A Note on Common Fixed Point Results in Uniformly Convex Hyperbolic Spaces

It is shown that the notion ofmappings satisfying condition (K) introduced byAkkasriworn et al. (2012) is weaker than the notion of asymptotically quasi-nonexpansivemappings in the sense ofQihou (2001) and isweaker than the notion of pointwise asymptotically nonexpansivemappings in the sense of Kirk andXu (2008).We also obtain a commonfixed point for a commuting pair of amapping satisfying condition (K) and a multivalued mapping satisfying condition (Cλ) for some λ ∈ (0, 1). Our results properly contain the results of Abkar and Eslamian (2012), Akkasriworn et al. (2012), and many others.

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][3][4][5][6][7][8][9][10][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  ∈ (0, 1).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  ∈ (0, 1).
In this note, motivated by the above results, we prove that the condition () is even weaker than asymptotically quasinonexpansiveness 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.
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  0 ∈  such that The following lemma, which is proved by Khamsi and Khan [22], is also needed.Lemma 4. Fix  ∈ .For each  > 0 and for each  ∈ (0, 2], set where the infimum is taken over all ,  ∈  such that (, ) ≤ , (, ) ≤  and (, ) ≥ .Then Ψ(, ) > 0 for any  > 0 and  ∈ (0, 2].Moreover, for each fixed  > 0, we have We shall denote by 2  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, (ii) condition (  ) if there exists  ∈ (0, 1) such that for each ,  ∈ ,   (,  ()) ≤  (, ) implies We say that  −  is strongly demiclosed if for every sequence {  } in  which converges to  ∈  and such that lim  → ∞ (  , (  )) = 0, we have  ∈ ().
We note that for every continuous mapping  :  → 2  ,  −  is strongly demiclosed but the converse is not true (see [13,Example 5]).Notice also that if  satisfies condition (), then  −  is strongly demiclosed (see [

Main Results
We begin this section by proving that every quasi-nonexpansive mapping 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].
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  = () ∈ ().
As consequences of Proposition 7, Proposition 8, and Theorem 12, we obtain the following.
Finally, we show that the strongly demiclosedness of  −  in Theorem 12 cannot be removed.
It is easy to see that  and  commute.In [13], the authors prove that either This implies that  satisfies condition ( (3/4)+ ) for all  ∈ (0, 1/4).Let {  } = {1/} ∞ =1 , then {  } is an approximate fixed point sequence for  which converges to 0. But 0 is not a fixed point of .This shows that  −  is not strongly demiclosed.Obviously,  does not have a fixed point.

Example 15 .
Put  = R and  = [−1/4, 1].Let  be the identity mapping on  and let  be the mapping on  defined by
Definition 6.A single-valued mapping  :  →  is said to (i) satisfy condition () if Fix() is nonempty closed and convex, and for each  ∈ Fix() and any closed convex subset  with () ⊆ , the nearest point of  in  must be contained in Fix();(ii)