A note on fixed point sets in CAT(0) spaces

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1 J. Math. Anal. Appl. 320 (2006) Note A note on fied point sets in CAT(0) spaces P. Chaoha,1, A. Phon-on Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand Received 7 July 2005 Available online 2 September 2005 Submitted by G. Jungck Abstract We show that the fied point set of a quasi-nonepansive selfmap of a nonempty conve subset of a CAT(0) space is always closed, conve and contractible. Moreover, we give a construction of a continuous selfmap of a CAT(0) space whose fied point set is prescribed Elsevier Inc. All rights reserved. Keywords: Fied point set; CAT(0) spaces; Nonepansive maps Introduction Geometric and topological properties of the fied point set have been studied to some degree for nonepansive maps of certain kinds of metric spaces. For eample, it is known that the fied point set of a nonepansive selfmap of a bounded hyperconve space X is also hyperconve and hence a nonepansive retract of X (see [3]). Recently, in [4] and [5], W.A. Kirk developed the fied point theory for CAT(0) spaces and proved an interesting fact about the fied point set: Theorem. [5] Suppose X is a nonempty bounded closed conve subset of a complete CAT(0) space, and suppose f : X X is nonepansive. Then the fied point set of f is nonempty, closed and conve. * Corresponding author. addresses: phichet.c@chula.ac.th (P. Chaoha), akez424@hotmail.com (A. Phon-on). 1 Supported by Thailand Research Fund Grant No. BRG X/$ see front matter 2005 Elsevier Inc. All rights reserved. doi: /j.jmaa

2 984 P. Chaoha, A. Phon-on / J. Math. Anal. Appl. 320 (2006) Motivated by the theorem above in the situation where the eistence of a fied point is assumed, we try to eplore the connection between fied point sets and closed conve subsets in CAT(0) spaces in greater generality. In Section 1, we show that the fied point set (if nonempty) of a quasi-nonepansive selfmap (which may not even be continuous, see below) of a conve subset of a CAT(0) space is always closed, conve and hence contractible. Therefore, we have a complete homotopical/homological description of the fied point set of a quasi-nonepansive selfmap of a nonempty conve subset of a CAT(0) space. In Section 2, we try to do something opposite. It is also known that, for a nonempty complete conve subset K of a CAT(0) space X, we can always find a nonepansive selfmap of X (for eample, the nearest point projection in [2]) whose fied point set is precisely K. Hence, for a complete CAT(0) space X, there is a nonepansive map whose fied point set is any prescribed nonempty closed conve subset of X. However, the conveity assumption here is very crucial because there is no nonepansive selfmap of C whose fied point set is eactly the unit circle (by considering where 0 should be mapped). Interestingly, if we only require the continuity of the map, the conveity assumption can be dropped. In fact, we will give an eplicit construction a continuous selfmap of a CAT(0) space X whose fied point set is any prescribed nonempty closed subset of X. Throughout this paper, for a space X, the fied point set of f : X X will be denoted by F(f). Since we are interested in properties of F(f), we will always assume that F(f). According to [1], the map f will be called quasi-nonepansive if d(f(),p) d(,p) for each X and p F(f). Clearly, a nonepansive map is quasi-nonepansive, but not vice versa. In fact, it is easy to see that a quasi-nonepansive map needs not be continuous (e.g., f() 2 for all [0, 1) and f(1) 0). A CAT(0) space is simply a geodesic metric space whose each geodesic triangle is at least as thin as its comparison triangle in the euclidean plane (see [2,4,5] for precise definitions and properties). Since a CAT(0) space (X, d) is always uniquely geodesic, we will use [,y] to denote the geodesic segment joining and y, and (1 t) ty to denote the unique point z [,y] such that d(,z) td(,y) for any t [0, 1]. With this notation, we immediately obtain the following facts: for any,y,z (X, d) and s,t [0, 1], (1) d((1 t) ty,(1 s) sy) t s d(,y), (2) d((1 t) ty,(1 t) tz) d(y,z). One nice thing about a CAT(0) space is that it is always contractible. Moreover, one can easily prove that a subset of a CAT(0) space, equipped with the induced metric, is a CAT(0) space if and only if it is conve. Therefore, a conve subset of a CAT(0) space is always contractible. 1. Fied point sets of quasi-nonepansive maps Lemma 1.1. The fied point set of a quasi-nonepansive selfmap of a metric space is always closed.

3 P. Chaoha, A. Phon-on / J. Math. Anal. Appl. 320 (2006) Proof. Suppose f : (X, d) (X, d) is a quasi-nonepansive map and ( n ) a sequence in F(f) converging to. Then, for each ε>0, there is N N such that d(, N )<ε/2 and hence d(,f()) d(, N ) + d( N, f ()) 2d(, N )<ε. Since ε is arbitrary, we must have d(,f()) 0; i.e., F(f). Therefore, F(f)is closed. Lemma 1.2. Let (X, d) be a CAT(0) space and,y,z X. If d(,z)+ d(y,z) d(,y), then z [,y]. Proof. Let Δ(,y,z) be the comparison triangle (up to isomorphism) in R 2 of the geodesic triangle Δ(,y,z), and w [,y] be such that d(,w) d(,z). By the above assumption, it follows that [,y] is simply a straight line with the point z in between; i.e., z [,y]. Moreover, since d(,w) d(,z),wemusthaved(,w) d(,z) and hence w z. Then, by the CAT(0) inequality, we have d(w,z) d(w,z) 0, which implies z w [,y]. Theorem 1.3. Let (X, d) be a conve subset of a CAT(0) space and f : X X a quasinonepansive map whose fied point set is nonempty. Then F(f) is closed, conve and hence contractible. Proof. Since (X, d) is conve in a CAT(0) space, it is also a CAT(0) space. Let,y F(f) and z [,y]. Since f is quasi-nonepansive, we have d(,f(z)) d(,z) and d(y,f(z)) d(y,z). Hence, we obtain the inequalities d(,y) d (,f(z) ) + d ( f(z),y ) d(,z)+ d(z,y) d(,y), which implies d(,f(z)) + d(f(z),y) d(,y). In fact, it is not difficult to see that d(,z) d(,f(z)) and d(z,y) d(f(z),y).forifd(, z) < d(, f (z)) or d(y,z) < d(y,f(z)), the above inequalities will give d(,y) d (,f(z) ) + d ( f(z),y ) <d(,z)+ d(z,y) d(,y), which is a contradiction. Now, by the previous lemma, we have f(z) [,y], and hence z f(z) because z is the only point in [,y] satisfying d(,z) d(,f(z)). Therefore, [,y] F(f); i.e., F(f)is conve and hence contractible. 2. Maps with prescribed fied point sets Theorem 2.1. Let A be a nonempty subset of a CAT(0) space (X, d). Then there eists a continuous map f : X X such that F(f) A. Proof. For each X, let k 1+ [0, 1]. First, note that for each,y X,

4 986 P. Chaoha, A. Phon-on / J. Math. Anal. Appl. 320 (2006) k k y 1 + d(y,a) 1 + d(y,a) d(y,a) d(y,a) d(y,a) (1 + d(y, A))(1 + ) d(y,a). Now, fi 0 A and define f : X X by f() (1 k ) k 0 for all X. ɛ To see that f is continuous, we let X, ɛ>0 and δ d(, 0 )+1 > 0. Then, for each y B d (, δ), wehave d ( f(),f(y) ) d ( ) (1 k ) k 0,(1 k y )y k y 0 d ( ) (1 k ) k 0,(1 k y ) k y 0 + d ( ) (1 k y ) k y 0,(1 k y )y k y 0 k k y d(, 0 ) + d(,y) d(y,a) d( 0,)+ d(,y) d(,y) ( d( 0,)+ 1 ) <δ ( d( 0,)+ 1 ) ɛ. Finally, it is easy to see that f() if and only if (1 k ) k 0 if and only if k 0 if and only if 1+ 0 if and only if 0 if and only if A. Therefore, F(f) A as desired. Corollary 2.2. Let K be a nonempty closed subset of a CAT(0) space (X, d). Then there eists a continuous map f : X X whose fied point set is precisely K. Eample 2.3. Let X [0, 1], K {0, 1, 1/2, 1/3,...} a closed subset of X and 0 0. It is now easy to verify that the map f : X X from the previous theorem is of the form f() 1+d(,K) for each X. Notice that if we let f 0(0) 0 and, for each n N, f n : [ 1 n+1, 1 n ] [ 1 n+1, 1 n ] be defined by, 2n+1 1+ n f n () 1 2n(n+1) n 1, 2n+1 2n(n+1), 1 1 n+1 +, 1 n+1 then f n0 f n which is clearly continuous. Moreover, since F(f 0 ) {0} and F(f n ) { 1 n+1, 1 n }, it follows that F(f) n0 F(f n ) K.

5 P. Chaoha, A. Phon-on / J. Math. Anal. Appl. 320 (2006) References [1] M.A. Ahmed, F.M. Zeyada, On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonepansive mappings, J. Math. Anal. Appl. 274 (2002) [2] M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, [3] M.A. Khamsi, W.A. Kirk, An Introduction to Metric Spaces and Fied Point Theory, Wiley, New York, [4] W.A. Kirk, Geodesic geometry and fied point theory, in: Seminar of Mathematical Analysis, Universidad de Sevilla, Sevilla, 2003, pp [5] W.A. Kirk, Geodesic geometry and fied point theory II, in: Proceedings of the International Conference in Fied Point Theory and Applications, Valencia, Spain, 2003, pp