Viscosity approximation methods for nonexpansive semigroups in spaces
© Wangkeeree and Preechasilp; licensee Springer. 2013
Received: 22 December 2012
Accepted: 14 May 2013
Published: 19 June 2013
In this paper, we study the strong convergence of Moudafi’s viscosity approximation methods for approximating a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings in spaces. We prove that the proposed iterative scheme converges strongly to a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings which is also a unique solution of the variational inequality. The results presented in this paper extend and enrich the existing literature.
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 space if all geodesic triangles of appropriate size satisfy the following comparison axiom.
It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a 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 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 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 metric D, which is called the dual metric space of .
Recently, Dehghan and Rooin  introduced the duality mapping in spaces and studied its relation with subdifferential, by using the concept of quasilinearization. Then they presented a characterization of metric projection in spaces as follows.
Theorem 1.2 [, Theorem 2.4]
- (i)for each , is a nonexpansive mapping on C, i.e.,
for all ;
for each , the mapping from into C is continuous.
for all .
where is an arbitrary fixed element. Banach’s contraction mapping principle guarantees that has a unique fixed point in C. It is unclear, in general, what the behavior of is as , even if T has a fixed point. However, in the case of T having a fixed point, Browder  proved that converges strongly to a fixed point of T that is nearest to u in the framework of Hilbert spaces. Reich  extended Browder’s result to the setting of Banach spaces and proved, in a uniformly smooth Banach space, that converges strongly to a fixed point of T and the limit defines the (unique) sunny nonexpansive retraction from C onto .
He proved that the sequence generated by (1.5) converges to a fixed point of T.
where C is a nonempty closed convex subset of a real Hilbert space H, , is a sequence in , is a sequence of positive real numbers divergent to ∞. Under suitable conditions, they proved strong convergence of to a member of ℱ.
They proved that converges to the same point of ℱ in a reflexive strictly Banach space with a uniformly Gâteaux differentiable norm.
where C is a nonempty closed convex subset of a complete space X, , and are sequences of real numbers satisfying , , and . The proved that and converges to the element of ℱ nearest to u. For other related results, see [15, 16].
Furthermore, they also obtained that defined by (1.11) converges strongly as to under certain appropriate conditions imposed on .
By using the concept of quasilinearization, Wangkeeree and Preechasilp  improved Shi and Chen’s results. In fact, they proved the strong convergence theorems for two given iterative schemes (1.10) and (1.11) in a complete space without the property .
We also denote by the geodesic segment joining from x to y, that is, . A subset C of a 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 spaces to be very similar to the weak convergence in 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 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 space always has a Δ-convergent subsequence.
Lemma 2.5 
If C is a closed convex subset of a complete 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.
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 space, be a sequence in X and . Then -converges to x if and only if for all .
Lemma 2.8 [, Lemma 2.1]
Then converges to zero as .
3 Viscosity approximation methods
In this section, we present the strong convergence theorems of Moudafi’s viscosity approximation methods for a one-parameter continuous semigroup of nonexpansive mappings in spaces. Before proving main results, we need the following two vital lemmas.
which is the desired result. □
(ii) The proof is similar to (i). □
Now we are in a position to state and prove our main results.
Since , we have that , and so . Hence the sequence converges strongly to , which is the unique solution to the variational inequality (3.3). This completes the proof. □
If , then the following result can be obtained directly from Theorem 3.3.
Then converges strongly as to such that , which is equivalent to the variational inequality (3.3).
Applying Lemma 2.8, we can conclude that . This completes the proof. □
If , then the following corollary can be obtained directly from Theorem 3.5.
Then converges strongly as to such that , which is equivalent to the variational inequality (3.9).
The first author is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.
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