RESEARC H Open Access Rodé’s theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0) spaces Watcharapong Anakkamatee 1 and Sompong Dhompongsa 1,2* * Correspondence: sompongd@chiangmai.ac.th 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand Full list of author information is available at the end of the article Abstract We extend Rodé’s theorem on common fixed points of semigroups of nonexpansive mappings in Hilbert spaces to the CAT(0) space setting. 2000 Mathematics Subject Classification: 47H09; 47H10. Keywords: CAT(0) space, semigroup of nonexpansive mappings, Δ?Δ?-convergence 1 Introduction In 1976, Lim [1] introduced a co ncept of convergence in a general metric space, called strong Δ-convergence. In [2], Kirk and Panyanak introduced a concept of co nvergence in a CAT(0) space, called Δ-convergence (see Section 2 for the definition). Moreover, they showed that many Banach space concepts and results which involve weak conver- gence can be extended to the CAT(0) space setting by using the Δ-convergence. For each semigroup S,letB(S) be the Banach space of all bounded real-valued map- pings on S with s upremum norm. A continuous linear functional μ Î B(S)* (the dual space o f B(S)) is called a mean on B(S)if||μ || = μ(1). For any f Î B(S), we use the following notation: μ ( f ) = μ s ( f ( s )). A mean μ on B(S) is said to be left invariant [respectively, right invariant]ifμ s (f(ts)) = μ s (f(s )) [respectively, μ s (f(st)) = μ s (f(s ))] for all f Î B(S)andforallt Î S. We will say that μ is an invariant mean if it is both left and right invariants. I f B(S)hasan invariant mean, we call S an amenable semigroup. It is well known that every commu- tative semigroup is amenable [3]. For each s Î S and f Î B(S), we de fine elements l s f and r s f in B(S)by(l s f)(t)=f (st)and(r s f)(t)=f (ts)foranyt Î S, respectively. A net {μ a } of means on B(S) is said to be asymptotically invariant if lim α (μ α (l s f ) − μ α (f ))=0=lim α (μ α (r s f ) − μ α (f )) . In [4], Rodé proved the following: Theorem 1.1.[4]If S is an amenable semigro up, C is a closed convex subset of a Hil- bert space H, S = { T s : s ∈ S } is a nonexpansive semigroup on C such that the set F (S) of common fixed points of S is nonempty and {μ a } is an asymptotically invariant net of means, then for each x Î C, {T μ α x } converges weakly to an element of F ( S ) . Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 © 2011 Anakkamatee and Dhompongsa; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http ://creativecommons. org/licenses/by/2.0), which permits unrestricted use, distribut ion, and reproduction in any medium, provided the original work is properly cited. Further, for each x Î C, the limit point of {T μ α x } is the same for all asymptotically invariant nets of means {μ a }. It is remarked that if S is amenable, then there is always an asymptotically str ong invariant net of finite means, i.e., means that are convex combination of point evalua- tions. This follows from Proposition 3.3 in [5]. Development of this subject had been made by several authors [1,6-8]. The main purpose of this a rticle is to extend this result of Rodé for a nonexpansive semigroup on a CAT(0) space in which the Δ-convergence plays the role of weak convergence. 2 Preliminaries Let (X, d)beametricspace.Ageodesic joining x Î X to y Î X is a mapping c from a closed interval [0, l] ⊂ ℝ to X such that c(0) = x, c(l)=y and d(c(t), c(t’)) = |t-t’| for all t, t’Î [0, l]. In particular, c is an isometry and d(x, y)=l.Theimageg of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denote d [x, y]. Write c(a 0 + (1 - a)l)=ax ⊕(1 - a)y, and for α = 1 2 ,wewrite 1 2 x ⊕ 1 2 y as x⊕y 2 , the midpoint of x and y. The space X is said to be a geodesic space if every two points of X are joined by a geodesic. Following [2], a metric space X is said to be a CAT(0) space if it is geodesically con- nected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. This latter property, which is what we referred to as the (CN) inequality, enables one to define the concept of nonpositive curvature in this situation, generalizing the same concept in Riemannian geometry. In fact (cf. [[9], p. 163]), the following are equivalent for a geodesic space X: (i) X is a CAT(0) space. (ii) X satisfies the (CN) inequality:Ifx 0 , x 1 Î X and x 0 ⊕ x 1 2 is the midpoint of x 0 and x 1 , then d 2 (y, x 0 ⊕ x 1 2 ) ≤ 1 2 d 2 (y, x 0 )+ 1 2 d 2 (y, x 1 ) − 1 4 d 2 (x 0 , x 1 ), for all y ∈ X . (iii) X satisfies the Law of cosine:Ifa = d(p, q), b = d(p, r), c = d(q, r) and ξ is the Alexandrov angle at p between [p, q] and [p, r], then c 2 ≥ a 2 +b 2 -2ab cos ξ. For any subset C of X, let π = π D be a nearest point projection mappi ng from C to a subset D. It is known by [[9], pp. 176-177] (see also [[10], Proposition 2.6]) that if D is closed and convex, the mapping π is well-defined, nonexpansive, and satisfies d 2 ( x, y ) ≥ d 2 ( x, π x ) + d 2 ( πx, y ) for all x ∈ Candy∈ D . (1) Definition 2.1. [[11], Definition 5.13] A complete CAT(0) space X has the property of the nice projection onto geodesics (property (N) for short) if, given any geodesic segment I ⊂ X, it is the case that π I (m) Î [π I (x), π I (y)] for any x, y in X and m Î [x, y]. As noted in [11], we do not know of any example of a CAT() space which does not enjoy the property (N). Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 2 of 14 Let S be a semigroup, C be a closed convex subset of a Hilbert space H, and for each s in S, T s is a mapping from C into itsel f. Suppose {T s x : s Î S} is bounded for al l x Î C. Let x Î C and μ be a mean on B(S). By Riesz’s representation theorem, there exists a unique x 0 Î C such that μ s T s x, y  = x 0 , y (2) for all y Î H. Here 〈 , 〉 denotes the inner product on H. The following result is a mild generalization of a result of Kakavandi and Amini [[12], Lemma 2.1]. Lemma 2.2. LetCbeaclosedconvexsubsetofaCAT(0)spaceXandμ be a mean on B(S). For a bounded function h : S ® C, define ϕ μ (y):=μ s (d 2 (h(s), y) ) for all y Î X. Then,  μ attains its unique minimum at a point of co{h ( s ) : s ∈ S } . For each x Î C,denote S ( x ) := {T s x : s ∈ S } .If S ( x ) is bounded, then by Lemma 2.2 we put T μ (h):=argmin{y → μ s (d 2 (h(s), y))} , and for h(s) of the form T s x, we write T μ (h) shortly as T μ x. Remark 2.3. If X is a Hilbert space, then (i) T μ x = x 0 where x 0 verifies (2), and (ii) ||x 0 || 2 = sup yÎX (2〈x 0 , y〉 -||y|| 2 ). Proof. (i): Let x 0 be such that μ s 〈T s x, y〉 = 〈x 0 , y〉 for all y Î X. We have  μ (x 0 )= μ (0) +||x 0 || 2 -2〈x 0 , x 0 〉 =  μ (0) - ||x 0 || 2 ≤  μ (0) + ||T μ x|| 2 -2〈x 0 , T μ x〉 =  μ (T μ x). There- fore, x 0 = T μ x. (ii) : For any x, y Î X, we know that ||T s x - y|| 2 =||T s x|| 2 -2〈T s x, y〉 +||y|| 2 .Bythe linearity of μ and (2), we have μ s (||T s x - y|| 2 )=μ s (||T s x|| 2 )-2〈x 0 , y〉 +||y|| 2 .Thus, inf yÎX μ s (||T s x - y|| 2 )=μ s (||T s x|| 2 )-sup yÎX (2〈x 0 , y〉 -||y|| 2 ). On the other hand, by (i), inf yÎX μ s (||T s x - y|| 2 )=μ s (||T s x - x 0 || 2 )=μ s (||T s x|| 2 )-2μ s 〈T s x, x 0 〉 +||x 0 || 2 = μ s (|| T s x|| 2 )-||x 0 || 2 . Hence, ||x 0 || 2 = sup yÎX (2〈x 0 , y〉 -||y|| 2 ). ■ Let C be a closed convex subset of a CAT(0) space X and S asemigroup.Wesay that the set S( S ) := {T s : s ∈ S } is a nonexpansive semigroup on C if (i) T s : C ® C is a nonexpansive mapping, i.e., d(T s x, T s y) ≤ d(x, y) for all x, y Î X, for all s Î S, (ii) the mapping s ® T s x is bounded for all x Î C, and (iii) T ts = T t T s , for all s, t Î S. We denote by F ( S ) the set of a ll common fixed points of mappings in S ( S ) , i.e., F(S ):=  s ∈ s F( T s ) , where F (T s ):={x Î C : T s x = x} is the set of fixed points of T s . For any bounded net {x a } in a closed convex subset C of a CAT(0) space X, put r(x, {x α }) = lim sup α d(x, x α ) Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 3 of 14 for each x Î C. The asymptotic radius of {x a }onC is given by r(C, {x α })=inf x ∈ C r(x, {x α }) , and the asymptotic center of {x a }inC is the set A ( C, {x α } ) = {x ∈ C : r ( x, {x α } ) = r ( C, {x α } ) } . It is known that in a complete CAT(0) space, A(C,{x a }) consists of exactly one point and A(X,{x a }) = A(C,{x a }) (cf. [2]). Remark 2.4. (i) Let D, E be directions and ν : E ® D. If {x ν(b) : b Î E} is a subnet of a bounded net {x a : a Î D}, then r (C,{x ν(b) }) ≤ r(C,{x a }). (ii) Let C be a closed convex subset of a CAT(0 ) space X, T : C ® C a nonexpansive mapping and x Î C. If {T n x} is bounded and if z Î A(C,{T n x}), then z Î F(T). Proof.(i)Leta 0 Î D. By the definition of subnets, there exists b 0 Î E such that ν(b) ≽ a 0 for all b ≽ b 0 . For each x Î C,wehave sup αα 0 d(x, x α ) ≥ sup β  β 0 d(x, x ν(β) ) . Thus, sup αα 0 d(x, x α ) ≥ inf β 1 sup β  β 1 d(x, x ν(β) ) ,andthisholdsforalla 0 . Hence, r(x, {x α })=inf α 0 sup α  α 0 d(x, x α ) ≥ r(x, x ν(β) ) ,andthisholdsforallx Î C.Conse- quently, r(C,{x a }) = inf xÎC r(x,{x a }) ≥ inf xÎC r(x, x ν(b) )=r(C,{x ν(b) }). (ii) Since T is nonexpansive, lim sup n d 2 (T n x, Tz) = lim sup n d 2 (TT n x, Tz) ≤ lim sup n d 2 (T n x, z). As every asymptotic center is unique, we have z = Tz. □ Definition 2.5. [[2], Definition 3.3] A net {x a } in X is said to Δ-converge to x Î Xifx istheuniqueasymptoticcenterof{u b } for every subnet {u b } of {x a }. In this case, we write Δ - lim a x a = x and call x the Δ-limit of {x a }. Proposition 2.6. [[2], Proposition 3.4] Every b ound ed net i n X has a Δ-convergent subnet. Remark 2.7. (i) Let D be a direction,{x a : a Î D} a net in X and x Î X. If lim sup a d(x, x a ) > r for some r >0, then there exists a subnet {x β α } of {x a } such that d(x, x β α ) ≥ ρ for all a. (ii) Let {x a } be a net in X. Then,{x a } Δ-converges to x Î X if and only if every subnet {x a’ } of {x a } has a subnet {x a“ } which Δ-converges to x. Proof. (i): For each a Î D,wehavesup a’ ≽a d(x, x a’ ) > r. Thus there ex ists b a ≻ a such that d(x, x β α ) ≥ ρ , and this holds for all a. Set a set E ={b a : a Î D}. Clearly, E is a direction, and define ν : E ® D by ν (b a )=b a .Leta 0 Î D, thus ν(b a ) ≽ a 0 for all β α  β α 0 and this shows that {x β α } is a subnet of {x a } satisfying d(x, x β α ) ≥ ρ for all a. ii): It is easy to see that if {x a } Δ-converges to x , then every subnet of {x a }also Δ-converges to x. On the other hand, suppose {x a } does not Δ-converge to x.Thus, there exists a subnet {x b }of{x a } such that x ∉ A (C,{x b }), and so lim sup b d(x, x b )>r >r(C,{x b }) for some r >0.By(i),thereexistsasubnet {x γ β } of {x b } satisfying d(x, x γ β ) ≥ ρ for all b. By assumption, there exists a subnet {x (γ β ) η } of {x γ β } Δ-conver- ging to x. Using Remark 2.4, ρ ≤ lim sup γ d(x, x (γ β ) η )=r(C, {x (γ β ) η }) ≤ r(C, {x γ β }) ≤ r(C, {x β } ) , a contradiction. □ In [13], Berg and Nikolaev introduced a concept of quasilinearization. Let us formally denote a pair (a, b) Î X × X by −→ ab and call it a vector. Then, quasilinearization is defined as a map 〈, 〉 :(X × X)×(X × X)by Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 4 of 14  −→ ab, −→ cd  = 1 2 d 2 (a, d)+ 1 2 d 2 (b, c) − 1 2 d 2 (a, c) − 1 2 d 2 (b, d ) for all a, b, c, d Î X. Recently, Kakavandi and Amini [14] introduced a concept of w- conv ergence: a sequence {x n }issaidtow-converge to x Î X if lim n→∞  −→ x n x, −→ ab  = 0 for all a, b Î X. Proposition 2.8. [[14], Proposition 2.5] For sequences in a complete CAT(0) space X, w-convergence implies Δ-convergence (to the same limit). A simple example shows that the converse of this proposition does not hold: Example 2.9. Consider an ℝ-tree in ℝ ∞ defined as follow: Let {e n } be the standard basis, x 0 = e 1 = (1, 0, 0, 0, ), and for each n, let x n = x 0 + e n+1 . An ℝ-tree is formed by the segments [x 1 , x n ] for n ≥ 0. It is easy to see that {x n } Δ-converges to x 1 . But {x n } does not w-converge to x 1 since  −−→ x n x 1 , −−→ x 0 x 1  = − 1 for all n ≥ 2. Thus, a bounded sequence does not necessary contain an w-convergent subsequence. 3 Main results 3.1 Δ-convergence Lemma 3.1. [[12], Lemma 3.1] If C is a closed convex subset of a CAT(0) space X and T : C ® C is a nonexpansive mapping, then F(T) is closed and convex. Lemma 3.2. [[12], Proposition 3.2] Let C be a closed convex subset of a CAT(0) space X and S an amenable semigroup. If S( S ) is a nonexpansive semigroup on C, then the fol- lowing conditions are equivalent. (i) S( x ) is bounded for some x Î C; (ii) S ( x ) is bounded for all x Î C; (iii) F ( S ) = ∅ . Proposition 3.3. [[12], Theorem 3.3] LetCbeaclosedconvexsubsetofacomplete CAT(0) space X, S an amenable semi group, and S ( S ) a nonexpansive semigroup on C with F (S) = ∅ . Then, T μ x ∈ F(S ) for any invariant mean μ on B(S). We now let S be a commutative semigroup and define a partial order ≽ on S by s ≽ t if s = t or there exists u Î S such that s = ut. When s ≽ t but s ≠ t, we simply write s ≻ t. We can see that (S, ≽) is a directed set. Examples of such S are the usual ordered sets (N ∪ {0}, +, ≥) and (ℝ + ∪ {0}, +, ≥). The following fact is well known: Proposition 3.4. Let μ be a right invariant mean on B(S). Then, sup s inf t f (ts) ≤ μ( f (s)) ≤ inf s sup t f (ts ) for each f Î B(S). Similarly, let μ be a left invariant mean on B(S). Then, sup s inf t f (st) ≤ μ(f (s)) ≤ inf s sup t f (st ) for each f Î B(S). Remark 3.5. If lim s f (s)=a for some a Î ℝ and {s’} is a subnet of {s} satisfying s’ ≻ s for each s, then μ s  ( f ( s  )) = a . Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 5 of 14 Proof. This is an easy consequence of Proposition 3.4 since μ s’ (f (s’ )) = μ s (f (s’ )) = lim s f (s’)=a. ■ Proposition 3.6. [[12], Proposition 4.1] Let C be a closed convex subset of a complete CAT(0) space X, S a commutative semigroup, and S ( S ) a nonexpansive semigrou p on C with F (S) = ∅ . Then, for each x Î C,, the net {πT s x} sÎS converges to a point Px in F ( S ) , where π = π F ( S ) : C → F( S) is the nearest point projection. Propos ition 3.7. Let C be a closed convex subset o f a complete CAT(0) space X, S a commutat ive semigroup, and S ( S ) a nonexpansive semigroup on C with F ( S ) = ∅ . Then, for any invariant mean μ on B(S), T μ x = lim s πT s x = Px for all x Î C. Proof.Fixx Î C and let ε >0. From Proposition 3.6, we see that there exists s 0 Î S such that d(πT s x, Px) < ε for all s ≽ s 0 .WeknowbyProposition3.3that T μ x ∈ F(S ) . So, d(Px, T s x) ≤ d (Px, πT s x)+d(π T s x, T s x) <d(πT s x, T s x)+ε ≤ d(T μ x, T s x)+ε for all s ≽ s 0 .Since{T s x : s Î S} is b ounded by Lemma 3. 2, there exis ts M>0 such that d (T μ x, T s x) <Mfor all s Î S. Therefore, d 2 (Px, T s x) ≤ d 2 (T μ x, T s x)+2Mε + ε 2 for each s ≽ s 0 .Sinceμ is an invariant mean, we have μ s (d 2 (Px, T s x)) = μ s (d 2 (Px, T ss 0 x)) ≤ μ s (d 2 (T μ x, T ss 0 x))+2Mε+ε 2 = μ s (d 2 (T μ x, T s x))+2Mε+ε 2 for any ε >0. By the argminimality of T μ x (see Lemma 2.2), T μ x = Px. □ In order to obtain the Rodé’s theorem (Theorem 1.1) in the framework of CAT(0) spaces, we need to restrict the asymptotically invariant nets of means {μ a }tothose that satisfy an additional condition: for each t Î S, μ α s (d 2 (T s x, y)) − μ α s (d 2 (T st x, y)) → 0 uniformly for y ∈ C . (3) In the Hilbert space setting, condition (3) is not required because the weak conver- gence can obtain from (2) directly. Lemma 3.8. Let X be a comp lete CAT(0) space th at has property (N), C be a closed convex s ubset of X, S a commutative semigroup, and S ( S ) a nonexpansive semigroup on Cwith F ( S ) = ∅ . Suppose {μ a } is an asymptotically invariant nets of means on B(S) satisfying condition (3). If { T μ α x } Δ-converges to x 0 , then x 0 ∈ F (S) . Proof. First, we show that, for each r Î S, lim α d(T μ α x, T r T μ α x)=0. (4) If this is not the case, there must be some δ >0sothatforeacha, there exist s a’ ≻ a satisfying d(T μ α  x, T r T μ α  x) ≥ δ .Put ε = δ 2 2 . Since the asymptoticall y invariant net {μ a } satisfies (3), there exists a 0 for which for each a ≽ a 0 , ϕ μ α (T r T μ α x)=μ α s (d 2 (T s x, T r T μ α x)) <μ α s (d 2 (T r T s x, T r T μ α x)) + ε ≤ μ α s (d 2 (T s x, T μ α x))+ε = ϕ μ α (T μ α x)+ ε .Set w = T μ α  0 x ⊕ T r T μ α  0 x 2 . B y the (CN ) inequality, the foll owing in equalities hold for each s Î S: d 2 (T s x, w) ≤ 1 2 d 2 (T s x, T μ α  0 x)+ 1 2 d 2 (T s x, T r T μ α  0 x) − 1 4 d 2 (T μ α  0 x, T r T μ α  0 x ) ≤ 1 2 d 2 (T s x, T μ α  0 x)+ 1 2 d 2 (T s x, T r T μ α  0 x) − δ 2 4 . Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 6 of 14 Therefore, ϕ μ α  0 (w) ≤ 1 2 ϕ μ α  0 (T μ α  0 x)+ 1 2 ϕ μ α  0 (T r T μ α  0 x) − δ 2 4 <ϕ μ α  0 (T μ α  0 x)+ ε 2 − δ 2 4 = ϕ μ α  0 (T μ α  0 x), which is a contradiction and thus (4) holds. Next, we show that x 0 ∈ F ( S ) . We suppose on the contrary that x 0 /∈ F ( S ) . Thus, for some r Î S, T r x 0 ≠ x 0 ,i.e.,d(x 0 , T r x 0 ):=g >0. Since {T μ α x}⊂co{T s x } ,itisbounded. We can get an M>0sothat d(T μ α x, x 0 ) ≤ M for all a.Welet 0 <ε<min{ γ 2 1 6 M ,2M } .From(4),thereexistsa 0 with the pro perty that d(T r T μ α x, T μ α x) < ε for all a ≽ a 0 .Now,foreacha ≽ a 0 , d(T μ α x, T r x 0 ) ≤ d(T μ α x, T r T μ α x)+d(T r T μ α x, T r x 0 ) < d(T μ α x, x 0 )+ ε .Thus, d 2 (T μ α x, T r x 0 ) < d 2 (T μ α x, x 0 )+2εd(T μ α x, x 0 )+ε 2 .Let w = x 0 ⊕T r x 0 2 .Usingthe(CN) inequality, we see that d 2 (T μ α x, w) ≤ 1 2 d 2 (T μ α x, x 0 )+ 1 2 d 2 (T μ α x, T r x 0 ) − 1 4 d 2 (x 0 , T r x 0 ) ≤ 1 2 d 2 (T μ α x, x 0 )+ 1 2 (d 2 (T μ α x, x 0 )+2εM + ε 2 ) − γ 2 4 = d 2 (T μ α x, x 0 )+εM + ε 2 2 − γ 2 4 for all a ≽ a 0 . Consequently, lim sup α d 2 (T μ α x, w) ≤ lim sup α d 2 (T μ α x, x 0 )+εM + ε 2 2 − γ 2 4 < lim sup α d 2 (T μ α x, x 0 ), contradicting to the fact that {x 0 } is the center of { T μ α x } . Therefore, T r x 0 = x 0 for all r Î S, and this shows that x 0 ∈ F (S) as desired. □ Theorem 3.9. Let X be a complete CAT(0) space that has Property (N), C be a closed convex s ubset of X, S a commutative semigroup, and S ( S ) a nonexpansive semigroup on C with F (S) = ∅ . Suppose {μ a }is an asymptotically invariant net of means on B(S) satis- fying condition (3). Then, {T μ α x } Δ-converges to Px ∈ F ( S ) for all x Î C. Here, Px is defined in Proposition 3.6. Proof.Letx Î C and {μ a’ } be any subnet of {μ a }. There exists a subn et {μ a” }of{μ a’ } such that {μ a” }w*-convergestoμ for some invariant mean μ on B(S). By Proposition 3.7, T μ x = Px. Since the net {T μ α  x}⊂co{T s x : s ∈ S } ,itisbounded.ThenbyProposi- tion 2.6, there exists a subnet {μ α β } of {μ a” } such that {T μ α β x } Δ-converges to some x 0 Î C. By Lemma 3.8, x 0 ∈ F ( S ) . We show x 0 = T μ x by splitting the proof into three steps. Step 1.If T μ α β x := argmin{y → μ β s (d 2 (T ss 0 x, y)) } , then T μ α β x ∈ co{T s x} ss 0 . Suppose T μ α β x /∈ co{T s x} ss 0 ,by(1), d 2 (T ss 0 x, T μ α β x) ≥ d 2 (T ss 0 x, π T μ α β x)+d 2 (T μ α β x, π T μ α β x ) for each s Î S where π : C → co{T s x} s  s 0 is the nearest point projection. Thus, Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 7 of 14 μ α β (d 2 (T ss 0 x, T μ α β x)) ≥ μ α β (d 2 (T ss 0 x, πT μ α β x))+d 2 (T μ α β x, πT μ α β x) >μ α β (d 2 (T ss 0 x, πT μ α β x) ) .This impossibility shows that T μ α β x ∈ co{T s x} ss 0 . Step 2. lim β d(T μ α β x, T μ α β x)= 0 . If this does not hold, there must be some h >0 so that for each b, there exists b’ ≻ b satisfying d(T μ α β  x, T μ α β  x) ≥ η .Put ε = η 2 2 . Since the asymptotically invariant net {μ b } satisfies ( 3), there exists b 0 such that |μ α β s (d 2 (T s x, T μ α β x)) − μ α β s (d 2 (T ss 0 x, T μ α β x))| < ε for each b ≽ b 0 . We suppose first that μ α β  0 s (d 2 (T ss 0 x, T μ α β  0 x)) ≤ μ α β  0 s (d 2 (T s x, T μ α β  0 x) ) .Set w = T μ α β  0 x ⊕ T μ α β  0 x 2 .By(CN) inequality, the following inequalities hold for each s Î S: d 2 (T s x, w) ≤ 1 2 d 2 (T s x, T μ α β  0 x)+ 1 2 d 2 (T s x, T μ α β  0 x) − 1 4 d 2 (T μ α β  0 x, T μ α β  0 x ) ≤ 1 2 d 2 (T s x, T μ α β  0 x)+ 1 2 d 2 (T s x, T μ α β  0 x) − η 2 4 . Therefore, ϕ μ α β  0 (w) ≤ 1 2 ϕ μ α β  0 (T μ α β  0 x)+ 1 2 ϕ μ α β  0 (T μ α β  0 x) − η 2 4 < 1 2 ϕ μ α β  0 (T μ α β  0 x)+ 1 2 μ α β  0 s (d 2 (T ss 0 x, T μ α β  0 x)) + ε 2 − η 2 4 ≤ ϕ μ α β  0 (T μ α β  0 x)+ ε 2 − η 2 4 = ϕ μ α β  0 (T μ α β  0 x), contradicting to the argminimality of T μ α β  0 x . In case μ α β  0 s (d 2 (T s x, T μ α β  0 x)) <μ α β  0 s (d 2 (T ss 0 x, T μ α β  0 x) ) ,wecanshowinthesamewaythat μ α β  0 s (d 2 (T ss 0 x, w)) <μ α β  0 s (d 2 (T ss 0 x, T μ α β  0 x) ) for some w which also leads to a contradiction. Step 3. x 0 = T μ x. We suppose on the contrary and l et h := d(x 0 , T μ x) >0. Let I =[T μ x, x 0 ]andπ I : C ® I be the nearest point projection onto I. S ince {T s x} is bounded, there exists M>0 such that d(T s x, π I (T s x)) ≤ M for all s Î S.Set N 0 > 4(M+η) 5 η and ρ = η 5N 0 .Suppose there exists s 0 Î S such that d(π I (T s x), x 0 ) ≥ 2r for all s ≽ s 0 . We know, by Step 1, that T μ α β x ∈ co{T s x} ss 0 .LetA := {y Î C: d(π I (y), x 0 )>2r}. Using property (N), A i s convex and co{T s x} s  s 0 ⊂ ¯ A ⊂{y ∈ C : d(π 1 (y), x 0 ) ≥ 2ρ } and thus d(π I (T μ α β x), x 0 ) ≥ 2 ρ .By Step 2, lim β d(T μ α β x, T μ α β x)= 0 . Choose b 0 , using the nonexpansiveness of π I ,sothat d(π I (T μ α β x), π I (T μ α β x)) < ρ for all b ≽ b 0 . Thus, d(π I (T μ α β x), x 0 ) > ρ for all b ≽ b 0 . But then x 0 /∈ co{T μ α β x} β β 0 which contradicts to the fact that x 0 is the Δ - limit of {T μ α β x } . Therefore, there must be a subnet {s’}ofS such that s’ ≻ s for all s and d(π I (T s  x), x 0 ) < 2ρ = 2 η 5N 0 (5) Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 8 of 14 for all s’. Hence, d(π I (T s  x), T μ x)=η − d(π I (T s  x), x 0 ) >η− 2 η 5N 0 . (6) By the property of N 0 ,5h 2 N 0 >4hM +4h 2 and so η 2 − 4η 2 5N 0 > 4ηM 5N 0 . (7) From (5), (6), and (7), d 2 (π I (T s  x), T μ x) >η 2 − 4η 2 5N 0 +( 2η 5N 0 ) 2 > 4ηM 5N 0 +( 2η 5N 0 ) 2 > 2d ( x 0 , π I ( T s  x )) d ( T s  x, π I ( T s  x )) + d 2 ( x 0 , π I ( T s  x )). Using (1), d 2 (T s  x, T μ x) ≥ d 2 (π I (T s  x), T μ x)+d 2 (π I (T s  x), T s  x) > d 2 (x 0 , π I (T s  x)) + 2d(x 0 , π I (T s  x))d(T s  x, π I (T s  x)) + d 2 (π I (T s  x), T s  x ) =(d(x 0 , π I (T s  x)) + d(T s  x, π I (T s  x))) 2 ≥ d 2 ( T s  x, x 0 ) for all s’.Sincethepointsx 0 and T μ x belong to the set F ( S ) ,thenets{d 2 (T s x, x 0 )} and {d 2 (T s x, T μ x)} are decreasing. So, lim s d 2 (T s x, x 0 )andlim s d 2 (T s x, T μ x)exist. Hence,  μ (T μ x )=lim s d 2 (T s x, T μ x ) = lim s’ d 2 (T s’ x , T μ x)=μ s’ (d 2 ( T s’ x, T μ x )) ≥ μ s’ (d 2 (T s’ x, x 0 )) = lim s’ d 2 (T s’ x, x 0 )=lim s d 2 (T s x, x 0 )= μ (x 0 ), a contradiction. Thus, x 0 = T μ x. The above argument shows that, for every subnet {μ a’ }of{μ a }, there exists a subnet {μ α β } of {μ a’ }suchthat {T μ α β x } Δ-converges to T μ x(= Px). By Remark 2.7 (ii), {T μ α x } Δ-converges to Px. □ It is an interesting open problem to determine whether Theorem 3.9 remains valid when the semigroup is amenable but not commutative. 3.2 Applications Proposition 3.10. Let C be a closed convex subset of a complete CAT(0) space X and T : C ® CbeanonexpansivemappingwithF(T) ≠ ∅. Let S =(N ∪ {0}, +), S ( S ) = {T n : n ∈ S } , Λ = N or ℝ + and b lk ≥ 0 be such that  k ∈ S β λk = 1 for all l Î Λ. Suppose for all k Î S, lim λ → ∞ β λk =0 (8) and for each m Î S, lim λ →∞ ∞  k = m |β λk − β λ(k−m) | =0 . (9) For any f =(a 0 , a 1 , ) Î B(S) let μ λ (f )=  ∞ k = 0 β λk a k . Then for each x Î C, {T μ λ x } Δ-converges to z for some z in F(T). In particular, if X is a H ilbert space, we have  ∞ k = 0 β λk T k x converges weakly to z for some z in F(T) as l ® ∞. Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 9 of 14 Proof. For each m Î S, |μ λ (f )−μ λ (r m f )| =     ∞ k=0 β λk a k −  ∞ k=0 β λk a k+m    ≤  m− 1 k=0 | β λk || a k | +  ∞ k=m   β λk − β λ(k−m)   | a k | . By (8) and (9), we have lim λ→∞ |μ λ ( f ) − μ λ ( r m f ) | = 0 , and this shows that the net {μ l }is asymptotically invariant. Let x Î C and consider a k of the form a k = d 2 (T k x, y) where y Î C. We see that {μ l } sati sfies (3). By Theorem 3.9, we have {T μ λ x } Δ -convergestoz for some z in F(T). In Hilbert spaces, by a well-known result in probability theory, we know that ∞  k = 0 β λk      T k x − ∞  k = 0 β λk T k x      2 ≤ ∞  k = 0 β λk    T k x − y    2 for all y Î C. So we have T μ λ x =  ∞ k = 0 β λk T k x . □ Corollary 3.11 (Bailo n Ergodic Theorem). LetCbeaclosedconvexsubsetofaHil- bert space H and T : C ® C be a nonexpansive mapping with F(T) ≠ ∅. Then, for any x Î C, S n x = 1 n n− 1  k = 0 T k x converges weakly to z for some z in F(T) as n ® ∞. Proof. Let Λ = N and put, for l Î Λ and k Î S =(N ∪ {0}, +), β λk =  1 λ , k ≤ λ − 1, 0, k >λ− 1 . The result now follows from Proposition 3.10. □ Corollary 3.12. [[15], Theorem 3.5.1] LetCbeaclosedconvexsubsetofaHilbert space H and T : C ® C be a nonexpansive mapping with F(T) ≠ ∅. Then, for any x Î C, S r x =(1− r)  ∞ k = 0 r k T k x converges weakly to z for some z in F(T) as r ↑ 1. Proof. Let Λ = ℝ + and put, for l Î Λ and k Î S =(N ∪{0}, +), β λk = (λ − 1) k λ k+1 . Taking r = λ−1 λ , Proposition 3.10 implies the desired result. Let S =(ℝ + ∪ {0}, +) and C be a closed convex subset of a Hilbert space H.Then,a family S ( S ) = {T ( s ) : s ∈ S } is said to be a continuous nonexpansive semigroup on C if S ( S ) satisfies the following: (i) T(s):C ® C is a nonexpansive mapping for all s Î S, (ii) T(t + s)x = T(t)T(s)x for all x Î C and t, s Î S, (iii) for each x Î C, the mapping s ® T(s)x is continuous, and (iv) T(0)x = x for all x Î C. Proposition 3.13. Let C be a closed convex subset of a Hilbert space H. Let S =(ℝ + ∪ {0}, + ), S ( S ) be a continuous nonexpa nsive semigroup on C with F ( S ) = ∅ , Λ = ℝ + and g l be a density function on S,i.e., g l ≥ 0 and  ∞ 0 g λ (s)ds = 1 for all l Î Λ. Suppose g l satisfies the following properties. for each h Î S , Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 10 of 14 [...]... Amini, M: Duality and subdifferential for convex functions on complete CAT(0) metric spaces Nonlinear Anal 73, 3450–3455 (2010) doi:10.1016/j.na.2010.07.033 15 Takahashi, W: Nonlinear Functional Analysis Yokohama Publisher, Yokohama (2000) doi:10.1186/1687-1812-2011-34 Cite this article as: Anakkamatee and Dhompongsa: Rodé’s theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0). .. AT: Semigroup of nonexpansive mappings on a Hilbert space J Math Anal Appl 105, 514–522 (1985) doi:10.1016/ 0022-247X(85)90066-6 8 Lau, AT, Shioji, N, Takahashi, W: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces J Funct Anal 191, 62–75 (1999) 9 Bridson, M, Haeiger, A: Metric Spaces of Non-Positive Curvature Springer,... λ □ Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 Page 12 of 14 By using Lemma 2.2, we can obtain a strong convergence theorem in Hilbert spaces stated as Theorem 3.17 below Proposition 3.16 Let C be a closed convex subset of a Hilbert space H and T : C ® C be a nonexpansive mapping with F(T) ≠ ∅ Given x... G: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space J Math Anal Appl 85, 172–178 (1982) doi:10.1016/0022-247X(82)90032-4 5 Kaniuth, E, Lau, AT, Pym, J: On character amenability of Banach algebras J Math Anal Appl 34, 942–955 (2008) 6 Kada, O, Lau, AT, Takahashi, W: Asymptotically invariant nets and fixed point sets for semigroups of nonexpansive mappings Nonlinear Anal 29,... implies r2 ≤ V Theorem 3.17 Let C be a closed convex subset of a Hilbert space H and T : C ® C ¯ be a nonexpansive mapping with F(T) ≠ ∅, Suppose z, Πn, V, and xnbe defined as in x Proposition 3.16 If the sequence {¯ n }satisfies Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 ⎛ ⎞ lim ⎝ n→∞ Page 13 of 14 ¯ βnk... doi:10.1016/j.jmaa.2008.12.015 12 Kakavandi, BA, Amini, M: Non-linear ergodic theorem in complete non-positive curvature metric spaces Bull Iran Math Soc (in press) Anakkamatee and Dhompongsa Fixed Point Theory and Applications 2011, 2011:34 http://www.fixedpointtheoryandapplications.com/content/2011/1/34 13 Berg, ID, Nikolaev, IG: Quasilinerization and curvature of Alexandrov spaces Geom Dedicata 133, 195–218... CAT(0) spaces Fixed Point Theory and Applications 2011 2011:34 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Page 14 of 14 ... Proposition 3.13 □ Corollary 3.15 [[15], Theorem 3.5.3] Let C be a closed convex subset of a Hilbert space H Suppose S = (ℝ+ ∪ {0}, +) and S (S)be a continuous nonexpansive semi-group on C with F(S ) = ∅ Then, for any x Î C, ∞ r e−rs T(s)xds 0 converges weakly to some z ∈ F(S )as r ↓ 0 1 Proof Let Λ = ℝ+ and put, for l Î Λ and s Î S, g (s) = 1 e− λ s Again, we can then λ λ 1 apply Proposition 3.13 by taking... ϕnl (¯ nl ) x 2 2 4 x Using Lemma 2.2, we see that this contradicts to the minimality of ϕnl (¯ nl ) □ Acknowledgements The authors would like to thank Anthony To-Ming Lau for drawing the problem into our attention and also for giving valuable advice during the preparation of the manuscript We thank the referee for valuable and useful comments We also wish to thank the National Research University... competing interests Received: 1 February 2011 Accepted: 15 August 2011 Published: 15 August 2011 References 1 Lim, TC: Remarks on some fixed point theorems Proc Am Math Soc 60, 179–182 (1976) doi:10.1090/S0002-9939-19760423139-X 2 Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces Nonlinear Anal 68, 3689–3696 (2008) doi:10.1016/ j.na.2007.04.011 3 Day, MM: Amenable semigroups Illinois . Open Access Rodé’s theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0) spaces Watcharapong Anakkamatee 1 and Sompong Dhompongsa 1,2* * Correspondence: sompongd@chiangmai.ac.th 1 Department. article as: Anakkamatee and Dhompongsa: Rodé’s theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0) spaces. Fixed Point Theory and Applications 2011 2011:34. Submit your. theorem on common fixed points of semigroups of nonexpansive mappings in Hilbert spaces to the CAT(0) space setting. 2000 Mathematics Subject Classification: 47H09; 47H10. Keywords: CAT(0) space, semigroup