#### Abstract

It is shown that the notion of mappings satisfying condition introduced by Akkasriworn et al. (2012) is weaker than the notion of asymptotically quasi-nonexpansive mappings in the sense of Qihou (2001) and is weaker than the notion of pointwise asymptotically nonexpansive mappings in the sense of Kirk and Xu (2008). We also obtain a common fixed point for a commuting pair of a mapping satisfying condition and a multivalued mapping satisfying condition for some . Our results properly contain the results of Abkar and Eslamian (2012), Akkasriworn et al. (2012), and many others.

#### 1. Introduction

Let be a metric space. A mapping is said to be *nonexpansive* if

A point is called a *fixed point* of if . We shall denote by the set of fixed points of . The mapping is said to be *quasi-nonexpansive* if and

A single-valued mapping and a multivalued mapping are said to be *commute* if

The first result concerning to the existence of common fixed points for a commuting pair of a single-valued quasi-nonexpansive mapping and a multivalued nonexpansive mapping was established in Hilbert spaces by Itoh and Takahashi [1]. Since then the common fixed point theory for commuting pairs of single-valued and multivalued mappings has been rapidly developed and many of papers have appeared (see, e.g., [2–11] and the references therein).

In 2008, Suzuki [12] introduced a condition on mappings, which is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness and called it condition . Later on, García-Falset et al. [13] introduced two generalizations of condition , namely, conditions and , and studied the existence of fixed points for mappings satisfying such conditions. These conditions were extended to the multivalued case by Abkar and Eslamian [11] and Espínola et al. [14]. However, these conditions still lie between nonexpansiveness and quasi-nonexpansiveness in both single-valued and multivalued cases. On the other hand, Qihou [15] introduced the notion of asymptotically quasi-nonexpansive mappings and Kirk and Xu [16] introduced the notion of pointwise asymptotically nonexpansive mappings. Both of them generalize the notion of asymptotically nonexpansive mappings in the sense of Goebel and Kirk [17].

Recently, Abkar and Eslamian [18] studied the existence of common fixed points for three different classes of generalized nonexpansive mappings including a quasi-nonexpansive single-valued mapping, a pointwise asymptotically nonexpansive single-valued mapping, and a multivalued mapping satisfying conditions and for some . Very recently, Akkasriworn et al. [19] introduced a condition on mappings, namely, condition , which is weaker than both quasi-nonexpansiveness and asymptotically nonexpansiveness and proved the existence of common fixed points for a commuting pair of a single-valued mapping satisfying condition and a multivalued mapping satisfying conditions and for some .

In this note, motivated by the above results, we prove that the condition is even weaker than asymptotically quasi-nonexpansiveness and is weaker than pointwise asymptotically nonexpansiveness in the setting of uniformly convex hyperbolic spaces. Moreover, we also obtain a common fixed point theorem with some weaker assumptions.

#### 2. Preliminaries

*Definition 1 (see [20]). *A *hyperbolic space* is a triple where is a metric space and is such that for all and , we have(W1);
(W2);
(W3);
(W4).

If , and , then we use the notation for . It is easy to see that for any , and , we have

We shall denote by the set . A nonempty subset of is said to be *convex* if for all .

*Definition 2 (see [20]). *The hyperbolic space is called *uniformly convex* if for any , and there exists a such that for all with , , and , it is the case that

A function providing such a for given and is called *a modulus of uniform convexity. *

Obviously, uniformly convex Banach spaces are uniformly convex hyperbolic spaces. CAT(0) spaces are also uniformly convex hyperbolic spaces, see [20, Proposition 8]. From now on, stands for a complete uniformly convex hyperbolic space having a modulus of uniform convexity such that for a fixed , is a constant function on .

The following lemma can be found in [21].

Lemma 3. *Let be a nonempty closed convex subset of and . Then there exists a unique point such that
*

The following lemma, which is proved by Khamsi and Khan [22], is also needed.

Lemma 4. *Fix . For each and for each , set
**
where the infimum is taken over all such that , and . Then for any and . Moreover, for each fixed , we have*(i)*;
*(ii)* is a nondecreasing function of ;*(iii)*If , then .*

We shall denote by the family of nonempty subsets of , by the family of nonempty closed and bounded subsets of , by the family of nonempty compact subsets of , and by the family of nonempty compact convex subsets of . Let be the Hausdorff distance on , that is,

*Definition 5. *A multivalued mapping is said to satisfy (i)*condition * if there exists such that for each ,
(ii)*condition * if there exists such that for each ,

We say that is *strongly demiclosed* if for every sequence in which converges to and such that , we have .

We note that for every continuous mapping , is strongly demiclosed but the converse is not true (see [13, Example 5]). Notice also that if satisfies condition , then is strongly demiclosed (see [23, Proposition 2.10]).

*Definition 6. *A single-valued mapping is said to(i) satisfy *condition * if Fix is nonempty closed and convex, and for each and any closed convex subset with , the nearest point of in must be contained in Fix;(ii) be *asymptotically nonexpansive* if there exists a sequence of positive numbers with and such that
(iii) be *pointwise asymptotically nonexpansive* if there exists a sequence of mappings with and such that
(iv) be *asymptotically quasi-nonexpansive* if Fix is nonempty and there exists a sequence of positive numbers with and such that

#### 3. Main Results

We begin this section by proving that every quasi-nonexpansive mapping satisfies condition .

Proposition 7. *Let be a nonempty convex subset of . If is a quasi-nonexpansive mapping, then satisfies condition .*

*Proof. *By [24, Theorem 4.2], is closed and convex. Let and be a closed convex subset of with . Let be such that . Since is quasi-nonexpansive, . By the uniqueness of , we have . Therefore satisfies condition .

The following two propositions show that the notion of mappings satisfying condition is weaker than the notion of pointwise asymptotically nonexpansive mappings and weaker than the notion of asymptotically quasi-nonexpansive continuous mappings. For a mapping that satisfies condition but is neither pointwise asymptotically nonexpansive nor asymptotically quasi-nonexpansive, see [19].

Proposition 8. *Let be a nonempty bounded closed convex subset of . If is a pointwise asymptotically nonexpansive mapping, then satisfies condition .*

*Proof. *By [9, Theorem 3.11], is nonempty closed and convex. Since is bounded, there exists such that for all . We now let and be a closed convex subset of with . Let be such that . Since is uniformly convex, then by Lemma 4 for each integers , we have

Since and is convex, we have

This, together with (14), we get

Consequently, . By Lemma 4, . Hence is a Cauchy sequence. Let . Now, letting in (14), we get that

Since is continuous,

By (17), (18), and the uniqueness of , we get .

Proposition 9. *Let be a nonempty bounded closed convex subset of . If is continuous and asymptotically quasi-nonexpansive, then satisfies condition .*

* Proof. *Since is continuous, is closed. Next, we show that is convex. Let be two different points in and let . It is enough to show that . Since is asymptotically quasi-nonexpansive,

Let . Then, for each there exists such that if then

We will show that the diameters of the sets tend to as tends to and so , which proves that . Let , and . Then as . Assume that and let . Thus, for each there exist such that . Since , and , we get that for each ,

By letting , we get that , which is a contradiction. Hence is convex. The proof of the remaining part closely follows the proof of Proposition 8, upon replacing with .

*Remark 10. *Continuity seems essential to the proof of Proposition 9. We do not have an example to show that it is necessary.

The following result is a consequence of [23, Theorem 3.2].

Theorem 11. *Let be a nonempty bounded closed convex subset of . Suppose that satisfies condition and is strongly demiclosed. Then has a fixed point. *

Now, we are ready to prove our main theorem.

Theorem 12. *Let be a nonempty bounded closed convex subset of and be a mapping satisfying condition . Suppose that satisfies condition and is strongly demiclosed. If and commute, then there exists such that .*

*Proof. *This proof is patterned after the proof of [25, Theorem 3.1]. Commutative of and implies that for all . Then we have Fix for all since satisfies condition . Therefore, the mapping is well defined. Since is strongly demiclosed, then is strongly demiclosed. Next, we show that satisfies condition . Let be such that

This implies that and hence since satisfies condition . We claim that for all . Let be the point in such that . Again by the condition , we have . This shows that . Now, for each satisfying (22), we have

By Theorem 11, there exists such that . As a result, we have .

As consequences of Proposition 7, Proposition 8, and Theorem 12, we obtain the following.

Corollary 13 (see [18, Theorem 3.2]). *Let be a nonempty bounded closed convex subset of a complete CAT(0) space . Let be a pointwise asymptotically nonexpansive mapping, and a quasi-nonexpansive mapping, and let be a multivalued mapping satisfying conditions and for some . If , and are pairwise commuting, then there exists a point such that .*

Corollary 14 (see [19, Theorem 3.3]). *Let be a nonempty bounded closed convex subset of a uniformly convex Banach space . Let be a mapping satisfying condition , and let be a multivalued mapping satisfying conditions and for some . If and commute, then there exists such that .*

Finally, we show that the strongly demiclosedness of in Theorem 12 cannot be removed.

*Example 15. *Put and . Let be the identity mapping on and let be the mapping on defined by

It is easy to see that and commute. In [13], the authors prove that either

We now let , then either

This implies that satisfies condition () for all . Let , then is an approximate fixed point sequence for which converges to . But is not a fixed point of . This shows that is not strongly demiclosed. Obviously, does not have a fixed point.

#### Acknowledgments

The authors are grateful to Professor Sompong Dhompongsa for his suggestions and advices during the preparation of the paper. This paper was supported by the Commission on Higher Education and Thailand Research Fund.