Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications

: In this article, we prove ergodic convergence for a sequence of nonexpansive mappings in Hadamard (complete ( ) CAT 0 ) spaces. Our result extends the nonlinear ergodic theorem by Baillon for non-expansive mappings from Hilbert spaces to Hadamard spaces. We further establish the standard nonlinear ergodic theorem and apply our results to the problem of ﬁ nding a common ﬁ xed point of a countable family of nonexpansive mappings. Finally, we propose some applications of our results to solve convex optimization problems in Hadamard spaces


Introduction
Let H be a real Hilbert space and T be a mapping from H into H . Then T is called a nonexpansive mapping if ‖ ‖ ‖ ‖ − ≤ − Tx Ty x y for all ∈ x y H , .
x H Tx x Fix denotes the set of all fixed points of the mapping T and represents the set of all natural numbers.The mean ergodic theorem was studied by von Neumann [1] in Hilbert spaces and Brikhoff [2] in Banach spaces.Later in 1975, Baillon [3] proved the first nonlinear ergodic theorem as follows: Theorem 1.1.[3] Let H be a real Hilbert space, D be a nonempty closed convex subset of H, and → T D D : be a nonexpansive mapping.Suppose that ( ) T Fix is nonempty.Define a sequence { } z n by ∈ x D and for all ∈ n .Then { } z n converges weakly to an element ( ) ∈ z T Fix .
Recently, several researchers have been studying and applying the mean ergodic theory to other more general spaces [15][16][17].One of the natural generalizations is to Hadamard spaces as many interesting examples and applications have been so far confirmed [18][19][20].Even though there have already been several developments in fixed point theory and nonlinear analysis in Hadamard spaces, the study mean ergodic theory remains a challenge.This is because the ergodic mean that is used in { } z n as in Theorems 1.1 and 1.2 cannot be straightforwardly defined.As a consequence, a number of distinct definitions of an erogodic mean were considered in Hadamard spaces.In 2011, Ahmadi Kakavandi and Amini [21] extended the Baillon nonlinear ergodic theorem for nonexpansive mappings on Hilbert spaces to amenable semigroups and commutative semigroups of nonexpansive mappings on Hadamard spaces by using the concept of invariant mean, which is a specific case of mean for semitopological semigroups (see [21] for more details).Later in 2015, Ahmadi Kakavandi [22] also proved the convergence of asymptotically invariant means for amenable semigroups of nonexpansive mappings in Hadamard spaces by assuming that the mean satisfies a certain condition, that is, the property [22].On the other hand, Anakkamatee and Dhompongsa [23] proposed the ergodic convergence of nonexpansive mappings on compact Hadamard spaces [23,Theorem 3.6] by using the nonlinear combination for the orbit of multi-valued nonexpansive mappings on nonempty bounded closed convex subset of .One may also consult the previous studies [24][25][26] for other developments in this direction.
For general Hadamard spaces, the standard mean ergodic theorem for a general nonexpansive mapping is still an open problem due to a technical detail.To be more precise, the set of Δ-cluster points of the mean ergodic sequence might not be a singleton.Likewise, even though there are many definitions of a "mean" that were provided earlier, none of them reduces to the linear mean in general Hilbert spaces.Recently, in 2021, Khatibzadeh and Pouladi [27] established a mean ergodic theorem for general nonexpansive mappings on Hadamard spacecs with the ( ) Q 4 condition by applying the concept of a Karcher mean (see Section 2 for definitions of Karcher means and the ( ) Q 4 condition).This approach is considered the most natural since a Karcher mean reduces, without any artificial modification, to a linear mean whenever the space in question is Hilbert.Motivated by the aforementioned studies, we show in this article the mean ergodic convergence of a sequence of nonexpansive mappings in the setting of an Hadamard space satisfying the ( ) Q 4 condition.Our main result generalizes a result of Khatibzadeh and Pouladi (Corollary 3.5) and extends Theorem 1.1.In addition, our results extends Theorem 1.2 when the sequence of nonexpansive mappings uniformly converges to some mapping on each nonempty bounded subset of such spaces.
This article is structured as follows.In Section 2, we recall the definitions and backgrounds of Hadamard spaces that are necessary for our results in the subsequent sections.In Section 3, we present auxiliary lemmas, followed by our main result (i.e., Δ-convergence theorem) for a sequence of nonexpansive mappings, and then we demonstrate a Δ-convergence-version generalization of the standard nonlinear ergodic theorem for non- expansive mappings on Hadamard spaces.Further, we also apply our results to the common fixed point problem of a countable family of nonexpansive mappings.In Section 4, we propose and describe some applications for our main results to solve convex optimization problems on Hadamard spaces.Finally, in Section 5, we provide the conclusions of this research.

Preliminaries
In this section, some basic notions and necessary lemmas for the subsequent results are given.Throughout this article, the set of all real numbers is denoted by and suppose that 2 denotes the Euclidean plane equipped with the Euclidean norm.
Let ( ) X ρ , be a metric space and ∈ x y X , .A geodesic path joining x and y is a mapping , and , in ( ) x x x Δ , , 1 2 3 and their comparison points p q , in ( ) , be a uniquely geodesic metric space and For an arbitrary subset C of X , the closed convex hull of C is the smallest closed and convex set that contains C, which is denoted by ( ) co C .Recall that the effective domain of f is defined by The concept of Δ-convergence in a ( ) CAT 0 space was introduced by Lim [31] to replace the weak conver- gence of a Hilbert space.Later in 2008, Kirk and Panyanak [32] showed that various properties of the weak convergence in linear spaces are shared by this notion of convergence in Hadamard spaces (see also [33]).
Let ( ) ρ , be an Hadamard space and { } x n be a sequence of .The asymptotic center ({ }) x n of sequence { } x n is defined by x n is a singleton in Hadamard space [34, Proposition 7].The sequence { } x n is said to be Δ-convergent to an element z of if In this case, we say that z is the Δ-limit of { } x n .If { } x n is convergent to z in its metric d, then it is also Δ-convergent to z.Every Δ-convergent sequence is bounded, and all of its subsequences are again Δ-convergent to the same limit.The notation ({ }) ω x n Δ denotes the set of all ∈ z ¯in which z ¯is a Δ-limit of some subsequence { } [32,35,36] for more details about Δ-convergence.The following results were proved in the study by Kirk and Panyanak [32] and are necessary for the proofs of our main results.
, be an Hadamard space and { } x n be a bounded sequence in .Then the sequence { } , be an Hadamard space, D be a nonempty closed convex subset of and Recently, in 2021, Chaipunya et al. [37] established the following proposition, which is very helpful for proving Δ-convergence on Hadamard spaces.Proposition 2.4.[37] Let ( ) ρ , be an Hadamard space and { } x n be a bounded sequence in .Suppose that Next, let us recall the notion of the ( ) Q 4 condition of Kirk and Panyanak [32].
Definition 2.5.[32] Let ( ) ρ , be an Hadamard space.We say that satisfies the ( ) This condition is known to hold in several ( ) CAT 0 spaces including Hilbert spaces and -trees.Later, Espínola and Fernández-León [35] proved that any ( ) CAT 0 space of constant curvature satisfies the ( ) Q 4 condition [35,Theorem 5.7].They also showed that a ( ) CAT 0 space obtained by gluing two spaces of different constant curvatures fails the condition [35,Theorem 5.11].This condition was modified by Ahmadi Kakavandi [38] into the ( ) Q 4 condition, which allows more fruitful applications especially in convex analysis.
Definition 2.6.[38] Let ( ) ρ , be an Hadamard space.We say that satisfies the ( ) One may simply observe that this condition implies that the half space given by ( From [38], we know that any Hilbert spaces, -trees and ( ) CAT 0 space of constant curvature satisfy also the ( ) Q 4 condition.Now, we recall a Karcher mean (or a Fréchet mean) for a given sequence.This notion finally allows us to study mean ergodic convergence of an algorithm.Let ( ) ρ , be an Hadamard space and { } x n be a sequence in .For any for all ∈ y .As stated earlier, the Karcher means of x x x , ,…, n 1 2 and The following useful lemma was given in [27].
[27] Let ( ) ρ , be an Hadamard space and { } x n be a sequence in .Then for each ∈ y and ≥ k 1, we have ( .
We finish the section with the propositions and lemmas from [18,39] that are required in our proofs.
for every four points ∈ x y z w , , , . Lemma . Define a mapping → U : by Then U is a nonexpansive mapping.
[39] Let ( ) ρ , be an Hadamard space and → S : n be nonexpansive mappings for all

Main results
The main objective of this section is to prove the mean ergodic theorems for a sequence of nonexpansive mappings in Hadamard spaces.In addition, we would like to address the problem of finding common fixed points of a countable family of nonexpansive mappings in such spaces using our results.

Auxiliary lemmas
Let us first derive some preliminary facts for our main results.x n , we have

S n S n S n n S n n S n n S n n
Then there is a ∈ k with ≥ ′ k n n , .Thus, it follows from (3.1) that

S n S n S n S k S k S n
This implies that the sequence { } P x S n is a Cauchy sequence.Since S is closed in , the sequence { } P x S n converges strongly to some element ∈ z S. □ The next lemma is well known in the Hilbert setting [4].However, it has not been pointed out for an Hadamard space.
be an Hadamard space and → T : is nonempty and the sequence { } T n pointwise converges to a mapping T. Then the mapping T is nonexpansive and  is bounded; for all ≥ n 1.
Proof.Firstly, we will prove the conclusion (1).Fix ∈ x and define the sequence { } x n by (2.3).Since by Lemma 2.7 (2), the definition of x n and the nonexpansivity of T n , we see that is bounded.Next, we shall show the conclusion (2).By Lemma 2.7 (1), we obtain .
Now, since and the conclusion (1), and hence, the sequence { } x By Lemma 2.7 (1), we have and by the definition of x n and the nonexpansivity of T n , we obtain , .
. Since { } T n converges uniformly to T on each nonempty bounded subset of , it follows for any given ε there exists 2), we obtain are bounded for all ∈ j , so we obtain Mean Ergodic Theorems  9 which is a contradiction.Therefore, we obtain From the definition of ( ) and the nonexpansivity of T , we have .
Finally, we shall show conclusion (4).Let m k m and P F be the metric projection mapping from onto F .By inequality (2.2), we have for all ∈ i n , .On the other hand, by the definition of ( ) , we obtain The last two inequalities allow us to conclude that This implies that ( ( ) , which completes the proof.□

A mean ergodic theorem for a uniformly convergent family
With all the auxiliary tools laid out, we are now in the position to state and prove our first main result.
Theorem 3.4.Let ( ) ρ , be an Hadamard space and → T : is nonempty and the sequence { } T n uniformly converges to a mapping T on each nonempty bounded subset of .Then the following hold: and satisfies the ( ) Proof.We first prove the conclusion (1).Let is closed and convex.From Lemmas 3.1 (2) and (3.3), we obtain that . By the definition of the sequence { } x n and Lemma 3.1 (1), { ( ) } ρ P x x , F n n is nonincreasing.By Lemma 3.3 (1), we see that the sequence { ( )} σ x n is bounded, and hence, by Lemma 2.2, we know that there exists a subsequence ( ) Suppose to contrary that there is a > δ 0 such that By the definition of a quasi-inner product and inequality (2.2), we obtain that for all is nonincreasing, by the aforementioned inequality, Lemma 2.1, the Cauchy-Schwarz inequality and the nonexpansivity of T n , we have x P x zz x P x P x z ρ x P x ρ P x z ρ x P x ρ P x z ρ x P x ρ P x z , we obtain a positive number k 0 such that , is closed by continuity of the metric function.Also, by Lemma 2.2, ( ) An immediate consequence of the above theorem is stated as the following corollary, which is also the main result of the study by Khatibzadeh and Pouladi [27].Further motivated by Reich [5], we also obtain the following consequence that provides another generalization of the aforementioned corollary.Mean Ergodic Theorems  11 , .
and by the aforementioned inequality and (2.1), we obtain this means that f is strongly convex with = k 1.This completes the proof.□ Lemma 3.8.Let ( ) ρ , be an Hadamard space and → S : Then U is a nonexpansive mapping.and hence, By dividing by − t 1 and letting → t 1, we have Thus, we have  Since S n is nonexpansive, Therefore, we have U , which is nonexpansive.□ Lemma 3.9.Let ( ) ρ , be an Hadamard space and → S : . Thus, we have . □ Finally, we can now apply Theorem 3.4 to the problem of finding a common fixed point of a countable family of nonexpansive mappings in Hadamard spaces.Note that our result is an extension of Akatsuka et al. , where the mapping T is defined as follows: Proof.First, we will prove conclusion (1).By Lemmas 2.12 and 3.7, we know that mappings T n and T are well defined for all ∈ n .Moreover, by Lemmas 2.13 and 3.8, we know that mappings T n and T are nonexpansive for all ∈ n .From Lemma 2.14, we obtain Next, we shall show conclusion (2).Let D be a nonempty bounded subset of and ∈ y D. By applying a similar proof technique as in Lemma 2.7, we obtain the following two inequalities: By combination of these inequalities (3.9) and (3.10), we obtain that  .By conclusion (1), we know that ( ) , so we see that .This implies that the sequence { } T n uni- formly converges to the mapping T on each nonempty bounded subset of .Finally, we shall show conclusion (3).By conclusion (2), we know that { } T n converges uniformly to the mapping T on each nonempty bounded subset of , and combination with conclusion (1) and Theorem 3.4, we obtain conclusion (3).We have thus arrived at the desired conclusion.□ 3.4, we have desired result.□

. 1 . ( 2 )
Then the following holds: (1) → T : n are nonexpansive mappings for all ∈ n and The sequence { } T n uniformly converges to the mapping T on each nonempty bounded subset of .(3) The sequence { ( )} σ x n is Δ-convergent to ( ) Fix and { ( )} σ x n is defined by (2.4).

2 ) 1 ,
The sequence { } T n converges uniformly to the mapping T on each nonempty bounded subset of .(3) The sequence { ( )} σ x n is Δ-convergent to ( ) where = →∞ z Px lim n F n and { ( )} σ x n is defined by (2.4).
is called the comparison point of u, where γ is the geodesic joining x i to x j .If every two points p q , where x n is defined by(2.3).We first prove the following lemmas, which are important for our main results.
is defined by(3.5).Therefore, on the basis of Lemma 3.7, we can conclude that U is well defined and single valued.Let