ona`proximationfixedpointsfornonexpansivemaps , Indian J. math., 34(1992), 91-98. 8] S. Nadler, Multivaluedcontractionmappings ,Pacic J. Math., 30(1969), 475-488. 9] Z. Opial, Weakconvegenceofthesequenceofsuccessiv eapp roximationsfornonexpans ive mappi ngs , Bull. Amer. Math. So c., 73(1973), 591-597. 10] P. Pietramala, Convergenceofapproximatingfixedpointssetsformultivaluednonexpansive mappings , Comment. Math. Univ. Carolinae, 32(1991), 697-701. 11] S. Reich, StrongconvergencetheoremsforresolventsofaccretiveoperatorsinBanachspaces ,J.Math. Anal. Appl., 75(1980), 287-292. 12] S. P. Singh and B. Watson, OnapproximatingfixedpointsProc.SymposiaPureMath. , 45, pt. 2(1986), 393-395. 13] H. K. Xu, Onweaklynonexpansiveand*-nonexpansivemultivaluedmappings , Math. Jap onica, 39(1991), 441-445. 14] H. K. Xu and X. M. Yin, Strongconvergencetheoremsfornonexpansivenon-selfmappings,NonlinearAnalysis , to app ear. Departamento de Analisis Matematico, Facultad de Matematicas,Universidad de Sevilla, Ap do. 1160, 41080-Sevilla, Spain. Institute of Applied Mathematics, East China University of Science and Technology, Shanghai 200237, China.