Existence and Approximation of Attractive Points of the Widely More Generalized Hybrid Mappings in Hilbert Spaces

and Applied Analysis 3 Lemma 4 (Aoyama et al. [24]). Let {s n } be a sequence of nonnegative real numbers, let {α n } be a sequence of [0, 1] with

A mapping  :  →  is called nonexpansive [1] if ‖− ‖ ≤ ‖ − ‖ for all ,  ∈ .A mapping  :  →  is called nonspreading [2], hybrid [3] for all ,  ∈ , respectively; see also [4,5].These three terms are independent, and they are deduced from the notion of firmly nonexpansive mapping in a Hilbert space; see [3].A mapping  :  →  is said to be firmly nonexpansive if      −      2 ≤ ⟨ − ,  − ⟩ (2) for all ,  ∈ ; see, for instance, Goebel and Kirk [6].The class of nonspreading mappings was first defined in a strictly convex, smooth, and reflexive Banach space.The resolvents of a maximal monotone operator are nonspreading mappings; see [2] for more details.These three classes of nonlinear mappings are important in the study of the geometry of infinite dimensional spaces.Indeed, by using the fact that the resolvents of a maximal monotone operator are nonspreading mappings, Takahashi et al. [7] solved an open problem which is related to Ray's theorem [8] in the geometry of Banach spaces.Motivated by these mappings, Kocourek et al. [9] introduced a broad class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings, and hybrid mappings.A mapping  :  →  is said to be generalized hybrid if there exist ,  ∈ R such that       −      2 + (1 − )      −      An (, )-generalized hybrid mapping is nonexpansive for  = 1 and  = 0, nonspreading for  = 2 and  = 1, and hybrid for  = 3/2 and  = 1/2.They proved fixed point theorems for such mappings; see also Kohsaka and Takahashi [10] and Iemoto and Takahashi [4].Moreover, they proved the following nonlinear ergodic theorem which generalizes Baillon's theorem [11].
Theorem 1 (see [9]).Let  be a real Hilbert space, let  be a nonempty closed convex subset of , let  be a generalized hybrid mapping from  into itself with () ̸ = 0, and let  be the metric projection of  onto ().Then for any  ∈ , converges weakly to  ∈ (), where  = lim  → ∞   .
We see that the set  needs to be closed and convex in Theorem 1.As a contrast, Takahashi and Takeuchi [12] proved the following theorem which establishes the existence of attractive point and mean convergence property without the convexity assumption in a Hilbert space; see also Lin and Takahashi [13] and Takahashi et al. [14].
Theorem 2. Let  be a real Hilbert space, and let  be a nonempty subset of .Let  be a generalized hybrid mapping from  into itself.Let {V  } and {  } be sequences defined by for all  ∈ N. If {V  } is bounded, then the followings hold: (1) () is nonempty, closed, and convex; (2) {  } converges weakly to  0 ∈ (), where  0 = lim  → ∞  () V  and  () is the metric projection of  onto ().
Very recently Kawasaki and Takahashi [15] introduced a class of nonlinear mappings in a Hilbert space which covers contractive mappings [16] for any ,  ∈ ; see also Kawasaki and Takahashi [17].
A mapping  :  →  is called quasi-nonexpansive if () ̸ = 0 and ‖ − ‖ ≤ ‖ − ‖ for all  ∈  and  ∈ ().It is well known that if  is closed and convex and  :  →  is quasi-nonexpansive, then () is closed and convex; see Itoh and Takahashi [18].For a simpler proof of such a result in a Hilbert space, see, for example, [19].A generalized hybrid mapping with a fixed point is quasinonexpansive.However, a widely more generalized hybrid mapping is not quasi-nonexpansive generally even if it has a fixed point.In [15], they proved fixed point theorems and nonlinear ergodic theorems of Baillon's type for such new mappings in a Hilbert space.
In this paper, motivated by these results, we establish the attractive point theorem and mean convergence theorem without the commonly required convexity for the widely more generalized hybrid mappings in a Hilbert space.Moreover, we prove a weak convergence theorem of Mann's type [20] and a strong convergence theorem of Shimizu and Takahashi's type [21] for such a class of nonlinear mappings in a Hilbert space which generalize Kocourek et al. [9] and Hojo and Takahashi [22] for generalized hybrid mappings, respectively.

Preliminaries
Throughout this paper, we denote by N the set of positive integers.Let  be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖.We denote the strong convergence and the weak convergence of {  } to  ∈  by   →  and   ⇀ , respectively.Let  be a nonempty subset of .We denote by co the closure of the convex hull of .In a Hilbert space, it is known [1] that for any ,  ∈  and  ∈ R, Furthermore, we have that for any , , ,  ∈ .
Let  be a nonempty closed convex subset of  and  ∈ .Then we know that there exists a unique nearest point  ∈  such that ‖ − ‖ = inf ∈ ‖ − ‖.We denote such a correspondence by  =   .The mapping   is called the metric projection of  onto .It is known that   is nonexpansive and for any  ∈  and  ∈ ; see [1] for more details.For proving a nonlinear ergodic theorem in this paper, we also need the following lemma proved by Takahashi and Toyoda [23].
Lemma 3. Let  be a nonempty closed convex subset of .
To prove a strong convergence theorem in this paper, we need the following lemma.
Lemma 5. Let  be a real Hilbert space, let {  } be a bounded sequence in , and let  be a mean on  ∞ .Then there exists a unique point  0 ∈ co{  |  ∈ N} such that for any  ∈ .
The following result obtained by Takahashi and Takeuchi [12] is important in this paper.Lemma 6.Let  be a Hilbert space, let  be a nonempty subset of , and let  be a mapping from  into .Then () is a closed and convex subset of .
We also know the following result from [14].

Lemma 7.
Let  be a Hilbert space, let  be a nonempty subset of , and let  be a quasi-nonexpansive mapping from  into .Then () ∩  = ().

Attractive Point Theorems
Let  be a real Hilbert space, and let  be a nonempty subset of .Recall that a mapping  from  into  is said to be widely more generalized hybrid [15]  for any ,  ∈ .Such a mapping  is called (, , , , , , )widely more generalized hybrid.An (, , , , , , )-widely more generalized hybrid mapping is generalized hybrid in the sense of Kocourek et al. [9] if  +  = − −  = 1 and  =  =  = 0. We first prove an attractive point theorem for widely more generalized hybrid mappings in a Hilbert space.
Then  has an attractive point if and only if there exists  ∈  such that {   |  = 0, 1, . ..} is bounded.
Conversely suppose that there exists  ∈  such that {   |  = 0, 1, . ..} is bounded.Since  is an (, , , , , , )-widely more generalized hybrid mapping from  into itself, we obtain that for any  ∈ N ∪ {0} and  ∈ .By ( 9) we obtain that Thus we have that From we have that and hence By  +  +  +  ≥ 0, we have that From this inequality and  +  ≥ 0 we obtain that Applying a Banach limit  to both sides of this inequality, we obtain that and hence Since there exists  ∈  by Lemma 5 such that for any  ∈ , we obtain from (24) that We obtain from (9) that and hence Since  +  ≥ 0, we obtain that Since  +  > 0, we obtain that This implies that  ∈  is an attractive point.
Using Theorem 8, we can show the following attractive point theorem for generalized hybrid mappings in a Hilbert space.
Theorem 9 (Takahashi and Takeuchi [12]).Let  be a Hilbert space, let  be a nonempty subset of , and let  be a generalized hybrid mapping from  into ; that is, there exist real numbers  and  such that Proof.An (, )-generalized hybrid mapping  is an (, 1 − , −, −(1 − ), 0, 0, 0)-widely more generalized hybrid mapping.Furthermore, the mapping satisfies the condition (2) in Theorem 8, that is, Then we have the desired result from Theorem 8.

Nonlinear Ergodic Theorems
In this section, using the technique developed by Takahashi [26], we prove a mean convergence theorem without convexity for widely more generalized hybrid mappings in a Hilbert space.Before proving the result, we need the following two lemmas.
As the proof of Theorem 9, we can prove Takahashi and Takeuchi's mean convergence theorem for generalized hybrid mappings in a Hilbert space.
Theorem 13.Let  be a Hilbert space, let  be a nonempty subset of , and let  be a generalized hybrid mapping from  into itself; that is, there exist ,  ∈ R such that for all ,  ∈ .Suppose that () ̸ = 0, and let  be the metric projection from  onto ().Then for any  ∈ , converges weakly to  ∈ (), where  = lim  → ∞   .

Weak Convergence Theorems of Mann's Type
In this section, we prove a weak convergence theorem of Mann's type [20] for widely more generalized hybrid mappings in a Hilbert space by using Lemma 11 and the technique developed by Ibaraki and Takahashi [27,28].
Using Theorem 14, we can show the following weak convergence theorem of Mann's type for generalized hybrid mappings in a Hilbert space.
Proof.As in the proof of Theorem 9, a generalized hybrid mapping is a widely more generalized hybrid mapping.Since {  } ⊂  and  is closed and convex, we have from Theorem 14 that V ∈ () ∩ .A generalized hybrid mapping with () ̸ = 0 is quasi-nonexpansive, we have from Lemma 7 that () ∩  = ().Thus {  } converges weakly to an element V ∈ ().
Using Theorem 16, we can show the following result obtained by Kurokawa and Takahashi [30].
Theorem 17 (Hojo and Takahashi [22]).Let  be a nonempty closed convex subset of a real Hilbert space .Let  be a generalized hybrid mapping of  into itself.Let  Proof.As in the proof of Theorem 9, a generalized hybrid mapping is a widely more generalized hybrid mapping.Since {  }, {  } ⊂  and  is closed and convex, we have from Theorem 16 that  ∈ () ∩ .A generalized hybrid mapping with () ̸ = 0 is quasi-nonexpansive, we have from Lemma 7 that () ∩  = ().Thus {  } and {  } converge strongly to an element  ∈ ().