#### Abstract

We establish theorems of strong convergence, for the Ishikawa-type (or two step; *cf*. Ishikawa, 1974) iteration scheme, to a fixed point of a uniformly *L*-Lipschitzian asymptotically demicontractive mapping and a uniformly *L*-Lipschitzian hemicontractive mapping in *CAT*(0) space. Moreover, we will propose some open problems.

#### 1. Introduction

Let be a metric space. One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [1] introduced a convex structure in a metric space . A mapping is a *convex structure* in if
for all and . A metric space together with a convex structure is known as a convex metric space. A nonempty subset of a convex metric space is said to be *convex* if
for all and . In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [1]).

*Example 1 (see [2]). *Let , for all , , and . We define a mapping by
and define a metric by
Then we can show that is a convex metric space, but it is not a normed linear space.

A metric space is a space (the term is due to Gromov [3] and it is an acronym for E. Cartan, A. D. Aleksandrov, and V. A. Toponogov) if it is geodesically connected and if every geodesic triangle in is at least as “thin” as its comparison triangle in the Euclidean plane (see, e.g., [4], page 159). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [4] or Burago et al. [5].

Let be a metric space. A *geodesic path* joining to (or, more briefly, a *geodesic* from to ) is a mapping from a closed interval to such that , , and , for all . In particular, is an isometry and . The image of is called a *geodesic* (or *metric*) *segment* joining and . When it is unique, this geodesic is denoted by . The space is said to be a *geodesic space* if every two points of are joined by a geodesic, and is said to be *uniquely geodesic* if there is exactly one geodesic joining and for each . A subset is said to be *convex* if includes every geodesic segment joining any two of its points.

A* geodesic triangle* is a geodesic metric space that consists of three points (the *vertices* of ) and is a geodesic segment between each pair of vertices (the *edges* of ). A *comparison triangle* for the geodesic triangle in is a triangle in such that for . Such a triangle always exists (see, [4]).

A geodesic metric space is said to be a *space* if all geodesic triangles of appropriate size satisfy the following comparison axiom.

Let be a geodesic triangle in and let be a comparison triangle for . Then is said to satisfy the *inequality* if for all and all comparison points ,
Complete spaces are often called *Hadamard spaces* (see, [6]). If are points of a space and if is the midpoint of the segment , which we will denote by , then the inequality implies
This inequality is the (CN) inequality of Bruhat and Tits [7]. In fact, a geodesic space is a space if and only if it satisfies the (CN) inequality (*cf.* [4], page 163). The previous inequality has been extended by Khamsi and Kirk [8] as
for any and . The inequality also appeared in [9].

Let us recall that a geodesic metric space is a space if and only if it satisfies the inequality (see, [4], page 163). Moreover, if is a metric space and , then for any , there exists a unique point such that for any and . In view of the previous inequality, space has Takahashi's convex structure . It is easy to see that for any and , As a consequence, Moreover, a subset of space is convex if for any , we have .

*Definition 2. *Let be a nonempty subset of a metric space . Let denote the fixed point set of . Let .(1) A mapping is said to be -*strict asymptotically pseudocontractive* with sequence if for some constant , and
for all , . If , then is said to be *asymptotically nonexpansive* with sequence , that is,
(2) A mapping is said to be *asymptotically demicontractive* with sequence if for some constant , , and
for all , . If , then is said to be *asymptotically quasi-nonexpansive* with sequence , that is,
(3) A mapping is said to be *asymptotically pseudocontractive* with sequence if and
for all , .(4) A mapping is said to be *asymptotically hemicontractive* with sequence if and
for all , .(5) A mapping is said to be *uniformly* -Lipschitzian if for some constant ,
for all .

Liu [10] has proved the convergence of Mann and Ishikawa iterative sequence for uniformly -Lipschitzian asymptotically demicontractive and hemicontractive mappings in Hilbert space (*cf*. [11]). The existence of (common) fixed points of one mapping (or two mappings or family of mappings) is not known in many situations. So the approximation of fixed points of one or more nonexpansive, asymptotically nonexpansive, or asymptotically quasi-nonexpansive mappings by various iterations have been extensively studied in Banach spaces, convex metric spaces, spaces, and so on (see, [2, 6, 8, 9, 12–27]).

In this paper, we establish theorems of strong convergence for the Ishikawa-type (or two step, *cf.* [28]) iteration scheme to a fixed point of a uniformly -Lipschitzian asymptotically demicontractive mapping and a uniformly -Lipschitzian asymptotically hemicontractive mapping in space. Moreover, we will propose some open problems.

#### 2. Preliminaries

We introduce the following iteration process.

Let be a nonempty convex subset of a space and let be a given mapping. Let be a given point.

*Algorithm 3. *The sequences and defined by the iterative process
is called an *Ishikawa-type* iterative sequence (*cf.* [28]).

If , then Algorithm 3 reduces to the following

*Algorithm 4. *The sequence defined by the iterative process
is called a *Mann-type* iterative sequence (*cf.* [29]).

Lemma 5 (see [10]). *Let sequences , satisfy that
**, for all , is convergent, and has a subsequence converging to . Then, we must have
*

#### 3. Convergence Theorems

Lemma 6. *Let be a space and let be a nonempty convex subset of . Let be an uniformly -Lipschitzian mapping and let , be sequence in . Define the iteration scheme as Algorithm 3. Then
**
for all .*

*Proof. *Let . We have
From (22), we get
From (22) and (23), we get
This completes the proof of Lemma 6.

Theorem 7. *Let be a complete space, let be a nonempty bounded closed convex subset of , and let be a completely continuous and uniformly -Lipschitzian and asymptotically demicontractive with sequence , , , , for all and some . Given , define the iteration scheme by
**
Then converges strongly to some fixed point of .*

*Proof. *Since is a completely continuous mapping in a bounded closed convex subset of complete metric space, from Schauder's theorem, is nonempty. It follows from inequality that
for all . Since is a asymptotically demicontractive, we get
Since , we have . Thus,
From (27), we have
for all . Since is bounded and is self-mapping in , there exist some so that , for all . Since , it follows from (29) that
Therefore,
So
for all . Since , we get
Therefore,
Since is a uniformly -Lipschitzian, it follows from Lemma 6 that
Since is a bounded sequence and is completely continuous, there exist a convergent subsequence of . Therefore, from (35), has a convergent subsequence . Let . It follows from the continuity of and (35), we have . Therefore, has a subsequence which converges to the fixed point of . Let in the inequality (30). Since and , from (30) and Lemma 5, we have
Therefore,
This completes the proof of Theorem 7.

Corollary 8. *Let be a complete space, let be a nonempty bounded closed convex subset of , and let be a completely continuous and uniformly -Lipschitzian and -strict asymptotically pseudocontractive with sequence , , , and , for all and some . Given , define the iteration scheme by
**
Then converges strongly to some fixed point of .*

*Proof. *By Definition 2, is -strict asymptotically pseudocontractive; then must be asymptotically demicontractive. Therefore, Corollary 8 can be proved by using Theorem 7.

Lemma 9. *Let be a space and let be a nonempty convex subset of . Let be an uniformly -Lipschitzian and asymptotically hemicontractive with sequence , for all , and is nonempty. Define the iteration scheme as follows:
**
Then the following inequality holds:
**
for all .*

*Proof. *It follows from inequality that
for all . Since is asymptotically hemicontractive, we get
From (42) and (44), we have
From inequality, we have
Substituting (45) and (46) into (43), we get
From (41) and (47), we obtain
Since is uniformly -Lipschitzian, we have
Substituting (49) into (48), we obtain
This completes the proof of Lemma 9.

Lemma 10. *Let be a space and let be a nonempty bounded convex subset of . Let be a uniformly -Lipschitzian and asymptotically hemicontractive with sequence , for all and . Let be nonempty. Given , define the iteration scheme by
**
If for some and , then
*

*Proof. *First, we will prove . From Lemma 9 and , we have
Thus
Since , we have . Hence, is bounded. By boundedness of and , we obtain that is bounded. Therefore, there exists a constant such that
From (54) and (55), we get

Let . Since , there exists such that
for all . Suppose that , then there exist a and a subsequence of such that
Without loss of generality, we let . From (56), we have
so
From (57)–(60) and , we obtain
Since and the boundedness of , the right side of (61) is bounded. However, if we have , then the left side of (61) is unbounded. This is a contradiction. Therefore,
Since is a uniformly -Lipschitzian, from Lemma 6, we get
This completes the proof of Lemma 10.

Theorem 11. *Let be a complete space, let be a nonempty bounded closed convex subset of , and let be a completely continuous and uniformly -Lipschitzian and asymptotically hemicontractive with sequence satisfying , for all . Given , define the iterative scheme by
**
If with for some and , then converges strongly to some fixed point of .*

*Proof. *Since is a completely continuous mapping in a bounded closed convex subset of complete metric space, from Schauder's theorem, is nonempty. Since is completely continuous, there exist a convergent subset of . Let
Since , from Lemma 10, we have
On the other hand, from the continuity of , (66), and Lemma 10, we have
This means that is a fixed point of . From (55), (57), and , we obtain Lemma 9 that
From (66), there exists a subsequence of which converges to . Therefore, from Lemma 5 and (68),
Hence,
This completes the proof of Theorem 11.

Corollary 12. *Let be a complete space, let be a nonempty bounded closed convex subset of , and let be a completely continuous and uniformly -Lipschitzian and asymptotically pseudocontractive with sequence satisfying , for all . Given , define the iterative scheme by
**
If with for some and , then converges strongly to some fixed point of .*

*Proof. *By Definition 2, is an asymptotically pseudocontractive mapping, then is an asymptotically hemicontractive mapping. Since , we have . Obviously, . Therefore, Corollary 12 can be proved by using Theorem 11.

#### 4. Some Remarks and Open Problems

Let be a semigroup. We denote by the space of all bounded real-valued functions defined on with supremum norm. For each , we define the left and right translation operators and on by
for each and , respectively. Let be a subspace of containing . An element in the dual space of is said to be a * mean* on if . For , we can define a point evaluation by for each . It is well known that is mean on if and only if
for each . Each mean on is the weak*-limit of convex combination of point evaluations.

Let be a translation invariant subspace of (i.e., and for each ) containing . Then a mean on is said to be *left invariant* (resp., *right invariant*) if
for each and . A mean on is said to be *invariant* if is both left and right invariant ([30–34]). is said to be *left* (resp., *right*) *amenable* if has a left (resp., right) invariant mean. is amenable if is left and right amenable. In this case, we say that the semigroup is an amenable semigroup (see [35, 36]). Moreover, is amenable when is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable.

A net of means on is said to be *asymptotically left* (resp., *right*) *invariant* if
for each and , and it is said to be *left* (resp., *right*) *strongly asymptotically invariant* (or *strong regular*) if
for each , where and are the adjoint operators of and , respectively. Such nets were first studied by Day in [35] where they were called *weak*** invariant* and * norm invariant*, respectively.

It is easy to see that if a semigroup is left (resp., right) amenable, then the semigroup , where for all is also left (resp., right) amenable and conversely.

A semigroup is called *left reversible* if any two right ideals of have nonvoid intersection, that is, for . In this case, is a directed system when the binary relation “” on is defined by if and only if for . It is easy to see that for all . Further, if , then for all . The class of left reversible semigroup includes all groups and commutative semigroups. If a semigroup is left amenable, then is left reversible. But the converse is not true ([31, 37–41]).

Let be a semigroup and denote the fixed point set of . Then is called a *representation of * if and for each . We denote by the set of common fixed points of , that is,

*Open Problem 1. *It will be interesting to obtain a generalization of both Theorems 7 and 11 to commutative, amenable, and reversible semigroups as in the case of Hilbert spaces or some Banach spaces (*cf.* [8, 30, 32, 42–45]).

For a real number , a space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding triangle in a model space with curvature .

For , the 2-dimensional model space is the Euclidean space with the metric induced from the Euclidean norm. For , is the 2-dimensional sphere whose metric is length of a minimal great arc joining each two points. For , is the 2-dimensional hyperbolic space with the metric defined by a usual hyperbolic distance. For more details about the properties of spaces, see [4, 46–48].

*Open Problem 2. *It will be interesting to obtain a generalization of both Theorems 7 and 11 to space.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author would like to thank Professor A. T.-M Lau and Professor J. K. Kim for their helpful suggestions. Also, special thanks are due to the referees for their deep insight which improved the presentation of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01010526).