Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169837, 16 pages doi:10.1155/2010/169837 Research Article Fixed Points of Singleand Set-Valued Mappings in Uniformly Convex Metric Spaces with No Metric Convexity Rafa Esp´ ınola,1Aurora Fern ´ andez-Le ´ on,1and Bo˙ zena Pia¸tek2 1Departamento de An´ alisis Matem´ atico, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain 2Institute of Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland Correspondence should be addressed to Rafa Esp´ ınola, [email protected] Received 20 April 2009; Accepted 28 May 2009 Academic Editor: Mohamed A. Khamsi Copyright q2010 Rafa Esp´ ınola et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for setvalued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings. 1. Introduction This paper is motivated by the recent paper 1.In1the authors study different questions related to fixed points of asymptotic pointwise contractive/nonexpansive mappings in CAT0spaces. CAT0spaces are studied in 1as a very significant example within the class of uniformly convex metric spaces the reader can consult 2for details on CAT0 spaces. In our present paper we propose to consider similar questions on uniformly convex metric spaces under the mildest additional conditions we may impose. More precisely, we will work with uniformly convex metric spaces with either a monotone modulus of convexity in the sense first given in 3or a lower semicontinuous from the right modulus of convexity see Section 2 for proper definitions. For a recent survey on the existence of fixed points in geodesic spaces, the reader may check 4, for recent achievements on related topics the reader may also check 5. The notion of asymptotic pointwise contractions was introduced in 6. Then it was also studied in 7where, by means of ultrapower techniques, different results about the 2 Fixed Point Theory and Applications existence of fixed points and convergence of iterates were proved. In 8new proofs were presented but this time after applying only elementary techniques. Very recently, in 1, these techniques were applied in CAT0, where the authors attend to the Bruhat-Tits inequality for CAT0spaces in order to obtain such results. In the present paper we show that actually most of those results still hold for general uniformly convex metric spaces under mild conditions on the modulus of convexity. In Section 3 we focus on single-valued mappings and, in particular, on mappings which are asymptotically pointwise contractive/nonexpansive to study the existence of fixed points, convergence of Picard’s iterates, and the structure of their sets of fixed points. As a technical result we need to show that bounded sequences in these spaces have a unique asymptotic center which, as a by-product, leads to Kirk’s Fixed Point Theorem. In Section 4 we study different problems regarding set-valued mappings in these spaces. The main technical difficulty to achieve similar results to those shown in 1is that now we cannot count on the existence of fixed points for nonexpansive setvalued mappings for the kind of spaces we deal with. Finding fixed point for set-valued nonexpansive mappings in uniformly convex metric spaces was first studied by Shimizu and Takahashi 9, where the existence of fixed points was guaranteed under stronger conditions on the modulus of convexity and the additional condition of metric convexity of the space. The fact that we do not have that the metric are convex will make the problem more complicated and this will take us to impose new conditions on the modulus of convexity which we will relate with the geometry of the space. 2. Basic Definitions and Results We introduce next some basic definitions. Definition 2.1. Let X, dbe a metric space. A mapping T:X→Xis called a pointwise contraction if there exists a mapping α:X→0,1such that dTx,Ty≤αxdx,y2.1 for any y∈X. It is proved in 8see also 6 that if Kis a weakly compact convex subset of a Banach space and T:K→Kis a pointwise contraction, then Thas a unique fixed point and the sequence of the iterates of Tconverges to the fixed point for any x∈K. As it is pointed out in 1, the uniqueness of fixed points and convergence of iterates for these mappings directly follow if existence is guaranteed. Definition 2.2. Let X, dbe a metric space. Let T:X→Xbe a mapping, and let αn:X→ 0,∞for each n∈Nbe such that dTnx,Tny≤αnxdx, yfor any y∈X.2.2 Then iTis called an asymptotic pointwise contraction if {αn}converges pointwise to α: X→0,1; Fixed Point Theory and Applications 3 iiTis called an asymptotic pointwise nonexpansive mapping if lim sup αnx≤1for any x∈X; iiiTis called a strongly asymptotic pointwise contraction if lim sup αnx≤k,with 0<k<1, for any x∈X. In this paper we will mainly work with uniformly convex geodesic metric space. Since the definition of uniform convexity requires the existence of midpoints, the word geodesic is redundant and so, for simplicity, we will usually omit it. Definition 2.3. A geodesic metric space X, dis said to be uniformly convex if for any r>0and any ε∈0,2there exists δ∈0,1such that for all a, x, y ∈Xwith dx,a≤r,dy,a≤r and dx,y≥εr it is the case that dm, a≤1−δr, 2.3 where mstands for any midpoint of any geodesic segment x, y. A mapping δ:0,∞× 0,2→0,1providing such a δδr, εfor a given r>0andε∈0,2is called a modulus of uniform convexity. Notice that this definition of uniform convex metric spaces is weaker than the one used in 9in two ways. First, we do not impose that the metric is convex and, second, our modulus of convexity does depend on the two variables rand εwhile it is assumed to depend only on εin 9. Definition 2.4. Let X, dbe a metric space, then the metric is said to be convex if for any x, y and zin X,andma midpoint in between xand y, dz, m≤1/2dz, xdz, y.2.4 It is easy to see that uniformly convex metric spaces are uniquely geodesic, that is, for each two points there is just one geodesic joining them. Therefore midpoints and geodesic segments x,yjoining two points are unique. In this case there is a natural way to define convexity. A subset Cof a uniquelygeodesic space is said to be convex if x, y⊆Cfor any x,y ∈C. For more about geodesic spaces the reader may check 2. To obtain our results we will need to impose additional conditions on the modulus of convexity. Following 3,10we consider the notion of monotone modulus of convexity as follows. Definition 2.5. If a uniformly convex metric space Xadmits a modulus of convexity δsuch that it decreases with rfor each fixed εthen we say that δis a monotone modulus of convexity for X. In the same way we define a lower semicontinuous from the right modulus of convexity as follows. Definition 2.6. If a uniformly convex metric space Xadmits a modulus of convexity δsuch that it is lower semicontinuous from the right with respect to rfor each fixed εthen we say δis a lower semicontinuous from the right modulus of convexity for X. 4 Fixed Point Theory and Applications Let Xbe a metric space and Fa family of subsets of X. Then, following 1, we say that Fdefines a convexity structure on Xif it contains the closed balls and is stable by intersection. Let Xbe a metric space and Fa convexity structure on X.GivenΦ:X→0,∞,we say that Φis F-convex if {x:Φx≤r}∈Ffor any r≥0. If we consider a bounded sequence {xn}in X,weareabletodefineafunctionr·,x n, called type, such that for each x rx,xnlim sup n→∞ dxn,x.2.5 The asymptotic center of a bounded sequence with respect to a subset Cof Xis then defined as AC{xn}x∈X:rx,xn≤ry,xnfor any y∈C.2.6 If the asymptotic center is taken with respect to Xthen it is simply denoted by A{xn}. Definition 2.7. We say that a convexity structure is T-stable if types are F-convex. In 1the following definition of compactness for convexity structure was considered. Definition 2.8. Given Fa convexity structure, we will say that Fis compact if any family Aαα∈Γof elements of Fhas nonempty intersection provided ∩α∈F/ ∅foranyfinitesubset F⊂Γ. In our paper we will rather use the idea of compactness given in 11. Notice that this second notion of compactness is weaker than the previous one. Definition 2.9. Given Fa convexity structure, we will say that Fis nested compact if any decreasing chain Aαα∈Γof nonempty bounded elements of Fhas nonempty intersection. A very important property given in 3about complete uniformly convex metric spaces with monotone modulus of convexity is that decreasing sequences of nonempty bounded closed and convex subsets of these spaces have nonempty intersection. As a consequence, we have that if Fstands for the collection of nonempty closed and convex subsets of a complete uniformly convex metric space with monotone modulus of convexity, then Fis a nested compact convexity structure. Remark 2.10. It is not hard to see that the same remains true if the monotone condition on the modulus is replaced by lower semicontinuity from the right. 3. Asymptotic Pointwise Contractions in Uniformly Convex Metric Spaces In this section we give different results for the above defined mappings in uniformly convex metric spaces. Although, for expository reasons, our results will be usually proved only for uniformly convex metric spaces with a monotone modulus of convexity, they also hold when there is a lower semicontinuous modulus of convexity. Some indications about differences in both cases will be given. We begin with a technical result. Fixed Point Theory and Applications 5 Proposition 3.1. Let X, dbe a complete uniformly convex metric space with a monotone (or lower semicontinuous from the right) modulus of convexity δr, ε. Consider the family Fof all nonempty closed and convex subsets of X.ThenFdefines a nested compact and T-stable convexity structure on X. Proof. It only remains to be proved that Fis T-stable. Let {xn}be a bounded sequence in X and consider the type defined by {xn}. We need to show that Cr{x:rx,xn≤r}∈Ffor any positive r. It is immediate to see that Cris closed and nonempty. To see that Cr∈Fis also convex, consider xand yto be two different points in Cr. There is no restriction if we assume that lim sup dy,xn≤lim sup dx, xnr1≤r.Letmbe the midpoint of the segment x,y and take ε1dx, y/r1, then, by uniform convexity, we have that dm, xn≤1−δmaxdx,xn,dy,xn,ε 1maxdx, xn,dy,xn <maxdx, xn,dy,xn, 3.1 and so, lim sup dm, xn≤lim sup maxdx, xn,dy,xnr1≤r. 3.2 Hence, m∈Cr. The following theorems were proved in 1under the hypothesis of compactness on the convexity structure. We state it, however, under the hypothesis of nested compactness since this is all it is actually required in the proofs given in 1. Theorem 3.2. Let Xbe a bounded metric space. Assume that the convexity structure AMis nested compact. Let T:X→Xbe a pointwise contraction. Then Thas a unique fixed point x0. Moreover the orbit {Tnx}converges to x0, for each x∈X. Theorem 3.3. Let Xbe a bounded metric space. Assume that the convexity structure AMis nested compact. Let T:X→Xbe a strongly asymptotic pointwise contraction. Then Thas a unique fixed point x0. Moreover the orbit {Tnx}converges to x0, for each x∈X. Now the next corollary follows. Corollary 3.4. The above theorems hold for complete bounded uniformly convex metric spaces with either monotone or lower semicontinuous from the right modulus of convexity. The following lemma is immediate. Lemma 3.5. Let Xbe a metric space and Fa nested compact convexity structure on Xwhich is T-stable. Then for any type r·,x n,thereexistsx0∈Xsuch that rx0,x ninf{rx, xn:x∈X}.3.3 As a direct consequence of Proposition 3.1 and the previous lemma we get the following result for asymptotic pointwise contractions. We omit the details of its proof as it follows similar patterns as in 1, Theorem 4.2. 6 Fixed Point Theory and Applications Theorem 3.6. Let X, dbe a complete uniformly convex metric space with a monotone (or lower semicontinuous from the right) modulus of convexity δr, ε. Suppose Xis bounded. Then every T: X→Xasymptotic pointwise contraction has a unique fixed point x0. Moreover, the orbit {Tnx} converges to x0for each x∈X. Next we show some consequences of Proposition 3.1 and Lemma 3.5. The cases for monotone and lower semicontinuous from the right modulus of convexity are shown separately as they require different proofs. Corollary 3.7. Let Xbe a complete uniformly convex metric space with a monotone modulus of convexity and {xn}a bounded sequence in X. Then the set of asymptotic centers of {xn}is a singleton. Proof. Let uand vbe two different points in A{xn},and let mbe the midpoint of u, v.Let rru, xnru, xn,cr1,and ε1du, v/c. By the uniform convexity, there exists N∈Nsuch that for every n≥N, dm, xn≤1−δmax{du, xn,dv, xn},ε 1 max{du, xn,dv, xn} ≤1−δc,ε1 max{du, xn,dv,xn}. 3.4 If we let ngo to infinite, we obtain that rm, xn≤1−δc,ε1r<r,which is clearly a contradiction. Remark 3.8. This corollary has been first proved in 12, Proposition 3.3for a certain class of uniformly convex hyperbolic spaces with monotone modulus of convexity. Now we show the lower semicontinuous case. Corollary 3.9. Let Xbe a complete uniformly convex metric space with a lower semicontinuous from the right modulus of convexity and {xn}a bounded sequence in X. Then the set of asymptotic centers of {xn}is a singleton. Proof. Let uand vbe two different points in A{xn}and let mbe the midpoint of u, v.Let rru, xnrv,xn,εdu, v/r1,and let us fix p∈N. Then max{du, xn,dv,xn}≤ rp−1for each nlarge enough. By the uniform convexity, dm, xn≤1−δrp−1,εrp−13.5 for the same nas above and finally rm, xn≤1−δrp−1,εrp−1.3.6 Now it suffices to observe that δrp−1,ε≥1 2δr, ε, 1−δrp−1,ε≤1−1 2δr, ε. 3.7 Fixed Point Theory and Applications 7 for plarge enough. Combining it with 3.6and taking limp→∞ we obtain rm, xn<ras in the former corollary, and thus the contradiction. Another consequence is Kirk Fixed Point Theorem in uniformly convex metric spaces. Corollary 3.10. Let Xbe a complete uniformly convex geodesic metric space with a monotone (or lower semicontinuous from the right) modulus of convexity. Suppose Xis bounded, then any nonexpansive mapping T:X→Xhas a fixed point. Proof. Consider x∈Xand {Tnx}the sequence of its iterates. Let ωbe the only asymptotic center of {Tnx}in X. Then, by the nonexpansiveness of T, it follows that rTω,Tnx ≤ rω,Tnx and so, Tωω. Now we present a counterpart for 1, Theorem 5.1. Theorem 3.11. Let X, dbe a complete uniformly convex metric space with a monotone (or lower semicontinuous from the right) modulus of convexity δr, ε.LetCbe a bounded closed convex nonempty subset of X. Then any T:C→Casymptotic pointwise nonexpansive mapping has a fixed point, and the set of fixed points of T,FixT, is closed and convex. Proof. Let x∈Cand consider xnTnx.FromCorollary 3.7, we know that AC{xn}is a singleton. Let ωbe the only point in that set, that is, ωis such that rω, xninf{ru, xn:u∈ C}. We want to show that {Tmω}is a Cauchy sequence. Suppose this is not the case. Then there exists a separated subsequence {Tmiω}of {Tmω}, that is, there exists ε>0 such that dTmkω,Tmhω ≥εfor every k/ hin N. Let mkh be the midpoint of the segment Tmkω,Tmhω,cdiamCand ε1ε/c. The uniform convexity of the space, together with its monotone character, implies that for every kand hin N dmkh,x n≤1−δmax{dTmhω,x n,dTmkω,x n},ε 1 max{dTmhω,x n,dTmkω,x n } ≤1−δc,ε1 max{dTmhω,x n,dTmkω,x n}. 3.8 Notice that, by definition of T, rTmω,x n≤αmωrω,xn.3.9 Then, if we let ngo to infinity, rω,xn≤rmkh,x n ≤1−δc,ε1 max{rTmkω,x n,rTmhω,x n} ≤1−δc,ε1 max{αmkωrω,xn,α mhωrω,xn}. 3.10 Since Tis pointwise asymptotic nonexpansive, then rω,xn≤1−δc,ε1rω, xn,3.11 8 Fixed Point Theory and Applications and so rω,xn0, which is a contradiction since, in virtue of 3.9,thisimpliesthatTmω converges to ω. Therefore, {Tmω}is a Cauchy sequence and its limit, again by 3.9,isω. Then, from the continuity of T,Tωω. In consequence, FixTis nonempty. Now, since Tis continuous, FixTis closed. We show next that FixTis also convex. Let u, v be two different points in FixTand wthe midpoint of the segment u, v. We need to show that w∈FixT. Now, since Tis pointwise asymptotic nonexpansive, du, Tnw dTnu,Tnw ≤αnwdu, wαnwdu, v 23.12 and, equally, dv,Tnw dTnv,Tnw ≤αnwdv,wαnwdu, v 2.3.13 Therefore, for ε>0,there exists n0such that if n≥n0then Tnw∈Bu, du, v 2ε∩Bv, du, v 2εDε,3.14 but, from the proof of Proposition 2.2 in 3, the diameters of the sets Dεtend to 0 as εtends to 0 and so lim Tnww, which proves wis a fixed point of T. Remark 3.12. The proof for the lower semicontinuous case follows in a similar way but following the reasoning of Corollary 3.9. In 1a demiclosed principle is also given for asymptotic pointwise nonexpansive mappings in CAT0spaces. Next we show that an equivalent result is also possible for uniformly convex metric spaces. Following 1we define {xn}Cωif and only if rω, xninf x∈Crx,xn,3.15 where Cis a closed and convex subset of a uniformly convex metric space containing the bounded sequence {xn}. Notice that this definition does not depend on the set Cwhen the space Xis a complete CAT0space. This is due to the fact that the asymptotic center of a bounded sequence of a complete CAT0space belongs to the closed convex hull of the sequence, which easily follows from the very well-known fact that the metric projection onto closed convex subsets of a complete CAT0space is nonexpansive see 2for details. Recall that the existence and uniqueness of such a ω∈Cin a complete uniformly convex metric spaces with monotone modulus of convexity is guaranteed by Corollary 3.7. Proposition 3.13. Let X, dbe a complete uniformly convex metric space with a monotone modulus of convexity δr, ε.LetCbe a bounded closed convex nonempty subset of X.LetT:C→Can asymptotic pointwise nonexpansive mapping. Let {xn}∈Cbe an approximate fixed point sequence, that is, limn→∞dxn,Txn  0, and such that xnωfor a certain ω∈C.ThenTωω. Fixed Point Theory and Applications 9 Proof. Since {xn}is an approximate fixed point sequence, then we have that rx,xnlim sup n→∞ dx,Tmxn rx,Tmxn 3.16 for any m≥1see Note Added in Proof at the end of the paper. In consequence, since rTmx,Tmxn ≤αmxrx,xnfor x∈C,3.9holds for any x. Therefore, particularizing for ω, we have that lim supm→∞rTmω,x nrω, xn. NowweclaimthatTmω→ωas m→∞. Suppose on the contrary that there exist an ε>0 and a subsequence {Tmkω}of {Tmω}such that dTmkω,ω≥εfor every k∈N.Letωmk be the midpoint of the geodesic segment Tmkω,ω,cdiamCand ε1ε/c. By uniform convexity, for every k, we have that dωmk,x n≤1−δmax{dω, xn,dTmkω,x n},ε 1 max{dω, xn,dTmkω,x n} ≤1−δc,ε1 max{dω,xn,dTmkω,x n}. 3.17 If we consider the upper limit of the above inequality when n→∞,weget rω,xn≤rωmk,x n≤1−δc,ε1 max{rω,xn,rTmkω,x n}.3.18 If we do the same when k→∞, we finally obtain that rω, xn≤1−δc,ε1rω,xn. Therefore rω,xn0,and the existence of fixed point follows the same as in Theorem 3.11. Remark 3.14. The proof for the lower semicontinuous case follows in a similar way but following the reasoning of Corollary 3.9. 4. Fixed Points of Set-Valued Mappings In this section we present fixed points theorems for set-valued mappings defined on uniformly convex metric spaces. Results stated for uniformly convex metric space with a monotone modulus of convexity also hold if there is a lower semicontinuous from the right modulus of convexity. Proofs of this second case will be omitted as they are based on technical results already proved for both kinds of modulus in Section 3. The Hausdorffmetric on the closed and bounded parts of a metric space Xis defined as follows. If Uand Vare bounded and closed subsets of a metric space X, then HU, V inf{ε>0:U⊆NεV,V⊆NεU},4.1 where NεV{y∈X:disty,Vinf{dy,x:x∈V}<ε}.LetCbe a subset of a metric space X. A mapping T:C→2Xwith nonempty bounded closed values is nonexpansive if HTx,Ty≤dx, y4.2 for all x, y ∈C. Our main goal in this section is to study if given Xis a bounded uniformly convex metric space with monotone modulus of convexity, then every nonexpansive 16 Fixed Point Theory and Applications 19A. Kaewcharoen and W. A. Kirk, “Proximinality in geodesic spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 43591, 10 pages, 2006. 20T. Zamfirescu, “On the cut locus in Alexandrov spaces and applications to convex surfaces,” Pacific Journal of Mathematics, vol. 217, no. 2, pp. 375–386, 2004.