Rodé's theorem on common fixed points of semigroup of nonexpansive mappings in CAT(0) spaces

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.

In 1976, Lim [1] introduced a concept of convergence in a general metric space, called strong Δ-convergence. In [2], Kirk and Panyanak introduced a concept of convergence 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 convergence can be extended to the CAT(0) space setting by using the Δ-convergence.

For each semigroup S, let B(S) be the Banach space of all bounded real-valued mappings on S with supremum norm. A continuous linear functional μ ∈ B(S)* (the dual space of B(S)) is called a mean on B(S) if || μ || = μ(1). For any f ∈ B(S), we use the following notation:

A mean μ on B(S) is said to be left invariant [respectively, right invariant] if μ_s (f(ts)) = μ_s (f(s)) [respectively, μ_s (f(st)) = μ_s (f(s))] for all f ∈ B(S) and for all t ∈ S. We will say that μ is an invariant mean if it is both left and right invariants. If B(S) has an invariant mean, we call S an amenable semigroup. It is well known that every commutative semigroup is amenable [3]. For each s ∈ S and f ∈ B(S), we define elements l_s f and r_s f in B(S) by (l_sf)(t) = f (st) and (r_sf)(t) = f (ts) for any t ∈ S, respectively. A net {μ_α} of means on B(S) is said to be asymptotically invariant if

In [4], Rodé proved the following:

Theorem 1.1. [4]If S is an amenable semigroup, C is a closed convex subset of a Hilbert space is a nonexpansive semigroup on C such that the set of common fixed points of is nonempty and {μ_α} is an asymptotically invariant net of means, then for each x ∈ C,converges weakly to an element of .

Further, for each x ∈ C, the limit point of is the same for all asymptotically invariant nets of means {μ_α}.

It is remarked that if S is amenable, then there is always an asymptotically strong invariant net of finite means, i.e., means that are convex combination of point evaluations. 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 article 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.

Let (X, d) be a metric space. A geodesic joining x ∈ X to y ∈ 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'∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image γ of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted [x, y]. Write c(α 0 + (1 - α)l) = αx ⊕(1 - α)y, and for , we write as , 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 connected 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: If x₀, x₁ ∈ X and is the midpoint of x₀ and x₁, then

(iii) X satisfies the Law of cosine: If a = 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 c² ≥ a²+b² -2ab cos ξ.

For any subset C of X, let π = π_D be a nearest point projection mapping 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

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) ∈ [π_I(x), π_I(y)] for any x, y in X and m ∈ [x, y].

As noted in [11], we do not know of any example of a CAT(κ) space which does not enjoy the property (N).

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 itself. Suppose {T_sx : s ∈ S} is bounded for all x ∈ C. Let x ∈ C and μ be a mean on B(S). By Riesz's representation theorem, there exists a unique x₀ ∈ C such that

for all y ∈ 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. Let C be a closed convex subset of a CAT(0) space X and μ be a mean on B(S). For a bounded function h : S → C, define

for all y ∈ X. Then, φ_μ attains its unique minimum at a point of .

For each x ∈ C, denote . If is bounded, then by Lemma 2.2 we put

and for h(s) of the form T_sx, we write T_μ(h) shortly as T_μx.

Remark 2.3. If X is a Hilbert space, then

(i) T_μx = x₀ where x₀ verifies (2), and

(ii) ||x₀||² = sup_y∈X (2〈x₀, y〉 - ||y||²).

Proof. (i): Let x₀ be such that μ_s〈T_sx, y〉 = 〈x₀, y〉 for all y ∈ X. We have φ_μ(x₀) = φ_μ(0) + ||x₀||² - 2〈x₀, x₀〉 = φ_μ(0) - ||x₀||² ≤ φ_μ(0) + ||T_μx||² - 2〈x₀, T_μx〉 = φ_μ (T_μx). Therefore, x₀ = T_μx.

(ii): For any x, y ∈ X, we know that ||T_sx - y||² = ||T_sx||² - 2〈T_sx, y〉 + ||y||². By the linearity of μ and (2), we have μ_s(||T_sx - y||²) = μ_s(||T_sx||²) - 2〈x₀, y〉 + ||y||². Thus, inf_y∈X μ_s(||T_sx - y||²) = μ_s(||T_sx||²) - sup_y∈X (2〈x₀, y〉 - ||y||²). On the other hand, by (i), inf_y∈X μ_s(||T_sx - y||²) = μ_s(||T_sx - x₀||²) = μ_s(||T_sx||²) - 2μ_s〈T_sx, x₀〉 + ||x₀||² = μ_s(||T_sx||²) - ||x₀||². Hence, ||x₀||² = sup_y∈X (2〈x₀, y〉 - ||y||²). ■

Let C be a closed convex subset of a CAT(0) space X and S a semigroup. We say that the set is a nonexpansive semigroup on C if

(i) T_s : C → C is a nonexpansive mapping, i.e., d(T_sx, T_sy) ≤ d(x, y) for all x, y ∈ X, for all s ∈ S,

(ii) the mapping s → T_sx is bounded for all x ∈ C, and

(iii) T_ts = T_tT_s, for all s, t ∈ S.

We denote by the set of all common fixed points of mappings in , i.e., , where F (T_s) := {x ∈ C : T_sx = x} is the set of fixed points of T_s.

For any bounded net {x_α} in a closed convex subset C of a CAT(0) space X, put

for each x ∈ C. The asymptotic radius of {x_α} on C is given by

and the asymptotic center of {x_α} in C is the set

It is known that in a complete CAT(0) space, A(C, {x_α}) consists of exactly one point and A(X, {x_α}) = A(C, {x_α}) (cf. [2]).

Remark 2.4. (i) Let D, E be directions and ν : E → D. If {x_ν(β) : β ∈ E} is a subnet of a bounded net {x_α : α ∈ D}, then r(C, {x_ν(β)}) ≤ r(C, {x_α}).

(ii) Let C be a closed convex subset of a CAT(0) space X, T : C → C a nonexpansive mapping and x ∈ C. If {Tⁿx} is bounded and if z ∈ A(C, {Tⁿx}), then z ∈ F(T).

Proof. (i) Let α₀ ∈ D. By the definition of subnets, there exists β₀ ∈ E such that ν(β) ≽ α₀ for all β ≽ β₀. For each x ∈ C, we have . Thus, , and this holds for all α₀. Hence, , and this holds for all x ∈ C. Consequently, r(C, {x_α}) = inf_x∈C r(x, {x_α}) ≥ inf_x∈C r(x, x_ν(β)) = r(C, {x_ν(β)}).

(ii) Since T is nonexpansive, lim sup_n d²(Tⁿx, Tz) = lim sup_n d²(TTⁿx, Tz) ≤ lim sup_n d²(Tⁿx, z).

As every asymptotic center is unique, we have z = Tz. □

Definition 2.5. [[2], Definition 3.3] A net {x_α} in X is said to Δ-converge to x ∈ X if x is the unique asymptotic center of {u_β} for every subnet {u_β} of {x_α}. In this case, we write Δ - lim_α x_α = x and call x the Δ-limit of {x_α}.

Proposition 2.6. [[2], Proposition 3.4] Every bounded net in X has a Δ-convergent subnet.

Remark 2.7. (i) Let D be a direction, {x_α : α ∈ D} a net in X and x ∈ X. If lim sup_α d(x, x_α) > ρ for some ρ > 0, then there exists a subnet of {x_α} such that for all α.

(ii) Let {x_α} be a net in X. Then, {x_α} Δ-converges to x ∈ X if and only if every subnet {x_α'} of {x_α} has a subnet {x_α"} which Δ-converges to x.

Proof. (i): For each α ∈ D, we have sup_α'≽α d(x, x_α') > ρ. Thus there exists β_α ≻ α such that , and this holds for all α. Set a set E = {β_α : α ∈ D}. Clearly, E is a direction, and define ν : E → D by ν (β_α) = β_α. Let α₀ ∈ D, thus ν(β_α) ≽ α₀ for all and this shows that is a subnet of {x_α} satisfying for all α.

ii): It is easy to see that if {x_α} Δ-converges to x, then every subnet of {x_α} also Δ-converges to x. On the other hand, suppose {x_α} does not Δ-converge to x. Thus, there exists a subnet {x_β} of {x_α} such that x ∉ A (C, {x_β}), and so lim sup_β d(x, x_β) > ρ > r(C, {x_β}) for some ρ > 0. By (i), there exists a subnet of {x_β} satisfying for all β. By assumption, there exists a subnet of Δ-converging to x. Using Remark 2.4, , a contradiction. □

In [13], Berg and Nikolaev introduced a concept of quasilinearization. Let us formally denote a pair (a, b) ∈ X × X by and call it a vector. Then, quasilinearization is defined as a map 〈, 〉 : (X × X) × (X × X) by

for all a, b, c, d ∈ X. Recently, Kakavandi and Amini [14] introduced a concept of w- convergence: a sequence {x_n} is said to w-converge to x ∈ X if for all a, b ∈ 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, x₀ = e₁ = (1, 0, 0, 0,...), and for each n, let x_n = x₀ + e_n+1. An ℝ-tree is formed by the segments [x₁, x_n] for n ≥ 0. It is easy to see that {x_n} Δ-converges to x₁. But {x_n} does not w-converge to x₁ since for all n ≥ 2.

Thus, a bounded sequence does not necessary contain an w-convergent subsequence.

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 is a nonexpansive semigroup on C, then the following conditions are equivalent.

(i) is bounded for some x ∈ C;

(ii) is bounded for all x ∈ C;

(iii) .

Proposition 3.3. [[12], Theorem 3.3] Let C be a closed convex subset of a complete CAT(0) space X, S an amenable semigroup, and a nonexpansive semigroup on C with . Then, 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 ∈ 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,

for each f ∈ B(S). Similarly, let μ be a left invariant mean on B(S). Then,

for each f ∈ B(S).

Remark 3.5. If lim_s f (s) = a for some a ∈ ℝ and {s'} is a subnet of {s} satisfying s' ≻ s for each s, then

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 a nonexpansive semigroup on C with . Then, for each x ∈ C,, the net {πT_sx}_s∈S converges to a point Px in , where is the nearest point projection.

Proposition 3.7. Let C be a closed convex subset of a complete CAT(0) space X, S a commutative semigroup, and a nonexpansive semigroup on C with . Then, for any invariant mean μ on B(S), T_μx = lim_s πT_sx = Px for all x ∈ C.

Proof. Fix x ∈ C and let ε > 0. From Proposition 3.6, we see that there exists s₀ ∈ S such that d(πT_sx, Px) < ε for all s ≽ s₀. We know by Proposition 3.3 that . So, d(Px, T_sx) ≤ d(Px, πT_sx) + d(πT_sx, T_sx) < d(πT_sx, T_sx) + ε ≤ d(T_μx, T_sx) + ε for all s ≽ s₀. Since {T_sx : s ∈ S} is bounded by Lemma 3.2, there exists M > 0 such that d(T_μx, T_sx) < M for all s ∈ S. Therefore, d²(Px, T_sx) ≤ d²(T_μx, T_sx)+2Mε + ε² for each s ≽ s₀. Since μ is an invariant mean, we have for any ε > 0. By the argminimality of T_μx (see Lemma 2.2), T_μx = 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_α} to those that satisfy an additional condition: for each t ∈ S,

In the Hilbert space setting, condition (3) is not required because the weak convergence can obtain from (2) directly.

Lemma 3.8. Let X be a complete CAT(0) space that has property (N), C be a closed convex subset of X, S a commutative semigroup, and a nonexpansive semigroup on C with . Suppose {μ_α} is an asymptotically invariant nets of means on B(S) satisfying condition (3). If Δ-converges to x₀, then .

Proof. First, we show that, for each r ∈ S,

If this is not the case, there must be some δ > 0 so that for each α, there exists α' ≻ α satisfying . Put . Since the asymptotically invariant net {μ_α} satisfies (3), there exists α₀ for which for each α ≽ α₀, . Set . By the (CN) inequality, the following in equalities hold for each s ∈ S:

Therefore,

which is a contradiction and thus (4) holds.

Next, we show that . We suppose on the contrary that . Thus, for some r ∈ S, T_rx₀ ≠ x₀, i.e., d(x₀, T_rx₀) := γ > 0. Since , it is bounded. We can get an M > 0 so that for all α. We let . From (4), there exists α₀ with the property that for all α ≽ α₀. Now, for each α ≽ α₀, .Thus, . Let . Using the (CN) inequality, we see that

for all α ≽ α₀. Consequently,

contradicting to the fact that {x₀} is the center of . Therefore, T_rx₀ = x₀ for all r ∈ S, and this shows that as desired. □

Theorem 3.9. Let X be a complete CAT(0) space that has Property (N), C be a closed convex subset of X, S a commutative semigroup, and a nonexpansive semigroup on C with . Suppose {μ_α}is an asymptotically invariant net of means on B(S) satisfying condition (3). Then, Δ-converges to for all x ∈ C. Here, Px is defined in Proposition 3.6.

Proof. Let x ∈ C and {μ_α'} be any subnet of {μ_α}. There exists a subnet {μ_α"} of {μ_α'} such that {μ_α"} w*-converges to μ for some invariant mean μ on B(S). By Proposition 3.7, T_μx = Px. Since the net , it is bounded. Then by Proposition 2.6, there exists a subnet of {μ_α"} such that Δ-converges to some x₀ ∈ C. By Lemma 3.8, .

We show x₀ = T_μx by splitting the proof into three steps.

Step 1. If , then .

Suppose , by (1), for each s ∈ S where is the nearest point projection. Thus, . This impossibility shows that .

Step 2. .

If this does not hold, there must be some η > 0 so that for each β, there exists β' ≻ β satisfying . Put . Since the asymptotically invariant net {μ_β} satisfies (3), there exists β₀ such that for each β ≽ β₀. We suppose first that . Set . By (CN) inequality, the following inequalities hold for each s ∈ S:

Therefore,

contradicting to the argminimality of . In case , we can show in the same way that for some w which also leads to a contradiction.

Step 3. x₀ = T_μx.

We suppose on the contrary and let η := d(x₀, T_μx) > 0. Let I = [T_μx, x₀] and π_I : C → I be the nearest point projection onto I. Since {T_sx} is bounded, there exists M > 0 such that d(T_sx, π_I (T_sx)) ≤ M for all s ∈ S. Set and . Suppose there exists s₀ ∈ S such that d(π_I(T_sx), x₀) ≥ 2ρ for all s ≽ s₀. We know, by Step 1, that . Let A := {y ∈ C: d(π_I(y), x₀) > 2ρ}. Using property (N), A is convex and and thus . By Step 2, . Choose β₀, using the nonexpansiveness of π_I, so that for all β ≽ β₀. Thus, for all β ≽ β₀. But then which contradicts to the fact that x₀ is the Δ - limit of . Therefore, there must be a subnet {s'} of S such that s' ≻ s for all s and

for all s'. Hence,

By the property of N₀, 5η²N₀ > 4ηM + 4η² and so

From (5), (6), and (7),

Using (1),

for all s'. Since the points x₀ and T_μx belong to the set , the nets {d²(T_sx, x₀)} and {d²(T_sx, T_μx)} are decreasing. So, lim_s d²(T_sx, x₀) and lim_s d²(T_sx, T_μx) exist. Hence, φ_μ(T_μx) = lim_s d²(T_sx, T_μx) = lim_s' d²(T_s'x, T_μx) = μ_s' (d²(T_s'x, T_μx)) ≥ μ_s'(d²(T_s'x, x₀)) = lim_s' d²(T_s'x, x₀) = lim_s d²(T_sx, x₀) = φ_μ(x₀), a contradiction. Thus, x₀ = T_μx.

The above argument shows that, for every subnet {μ_α'} of {μ_α}, there exists a subnet of {μ_α'} such that Δ-converges to T_μx(= Px). By Remark 2.7 (ii), 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 → C be a nonexpansive mapping with F(T) ≠ ∅. Let S = (ℕ ∪ {0}, +), , Λ = ℕ or ℝ⁺ and β_λk ≥ 0 be such that for all λ ∈ Λ. Suppose for all k ∈ S,

and for each m ∈ S,

For any f = (a₀, a₁,...) ∈ B(S) let . Then for each x ∈ C, Δ-converges to z for some z in F(T).

In particular, if X is a Hilbert space, we have converges weakly to z for some z in F(T) as λ → ∞.

Proof. For each m ∈ S, . By (8) and (9), we have , and this shows that the net {μ_λ} is asymptotically invariant. Let x ∈ C and consider a_k of the form a_k = d²(T^kx, y) where y ∈ C. We see that {μ_λ} satisfies (3). By Theorem 3.9, we have Δ - converges to z for some z in F(T).

In Hilbert spaces, by a well-known result in probability theory, we know that

for all y ∈ C. So we have . □

Corollary 3.11 (Bailon Ergodic Theorem). Let C be a closed convex subset of a Hilbert space H and T : C → C be a nonexpansive mapping with F(T) ≠ ∅. Then, for any x ∈ C,

converges weakly to z for some z in F(T) as n → ∞.

Proof. Let Λ = ℕ and put, for λ ∈ Λ and k ∈ S = (ℕ ∪ {0}, +),

The result now follows from Proposition 3.10. □

Corollary 3.12. [[15], Theorem 3.5.1] Let C be a closed convex subset of a Hilbert space H and T : C → C be a nonexpansive mapping with F(T) ≠ ∅. Then, for any x ∈ C, converges weakly to z for some z in F(T) as r ↑ 1.

Proof. Let Λ = ℝ⁺ and put, for λ ∈ Λ and k ∈ S = (ℕ ∪{0}, +),

Taking , 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 is said to be a continuous nonexpansive semigroup on C if satisfies the following:

(i) T(s) : C → C is a nonexpansive mapping for all s ∈ S,

(ii) T(t + s)x = T(t)T(s)x for all x ∈ C and t, s ∈ S,

(iii) for each x ∈ C, the mapping s → T(s)x is continuous, and

(iv) T(0)x = x for all x ∈ C.

Proposition 3.13. Let C be a closed convex subset of a Hilbert space H. Let S = (ℝ⁺ ∪ {0}, +), be a continuous nonexpansive semigroup on C with , Λ = ℝ⁺ and g_λ be a density function on S,i.e., g_λ ≥ 0 and for all λ ∈ Λ. Suppose g_λ satisfies the following properties. for each h ∈ S,

uniformly on [0, h] and

Then, for any x ∈ C,

converges weakly to some as λ → ∞.

Proof. For f ∈ B(S) we define for all λ > 0. Thus, μ_λ is a mean on B(S). For any h ∈ S we consider

By (10) and (11), . So, {μ_λ} is asymptotically invariant. For each z ∈ C, let f(s) = ||z - T(s)x||². We see that {μ_λ} satisfies (3). For each x ∈ C, we know that

for all y ∈ C. Thus, . By Theorem 3.9, we have converges weakly to some as λ → ∞. □

Corollary 3.14. [[15], Theorem 3.5.2] Let C be a closed convex subset of a Hilbert space H. Suppose S = (ℝ⁺ ∪ {0}, +) and be a continuous nonexpansive semigroup on C with . Then, for any x ∈ C,

converges weakly to some as λ → ∞.

Proof. Let Λ = ℝ⁺ and put, for λ ∈ Λ and s ∈ S, . The result now follows from 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 be a continuous nonexpansive semi-group on C with . Then, for any x ∈ C,

converges weakly to some as r ↓ 0.

Proof. Let Λ = ℝ⁺ and put, for λ ∈ Λ and s ∈ S, . Again, we can then apply Proposition 3.13 by taking . □

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 ∈ C and let r = r(C, {Tⁿx}). Let z be the unique asymptotic center of {Tⁿx}. For each n ∈ ℕ, define

and

where . If V := lim_n→∞ V_n, then V = r².

Proof. Given ε > 0. Since z ∈ A(C, {Tⁿx}), by Remark 2.4 (ii), z ∈ F(T). Choose n_ε ∈ ℕ such that ||Tⁿx - z|| < r + ε for all n ≥ n_ε. Fix n ≥ n_ε and let p = {β_nk}_k≥n ∈ Π_n. Thus,

So . Letting n → ∞, V = lim_{n → ∞} V_n ≤ (r + ε)² for any ε > 0. Hence, V ≤ r².

Next, we show that r² ≤ V. Indeed, since for all n ∈ ℕ, there exists a sequence with for each n and as n → ∞. Put . Since {Tⁿx} is bounded, there exists M > 0 such that . For each ε > 0, choose n_ε ∈ ℕ such that , V_n < V + ε for all n ≥ n_ε, and ||T^kx - z|| > r - ε for all k ≥ n_ε. Thus for any n ≥ n_ε,

Hence,

So (r - ε)² < V + ε + εM for any ε > 0. This 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, Π_n, V, and be defined as in Proposition 3.16. If the sequence satisfies

then converges (strongly) to z ∈ F (T ) as n → ∞.

Proof. Suppose for some ε > 0, there exists a subsequence of such that for all l ∈ ℕ. For each y ∈ C and n ∈ ℕ, define . Let . By (12) and z ∈ A(C, {T^kx}), we choose n_δ such that for all n_l ≥ n_δ and ||T^kx - z||² < r² + δ for all k ≥ n_δ. Fix l ≥ n_δ and let . By the Parallelogram law, we have for each k ≥ n_l,

Hence,

Using Lemma 2.2, we see that this contradicts to the minimality of . □

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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 Project under Thailand's Office of the Higher Education Commission for financial support.

1. Lim, TC: Remarks on some fixed point theorems. Proc Am Math Soc. 60, 179–182 (1976). Publisher Full Text

2. Kirk, WA, Panyanak, B: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008). Publisher Full Text

3. Day, MM: Amenable semigroups. Illinois J Math. 1, 509–544 (1957)

4. Rodé, G: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space. J Math Anal Appl. 85, 172–178 (1982). Publisher Full Text

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, 539–550 (1997). Publisher Full Text

7. Lau, AT: Semigroup of nonexpansive mappings on a Hilbert space. J Math Anal Appl. 105, 514–522 (1985). Publisher Full Text

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, Berlin (1999)

10. Sturm, KT: Probability measures on metric spaces of non-positive curvature. Heat Kernels and Analysis on Manifolds, Graphs and Metric Spaces (Paris, 2002), Contemp Math 338, Am Math Soc Providence, RI. 357–390 (2003)

11. Espínola, R, Fernández-León, A: CAT(κ) spaces, weak convergence and fixed points. J Math Anal Appl. 353, 410–427 (2009). Publisher Full Text

12. Kakavandi, BA, Amini, M: Non-linear ergodic theorem in complete non-positive curvature metric spaces. In: Bull Iran Math Soc

13. Berg, ID, Nikolaev, IG: Quasilinerization and curvature of Alexandrov spaces. Geom Dedicata. 133, 195–218 (2008). Publisher Full Text

14. Kakavandi, BA, Amini, M: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010). Publisher Full Text

15. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publisher, Yokohama (2000)