Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

AN ERGODIC THEOREM FOR

ASYMPTOTICALLY NONEXPANSIVE

MAPPINGS IN CAT(0) SPACES^∗

Hadi Khatibzadeh^†

Communicated by G. Moro¸sanu

DOI

10.56082/annalsarscimath.2026.2.95

Abstract

In the framework of CAT(0) spaces, we prove that the mean of

iterations of an asymptotically nonexpansive mapping with nonempty

ﬁxed point set converges weakly to a ﬁxed point. The result extends the

similar previous result for nonexpansive mappings in CAT(0) spaces

[13] and generalizes the results of [11] from Hilbert spaces to Hadamard

spaces.

Keywords: asymptotically nonexpansive, ergodic theorem, weak con-

vergence, ﬁxed point, CAT(0) space, Hadamard space.

MSC: 47H25, 47J25, 40A05.

1 Introduction

Let X be a normed linear space with norm k·k and let C ⊂ X be a nonempty,

closed, and convex set. A mapping T : C → C is called:

1. nonexpansive if for each x, y ∈ C, kTx − Tyk ≤ kx − yk

^∗Accepted for publication on November 28, 2025

^†hkhatibzadeh@znu.ac.ir, Department of Mathematics, University of Zanjan, Zanjan,

Iran

95

An ergodic theorem for asymptotically nonexpansive mappings

96

2. asymptotically nonexpansive if for each x, y ∈ C

kTⁿx − Tⁿyk ≤ (1 + α_n)kx − yk,

(1)

where α_nis a positive real sequence that converges to 0 as n → ∞.

For a mapping T : C → C, x ∈ C is called a ﬁxed point of T if Tx = x. The

set of all ﬁxed points of T is denoted by Fix(T). The asymptotic behavior of

iterations of nonexpansive mappings is a signiﬁcant issue in nonlinear anal-

ysis. A simple example in R illustrates that iterations of a nonexpansive

mapping is not generally convergent. However, Baillon [4] demonstrated

that if the mapping has a ﬁxed point, or equivalently, if the sequence of its

iterations is bounded, then the average of the iterations converges weakly

to a ﬁxed point of the mapping in a Hilbert space. This result is known as

Baillon’s nonlinear mean ergodic theorem, which serves as a nonlinear exten-

sion of the ergodic theorem of von Nuemann [23 ] for linear mappings. Since

then, several extensions of Baillon’s theorem have been proposed, including

extensions to Banach spaces by Beauzamy et al., Bruck, Reich and Hirano

in [5–7,12,21,22]. Another approach to extending this theorem involves re-

placing the nonexpansive mapping with a more general nonlinear mapping,

such as asymptotically nonexpansive mappings, which were ﬁrst deﬁned by

Goebel and Kirk in [10 ] to extend the Kirk ﬁxed point theorem. Baillion’s

ergodic theorem has been further extended for asymptotically nonexpansive

mappings by Hirano and Takahashi [11] in Hilbert spaces and by Oka [19]

in Banach spaces.

Recently, numerous results across various branches of nonlinear analysis

have been extended to nonlinear contexts such as Hadamard manifolds and

CAT(0) spaces. These include ﬁxed point theory for nonexpansive map-

pings, convex optimization, proximal algorithms, and nonlinear semigroup

theory. The ergodic theory of nonexpansive mappings has been investigated

in [1,2,13 –15,17] for CAT(0) spaces. In this paper, we establish an ergodic

theorem for asymptotically nonexpansive mappings in CAT(0) spaces.

Let (H, d) be a metric space. A geodesic between two points x₀and x₁in

H is a mapping γ : [0, d(x₀, x₁)] → H such that γ(0) = x₀, γ(d(x₀, x₁)) = x₁,

and d(γ(t), γ(t⁰)) = |t−t⁰|, ∀t, t⁰∈ [0, d(x₀, x₁)]. A metric space in which any

two points are connected by a geodesic is called a geodesic metric space. The

image of a geodesic connecting two points x₀and x₁is denoted by [x₀, x₁],

which is called a segment between x₀and x₁. Each point x_t∈ [x₀, x₁],

satisfying d(x_t, x₀) = td(x₀, x₁) and d(x_t, x₁) = (1 − t)d(x₀, x₁) for some

t ∈ [0, 1], is represented as (1 − t)x₀⊕ tx₁. A subset C of H is called convex

if for every x, y ∈ C, the segment [x, y] ⊂ C. For a subset C of H, the

H. Khatibzadeh

97

closed convex hull of C is denoted by coC, which is the intersection of all

closed and convex sets containing C. A geodesic metric space whose metric

d satisﬁes the following condition for each metric segment [x₀, x₁] and any

arbitrary point y ∈ H is known as a CAT(0) space.

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

ꢀ

ꢁ

2

d(x_t, y)

≤ (1 − t) d(x₀, y) ²+ t d(x₁, y)

− t(1 − t) d(x₀, x₁) .

(2)

ꢀ

ꢁ

2

From this point until the end of the paper, we will denote d(x, y)

by

d²(x, y). A complete CAT(0) space is referred to as a Hadamard space. A

Hadamard space H has the Q₄property if, for any arbitrary points x, y, p, q ∈

H and any point m in the segment [x, y], the following inequality holds:

d(p, x) ≤ d(q, x) & d(p, y) ≤ d(q, y) ⇒ d(p, m) ≤ d(q, m).

Hilbert spaces, R-trees, and CAT(0) spaces of constant curvature all possess

the Q₄property (see [2, 16]). It is easily seen that the Q₄property implies

that the set F(x, y) := {z ∈ H : d(x, z) ≤ d(y, z)} is convex for all x, y ∈ H.

Let (H, d) be a Hadamard space, and let {x_n} be a sequence in H. We

deﬁne

r(x, {x_n}) = lim sup d(x, x_n),

x ∈ H.

n→+∞

The asymptotic radius of the sequence {x_n} is given by

r({x_n}) = inf r(x, {x_n}),

x∈H

and the asymptotic center of {x_n} is the set

A({x_n}) = {x ∈ H : r({x_n}) = r(x, {x_n})}.

According to [8, Propsition 7], A({x_n}) is a singleton in Hadamard spaces.

A sequence {x_n} in a Hadamard space H is said to converge weakly to x ∈ H

if A({x_n}) = {x} for every subsequence {x_n} of {x_n} [8,9,16]. In this case,

k

we denote this weak convergence as x_n* x.

Let C be a closed convex subset of a Hadamard space H. For an arbitrary

mapping T : C → C and x ∈ C, the mean of the iterates T^kx, · · · , T^n+k−1

x

is denoted by σ_n^kx and is deﬁned as follows:

n+k−1

X

σ_n^kx := Argmin_y∈H

d²(Tⁱx, y),

σ_nx := σ⁰x.

(3)

n

i=k

Lemma 1. [2,13 ] For σ_n^kx deﬁned above and an arbitrary y ∈ H, we have:

An ergodic theorem for asymptotically nonexpansive mappings

98

P

ⁿ⁻¹d²(T^k+ix, y) −

ⁿ⁻¹d²(T^k+ix, σ_n^kx)

1

n

1

n

1. d²(σ_n^kx, y) ≤

i=0

P

2. d(σ_n^kx, y) ≤

ⁿ⁻¹d(T^k+ix, y).

1

i=0

n

Nonexpansive and asymptotically nonexpansive mappings are deﬁned

in metric spaces similarly to how they are deﬁned in normed spaces, by

replacing the norm with the corresponding metric.

2 Main result

The following theorem is the main result of the paper.

Theorem 1. Let H be a Hadamard space with the Q₄property, C be a

closed convex subset of H, and T : C → C be an asymptotically nonexpansive

mapping with Fix(T) = ∅. Then, for each x ∈ C, the sequence σ_nx (deﬁned

in (3)) converges weakly to a ﬁxed point of T, which is the asymptotic center

of the sequence {T^kx}.

To prove Theorem 1, we ﬁrst need to establish some lemmas.

Lemma 2. Assume that the conditions of Theorem 1 on C and T hold.

Then, for each x ∈ C and y ∈ Fix(T), we have the following:

1. lim_n→+∞d(Tⁿx, y) exists.

2. The sequence {σ_n^k}_nis bounded for k = 0, 1, · · · .

3. d(σ_nx, σ_n^kx) → 0 as n → +∞ for k = 1, 2, · · · .

4. σ_n^kx ∈ co{T^mx}_m≥kfor all n ≥ 1.

Proof. Part 1. For all y ∈ Fix(T), we have

d(T^m+nx, y) = d(T^mTⁿx, y) ≤ (1 + α_m)d(Tⁿx, y).

Taking the lim sup as m → +∞ and the lim inf as n → +∞, we conclude

that lim d(Tⁿx, y) exists.

Part 2. This follows directly from Part 1 of this lemma and Part 2 of

Lemma 1 .

Parts 3,4. The proofs for these parts are identical to those in Parts (iii)

and (iv) of [13, Lemma 2.3].

Lemma 3. Let C and T satisfy the assumptions of Theorem 1. Then we

have:

lim sup lim sup d(σ_nx, T^mσ_nx) = 0

(4)

m→+∞ n→+∞

H. Khatibzadeh

99

Proof. In Part 1 of Lemma 1 with k = 0, let y = T^mσ_nx, where m ≤ n.

Then, we have:

m−1

n−1

X

1

d²(σ_n, T^mσ_nx) ≤

d²(Tⁱx, T^mσ_nx) +

d²(Tⁱx, T^mσ_nx)

n

n _i=m

i=0

n−1

X

1

−

≤

−

≤

d²(Tⁱx, σ_nx)

n

i=0

m−1

n−m−1

X

1

d²(Tⁱx, T^mσ_nx) + (1 + α_m)²

d²(Tⁱx, σ_nx)

n

i=0

n−1

X

1

d²(Tⁱx, σ_nx)

n

i=0

m−1

n−m−1

X

1

d²(Tⁱx, T^mσ_nx) + (α²+ 2α_m)

d²(Tⁱx, σ_n)

m

n

i=0

Taking lim sup as n → +∞, we obtain

lim sup d²(σ_nx, T^mσ_nx) ≤ (α²+ 2α_m)D,

m

n→+∞

where D = sup_n,i≥0d²(Tⁱx, σ_nx) < +∞ (by Lemma 2, the sequences {T^kx}

and {σ_nx} are bounded). Now, taking the lim sup as m → ∞, we get (3).

Lemma 4. Let the assumptions of Theorem 1 regarding C and T be satisﬁed.

If the subsequence σ_nx of σ_nx converges weakly to a point y, then y ∈

j

Fix(T).

Proof. To prove this, it is suﬃcient to show that T^ky → y. By (2), the

triangle inequality, and asymptotic nonexpansiveness of T, we have:

1

d²(σ_nx, y ⊕ T^ky)

2

1

≤

d²(σ_nx, y) + d²(σ_nx, T^ky) − d²(y, T^ky)

2

1

2

1

4

ꢀ

ꢁ

1

2

d²(σ_nx, y) +

d(σ_nx, T^kσ_nx) + d(T^kσ_nx, T^ky)

− d²(y, T^ky)

2

1

2

1

4

ꢀ

ꢁ

1

2

d²(σ_nx, y) +

d(σ_nx, T^kσ_nx) + (1 + α_k)d(σ_nx, y)

− d²(y, T^ky).

2

4

Substituting n by n_jin the above expression and taking the limit superior

as j → +∞, we obtain:

An ergodic theorem for asymptotically nonexpansive mappings

100

1

lim sup d²(σ_nx, y) + d²(y, T^ky)

j

4

j→+∞

1

≤ lim sup d²(σ_nx, y ⊕ T^ky) + d²(y, T^ky)

j

2

4

j→+∞

ꢀ

ꢁ

1

2

1

≤

+ (1 + α_k)²lim sup d²(σ_nx, y)

j

2

j→+∞

1

2

+

lim sup d²(σ_nx, T^kσ_nx) + M(1 + α_k) lim sup d(σ_nx, T^kσ_nx),

n→+∞

where M := sup_n≥0d(σ_nx, y). Taking the limit superior as k → +∞, and

applying Lemma 3, we arrive at the desired result.

Proof of Theorem 1. Since, by Part 2 of Lemma 2, the sequence {σ_nx}

is bounded, there exists a subsequence σ_nx of σ_nx that converges weakly

i

to a point y ∈ Fix(T) (as stated in Lemma 4 ). We will prove that y is the

asymptotic center of the sequence {T^kx}. Let x₀be the asymptotic center

of {T^kx}. If y = x₀, then, by Part 1 of Lemma 2, we have:

lim sup d(T^kx, x₀) < lim sup d(T^kx, y) = lim d(T^kx, y).

k

Therefore, there exists k₀> 0 such that for each k ≥ k₀, we have T^kx ∈

F(x₀, y) = {z ∈ C : d(z, x₀) ≤ d(z, y)}. The closedness and convexity of

F(x₀, y) imply that co{T^kx : k ≥ k₀} ⊂ F(x₀, y). Since y does not belong

to F(x₀, y), it follows that y ∈ co{T^kx : k ≥ k₀}. However, by Lemma 4 and

by Parts 3 and 4 of Lemma 2, we know that y ∈ Fix(T) ∩ co{T^kx : k ≥ k₀}.

This leads to a contradiction, demonstrating that every weak clusters point

of {σ_nx} is the asymptotic center of {T^kx}. Therefore, we conclude that

σ_nx * y ∈ Fix(T) as n → ∞. ꢀ

Remark 1. If the conditions of Theorem 1 on T and C are satisﬁed, and if

the mapping T is asymptotically regular, meaning that d(Tⁿ⁺¹x, Tⁿx) → 0

as n → +∞ for each x ∈ C, then Tⁿx converges weakly to an element of

Fix(T).

To see why this is true, let y be a weak convergence point of a subsequence

Tⁿx of T x. By the asymptotic regularity of T, we have d(TT x, T x) →

0 as j → +∞. The demiclosed principle for asymptotically nonexpansive

mappings in CAT(0) spaces [18] implies that y ∈ Fix(T). Furthermore, by

Lemma 2 , lim_n→+∞d(Tⁿx, y) exists for each y ∈ Fix(T). Opial’s lemma

in CAT(0) spaces [20, Lemma 2.1] then shows that Tⁿx * y ∈ Fix(T) as

n → +∞.

n

n_j

j

H. Khatibzadeh

101

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