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

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.

1 Introduction

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:

$$\mu \left(f\right)={\mu }_s\left(f\left(s\right)\right).$$

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 _s f)(t) = f (st) and (r _s f)(t) = f (ts) for any t ∈ S, respectively. A net {μ _α } of means on B(S) is said to be asymptotically invariant if

$$\underset{\alpha }{\lim }\left({\mu }_{\alpha }\left(l_{s}f\right)-{\mu }_{\alpha }\left(f\right)\right)=0=\underset{\alpha }{\lim }\left({\mu }_{\alpha }\left(r_{s}f\right)-{\mu }_{\alpha }\left(f\right)\right).$$

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 $H,\mathcal{S}=\left\{T_s:s\in S\right\}$ is a nonexpansive semigroup on C such that the set$F\left(\mathcal{S}\right)$of common fixed points of$\mathcal{S}$is nonempty and {μ _α } is an asymptotically invariant net of means, then for each x ∈ C,$\left\{T_{{\mu }_{\alpha }}x\right\}$converges weakly to an element of$F\left(\mathcal{S}\right)$.

Further, for each x ∈ C, the limit point of$\left\{T_{{\mu }_{\alpha }}x\right\}$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.

2 Preliminaries

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 $\alpha =\frac{1}{2}$, we write $\frac{1}{2}x\oplus \frac{1}{2}y$ as $\frac{x\oplus y}{2}$, 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 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 x ₀, x ₁ ∈ X and $\frac{x_0\oplus x_1}{2}$ is the midpoint of x ₀ and x ₁, then

  $$d^2\left(y,\frac{x_0\oplus x_1}{2}\right)\le \frac{1}{2}{d}^2\left(y,x_0\right)+\frac{1}{2}{d}^2\left(y,x_1\right)-\frac{1}{4}{d}^2\left(x_0,x_1\right),forally\in X.$$

• (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 c ² ≥ a ²+b ² -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

$$d^2\left(x,y\right)\ge d^2\left(x,\pi x\right)+d^2\left(\pi x,y\right)forallx\in Candy\in D.$$

(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) ∈ [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 _s x : 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

$${\mu }_s⟨T_{s}x,y⟩=⟨x_0,y⟩$$

(2)

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

$${\phi }_{\mu }\left(y\right):={\mu }_s\left(d^2\left(h\left(s\right),y\right)\right)$$

for all y ∈ X. Then, φ _μ attains its unique minimum at a point of$\overline{co}\left\{h\left(s\right):s\in S\right\}$.

For each x ∈ C, denote $\mathcal{S}\left(x\right):=\left\{T_{s}x:s\in S\right\}$. If $\mathcal{S}\left(x\right)$ is bounded, then by Lemma 2.2 we put

$$T_{\mu }\left(h\right):=argmin\left\{y\mapsto {\mu }_s\left(d^2\left(h\left(s\right),y\right)\right)\right\},$$

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 = x₀where x₀verifies (2), and

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

Proof. (i): Let x₀ be such that μ _s 〈T _s x, 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 _μ x||² - 2〈x₀, T _μ x〉 = φ _μ (T _μ x). Therefore, x₀ = T _μ x.

(ii): For any x, y ∈ X, we know that ||T _s x - y||² = ||T _s x||² - 2〈T _s x, y〉 + ||y||². By the linearity of μ and (2), we have μ _s (||T _s x - y||²) = μ _s (||T _s x||²) - 2〈x₀, y〉 + ||y||². Thus, inf_y∈Xμ _s (||T _s x - y||²) = μ _s (||T _s x||²) - sup_y∈X(2〈x₀, y〉 - ||y||²). On the other hand, by (i), inf_y∈Xμ _s (||T _s x - y||²) = μ _s (||T _s x - x₀||²) = μ _s (||T _s x||²) - 2μ _s 〈T _s x, x₀〉 + ||x₀||² = μ _s (||T _s x||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 $\mathcal{S}\left(S\right):=\left\{T_s:s\in S\right\}$ 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 ∈ X, for all s ∈ S,

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

• (iii) T _ts = T _t T _s , for all s, t ∈ S.

We denote by $F\left(\mathcal{S}\right)$ the set of all common fixed points of mappings in $\mathcal{S}\left(S\right)$, i.e., $F\left(\mathcal{S}\right):={\bigcup }_{s\in s}F\left(T_s\right)$, where F (T _s ) := {x ∈ C : T _s x = 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

$$r\left(x,\left\{x_{\alpha }\right\}\right)=\underset{\alpha }{\lim \sup }d\left(x,x_{\alpha }\right)$$

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

$$r\left(C,\left\{x_{\alpha }\right\}\right)=\underset{x\in C}{\inf }r\left(x,\left\{x_{\alpha }\right\}\right),$$

and the asymptotic center of {x _α } in C is the set

$$A\left(C,\left\{x_{\alpha }\right\}\right)=\left\{x\in C:r\left(x,\left\{x_{\alpha }\right\}\right)=r\left(C,\left\{x_{\alpha }\right\}\right)\right\}.$$

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 ${\sup }_{\alpha \succcurlyeq {\alpha }_0}d\left(x,x_{\alpha }\right)\ge {\sup }_{\beta \succcurlyeq {\beta }_0}d\left(x,x_{\nu \left(\beta \right)}\right)$. Thus, ${\sup }_{\alpha \succcurlyeq {\alpha }_0}d\left(x,x_{\alpha }\right)\ge {\inf }_{{\beta }_1}{\sup }_{\beta \succcurlyeq {\beta }_1}d\left(x,x_{\nu \left(\beta \right)}\right)$, and this holds for all α₀. Hence, $r\left(x,\left\{x_{\alpha }\right\}\right)={\inf }_{{\alpha }_0}{\sup }_{\alpha \succcurlyeq {\alpha }_0}d\left(x,x_{\alpha }\right)\ge r\left(x,x_{\nu \left(\beta \right)}\right)$, and this holds for all x ∈ C. Consequently, r(C, {x _α }) = inf_x∈Cr(x, {x _α }) ≥ inf_x∈Cr(x, x_ν(β)) = r(C, {x_ν(β)}).

• (ii) Since T is nonexpansive, lim sup _n d ²(Tⁿx, Tz) = lim sup _n d ²(TT ⁿ x, Tz) ≤ lim sup _n d ²(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$\left\{x_{{\beta }_{\alpha }}\right\}$of {x _α } such that$d\left(x,x_{{\beta }_{\alpha }}\right)\ge \rho$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'≽αd(x, x_α') > ρ. Thus there exists β _α ≻ α such that $d\left(x,x_{{\beta }_{\alpha }}\right)\ge \rho$, 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 ${\beta }_{\alpha }\succcurlyeq {\beta }_{{\alpha }_0}$ and this shows that $\left\{x_{{\beta }_{\alpha }}\right\}$ is a subnet of {x _α } satisfying $d\left(x,x_{{\beta }_{\alpha }}\right)\ge \rho$ 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 $\left\{x_{{\gamma }_{\beta }}\right\}$ of {x _β } satisfying $d\left(x,x_{{\gamma }_{\beta }}\right)\ge \rho$ for all β. By assumption, there exists a subnet $\left\{x_{{\left({\gamma }_{\beta }\right)}_{\eta }}\right\}$ of $\left\{x_{{\gamma }_{\beta }}\right\}$ Δ-converging to x. Using Remark 2.4, $\rho \le {\lim \sup }_{\gamma }d\left(x,x_{{\left({\gamma }_{\beta }\right)}_{\eta }}\right)=r\left(C,\left\{x_{{\left({\gamma }_{\beta }\right)}_{\eta }}\right\}\right)\le r\left(C,\left\{x_{{\gamma }_{\beta }}\right\}\right)\le r\left(C,\left\{x_{\beta }\right\}\right)$, a contradiction. □

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

$$⟨\overrightarrow{ab},\overrightarrow{cd}⟩=\frac{1}{2}{d}^2\left(a,d\right)+\frac{1}{2}{d}^2\left(b,c\right)-\frac{1}{2}{d}^2\left(a,c\right)-\frac{1}{2}{d}^2\left(b,d\right)$$

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 ${\text{lim}}_{n\to \infty }〈\overrightarrow{x_{n}x},\overrightarrow{ab}〉=0$ 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 x₁since$⟨\overrightarrow{x_n{x}_1},\overrightarrow{x_0{x}_1}⟩=-1$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$\mathcal{S}\left(S\right)$is a nonexpansive semigroup on C, then the following conditions are equivalent.

(i)$\mathcal{S}\left(x\right)$is bounded for some x ∈ C;

(ii)$\mathcal{S}\left(x\right)$is bounded for all x ∈ C;

(iii)$F\left(\mathcal{S}\right)\ne \varnothing$.

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$\mathcal{S}\left(S\right)$a nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Then, $T_{\mu }x\in F\left(\mathcal{S}\right)$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,

$$\underset{s}{\sup }\underset{t}{\inf }f\left(ts\right)\le \mu \left(f\left(s\right)\right)\le \underset{s}{\inf }\underset{t}{\sup }f\left(ts\right)$$

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

$$\underset{s}{\sup }\underset{t}{\inf }f\left(st\right)\le \mu \left(f\left(s\right)\right)\le \underset{s}{\inf }\underset{t}{\sup }f\left(st\right)$$

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

$${\mu }_{s^{\prime }}\left(f\left(s^{\prime }\right)\right)=a.$$

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$\mathcal{S}\left(S\right)$a nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Then, for each x ∈ C,, the net {πT _s x}_s∈Sconverges to a point Px in$F\left(\mathcal{S}\right)$, where$\pi ={\pi }_{F\left(\mathcal{S}\right)}:C\to F\left(\mathcal{S}\right)$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$\mathcal{S}\left(S\right)$a nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Then, for any invariant mean μ on B(S), T _μ x = lim _s πT _s x = 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 _s x, Px) < ε for all s ≽ s₀. We know by Proposition 3.3 that $T_{\mu }x\in F\left(\mathcal{S}\right)$. So, d(Px, T _s x) ≤ d(Px, πT _s x) + d(πT _s x, T _s x) < d(πT _s x, T _s x) + ε ≤ d(T _μ x, T _s x) + ε for all s ≽ s₀. Since {T _s x : s ∈ S} is bounded by Lemma 3.2, there exists M > 0 such that d(T _μ x, T _s x) < M for all s ∈ S. Therefore, d²(Px, T _s x) ≤ d²(T _μ x, T _s x)+2Mε + ε² for each s ≽ s₀. Since μ is an invariant mean, we have ${\mu }_s\left(d^2\left(Px,T_{s}x\right)\right)={\mu }_s\left(d^2\left(Px,T_{s{s}_0}x\right)\right)\le {\mu }_s\left(d^2\left(T_{\mu }x,T_{s{s}_0}x\right)\right)+2M\epsilon +{\epsilon }^2={\mu }_s\left(d^2\left(T_{\mu }x,T_{s}x\right)\right)+2M\epsilon +{\epsilon }^2$ 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 _α } to those that satisfy an additional condition: for each t ∈ S,

$${\mu }_{{\alpha }_s}\left(d^2\left(T_{s}x,y\right)\right)-{\mu }_{{\alpha }_s}\left(d^2\left(T_{st}x,y\right)\right)\to 0uniformlyfory\in C.$$

(3)

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$\mathcal{S}\left(S\right)$a nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Suppose {μ _α } is an asymptotically invariant nets of means on B(S) satisfying condition (3). If$\left\{T_{{\mu }_{\alpha }}x\right\}$ Δ-converges to x₀, then$x_0\in F\left(\mathcal{S}\right)$.

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

$$\underset{\alpha }{\lim }d\left(T_{{\mu }_{\alpha }}x,T_r{T}_{{\mu }_{\alpha }}x\right)=0.$$

(4)

If this is not the case, there must be some δ > 0 so that for each α, there exists α' ≻ α satisfying $d\left(T_{{\mu }_{{\alpha }^{\prime }}}x,T_r{T}_{{\mu }_{{\alpha }^{\prime }}}x\right)\ge \delta$. Put $\epsilon =\frac{{\delta }^2}{2}$. Since the asymptotically invariant net {μ _α } satisfies (3), there exists α₀ for which for each α ≽ α₀, ${\phi }_{{\mu }_{\alpha }}\left(T_r{T_{\mu }}_{{}_{\alpha }}x\right)={\mu }_{{\alpha }_s}\left(d^2\left(T_{s}x,T_r{T}_{{\mu }_{\alpha }}x\right)\right)<{\mu }_{{\alpha }_s}\left(d^2\left(T_r{T}_{s}x,T_r{T}_{{\mu }_{\alpha }}x\right)\right)+\epsilon \le {\mu }_{{\alpha }_s}\left(d^2\left(T_{s}x,T_{{\mu }_{\alpha }}x\right)\right)+\epsilon ={\phi }_{{\mu }_{\alpha }}\left(T_{{\mu }_{\alpha }}x\right)+\epsilon$. Set $w=\frac{T_{{\mu }_{{\alpha }_0^{\prime }}}x\oplus T_r{T}_{{\mu }_{{\alpha }_0^{\prime }}}x}{2}$. By the (CN) inequality, the following in equalities hold for each s ∈ S:

$$\begin{array}{lll}\hfill d^2\left(T_{s}x,w\right)& \le \frac{1}{2}{d}^2\left(T_{s}x,T_{{\mu }_{{\alpha }_0^{\prime }}}x\right)+\frac{1}{2}{d}^2\left(T_{s}x,T_r{T}_{{\mu }_{{\alpha }_0^{\prime }}}x\right)-\frac{1}{4}{d}^2\left(T_{{\mu }_{{\alpha }_0^{\prime }}}x,T_r{T}_{{\mu }_{{\alpha }_0^{\prime }}}x\right)& \hfill \text{(1) }\\ \le \frac{1}{2}{d}^2\left(T_{s}x,T_{{\mu }_{{\alpha }_0^{\prime }}}x\right)+\frac{1}{2}{d}^2\left(T_{s}x,T_r{T}_{{\mu }_{{\alpha }_0^{\prime }}}x\right)-\frac{{\delta }^2}{4}.& \hfill \text{(2) }\\ \hfill \text{(3) }\end{array}$$

Therefore,

$$\begin{array}{ll}{\phi }_{{\mu }_{{{\alpha }^{\prime }}_0}}(w)\hfill & \le \frac{1}{2}{\phi }_{{\mu }_{{{\alpha }^{\prime }}_0}}(T_{{\mu }_{{{\alpha }^{\prime }}_0}}x)+\frac{1}{2}{\phi }_{{\mu }_{{{\alpha }^{\prime }}_0}}(T_r{T}_{{\mu }_{{{\alpha }^{\prime }}_0}}x)-\frac{{\delta }^2}{4}\hfill \\ <{\phi }_{{\mu }_{{{\alpha }^{\prime }}_0}}(T_{{\mu }_{{{\alpha }^{\prime }}_0}}x)+\frac{\epsilon }{2}-\frac{{\delta }^2}{4}\hfill \\ ={\phi }_{{\mu }_{{{\alpha }^{\prime }}_0}}(T_{{\mu }_{{{\alpha }^{\prime }}_0}}x),\hfill \end{array}$$

which is a contradiction and thus (4) holds.

Next, we show that $x_0\in F\left(\mathcal{S}\right)$. We suppose on the contrary that $x_0\notin F\left(\mathcal{S}\right)$. Thus, for some r ∈ S, T _r x₀ ≠ x₀, i.e., d(x₀, T _r x₀) := γ > 0. Since $\left\{T_{{\mu }_{\alpha }}x\right\}\subset \overline{co}\left\{T_{s}x\right\}$, it is bounded. We can get an M > 0 so that $d\left(T_{{\mu }_{\alpha }}x,x_0\right)\le M$ for all α. We let $0<\epsilon <min\left\{\frac{{\gamma }^2}{16M},2M\right\}$. From (4), there exists α₀ with the property that $d\left(T_r{T}_{{\mu }_{\alpha }}x,T_{{\mu }_{\alpha }}x\right)<\epsilon$ for all α ≽ α₀. Now, for each α ≽ α₀, $d\left(T_{{\mu }_{\alpha }}x,T_r{x}_0\right)\le d\left(T_{{\mu }_{\alpha }}x,T_r{T}_{{\mu }_{\alpha }}x\right)+d\left(T_r{T}_{{\mu }_{\alpha }}x,T_r{x}_0\right)<d\left(T_{{\mu }_{\alpha }}x,x_0\right)+\epsilon$.Thus, $d^2\left(T_{{\mu }_{\alpha }}x,T_r{x}_0\right)<d^2\left(T_{{\mu }_{\alpha }}x,x_0\right)+2\epsilon d\left(T_{{\mu }_{\alpha }}x,x_0\right)+{\epsilon }^2$. Let $w=\frac{x_0\oplus T_r{x}_0}{2}$. Using the (CN) inequality, we see that

$$\begin{array}{ll}{d}^2(T_{{\mu }_{\alpha }}x,w)\hfill & \le \frac{1}{2}{d}^2(T_{{\mu }_{\alpha }}x,x_0)+\frac{1}{2}{d}^2(T_{{\mu }_{\alpha }}x,T_r{x}_0)-\frac{1}{4}{d}^2(x_0,T_r{x}_0)\hfill \\ \le \frac{1}{2}{d}^2(T_{{\mu }_{\alpha }}x,x_0)+\frac{1}{2}(d^2(T_{{\mu }_{\alpha }}x,x_0)+2\epsilon M+{\epsilon }^2)-\frac{{\gamma }^2}{4}\hfill \\ =d^2(T_{{\mu }_{\alpha }}x,x_0)+\epsilon M+\frac{{\epsilon }^2}{2}-\frac{{\gamma }^2}{4}\hfill \end{array}$$

for all α ≽ α₀. Consequently,

$$\begin{array}{l}\underset{\alpha }{\lim \sup }{d}^2(T_{{\mu }_{\alpha }}x,w)\le \underset{\alpha }{\lim \sup }{d}^2(T_{{\mu }_{\alpha }}x,x_0)+\epsilon M+\frac{{\epsilon }^2}{2}-\frac{{\gamma }^2}{4}\\ <\underset{\alpha }{\lim \sup }{d}^2(T_{{\mu }_{\alpha }}x,x_0),\end{array}$$

contradicting to the fact that {x₀} is the center of $\left\{T_{{\mu }_{\alpha }}x\right\}$. Therefore, T _r x₀ = x₀ for all r ∈ S, and this shows that $x_0\in F\left(\mathcal{S}\right)$ 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$\mathcal{S}\left(S\right)$a nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Suppose {μ _α }is an asymptotically invariant net of means on B(S) satisfying condition (3). Then, $\left\{T_{{\mu }_{\alpha }}x\right\}$ Δ-converges to$Px\in F\left(\mathcal{S}\right)$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 _μ x = Px. Since the net $\left\{T_{{\mu }_{{\alpha }^{″}}}x\right\}\subset \overline{co}\left\{T_{s}x:s\in S\right\}$, it is bounded. Then by Proposition 2.6, there exists a subnet $\left\{{\mu }_{{\alpha }_{\beta }}\right\}$ of {μ _α" } such that $\left\{T_{{\mu }_{{\alpha }_{\beta }}}x\right\}$ Δ-converges to some x₀ ∈ C. By Lemma 3.8, $x_0\in F\left(\mathcal{S}\right)$.

We show x₀ = T _μ x by splitting the proof into three steps.

Step 1. If $\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}:=argmin\left\{y\mapsto {\mu }_{{\beta }_s}\left(d^2\left(T_{s{s}_0}x,y\right)\right)\right\}$, then $\overline{T_{{\mu }_{\alpha \beta }}x}\in \overline{co}{\left\{T_{s}x\right\}}_{s\succcurlyeq s_0}$.

Suppose $\overline{T_{{\mu }_{{\alpha }_{{}_{\beta }}}}x}\notin \overline{co}{\left\{T_{s}x\right\}}_{s\succcurlyeq s_0}$, by (1), $d^2\left(T_{s{s}_0}x,\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)\ge d^2\left(T_{s{s}_0}x,\pi \overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)+d^2\left(\overline{T_{{\mu }_{{\alpha }_{\beta }}}x},\pi \overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)$ for each s ∈ S where $\pi :C\to \overline{co}{\left\{T_{s}x\right\}}_{s\succcurlyeq s_0}$ is the nearest point projection. Thus, ${\mu }_{{\alpha }_{\beta }}\left(d^2\left(T_{s{s}_0}x,\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)\right)\ge {\mu }_{{\alpha }_{\beta }}\left(d^2\left(T_{s{s}_0}x,\pi \overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)\right)+d^2\left(\overline{T_{{\mu }_{{\alpha }_{\beta }}}x},\pi \overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)>{\mu }_{{\alpha }_{\beta }}\left(d^2\left(T_{s{s}_0}x,\pi \overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)\right)$. This impossibility shows that $\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\in \overline{co}{\left\{T_{s}x\right\}}_{s\succcurlyeq s_0}$.

Step 2. $li{m}_{\beta }d\left(T_{{\mu }_{{\alpha }_{\beta }}}x,\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)=0$.

If this does not hold, there must be some η > 0 so that for each β, there exists β' ≻ β satisfying $d\left(T_{{\mu }_{{\alpha }_{{\beta }^{\prime }}}}x,\overline{T_{{\mu }_{{\alpha }_{{\beta }^{\prime }}}}x}\right)\ge \eta$. Put $\epsilon =\frac{{\eta }^2}{2}$. Since the asymptotically invariant net {μ _β } satisfies (3), there exists β₀ such that $|{\mu }_{{\alpha }_{{\beta }_s}}\left(d^2\left(T_{s}x,\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)\right)-{\mu }_{{\alpha }_{{\beta }_s}}\left(d^2\left(T_{s{s}_0}x,\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)\right)|<\epsilon$ for each β ≽ β₀. We suppose first that ${\mu }_{{\alpha }_{{\beta }_{0_s}^{\prime }}}\left(d^2\left(T_{s{s}_0}x,\overline{T_{{\mu }_{{\alpha }_{{\beta }_0^{\prime }}}}x}\right)\right)\le {\mu }_{{\alpha }_{{\beta }_{0_s}^{\prime }}}\left(d^2\left(T_{s}x,T_{{\mu }_{{\alpha }_{{\beta }_0^{\prime }}}}x\right)\right)$. Set $w=\frac{T_{{\mu }_{{\alpha }_{{\beta }_0^{\prime }}}}x\oplus \overline{T_{{\mu }_{{\alpha }_{{\beta }_0^{\prime }}}}x}}{2}$. By (CN) inequality, the following inequalities hold for each s ∈ S:

$$\begin{array}{l}{d}^2(T_{s}x,w)\le \frac{1}{2}{d}^2(T_{s}x,T_{{\mu }_{{\alpha }_{{{\beta }^{\prime }}_0}}}x)+\frac{1}{2}{d}^2(T_{s}x,\overline{T_{{\mu }_{{\alpha }_{{{\beta }^{\prime }}_0}}}x})-\frac{1}{4}{d}^2(T_{{\mu }_{{\alpha }_{{{\beta }^{\prime }}_0}}}x,\overline{T_{{\mu }_{{\alpha }_{{{\beta }^{\prime }}_0}}}x})\\ \le \frac{1}{2}{d}^2(T_{s}x,T_{{\mu }_{{\alpha }_{{}_{{{\beta }^{\prime }}_0}}}}x)+\frac{1}{2}{d}^2(T_{s}x,\overline{T_{{\mu }_{{}_{{\alpha }_{{{\beta }^{\prime }}_0}}}}x})-\frac{{\eta }^2}{4}.\end{array}$$

Therefore,

$$\begin{array}{l}{\phi }_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}(w)\le \frac{1}{2}{\phi }_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}(T_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}x)+\frac{1}{2}{\phi }_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}(\overline{T_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}x})-\frac{{\eta }^2}{4}\\ <\frac{1}{2}{\phi }_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}(T_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}x)+\frac{1}{2}{\mu }_{{\alpha }_{{{\beta }^{\prime }}_{0_s}}}(d^2(T_{s{s}_0}x,\overline{T_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}x}))+\frac{\epsilon }{2}-\frac{{\eta }^2}{4}\\ \le {\phi }_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}(T_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}x)+\frac{\epsilon }{2}-\frac{{\eta }^2}{4}={\phi }_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}(T_{{\mu }_{\alpha }{}_{{}_{{{\beta }^{\prime }}_0}}}x),\end{array}$$

contradicting to the argminimality of $T_{{\mu }_{{\alpha }_{{\beta }_0^{\prime }}}}x$. In case ${\mu }_{{\alpha }_{{\beta }_{0_s}^{\prime }}}\left(d^2\left(T_{s}x,T_{{{\mu }_{\alpha }}_{{}_{{\beta }_0^{\prime }}}}x\right)\right)<{\mu }_{{\alpha }_{{\beta }_{0_s}^{\prime }}}\left(d^2\left(T_{s{s}_0}x,\overline{T_{{{\mu }_{\alpha }}_{{}_{{\beta }_0^{\prime }}}}x}\right)\right)$, we can show in the same way that ${\mu }_{{\alpha }_{{\beta }_{0_s}^{\prime }}}\left(d^2\left(T_{s{s}_0}x,w\right)\right)<{\mu }_{{\alpha }_{{\beta }_{0_s}^{\prime }}}\left(d^2\left(T_{s{s}_0}x,\overline{T_{{{\mu }_{\alpha }}_{{}_{{\beta }_0^{\prime }}}}x}\right)\right)$ for some w which also leads to a contradiction.

Step 3. x₀ = T _μ x.

We suppose on the contrary and let η := d(x₀, T _μ x) > 0. Let I = [T _μ x, x₀] and π _I : C → I be the nearest point projection onto I. Since {T _s x} is bounded, there exists M > 0 such that d(T _s x, π _I (T _s x)) ≤ M for all s ∈ S. Set $N_0>\frac{4\left(M+\eta \right)}{5\eta }$ and $\rho =\frac{\eta }{5N_0}$. Suppose there exists s₀ ∈ S such that d(π _I (T _s x), x₀) ≥ 2ρ for all s ≽ s₀. We know, by Step 1, that $\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\in \overline{co}{\left\{T_{s}x\right\}}_{s\succcurlyeq s_0}$. Let A := {y ∈ C: d(π _I (y), x₀) > 2ρ}. Using property (N), A is convex and $\overline{co}{\left\{T_{s}x\right\}}_{s\succcurlyeq s_0}\subset Ā\subset \left\{y\in C:d\left({\pi }_1\left(y\right),x_0\right)\ge 2\rho \right\}$ and thus $d\left({\pi }_I\left(\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right),x_0\right)\ge 2\rho$. By Step 2, $li{m}_{\beta }d\left(T_{{\mu }_{{\alpha }_{\beta }}}x,\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}\right)=0$. Choose β₀, using the nonexpansiveness of π _I , so that $d({\pi }_I(T_{{\mu }_{{\alpha }_{\beta }}}x),{\pi }_I(\overline{T_{{\mu }_{{\alpha }_{\beta }}}x}))<\rho$ for all β ≽ β₀. Thus, $d\left({\pi }_I\left(T_{{\mu }_{{\alpha }_{\beta }}}x\right),x_0\right)>\rho$ for all β ≽ β₀. But then $x_0\notin \overline{co}{\left\{T_{{\mu }_{{\alpha }_{\beta }}}x\right\}}_{\beta \succcurlyeq {\beta }_0}$ which contradicts to the fact that x₀ is the Δ - limit of $\left\{T_{{\mu }_{{\alpha }_{\beta }}}x\right\}$. Therefore, there must be a subnet {s'} of S such that s' ≻ s for all s and

$$d\left({\pi }_I\left(T_{s^{\prime }}x\right),x_0\right)<2\rho =\frac{2\eta }{5N_0}$$

(5)

for all s'. Hence,

$$d\left({\pi }_I\left(T_{s^{\prime }}x\right),T_{\mu }x\right)=\eta -d\left({\pi }_I\left(T_{s^{\prime }}x\right),x_0\right)>\eta -\frac{2\eta }{5N_0}.$$

(6)

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

$${\eta }^2-\frac{4{\eta }^2}{5N_0}>\frac{4\eta M}{5N_0}.$$

(7)

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

$$\begin{array}{l}{d}^2({\pi }_I(T_{s^{\prime }}x),T_{\mu }{x}_0)>{\eta }^2-=\frac{4{\eta }^2}{5N_0}+(\frac{2\eta }{5N_0})>\frac{4\eta M}{5N_0}+{(\frac{2\eta }{5N_0})}^2\\ >2d(x_0,{\pi }_I(T_{s^{\prime }}x))d(T_{s^{\prime }}x{\pi }_I(T_{s^{\prime }}x))+d^2(x_0,{\pi }_I(T_{s^{\prime }}x)).\end{array}$$

Using (1),

$$\begin{array}{c}{d}^2(T_{s^{\prime }}x,T_{\mu }x)\ge d^2({\pi }_I(T_{s^{\prime }}x),T_{\mu }x)+d^2({\pi }_I(T_{s^{\prime }}x),T_{s^{\prime }}x)\\ >d^2(x_0,{\pi }_I(T_{s^{\prime }}x))+2d(x_0,{\pi }_I(T_{s^{\prime }}x))d(T_{s^{\prime }}x,{\pi }_I(T_{s^{\prime }}x))+d^2({\pi }_I(T_{s^{\prime }}x),T_{s^{\prime }}x)\\ ={(d(x_0,{\pi }_I(T_{s^{\prime }}x))+d(T_{s^{\prime }}x,{\pi }_I(T_{s^{\prime }}x)))}^2\\ \ge d^2(T_{s^{\prime }}x,x_0)\end{array}$$

for all s'. Since the points x₀ and T _μ x belong to the set $F\left(\mathcal{S}\right)$, the nets {d²(T _s x, x₀)} and {d²(T _s x, T _μ x)} are decreasing. So, lim _s d²(T _s x, x₀) and lim _s d²(T _s x, T _μ x) exist. Hence, φ _μ (T _μ x) = lim _s d²(T _s x, T _μ x) = lim _s' d²(T _s' x, T _μ x) = μ _s' (d²(T _s' x, T _μ x)) ≥ μ _s' (d²(T _s' x, x₀)) = lim _s' d²(T _s' x, x₀) = lim _s d²(T _s x, x₀) = φ _μ (x₀), a contradiction. Thus, x₀ = T _μ x.

The above argument shows that, for every subnet {μ _α' } of {μ _α }, there exists a subnet $\left\{{\mu }_{{\alpha }_{\beta }}\right\}$ of {μ _α' } such that $\left\{T_{{\mu }_{{\alpha }_{\beta }}}x\right\}$ Δ-converges to T _μ x(= Px). By Remark 2.7 (ii), $\left\{T_{{\mu }_{\alpha }}x\right\}$ 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}, +), $\mathcal{S}\left(S\right)=\left\{T^n:n\in S\right\}$, Λ = ℕ or ℝ⁺and β _λk ≥ 0 be such that${\sum }_{k\in S}{\beta }_{\lambda k}=1$for all λ ∈ Λ. Suppose for all k ∈ S,

$$\underset{\lambda \to \infty }{\lim }{\beta }_{\lambda k}=0$$

(8)

and for each m ∈ S,

$$\underset{\lambda \to \infty }{\lim }\sum _{k=m}^{\infty }|{\beta }_{\lambda k}-{\beta }_{\lambda \left(k-m\right)}|=0.$$

(9)

For any f = (a₀, a₁,...) ∈ B(S) let${\mu }_{\lambda }\left(f\right)={\sum }_{k=0}^{\infty }{\beta }_{\lambda k}{a}_k$. Then for each x ∈ C, $\left\{T_{{\mu }_{\lambda }}x\right\}$Δ-converges to z for some z in F(T).

In particular, if X is a Hilbert space, we have${\sum }_{k=0}^{\infty }{\beta }_{\lambda k}{T}^{k}x$converges weakly to z for some z in F(T) as λ → ∞.

Proof. For each m ∈ S, $|{\mu }_{\lambda }\left(f\right)-{\mu }_{\lambda }\left(r_{m}f\right)|=\left|{\sum }_{k=0}^{\infty }{\beta }_{\lambda k}{a}_k-{\sum }_{k=0}^{\infty }{\beta }_{\lambda k}{a}_{k+m}\right|\le {\sum }_{k=0}^{m-1}\left|{\beta }_{\lambda k}\right|\left|a_k\right|+{\sum }_{k=m}^{\infty }\left|{\beta }_{\lambda k}-{\beta }_{\lambda \left(k-m\right)}\right|\left|a_k\right|$. By (8) and (9), we have $li{m}_{\lambda \to \infty }|{\mu }_{\lambda }\left(f\right)-{\mu }_{\lambda }\left(r_{m}f\right)|=0$, 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 $\left\{T_{{\mu }_{\lambda }}x\right\}$ Δ - converges to z for some z in F(T).

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

$$\sum _{k=0}^{\infty }{\beta }_{\lambda k}{∥T^{k}x-\sum _{k=0}^{\infty }{\beta }_{\lambda k}{T}^{k}x∥}^2\le \sum _{k=0}^{\infty }{\beta }_{\lambda k}{∥T^{k}x-y∥}^2$$

for all y ∈ C. So we have $T_{{\mu }_{\lambda }}x={\sum }_{k=0}^{\infty }{\beta }_{\lambda k}{T}^{k}x$. □

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,

$$S_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}{T}^{k}x$$

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}, +),

$${\beta }_{\lambda k}=\left\{\begin{array}{cc}\hfill \frac{1}{\lambda },\hfill & \hfill k\le \lambda -1,\hfill \\ \hfill 0,\hfill & \hfill k>\lambda -1.\hfill \end{array}\right.$$

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, $S_{r}x=\left(1-r\right){\sum }_{k=0}^{\infty }{r}^k{T}^{k}x$converges weakly to z for some z in F(T) as r ↑ 1.

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

$${\beta }_{\lambda k}=\frac{{\left(\lambda -1\right)}^k}{{\lambda }^{k+1}}.$$

Taking $r=\frac{\lambda -1}{\lambda }$, 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 $\mathcal{S}\left(S\right)=\left\{T\left(s\right):s\in S\right\}$ is said to be a continuous nonexpansive semigroup on C if $\mathcal{S}\left(S\right)$ 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}, +), $\mathcal{S}\left(S\right)$be a continuous nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$, Λ = ℝ⁺and g _λ be a density function on S,i.e., g _λ ≥ 0 and${\int }_0^{\infty }{g}_{\lambda }\left(s\right)ds=1$for all λ ∈ Λ. Suppose g _λ satisfies the following properties. for each h ∈ S,

$$\underset{\lambda \to \infty }{\lim }{g}_{\lambda }\left(s\right)=0$$

(10)

uniformly on [0, h] and

$$\underset{\lambda \to \infty }{\lim }{\int }_h^{\infty }|g_{\lambda }\left(s\right)-g_{\lambda }\left(s-h\right)|ds=0.$$

(11)

Then, for any x ∈ C,

$${\int }_0^{\infty }{g}_{\lambda }\left(s\right)T\left(s\right)xds$$

converges weakly to some$z\in F\left(\mathcal{S}\right)$as λ → ∞.

Proof. For f ∈ B(S) we define ${\mu }_{\lambda }\left(f\right)={\int }_0^{\infty }{g}_{\lambda }\left(s\right)f\left(s\right)ds$ for all λ > 0. Thus, μ _λ is a mean on B(S). For any h ∈ S we consider

$$\begin{array}{l}|{\mu }_{\lambda }(f)-{\mu }_{\lambda }(r_{h}f)|=\left|{\displaystyle {\int }_0^{\infty }{g}_{\lambda }}(s)f(s)ds-{\displaystyle {\int }_0^{\infty }{g}_{\lambda }}(s)f(s+h)ds\right|\\ \le {\displaystyle {\int }_0^h|}{g}_{\lambda }(s)\left|\right|f(s)|ds+{\displaystyle {\int }_h^{\infty }|}{g}_{\lambda }(s)-g_{\lambda }(s-h)||f(s)|ds.\end{array}$$

By (10) and (11), $li{m}_{\lambda }|{\mu }_{\lambda }\left(f\right)-{\mu }_{\lambda }\left(r_{h}f\right)|=0$. 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

$${\int }_0^{\infty }{g}_{\lambda }\left(s\right){∥{\int }_0^{\infty }{g}_{\lambda }\left(s\right)T\left(s\right)xds-T\left(s\right)x∥}^{2}ds\le {\int }_0^{\infty }{g}_{\lambda }\left(s\right){∥y-T\left(s\right)x∥}^{2}ds$$

for all y ∈ C. Thus, $T_{{\mu }_{\lambda }}x={\int }_0^{\infty }{g}_{\lambda }\left(s\right)T\left(s\right)xds$. By Theorem 3.9, we have ${\int }_0^{\infty }{g}_{\lambda }\left(s\right)T\left(s\right)xds$ converges weakly to some $z\in F\left(\mathcal{S}\right)$ 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$\mathcal{S}\left(S\right)$be a continuous nonexpansive semigroup on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Then, for any x ∈ C,

$$S_{\lambda }x=\frac{1}{\lambda }{\int }_0^{\lambda }T\left(s\right)xds$$

converges weakly to some$z\in F\left(\mathcal{S}\right)$as λ → ∞.

Proof. Let Λ = ℝ⁺ and put, for λ ∈ Λ and s ∈ S, $g_{\lambda }\left(s\right)=\frac{1}{\lambda }{\chi }_{\left[0,\lambda \right]}$. 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$\mathcal{S}\left(S\right)$be a continuous nonexpansive semi-group on C with$F\left(\mathcal{S}\right)\ne \varnothing$. Then, for any x ∈ C,

$$r{\int }_0^{\infty }{e}^{-rs}T\left(s\right)xds$$

converges weakly to some$z\in F\left(\mathcal{S}\right)$as r ↓ 0.

Proof. Let Λ = ℝ⁺ and put, for λ ∈ Λ and s ∈ S, $g_{\lambda }\left(s\right)=\frac{1}{\lambda }{e}^{-\frac{1}{\lambda }s}$. Again, we can then apply Proposition 3.13 by taking $r=\frac{1}{\lambda }$. □

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

$${\Pi }_n:=\left\{p={\left\{{\beta }_{nk}\right\}}_{k\ge n}\subset \left[0,1\right]:\sum _{k\ge n}{\beta }_{nk}=1\right\}$$

and

$$V_n:=\underset{p\in {\Pi }_n}{\sup }\sum _{k\ge n}{\beta }_{nk}{∥T^{k}x-{\bar{x}}_n∥}^2$$

where${\bar{x}}_n={\sum }_{k\ge n}{\beta }_{nk}{T}^{k}x$. 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,

$$\sum _{k\ge n}{\beta }_{nk}{∥T^{k}x-{\bar{x}}_n∥}^2\le \sum _{k\ge n}{\beta }_{nk}{∥T^{k}x-z∥}^2<{\left(r+\epsilon \right)}^2.$$

So $V_n=sup{}_{p\in {\Pi }_n}{\sum }_{k\ge n}{\beta }_{nk}{∥T^{k}x-{\bar{x}}_n∥}^2<{\left(r+\epsilon \right)}^2$. 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 $z\in \overline{co}{\left\{T^{k}x\right\}}_{k\ge n}$ for all n ∈ ℕ, there exists a sequence $\left\{{\bar{x}}_n\right\}$ with ${\bar{x}}_n\in co{\left\{T^{k}x\right\}}_{k\ge n}$ for each n and ${\bar{x}}_n\to z$ as n → ∞. Put ${\bar{x}}_n={\sum }_{k\ge n}{\beta }_{nk}{T}^{k}x$. Since {Tⁿx} is bounded, there exists M > 0 such that $∥T^{k}x-{\bar{x}}_n∥+∥T^{k}x-z∥\le M$. For each ε > 0, choose n _ε ∈ ℕ such that $∥{\bar{x}}_n-z∥<\epsilon$, 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 ,

$$\begin{array}{l}{\displaystyle \sum _{k\ge n}{\beta }_{nk}}|{\Vert T^{k}x-{\overline{x}}_n\Vert }^2-{\Vert T^{k}x-z\Vert }^2|\\ ={\displaystyle \sum _{k\ge n}{\beta }_{nk}}|\Vert T^{k}x-{\overline{x}}_n\Vert -\Vert T^{k}x-z\Vert |(\Vert T^{k}x-{\overline{x}}_n\Vert +\Vert T^{k}x-z\Vert )\\ ={\displaystyle \sum _{k\ge n}{\beta }_{nk}}\Vert {\overline{x}}_n-z\Vert (\Vert T^{k}x-{\overline{x}}_n\Vert +\Vert T^{k}x-z\Vert )\le \epsilon M.\end{array}$$

Hence,

$$\begin{array}{l}{(r-\epsilon )}^2<{\displaystyle \sum _{k\ge n}{\beta }_{nk}}{\Vert T^{k}x-z\Vert }^2\\ ={\displaystyle \sum _{k\ge n}{\beta }_{nk}}\Vert T^{k}x-z\Vert +{\displaystyle \sum _{k\ge n}{\beta }_{nk}}{\Vert T^{k}x-{\overline{x}}_n\Vert }^2-{\displaystyle \sum _{k\ge n}{\beta }_{nk}}{\Vert T^{k}x-{\overline{x}}_n\Vert }^2\\ \le V_n+{\displaystyle \sum _{k\ge n}{\beta }_{nk}}|{\Vert T^{k}x-{\overline{x}}_n\Vert }^2-{\Vert T^{k}x-z\Vert }^2|<V+\epsilon +\epsilon M.\end{array}$$

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${\bar{x}}_n$be defined as in Proposition 3.16. If the sequence$\left\{{\bar{x}}_n\right\}$satisfies

$$\underset{n\to \infty }{\lim }\left(\sum _{k\ge n}{\beta }_{nk}\left|\right|T^{k}x-{\bar{x}}_n|{|}^2\right)=V,$$

(12)

then$\left\{{\bar{x}}_n\right\}$converges (strongly) to z ∈ F (T ) as n → ∞.

Proof. Suppose for some ε > 0, there exists a subsequence $\left\{{\bar{x}}_{n_l}\right\}$ of $\left\{{\bar{x}}_n\right\}$ such that $∥{\bar{x}}_{n_l}-z∥\ge \epsilon$ for all l ∈ ℕ. For each y ∈ C and n ∈ ℕ, define ${\phi }_n\left(y\right):={\sum }_{k\ge n}{\beta }_{nk}\left|\right|T^{k}x-y|{|}^2$. Let $0<\delta <\frac{{\epsilon }^2}{8}$. By (12) and z ∈ A(C, {T^kx}), we choose n _δ such that $r^2-\delta =V-\delta <{\phi }_{n_l}\left({\bar{x}}_{n_l}\right)<V+\delta =r^2+\delta$ for all n _l ≥ n _δ and ||T^kx - z||² < r² + δ for all k ≥ n _δ . Fix l ≥ n _δ and let $\omega =\frac{{\bar{x}}_{n_l}+z}{2}$. By the Parallelogram law, we have for each k ≥ n _l ,

$$\left|\right|T^{k}x-\omega |{|}^2=\frac{1}{2}||T^{k}x-{\bar{x}}_{n_l}|{|}^2+\frac{1}{2}\left|\right|T^{k}x-z|{|}^2-\frac{1}{4}||{\bar{x}}_{n_l}-z|{|}^2.$$

Hence,

$${\phi }_{n_l}\left(\omega \right)<\frac{1}{2}\left(r^2+\delta \right)+\frac{1}{2}\left(r^2+\delta \right)-\frac{1}{4}{\epsilon }^2<r^2-\delta <{\phi }_{n_l}\left({\bar{x}}_{n_l}\right).$$

Using Lemma 2.2, we see that this contradicts to the minimality of ${\phi }_{n_l}\left({\bar{x}}_{n_l}\right)$. □