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

Abstract

It is shown that the notion of mappings satisfying condition (𝐾) 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 (𝐾) and a multivalued mapping satisfying condition (𝐶_𝜆) for some 𝜆∈(0,1). Our results properly contain the results of Abkar and Eslamian (2012), Akkasriworn et al. (2012), and many others.

1. Introduction

Let (𝑋,𝑑) be a metric space. A mapping 𝑡∶𝑋→𝑋 is said to be nonexpansive if 𝑑𝑡(𝑥),𝑡𝑦≤𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋. (1)

A point 𝑥∈𝑋 is called a fixed point of 𝑡 if 𝑥=𝑡(𝑥). We shall denote by Fix(𝑡) the set of fixed points of 𝑡. The mapping 𝑡 is said to be quasi-nonexpansive if Fix(𝑡)≠∅ and 𝑑𝑡(𝑥),𝑦≤𝑑𝑥,𝑦∀𝑥∈𝑋,𝑦∈Fix(𝑡). (2)

A single-valued mapping 𝑡∶𝑋→𝑋 and a multivalued mapping 𝑇∶𝑋→2^𝑋 are said to be commute if 𝑡(𝑇(𝑥))⊆𝑇(𝑡(𝑥))∀𝑥∈𝑋. (3)

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 (𝐶). Later on, García-Falset et al. [13] introduced two generalizations of condition (𝐶), namely, conditions (𝐸) and (𝐶_𝜆), 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 (𝐸) and (𝐶_𝜆) for some 𝜆∈(0,1). Very recently, Akkasriworn et al. [19] introduced a condition on mappings, namely, condition (𝐾), 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 (𝐾) and a multivalued mapping satisfying conditions (𝐸) and (𝐶_𝜆) for some 𝜆∈(0,1).

In this note, motivated by the above results, we prove that the condition (𝐾) 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 (𝑋,𝑑,𝑊) where (𝑋,𝑑) is a metric space and 𝑊∶𝑋×𝑋×[0,1]→𝑋 is such that for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝛼,𝛽∈[0,1], we have(W1)𝑑(𝑧,𝑊(𝑥,𝑦,𝛼))≤(1−𝛼)𝑑(𝑧,𝑥)+𝛼𝑑(𝑧,𝑦); (W2)𝑑(𝑊(𝑥,𝑦,𝛼),𝑊(𝑥,𝑦,𝛽))=|𝛼−𝛽|𝑑(𝑥,𝑦); (W3)𝑊(𝑥,𝑦,𝛼)=𝑊(𝑦,𝑥,1−𝛼); (W4)𝑑(𝑊(𝑥,𝑧,𝛼),𝑊(𝑦,𝑤,𝛼))≤(1−𝛼)𝑑(𝑥,𝑦)+𝛼𝑑(𝑧,𝑤).

If 𝑥,𝑦∈𝑋, and 𝛼∈[0,1], then we use the notation (1−𝛼)𝑥⊕𝛼𝑦 for 𝑊(𝑥,𝑦,𝛼). It is easy to see that for any 𝑥,𝑦∈𝑋, and 𝛼∈[0,1], we have 𝑑(),𝑑()=()𝑑.𝑥,1−𝛼𝑥⊕𝛼𝑦=𝛼𝑑𝑥,𝑦𝑦,1−𝛼𝑥⊕𝛼𝑦1−𝛼𝑥,𝑦 (4)

We shall denote by [𝑥,𝑦] the set {(1−𝛼)𝑥⊕𝛼𝑦∶𝛼∈[0,1]}. A nonempty subset 𝐶 of 𝑋 is said to be convex if [𝑥,𝑦]⊆𝐶 for all 𝑥,𝑦∈𝐶.

Definition 2 (see [20]). The hyperbolic space (𝑋,𝑑,𝑊) is called uniformly convex if for any 𝑟>0, and 𝜀∈(0,2] there exists a 𝛿∈(0,1] such that for all 𝑎,𝑥,𝑦∈𝑋 with 𝑑(𝑥,𝑎)≤𝑟, 𝑑(𝑦,𝑎)≤𝑟, and 𝑑(𝑥,𝑦)≥𝑟𝜀, it is the case that 𝑑121𝑥⊕2𝑦,𝑎≤(1−𝛿)𝑟. (5)

A function 𝜂∶(0,∞)×(0,2]→(0,1] providing such a 𝛿∶=𝜂(𝑟,𝜀) for given 𝑟>0 and 𝜀∈(0,2] 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, 𝑋 stands for a complete uniformly convex hyperbolic space having a modulus of uniform convexity 𝜂 such that for a fixed 𝜀∈(0,2], 𝜂(⋅,𝜀) is a constant function on (0,∞).

The following lemma can be found in [21].

Lemma 3. Let 𝐶 be a nonempty closed convex subset of 𝑋 and 𝑥∈𝑋. Then there exists a unique point 𝑥₀∈𝐶 such that 𝑑𝑥,𝑥₀=𝑑(𝑥,𝐶)∶=inf𝑑𝑥,𝑦∶𝑦∈𝐶. (6)

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

Lemma 4. Fix 𝑎∈𝑋. For each 𝑟>0 and for each 𝜀∈(0,2], set Ψ()1𝑟,𝜀=inf2𝑑²()+1𝑎,𝑥2𝑑²𝑎,𝑦−𝑑²1𝑎,21𝑥⊕2𝑦, (7) where the infimum is taken over all 𝑥,𝑦∈𝑋 such that 𝑑(𝑥,𝑎)≤𝑟, 𝑑(𝑦,𝑎)≤𝑟 and 𝑑(𝑥,𝑦)≥𝑟𝜀. Then Ψ(𝑟,𝜀)>0 for any 𝑟>0 and 𝜀∈(0,2]. Moreover, for each fixed 𝑟>0, we have(i)Ψ(𝑟,0)=0; (ii)Ψ(𝑟,𝜀) is a nondecreasing function of 𝜀;(iii)If  lim_𝑛→∞Ψ(𝑟,𝑡_𝑛)=0, then lim_𝑛→∞𝑡_𝑛=0.

We shall denote by 2^𝐶 the family of nonempty subsets of 𝐶, by 𝐶𝐵(𝐶) the family of nonempty closed and bounded subsets of 𝐶, by 𝐾(𝐶) the family of nonempty compact subsets of 𝐶, and by 𝐾𝐶(𝐶) the family of nonempty compact convex subsets of 𝐶. Let 𝐻 be the Hausdorff distance on 𝐶𝐵(𝐶), that is, 𝐻(𝐴,𝐵)∶=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴),𝐴,𝐵∈𝐶𝐵(𝐶). (8)

Definition 5. A multivalued mapping 𝑇∶𝐶→2^𝐶 is said to satisfy (i)condition (𝐸) if there exists 𝜇≥1 such that for each 𝑥,𝑦∈𝐶, 𝑑𝑥,𝑇𝑦≤𝜇𝑑(𝑥,𝑇(𝑥))+𝑑𝑥,𝑦; (9)(ii)condition (𝐶_𝜆) if there exists 𝜆∈(0,1) such that for each 𝑥,𝑦∈𝐶, ((𝑥𝐻𝑇(𝑥)𝑦.𝜆𝑑𝑥,𝑇))≤𝑑𝑥,𝑦implies,𝑇≤𝑑𝑥,𝑦 (10)

We say that 𝐼−𝑇 is strongly demiclosed if for every sequence {𝑥_𝑛} in 𝐶 which converges to 𝑥∈𝐶 and such that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇(𝑥_𝑛))=0, we have 𝑥∈𝑇(𝑥).

We note that for every continuous mapping 𝑇∶𝐶→2^𝐶,𝐼−𝑇 is strongly demiclosed but the converse is not true (see [13, Example 5]). Notice also that if 𝑇 satisfies condition (𝐸), then 𝐼−𝑇 is strongly demiclosed (see [23, Proposition 2.10]).

Definition 6. A single-valued mapping 𝑡∶𝐶→𝐶 is said to(i) satisfy condition (𝐾) if Fix(𝑡) is nonempty closed and convex, and for each 𝑥∈Fix(𝑡) and any closed convex subset 𝐴 with 𝑡(𝐴)⊆𝐴, the nearest point of 𝑥 in 𝐴 must be contained in Fix(𝑡);(ii) be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛} of positive numbers with lim_𝑛→∞𝑘_𝑛=1 and such that 𝑑𝑡^𝑛(𝑥),𝑡^𝑛𝑦≤𝑘_𝑛𝑑𝑥,𝑦∀𝑥,𝑦∈𝐶,𝑛∈ℕ; (11) (iii) be pointwise asymptotically nonexpansive if there exists a sequence of mappings 𝛼_𝑛∶𝐶→[0,∞) with lim_𝑛→∞𝛼_𝑛(𝑥)=1 and such that 𝑑𝑡^𝑛(𝑥),𝑡^𝑛𝑦≤𝛼_𝑛(𝑥)𝑑𝑥,𝑦∀𝑥,𝑦∈𝐶,𝑛∈ℕ; (12) (iv) be asymptotically quasi-nonexpansive if Fix(𝑡) is nonempty and there exists a sequence {𝑘_𝑛} of positive numbers with lim_𝑛→∞𝑘_𝑛=1 and such that 𝑑𝑡^𝑛(𝑥),𝑝≤𝑘_𝑛𝑑𝑥,𝑝∀𝑥∈𝐶,𝑝∈Fix(𝑡),𝑛∈ℕ. (13)

3. Main Results

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

Proposition 7. Let 𝐶 be a nonempty convex subset of 𝑋. If 𝑡∶𝐶→𝐶 is a quasi-nonexpansive mapping, then 𝑡 satisfies condition (𝐾).

Proof. By [24, Theorem 4.2], Fix(𝑡) is closed and convex. Let 𝑥∈Fix(𝑡) and 𝐴 be a closed convex subset of 𝐶 with 𝑡(𝐴)⊆𝐴. Let 𝑦∈𝐴 be such that 𝑑(𝑥,𝑦)=𝑑(𝑥,𝐴). Since 𝑡 is quasi-nonexpansive, 𝑑(𝑥,𝑡(𝑦))≤𝑑(𝑥,𝑦). By the uniqueness of 𝑦, we have 𝑡(𝑦)=𝑦. Therefore 𝑡 satisfies condition (𝐾).

The following two propositions show that the notion of mappings satisfying condition (𝐾) 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 (𝐾) but is neither pointwise asymptotically nonexpansive nor asymptotically quasi-nonexpansive, see [19].

Proposition 8. Let 𝐶 be a nonempty bounded closed convex subset of 𝑋. If 𝑡∶𝐶→𝐶 is a pointwise asymptotically nonexpansive mapping, then 𝑡 satisfies condition (𝐾).

Proof. By [9, Theorem 3.11], Fix(𝑡) is nonempty closed and convex. Since 𝐶 is bounded, there exists 𝑟>0 such that 𝑑(𝑎,𝑏)≤𝑟 for all 𝑎,𝑏∈𝐶. We now let 𝑥∈Fix(𝑡) and 𝐴 be a closed convex subset of 𝐶 with 𝑡(𝐴)⊆𝐴. Let 𝑦∈𝐴 be such that 𝑑(𝑥,𝑦)=𝑑(𝑥,𝐴). Since 𝑋 is uniformly convex, then by Lemma 4 for each integers 𝑙,𝑚≥1, we have 𝑑²1𝑥,2𝑡^𝑙𝑦⊕12𝑡^𝑚𝑦≤12𝑑²𝑥,𝑡^𝑙𝑦+12𝑑²𝑥,𝑡^𝑚𝑦1−Ψ𝑟,𝑟𝑑𝑡^𝑙𝑦,𝑡^𝑚𝑦≤12𝛼²_𝑙(𝑥)𝑑²+1𝑥,𝑦2𝛼²_𝑚(𝑥)𝑑²1𝑥,𝑦−Ψ𝑟,𝑟𝑑𝑡^𝑙𝑦,𝑡^𝑚𝑦. (14)
Since 𝑑(𝑥,𝑦)=𝑑(𝑥,𝐴) and 𝐴 is convex, we have 𝑑²𝑥,𝑦≤𝑑²1𝑥,2𝑡^𝑙𝑦⊕12𝑡^𝑚𝑦. (15)
This, together with (14), we get Ψ1𝑟,𝑟𝑑𝑡^𝑙𝑦,𝑡^𝑚𝑦≤12𝛼²_𝑙(𝑥)+12𝛼²_𝑚(𝑥)−1𝑑²𝑥,𝑦. (16)
Consequently, lim_{𝑙,𝑚→∞}Ψ(𝑟,(1/𝑟)𝑑(𝑡^𝑙(𝑦),𝑡^𝑚(𝑦)))=0. By Lemma 4, lim_{𝑙,𝑚→∞}𝑑(𝑡^𝑙(𝑦),𝑡^𝑚(𝑦))=0. Hence {𝑡^𝑛(𝑦)} is a Cauchy sequence. Let lim_𝑛→∞𝑡^𝑛(𝑦)=𝑧∈𝐴. Now, letting 𝑙,𝑚→∞ in (14), we get that 𝑑(𝑥,𝑧)≤𝑑𝑥,𝑦=𝑑(𝑥,𝐴). (17)
Since 𝑡 is continuous, 𝑑(𝑥,𝑡(𝑧))=lim_𝑛→∞𝑑𝑥,𝑡^𝑛+1𝑦=𝑑(𝑥,𝑧). (18)
By (17), (18), and the uniqueness of 𝑦, we get 𝑦=𝑧=𝑡(𝑧)∈Fix(𝑡).

Proposition 9. Let 𝐶 be a nonempty bounded closed convex subset of 𝑋. If 𝑡∶𝐶→𝐶 is continuous and asymptotically quasi-nonexpansive, then 𝑡 satisfies condition (𝐾).

Proof. Since 𝑡 is continuous, Fix(𝑡) is closed. Next, we show that Fix(𝑡) is convex. Let 𝑢,𝑣 be two different points in Fix(𝑡) and let 𝑤=(1/2)𝑢⊕(1/2)𝑣. It is enough to show that 𝑤∈Fix(𝑡). Since 𝑡 is asymptotically quasi-nonexpansive, 𝑑𝑡^𝑛(𝑤),𝑢≤𝑘_𝑛𝑑()=𝑘𝑤,𝑢^𝑛2𝑑(),𝑑𝑡𝑢,𝑣_𝑛(𝑤),𝑣≤𝑘_𝑛𝑑()=𝑘𝑤,𝑣^𝑛2𝑑(𝑢,𝑣). (19)
Let 𝑟=𝑑(𝑢,𝑣)/2. Then, for each 𝜀>0 there exists 𝑛₀∈ℕ such that if 𝑛≥𝑛₀ then 𝑡^𝑛(𝑤)∈𝐵(𝑢,𝑟+𝜀)∩𝐵(𝑣,𝑟+𝜀)∶=𝐷_𝜀. (20)
We will show that the diameters of the sets 𝐷_𝜀 tend to 0 as 𝜀 tends to 0 and so lim_𝑛→∞𝑡^𝑛(𝑤)=𝑤, which proves that 𝑤∈Fix(𝑡). Let 𝑟_𝜀=𝑑(𝑢,𝐷_𝜀), 𝑑_𝜀=diam(𝐷_𝜀) and 𝑑=lim_𝜀→0𝑑_𝜀=inf_𝜀>0𝑑_𝜀. Then 𝑟_𝜀→𝑟 as 𝜀→0. Assume that 𝑑>0 and let 𝜉∈(0,𝑑/2). Thus, for each 𝜀∈(0,𝜉) there exist 𝑥_𝜀,𝑦_𝜀∈𝐷_𝜀 such that 𝑑(𝑥_𝜀,𝑦_𝜀)≥𝑑_𝜀−𝜀≥𝑑−𝜀≥𝑑/2. Since 𝑑(𝑥_𝜀,𝑢)≤𝑟+𝜀, 𝑑(𝑦_𝜀,𝑢)≤𝑟+𝜀 and 𝑑(𝑥_𝜀,𝑦_𝜀)≥ 𝑑/2≥ (𝑟+𝜀)(𝑑/2(𝑟+1)), we get that for each 𝜀∈(0,𝜉), 𝑟_𝜀1≤𝑑2𝑥_𝜀⊕12𝑦_𝜀≤𝑑,𝑢1−𝜂𝑟+𝜀,2())=𝑑𝑟+1(𝑟+𝜀1−𝜂𝑟+1,2()().𝑟+1𝑟+𝜀 (21)
By letting 𝜀→0, we get that 𝑟≤(1−𝜂(𝑟+1,𝑑/2(𝑟+1)))𝑟<𝑟, which is a contradiction. Hence Fix(𝑡) is convex. The proof of the remaining part closely follows the proof of Proposition 8, upon replacing 𝛼_𝑚(𝑥) with 𝑘_𝑚.

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 𝐶 be a nonempty bounded closed convex subset of 𝑋. Suppose that 𝑇∶𝐶→𝐾(𝐶) satisfies condition (𝐶_𝜆) and 𝐼−𝑇 is strongly demiclosed. Then 𝑇 has a fixed point.

Now, we are ready to prove our main theorem.

Theorem 12. Let 𝐶 be a nonempty bounded closed convex subset of 𝑋 and 𝑡∶𝐶→𝐶 be a mapping satisfying condition (𝐾). Suppose that 𝑇∶𝐶→𝐾𝐶(𝐶) satisfies condition (𝐶_𝜆) and 𝐼−𝑇 is strongly demiclosed. If 𝑡 and 𝑇 commute, then there exists 𝑧∈𝐶 such that 𝑧=𝑡(𝑧)∈𝑇(𝑧).

Proof. This proof is patterned after the proof of [25, Theorem 3.1]. Commutative of 𝑡 and 𝑇 implies that 𝑡(𝑇(𝑥))⊆𝑇(𝑥) for all 𝑥∈Fix(𝑡). Then we have Fix(𝑡)∩𝑇(𝑥)≠∅ for all 𝑥∈Fix(𝑡) since 𝑡 satisfies condition (𝐾). Therefore, the mapping 𝐹(⋅)∶=𝑇(⋅)∩Fix(𝑡)∶Fix(𝑡)→𝐾𝐶(Fix(𝑡)) is well defined. Since 𝐼−𝑇 is strongly demiclosed, then 𝐼−𝐹 is strongly demiclosed. Next, we show that 𝐹 satisfies condition (𝐶_𝜆). Let 𝑥,𝑦∈Fix(𝑡) be such that 𝜆𝑑(𝑥,𝐹(𝑥))≤𝑑𝑥,𝑦. (22)
This implies that 𝜆𝑑(𝑥,𝑇(𝑥))≤𝑑(𝑥,𝑦) and hence 𝐻(𝑇(𝑥),𝑇(𝑦))≤𝑑(𝑥,𝑦) since 𝑇 satisfies condition (𝐶_𝜆). We claim that 𝑑(𝑢,𝐹(𝑣))=𝑑(𝑢,𝑇(𝑣)) for all 𝑢,𝑣∈Fix(𝑡). Let 𝑎 be the point in 𝑇(𝑣) such that 𝑑(𝑢,𝑎)=𝑑(𝑢,𝑇(𝑣)). Again by the condition (𝐾), we have 𝑎∈Fix(𝑡). This shows that 𝑑(𝑢,𝐹(𝑣))= 𝑑(𝑢,𝑇(𝑣)). Now, for each 𝑥,𝑦∈Fix(𝑡) satisfying (22), we have 𝐻𝐹(𝑥),𝐹𝑦=maxsup_{(𝑥)𝑢∈𝐹}𝑑𝑢,𝐹𝑦,sup_{𝑦𝑣∈𝐹}𝑑(𝑣,𝐹(𝑥))=maxsup_{(𝑥)𝑢∈𝐹}𝑑𝑢,𝑇𝑦,sup_{𝑦𝑣∈𝐹}𝑑(𝑣,𝑇(𝑥))≤maxsup_{(𝑥)𝑢∈𝑇}𝑑𝑢,𝑇𝑦,sup_{𝑦𝑣∈𝑇}𝑑((𝑥𝑇(𝑥)𝑦.𝑣,𝑇))=𝐻,𝑇≤𝑑𝑥,𝑦 (23)
By Theorem 11, there exists 𝑧∈Fix(𝑡) such that 𝑧∈𝐹(𝑧). As a result, we have 𝑡(𝑧)=𝑧∈𝑇(𝑧).

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

Corollary 13 (see [18, Theorem 3.2]). Let 𝐶 be a nonempty bounded closed convex subset of a complete CAT(0) space 𝑋. Let 𝑓∶𝐶→𝐶 be a pointwise asymptotically nonexpansive mapping, and 𝑔∶𝐶→𝐶 a quasi-nonexpansive mapping, and let 𝑇∶𝐶→𝐾𝐶(𝐶) be a multivalued mapping satisfying conditions (𝐸) and (𝐶_𝜆) for some 𝜆∈(0,1). If 𝑓,𝑔, and 𝑇 are pairwise commuting, then there exists a point 𝑧∈𝐶 such that 𝑧=𝑓(𝑧)=𝑔(𝑧)∈𝑇(𝑧).

Corollary 14 (see [19, Theorem 3.3]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝑋. Let 𝑡∶𝐶→𝐶 be a mapping satisfying condition (𝐾), and let 𝑇∶𝐶→𝐾𝐶(𝐶) be a multivalued mapping satisfying conditions (𝐸) and (𝐶_𝜆) for some 𝜆∈(0,1). If 𝑡 and 𝑇 commute, then there exists 𝑧∈𝐶 such that 𝑧=𝑡(𝑧)∈𝑇(𝑧).

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

Example 15. Put 𝑋=ℝ and 𝐶=[−1/4,1]. Let 𝑡 be the identity mapping on 𝐶 and let 𝑇 be the mapping on 𝐶 defined by 𝑇(𝑥)=−11𝑥=0,3−1𝑥𝑥∈4∪3,00,4,31−𝑥𝑥∈4,1. (24)
It is easy to see that 𝑡 and 𝑇 commute. In [13], the authors prove that either 𝑇(𝑥)𝑦≤3−𝑇𝑥−𝑦or4min|𝑥−𝑇(𝑥)|,𝑦−𝑇𝑦≥𝑥−𝑦. (25)

We now let 𝜀∈(0,1/4), then either 𝑇(𝑥)𝑦≤3−𝑇𝑥−𝑦or4+𝜀min|𝑥−𝑇(𝑥)|,𝑦−𝑇𝑦>𝑥−𝑦. (26)

This implies that 𝑇 satisfies condition (𝐶_(3/4)+𝜀) for all 𝜀∈(0,1/4). Let {𝑥_𝑛}={1/𝑛}^∞_𝑛=1, then {𝑥_𝑛} is an approximate fixed point sequence for 𝑇 which converges to 0. But 0 is not a fixed point of 𝑇. This shows that 𝐼−𝑇 is not strongly demiclosed. Obviously, 𝑇 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.