Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces
© Wangkeeree and Preechasilp; licensee Springer 2013
Received: 28 November 2012
Accepted: 9 February 2013
Published: 6 March 2013
The purpose of this paper is to study the strong convergence theorems of Moudafi’s viscosity approximation methods for a nonexpansive mapping T in CAT(0) spaces without the property . For a contraction f on C and , let be the unique fixed point of the contraction ; i.e.,
where is arbitrarily chosen and satisfies certain conditions. We prove that the iterative schemes and converge strongly to the same point such that , which is the unique solution of the variational inequality (VIP)
By using the concept of quasilinearization, we remark that the proof is different from that of Shi and Chen in J. Appl. Math. 2012:421050, 2012. In fact, strong convergence theorems for two given iterative schemes are established in CAT(0) spaces without the property .
Let be a metric space. A geodesic path joining to (or, more briefly, a geodesic from x to y) is a map c from a closed interval to X such that , , and for all . In particular, c is an isometry and . The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic segment is denoted by . The space is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each . A subset is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle in a geodesic metric space consists of three points , , in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle in is a triangle in the Euclidean plane such that for all .
A geodesic space is said to be a CAT(0) space if all geodesic triangles of an appropriate size satisfy the following comparison axiom.
It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include pre-Hilbert spaces, ℝ-trees (see ), Euclidean buildings (see ), the complex Hilbert ball with a hyperbolic metric (see ), and many others. Complete CAT(0) spaces are often called Hadamard spaces.
It is proved in  that a normed linear space satisfies the (CN)-inequality if and only if it satisfies the parallelogram identity, i.e., is a pre-Hilbert space; hence it is not so unusual to have an inner product-like notion in Hadamard spaces. Berg and Nikolaev  introduced the concept of quasilinearization as follows:
for all . It is known [, Corollary 3] that a geodesically connected metric space is a CAT(0) space if and only if it satisfies the Cauchy-Schwarz inequality.
Lemma 1.1 [, Lemma 2.1]
if and only if for all .
The set is a metric space with a metric D, which is called the dual metric space of .
Recently, Dehghan and Rooin  introduced the duality mapping in CAT(0) spaces and studied its relation with subdifferential by using the concept of quasilinearization. Then they presented a characterization of a metric projection in CAT(0) spaces as follows.
Theorem 1.2 [, Theorem 2.4]
Let C be a nonempty subset of a complete CAT(0) space X. Then the mapping T of C into itself is called nonexpansive iff for all . A point is called a fixed point of T if . We denote by the set of all fixed points of T. Kirk  showed that the fixed point set of a nonexpansive mapping T is closed and convex. A mapping f of C into itself is called contraction with coefficient iff for all . Banach’s contraction principle  guarantees that f has a unique fixed point when C is a nonempty closed convex subset of a complete metric space. The existence of fixed points and convergence theorems for several mappings in CAT(0) spaces have been investigated by many authors (see also [10–16]).
where . It is proved in  that converges strongly as to which is nearest to u (), and converges strongly as to which is nearest to u under certain appropriate conditions on , where is a metric projection from X onto C.
Furthermore, they also obtained that defined by (7) converges strongly as to under certain appropriate conditions imposed on .
All of the above bring us the following conjectures.
By using the concept of quasilinearization, we remark that the proof given below is different from that of Shi and Chen . In fact, strong convergence theorems for two given iterative schemes are established in CAT(0) spaces without the property .
We also denote by the geodesic segment joining from x to y, that is, . A subset C of a CAT(0) space is convex if for all .
The following lemmas play an important role in our paper.
Lemma 2.1 [, Proposition 2.2]
Lemma 2.2 [, Lemma 2.4]
Lemma 2.3 [, Lemma 2.5]
The concept of Δ-convergence introduced by Lim  in 1976 was shown by Kirk and Panyanak  in CAT(0) spaces to be very similar to weak convergence in the Banach space setting. Next, we give the concept of Δ-convergence and collect some basic properties.
Since it is not possible to formulate the concept of demiclosedness in a CAT(0) setting, as stated in linear spaces, let us formally say that ‘ is demiclosed at zero’ if the conditions Δ-converges to x and imply .
Lemma 2.4 
Every bounded sequence in a complete CAT(0) space always has a Δ-convergent subsequence.
Lemma 2.5 
If C is a closed convex subset of a complete CAT(0) space and if is a bounded sequence in C, then the asymptotic center of is in C.
Lemma 2.6 
If C is a closed convex subset of X and is a nonexpansive mapping, then the conditions Δ-converges to x and imply and .
Having the notion of quasilinearization, Kakavandi and Amini  introduced the following notion of convergence.
A sequence in the complete CAT(0) space w-converges to if , i.e., for all .
It is obvious that convergence in the metric implies w-convergence, and it is easy to check that w-convergence implies Δ-convergence [, Proposition 2.5], but it is showed in ([, Example 4.7]) that the converse is not valid. However, the following lemma shows another characterization of Δ-convergence as well as, more explicitly, a relation between w-convergence and Δ-convergence.
Lemma 2.7 [, Theorem 2.6]
Let X be a complete CAT(0) space, be a sequence in X, and . Then Δ-converges to x if and only if for all .
Lemma 2.8 [, Lemma 2.1]
Then converges to zero as .
The following two lemmas can be obtained from elementary computation. For convenience of the readers, we include the details.
(ii) The proof is similar to (i). □
3 Variational inequalities in CAT(0) spaces
In this section, we present strong convergence theorems of Moudafi’s viscosity methods in CAT(0) spaces. Our first result is the continuous version of Theorem 2.2 of Shi and Chen . By using the concept of quasilinearization, we note that the proof given below is different from that of Shi and Chen.
It follows from (13) that converges strongly to .
That is, solves the inequality (12).
Since , we have that , and so . Hence the net converges strongly to which is the unique solution to the variational inequality (12). This completes the proof. □
Remark 3.2 We give the different proof of [, Theorem 2.2]. In fact, the property imposed on a CAT(0) space X is removed.
If , then the following result can be obtained directly from Theorem 3.1.
Corollary 3.3 [, Lemma 2.2]
either or .
Then converges strongly as to such that which is equivalent to the variational inequality (12).
Applying Lemma 2.8, we can conclude that . This completes the proof. □
Remark 3.5 We give the different proof of [, Theorem 2.3]. In fact, the property imposed on a CAT(0) space X is removed.
If , then the following corollary can be obtained directly from Theorem 3.4.
Corollary 3.6 [, Theorem 2.3]
either or .
Then converges strongly as to which is nearest to u which is equivalent to the following variational inequality (18).
The first author is supported by Naresuan university.
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