Approximations of Fixed Points in the Hadamard Metric Space C AT p ( 0 )

: In this paper, we consider the recently introduced CAT p ( 0 ) , where the comparison triangles belong to (cid:96) p , for p ≥ 2. We ﬁrst establish an inequality in these nonlinear metric spaces. Then, we use it to prove the existence of ﬁxed points of asymptotically nonexpansive mappings deﬁned in CAT p ( 0 ) . Moreover, we discuss the behavior of the successive iteration introduced by Schu for these mappings in Banach spaces. In particular, we prove that the successive iteration generates an approximate ﬁxed point sequence.


Introduction
Hadamard metric spaces, also known as complete CAT(0) metric spaces, play an important role when dealing with the geometry of Bruhat-Tits building, metric trees, Hadamard manifolds, or simply connected nonpositively curved symmetric spaces. In fact, the power of Hadamard spaces goes beyond geometry. For example, CAT(0) geometry was used to solve an interesting problem in Dynamical billiards [1]. In a very simplistic way, Hadamard metric spaces are the nonlinear version of Hilbert vector spaces.
A metric Hadamard space (M, d) is characterized by the inequality (1) [2,3], known as the inequality of Bruhat and Tits, i.e., for any a, b, x ∈ M, there exists c ∈ M such that It is easy to check that c is a metric midpoint of a and b, i.e., d(c, a) = d(c, b) = d(a, b)/2. Note that in the linear Hilbert spaces, the inequality (1) becomes an equality. A Hadamard space for which the inequality (1) is an equality are known as flat Hadamard spaces. They are isomorphic to closed convex subsets of Hilbert spaces.
To understand this inequality, one has to look at the formal definition of a CAT(0) space with the comparison triangles taken in the Euclidean plane R 2 . In this case, the triangles in M are kind of slimmer than the comparison triangles in R 2 . This fundamental property motivated the authors of [4] to consider metric spaces for which the corresponding triangles are taken in general Banach spaces. The most natural example is to take these comparison triangles in p , for p > 1. The authors of [4] called these more general metric spaces "generalized CAT p (0)". In this work, we continue investigating the properties of these metric spaces and establish some existence fixed point results and their approximations.
For readers interested in metric fixed point theory, we recommend the book [5]. For more on geodesic metric spaces, we recommend the excellent book [2].

Basic Definitions and Preliminaries
Most of the terminology of geodesic metric spaces is taken from the work in [2]. Consider a metric space (M, d). A geodesic function ζ : [0, 1] → M is any function that satisfies d(ζ(α), ζ(β)) = |α − β|d(ζ(0), ζ(1)), for every α, β ∈ [0, 1]. (M, d) is said to be a geodesic space if, for every two points a, b ∈ M, there exists a function ζ : [0, 1] → M such that ζ(0) = a and ζ(1) = b and is geodesic. Throughout, we will use the notation ζ(α) = (1 − α) a ⊕ α b, for α ∈ (0, 1). (M, d) is said to be uniquely geodesic if any two points in M are connected by a unique geodesic. In this case, the range of the unique geodesic function connecting a and b will be denoted by [a, b], i.e., [a, b] Normed vector spaces are natural examples of geodesic metric spaces. Complete Riemannian manifolds, and polyhedral complexes of piecewise constant curvature are examples of nonlinear geodesic metric spaces. In these two examples, it is not obvious to show the existence of geodesics and show that they are unique. To determine when such spaces are uniquely geodesic is also a very hard task.
Geodesic triangles are naturally introduced in geodesic metric spaces. Indeed, let (M, d) be a geodesic metric space. Any three points-x, y, z ∈ M-will define a geodesic triangle ∆(x, y, z), which consists of the three given points called its vertices and the geodesic segments between each pair of vertices also known as the edges of ∆(x, y, z). Comparison triangles are crucial to the definition of CAT(0) spaces [6]. Given a geodesic triangle ∆(x 1 , holds for any i, j ∈ {1, 2, 3}. The pointx = αx i + (1 − α)x j is called a comparison point for x = α x i ⊕ (1 − α) x j , for any α ∈ [0, 1] and i = j.
Throughout (M, d) stands for a uniquely geodesic metric space.
(M, d) is said to be a CAT(0) space if all geodesic triangles satisfy the CAT(0) inequality. A Hadamard metric space is any complete CAT(0) space [7].
Let (M, d) be a CAT (0) space. Let x, y 1 , y 2 be in M. If m = 1 2 y 1 ⊕ 1 2 y 2 is the midpoint of y 1 and y 2 , then the CAT (0) inequality implies: Strictly convex Banach spaces are obviously uniquely geodesic. It is well known that a normed vector space is a CAT(0) space if and only if it is a pre-Hilbert space [2].
A recent extension to CAT(0) spaces was initiated in [4]. It is based on the idea that comparison triangles belong to a general Banach spaces instead of the Euclidean plane. Definition 2. [4] (E, . ) be a Banach space. The geodesic metric space (M, d) is said to be a CAT E (0) space if for any geodesic triangle ∆ in M, there exists a comparison triangle ∆ in E such that for any x, y ∈ ∆ and their comparison points x, y ∈ ∆. If E = p , for p ≥ 1, we say M is a CAT p (0) space.
In other words, Definition 2 gives a larger class of hyperbolic metric spaces. Throughout our work, we mainly focus on CAT p (0) metric spaces for p > 2. It is obvious that CAT 2 (0) space is exactly the classical CAT(0) space, which has been extensively studied.
The classical inequality (1) gives information about the middle point of two points. Except that many successive iterations, which present some interesting behavior, do not involve the middle point, but a convex combination of the given two points. Therefore, it is of utmost importance to prove or discover some metric properties of this kind of convex combinations.
Next, we will discuss a property of the convex combinations that holds in CAT p (0) spaces for p ≥ 2.
Proof. Note that this inequality is valid in p , for p ≥ 2. Indeed, Lim [8] proved the following inequality, for any β ∈ [0, 1] and x, y ∈ p , where for any β ∈ [0, 1] and x, y ∈ p . Next, we turn our attention to the proof of Lemma 1. Consider the geodesic triangle ∆(x, y 1 , The inequality (2) implies that which implies the conclusion of Lemma 1.
In the next section, we extend some known fixed point results in Banach spaces and CAT(0) spaces to the case of CAT p (0), for p > 2.

Fixed Point Results in CAT p (0)
Next, we investigate the fixed point problem for the class of asymptotically nonexpansive mappings. Note that this family of mappings was introduced by Goebel and Kirk [9] as a family of mappings that sits between the family of nonexpansive mappings [10] and the family of uniformly Lipschitzian mappings [11].
As we said before, throughout we consider (M, d) to be a geodesic metric space. (1) J is asymptotically nonexpansive if there exists {ρ n } such that lim n→∞ ρ n = 1 and d(J n (x), J n (y)) ≤ ρ n d(x, y), for any x, y ∈ M and n ∈ N. We can always assume that ρ n ≥ 1, for any n ∈ N. The fixed point problem for this class of mappings was extensively investigated [12][13][14][15]. It followed two directions: The first deals with the existence of a fixed point. The second deals with the approximation of the fixed points based on algorithms initiated by Schu [16]. In this work, we will follow the same directions as well.
A powerful tool used in investigating the existence of fixed points is the concept of type functions, which plays a major role in the study of metric fixed point theory in Banach spaces. Historically, it is also known as the asymptotic center. The following technical lemma shows why type functions are a powerful tool. Lemma 2. Let (M, d) be a complete CAT p (0) metric space, with p ≥ 2. Let C be a nonempty closed convex subset of M and θ : C → [0, +∞) be a type function generated by a bounded sequence {x n } ⊂ M. Then, the following hold.
(2) All minimizing sequences of θ converge to the same limit z ∈ C.
Proof. Set θ 0 = inf{θ(x); x ∈ C}. Without loss of generality, we assume θ 0 > 0. Let {z n } be a minimizing sequence of θ. Assume that {z n } is not Cauchy. As any subsequence of {z n } is also a minimizing sequence of θ, we may assume there exists ε 0 > 0 such that d(z n , z m ) ≥ ε 0 , for any n, m ∈ N. As C is convex, then 1 2 z m ⊕ 1 2 z m+1 ∈ C, for any m ∈ N. Lemma 1 implies for any n, m ∈ N. If we let n → ∞, we get which implies for any m ∈ N. If we let m → ∞, we get This contradiction shows that {z n } is Cauchy, which shows that (1) holds. To prove (2), let {z m } and {w m } be two minimizing sequences of θ. Consider the sequence {y m } defined by y 2m = z m and y 2m+1 = w m , for any m ∈ N. Then, {y m } is also a minimizing sequence of θ. From (1) we conclude that {y m } is convergent. As both {z m } and {w m } are subsequence of {y m } we conclude that {z m } and {w m } have the same limit. The conclusion of (3) follows from the simple fact that type functions are continuous.
In the first result, we discuss the existence of a fixed point for asymptotically nonexpansive mappings in CAT p (0) spaces. Proof. Let {ρ n } be the Lipschitz sequence associated to J. Fix x ∈ C. Consider the type function θ generated by {J n (x)}. Let z be the minimum point of θ which exists by using Lemma 2. Therefore, for any n, m ∈ N. If we let n → ∞, we will get θ(J m (z)) ≤ ρ m θ(z) = ρ m θ 0 , for any m ∈ N. If we let m → ∞, and using lim n→∞ ρ n = 1, we get lim m→∞ θ(J m (z)) = θ 0 , i.e., {J m (z)} is a minimizing sequence of θ. Using Lemma 2, we conclude that {J m (z)} converges to z. As J is continuous, we conclude that J(z) = z. The fact that Fix(J) is closed is obvious from the continuity of J. Let us prove that Fix(J) is convex. Let z 1 , z 2 ∈ Fix(J) be different. As Fix(J) is closed, we only need to prove that w = 1 2 z 1 ⊕ 1 2 z 2 ∈ Fix(J). Note that for n ∈ N and i = 1, 2. Therefore, for any n ∈ N. As lim n→∞ ρ n = 1, we conclude that Similarly, we can show that Using Lemma 1, we get for any n ∈ N. If we let n → ∞, we get lim n→∞ d(w, J n (w)) = 0. As J is continuous, then w ∈ Fix(J) as claimed, which completes the proof of Theorem 1.
Note that we may refine the boundedness assumption of C by assuming that an orbit of J is bounded. In this case, the above proof still holds. The convexity of the set of fixed points is a useful information, because it will allow us to prove the existence of a common fixed point for this class of mappings for example.
Next, we discuss the behavior of the successive iterations introduced by Schu [16] for asymptotically nonexpansive. In this case, Lemma 1 will prove to be crucial.
Recall that for a bounded nonempty subset C of a metric space (M, d), δ(C) denotes the diameter of C and is defined by  [16] is defined by for any n ∈ N, where x 0 ∈ C is a fixed arbitrary point. If z ∈ Fix(J), then lim n→∞ d(x n , z) exists.
Proof. First, note that which implies d(x n+1 , z) − d(x n , z) ≤ γ n (ρ n − 1)d(x n , z), for any n ∈ N. In particular, we have d(x n+1 , z) − d(x n , z) ≤ γ n (ρ n − 1)δ(C), for any n ∈ N. Therefore, for any n, m ∈ N. If we let m → ∞, we get for any n ∈ N. Using the assumption Therefore, lim sup is convergent.
In the next result, we show that the sequence generated by (3) almost gives a fixed point. where 0 < a ≤ b < 1. Fix x 0 ∈ C and consider the sequence {x n } generated by the iteration (3). Then, {x n } converges strongly to a fixed point z of J, i.e., lim n→∞ d(x n , z) = 0.
Proof. Let m ≥ 1 such that J m is compact. Then, there exists a subsequence {x φ(n) } such that {J m (x φ(n) )} converges to some point z ∈ C. Using Theorem 2, we know that lim n→∞ d(x n , J m (x n )) = 0, which implies {x φ(n) } also converges to z. Again, using Theorem 2, we know that lim n→∞ d(x n , J(x n )) = 0, which implies that {J(x φ(n) )} also converges to z. As J is continuous, we conclude that J(z) = z, i.e, z ∈ Fix(J). Moreover, Lemma 3 implies lim n→∞ d(x n , z) = r exists. As lim n→∞ d(x φ(n) , z) = 0, we conclude that r = 0, i.e., lim n→∞ d(x n , z) = 0. In other words, the sequence {x n } converges to z as claimed.
Therefore, we wonder whether a weaker convergence is happening if we relax the compactness assumption in Theorem 3. In the original work of Schu [16], the setting is a Banach space. Therefore, we may consider naturally the weak topology. In the nonlinear setting, it is still unknown what the weak topology looks like. Lim [17] introduced a convergence concept he called ∆-convergence based on the asymptotic center of a sequence. Except that this convergence does not capture the weak topology once we restrict ourselves to Banach spaces. It only happens if the Banach space enjoys the Opial property [18]. In the next result, we discard the compactness assumption. any x ∈ C. According to Lemma 2, the type function θ has a unique minimum point z, which is a fixed point of J. Let us prove that z is also the minimum point of any type function θ φ generated by a subsequence {x φ(n) } of {x n }. Again according to Lemma 2, there exists a unique minimum point z φ of θ φ , which is also a fixed point of J. Lemma 3 implies that lim n→∞ d(x n , z) and lim n→∞ d(x n , z φ ) exist. As z is the minimum point of θ, we get θ(z) ≤ θ(z φ ), which implies lim n→∞ d(x n , z) ≤ lim n→∞ d(x n , z φ ) or lim n→∞ d(x φ(n) , z) ≤ lim n→∞ d(x φ(n) , z φ ), i.e., θ φ (z) ≤ θ φ (z φ ). The uniqueness of the minimum point of θ φ implies that z = z φ , which completes the proof of Theorem 4.