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

Abstract

We study the widely more generalized hybrid mappings which have been proposed to unify several well-known nonlinear mappings including the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without the convexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type and a strong convergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our results can be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalized hybrid mappings.

1. Introduction

Let 𝐻 be a real Hilbert space, and let 𝐶 be a nonempty subset of 𝐻. For a mapping 𝑇∶𝐶→𝐻, we denote by 𝐹(𝑇) and 𝐴(𝑇) the sets of fixed points and attractive points of  𝑇, respectively, that is, (i)𝐹(𝑇)={𝑧∈𝐶∶𝑇𝑧=𝑧}; (ii)𝐴(𝑇)={𝑧∈𝐻∶‖𝑇𝑥−𝑧‖≤‖𝑥−𝑧‖,∀𝑥∈𝐶}.

A mapping 𝑇∶𝐶→𝐻 is called nonexpansive [1] if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝑇∶𝐶→𝐻 is called nonspreading [2], hybrid [3] if 2𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥²,3𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑇𝑥−𝑦²+𝑇𝑦−𝑥² (1) for all 𝑥,𝑦∈𝐶, 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 𝐹∶𝐶→𝐻 is said to be firmly nonexpansive if 𝐹𝑥−𝐹𝑦²≤𝑥−𝑦,𝐹𝑥−𝐹𝑦 (2) for all 𝑥,𝑦∈𝐶; 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 𝑇∶𝐶→𝐻 is said to be generalized hybrid if there exist 𝛼,𝛽∈ℝ such that 𝛼𝑇𝑥−𝑇𝑦²+(1−𝛼)𝑥−𝑇𝑦²≤𝛽𝑇𝑥−𝑦²+1−𝛽𝑥−𝑦² (3) for all 𝑥,𝑦∈𝐶, where ℝ is the set of real numbers. We call such a mapping an (𝛼,𝛽)-generalized hybrid mapping. An (𝛼,𝛽)-generalized hybrid mapping is nonexpansive for 𝛼=1 and 𝛽=0, nonspreading for 𝛼=2 and 𝛽=1, and hybrid for 𝛼=3/2 and 𝛽=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 𝐻 be a real Hilbert space, let 𝐶 be a nonempty closed convex subset of 𝐻, let 𝑇 be a generalized hybrid mapping from 𝐶 into itself with 𝐹(𝑇)≠∅, and let 𝑃 be the metric projection of 𝐻 onto 𝐹(𝑇). Then for any 𝑥∈𝐶, 𝑆_𝑛1𝑥=𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥 (4) converges weakly to 𝑝∈𝐹(𝑇), where 𝑝=lim_𝑛→∞𝑃𝑇^𝑛𝑥.

We see that the set 𝐶 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 𝐻 be a real Hilbert space, and let 𝐶 be a nonempty subset of 𝐻. Let 𝑇 be a generalized hybrid mapping from 𝐶 into itself. Let {𝑣_𝑛} and {𝑏_𝑛} be sequences defined by 𝑣₁∈𝐶,𝑣_𝑛+1=𝑇𝑣_𝑛,𝑏_𝑛=1𝑛^𝑛_𝑘=1𝑣_𝑘 (5) for all 𝑛∈ℕ. If {𝑣_𝑛} is bounded, then the followings hold: (1)𝐴(𝑇) is nonempty, closed, and convex; (2){𝑏_𝑛} converges weakly to 𝑢₀∈𝐴(𝑇), where 𝑢₀=lim_𝑛→∞𝑃_𝐴(𝑇)𝑣_𝑛 and 𝑃_𝐴(𝑇) is the metric projection of 𝐻 onto 𝐴(𝑇).

Very recently Kawasaki and Takahashi [15] introduced a class of nonlinear mappings in a Hilbert space which covers contractive mappings [16] and generalized hybrid mappings. A mapping 𝑇∶𝐶→𝐻 is called widely more generalized hybrid if there exist 𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂∈ℝ such that 𝛼𝑇𝑥−𝑇𝑦²+𝛽𝑥−𝑇𝑦²+𝛾𝑇𝑥−𝑦²+𝛿𝑥−𝑦²+𝜀‖𝑥−𝑇𝑥‖²+𝜁𝑦−𝑇𝑦²+𝜂(𝑥−𝑇𝑥)−𝑦−𝑇𝑦²≤0 (6) for any 𝑥,𝑦∈𝐶; see also Kawasaki and Takahashi [17].

A mapping 𝑇∶𝐶→𝐻 is called quasi-nonexpansive if 𝐹(𝑇)≠∅ and ‖𝑇𝑥−𝑦‖≤‖𝑥−𝑦‖ for all 𝑥∈𝐶 and 𝑦∈𝐹(𝑇). It is well known that if 𝐶 is closed and convex and 𝑇∶𝐶→𝐻 is quasi-nonexpansive, then 𝐹(𝑇) 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 quasi-nonexpansive. 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.

2. Preliminaries

Throughout this paper, we denote by ℕ the set of positive integers. Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. We denote the strong convergence and the weak convergence of {𝑥_𝑛} to 𝑥∈𝐻 by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively. Let 𝐴 be a nonempty subset of 𝐻. We denote by co𝐴 the closure of the convex hull of 𝐴. In a Hilbert space, it is known [1] that for any 𝑥,𝑦∈𝐻 and 𝛼∈ℝ, 𝑦²−‖𝑥‖²,()𝑦≤2𝑦−𝑥,𝑦𝛼𝑥+1−𝛼²=𝛼‖𝑥‖²+(1−𝛼)𝑦²−𝛼(1−𝛼)𝑥−𝑦². (7)(8)

Furthermore, we have that 2𝑥−𝑦,𝑧−𝑤=‖𝑥−𝑤‖²+𝑦−𝑧²−‖𝑥−𝑧‖²−𝑦−𝑤² (9) for any 𝑥,𝑦,𝑧,𝑤∈𝐻.

Let 𝐷 be a nonempty closed convex subset of 𝐻 and 𝑥∈𝐻. Then we know that there exists a unique nearest point 𝑧∈𝐷 such that ‖𝑥−𝑧‖=inf_𝑦∈𝐷‖𝑥−𝑦‖. We denote such a correspondence by 𝑧=𝑃_𝐷𝑥. The mapping 𝑃_𝐷 is called the metric projection of 𝐻 onto 𝐷. It is known that 𝑃_𝐷 is nonexpansive and 𝑥−𝑃_𝐷𝑥,𝑃_𝐷𝑥−𝑢≥0 (10) for any 𝑥∈𝐻 and 𝑢∈𝐷; 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 𝐷 be a nonempty closed convex subset of 𝐻. Let 𝑃 be the metric projection from 𝐻 onto 𝐷. Let {𝑢_𝑛} be a sequence in 𝐻. If ‖𝑢_𝑛+1−𝑢‖≤‖𝑢_𝑛−𝑢‖ for any 𝑢∈𝐷 and 𝑛∈ℕ, then {𝑃𝑢_𝑛} converges strongly to some 𝑢₀∈𝐷.

To prove a strong convergence theorem in this paper, we need the following lemma.

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 ∑^∞_𝑛=1𝛼_𝑛=∞, let {𝛽_𝑛} be a sequence of nonnegative real numbers with ∑^∞_𝑛=1𝛽_𝑛<∞, and let {𝛾_𝑛} be a sequence of real numbers with limsup_𝑛→∞𝛾_𝑛≤0. Suppose that 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛾_𝑛+𝛽_𝑛 (11) for all 𝑛=1,2,…. Then lim_𝑛→∞𝑠_𝑛=0.

Let 𝑙^∞ be the Banach space of bounded sequences with supremum norm. Let 𝜇 be an element of (𝑙^∞)^∗ (the dual space of 𝑙^∞). Then we denote by 𝜇(𝑓) the value of 𝜇 at 𝑓=(𝑥₁,𝑥₂,𝑥₃,…)∈𝑙^∞. Sometimes, we denote by 𝜇_𝑛(𝑥_𝑛) the value 𝜇(𝑓). A linear functional 𝜇 on 𝑙^∞ is called a mean if 𝜇(𝑒)=‖𝜇‖=1, where 𝑒=(1,1,1,…). A mean 𝜇 is called a Banach limit on 𝑙^∞ if 𝜇_𝑛(𝑥_𝑛+1)=𝜇_𝑛(𝑥_𝑛). We know that there exists a Banach limit on 𝑙^∞. If 𝜇 is a Banach limit on 𝑙^∞, then for 𝑓=(𝑥₁,𝑥₂,𝑥₃,…)∈𝑙^∞, liminf_𝑛→∞𝑥_𝑛≤𝜇_𝑛𝑥_𝑛≤limsup_𝑛→∞𝑥_𝑛. (12) In particular, if 𝑓=(𝑥₁,𝑥₂,𝑥₃,…)∈𝑙^∞ and 𝑥_𝑛→𝑎∈ℝ, then we have 𝜇(𝑓)=𝜇_𝑛(𝑥_𝑛)=𝑎. See [25] for the proof of existence of a Banach limit and its other elementary properties. Using means and the Riesz theorem, we can obtain the following result; see [25, 26].

Lemma 5. Let 𝐻 be a real Hilbert space, let {𝑥_𝑛} be a bounded sequence in 𝐻, and let 𝜇 be a mean on 𝑙^∞. Then there exists a unique point 𝑧₀∈co{𝑥_𝑛∣𝑛∈ℕ} such that 𝜇_𝑛𝑥_𝑛,𝑦=𝑧₀,𝑦 (13) for any 𝑦∈𝐻.

The following result obtained by Takahashi and Takeuchi [12] is important in this paper.

Lemma 6. Let 𝐻 be a Hilbert space, let 𝐶 be a nonempty subset of 𝐻, and let 𝑇 be a mapping from 𝐶 into 𝐻. Then 𝐴(𝑇) is a closed and convex subset of 𝐻.

We also know the following result from [14].

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

3. Attractive Point Theorems

Let 𝐻 be a real Hilbert space, and let 𝐶 be a nonempty subset of 𝐻. Recall that a mapping 𝑇 from 𝐶 into 𝐻 is said to be widely more generalized hybrid [15] if there exist 𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂∈ℝ such that 𝛼𝑇𝑥−𝑇𝑦²+𝛽𝑥−𝑇𝑦²+𝛾𝑇𝑥−𝑦²+𝛿𝑥−𝑦²+𝜀‖𝑥−𝑇𝑥‖²+𝜁𝑦−𝑇𝑦²+𝜂(𝑥−𝑇𝑥)−𝑦−𝑇𝑦²≤0 (14) for any 𝑥,𝑦∈𝐶. Such a mapping 𝑇 is called (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid. An (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping is generalized hybrid in the sense of Kocourek et al. [9] if 𝛼+𝛽=−𝛾−𝛿=1 and 𝜀=𝜁=𝜂=0. We first prove an attractive point theorem for widely more generalized hybrid mappings in a Hilbert space.

Theorem 8. Let 𝐻 be a real Hilbert space, let 𝐶 be a nonempty subset of 𝐻, and let 𝑇 be an (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping from 𝐶 into itself which satisfies either of the following conditions: (1)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0, 𝜀+𝜂≥0 and 𝜁+𝜂≥0; (2)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0, 𝜁+𝜂≥0 and 𝜀+𝜂≥0.
Then 𝑇 has an attractive point if and only if there exists 𝑧∈𝐶 such that {𝑇^𝑛𝑧∣𝑛=0,1,…} is bounded.

Proof. Suppose that 𝑇 has an attractive point 𝑧. Then ‖𝑇^𝑛+1𝑥−𝑧‖≤‖𝑇^𝑛𝑥−𝑧‖ for all 𝑥∈𝐶 and 𝑛∈ℕ. Therefore {𝑇^𝑛𝑧∣𝑛=0,1,…} is bounded.
Conversely suppose that there exists 𝑧∈𝐶 such that {𝑇^𝑛𝑧∣𝑛=0,1,…} is bounded. Since 𝑇 is an (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping from 𝐶 into itself, we obtain that 𝛼𝑇𝑥−𝑇^𝑛+1𝑧²+𝛽𝑥−𝑇^𝑛+1𝑧²+𝛾𝑇𝑥−𝑇^𝑛𝑧²+𝛿𝑥−𝑇^𝑛𝑧²+𝜀‖𝑥−𝑇𝑥‖²+𝜁𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²+𝜂(𝑥−𝑇𝑥)−𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²≤0 (15) for any 𝑛∈ℕ∪{0} and 𝑥∈𝐶. By (9) we obtain that (𝑥−𝑇𝑥)−𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²=‖𝑥−𝑇𝑥‖²+𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²−2𝑥−𝑇𝑥,𝑇^𝑛𝑧−𝑇^𝑛+1𝑧=‖𝑥−𝑇𝑥‖²+𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²+𝑥−𝑇^𝑛𝑧²+𝑇𝑥−𝑇^𝑛+1𝑧²−𝑥−𝑇^𝑛+1𝑧²−𝑇𝑥−𝑇^𝑛𝑧². (16) Thus we have that 𝛼+𝜂𝑇𝑥−𝑇^𝑛+1𝑧²+𝛽−𝜂𝑥−𝑇^𝑛+1𝑧²+𝛾−𝜂𝑇𝑥−𝑇^𝑛𝑧²+𝛿+𝜂𝑥−𝑇^𝑛𝑧²+𝜀+𝜂‖𝑥−𝑇𝑥‖²+𝜁+𝜂𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²≤0. (17) From 𝛾−𝜂𝑇𝑥−𝑇^𝑛𝑧²=𝛼+𝛾‖𝑥−𝑇𝑥‖²+𝑥−𝑇^𝑛𝑧²−2𝑥−𝑇𝑥,𝑥−𝑇^𝑛𝑧−𝛼+𝜂𝑇𝑥−𝑇^𝑛𝑧², (18) we have that 𝛼+𝜂𝑇𝑥−𝑇^𝑛+1𝑧²+𝛽−𝜂𝑥−𝑇^𝑛+1𝑧²+𝛼+𝛾‖𝑥−𝑇𝑥‖²+𝑥−𝑇^𝑛𝑧²−2𝑥−𝑇𝑥,𝑥−𝑇^𝑛𝑧−𝛼+𝜂𝑇𝑥−𝑇^𝑛𝑧²+𝛿+𝜂𝑥−𝑇^𝑛𝑧²+𝜀+𝜂‖𝑥−𝑇𝑥‖²+𝜁+𝜂𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²≤0, (19) and hence 𝛼+𝜂𝑇𝑥−𝑇^𝑛+1𝑧²−𝑇𝑥−𝑇^𝑛𝑧²+𝛽−𝜂𝑥−𝑇^𝑛+1𝑧²−2𝛼+𝛾𝑥−𝑇𝑥,𝑥−𝑇^𝑛𝑧+𝛼+𝛾+𝛿+𝜂𝑥−𝑇^𝑛𝑧²+𝛼+𝛾+𝜀+𝜂‖𝑥−𝑇𝑥‖²+𝜁+𝜂𝑇^𝑛𝑧−𝑇^𝑛+1𝑧²≤0. (20) By 𝛼+𝛽+𝛾+𝛿≥0, we have that −𝛽−𝜂=−𝛽+𝛿+𝛿+𝜂≤𝛼+𝛾+𝛿+𝜂. (21) From this inequality and 𝜁+𝜂≥0 we obtain that 𝛼+𝜂𝑇𝑥−𝑇^𝑛+1𝑧²−𝑇𝑥−𝑇^𝑛𝑧²+𝛽−𝜂𝑥−𝑇^𝑛+1𝑧²−𝑥−𝑇^𝑛𝑧²−2𝛼+𝛾𝑥−𝑇𝑥,𝑥−𝑇^𝑛𝑧+𝛼+𝛾+𝜀+𝜂‖𝑥−𝑇𝑥‖²≤0. (22) Applying a Banach limit 𝜇 to both sides of this inequality, we obtain that 𝛼+𝜂𝜇_𝑛𝑇𝑥−𝑇^𝑛+1𝑧²−𝜇_𝑛𝑇𝑥−𝑇^𝑛𝑧²+𝛽−𝜂𝜇_𝑛𝑥−𝑇^𝑛+1𝑧²−𝜇_𝑛𝑥−𝑇^𝑛𝑧²−2𝛼+𝛾𝜇_𝑛𝑥−𝑇𝑥,𝑥−𝑇^𝑛𝑧+𝛼+𝛾+𝜀+𝜂𝜇_𝑛‖𝑥−𝑇𝑥‖²≤0, (23) and hence −2𝛼+𝛾𝜇_𝑛𝑥−𝑇𝑥,𝑥−𝑇^𝑛𝑧+𝛼+𝛾+𝜀+𝜂‖𝑥−𝑇𝑥‖²≤0. (24) Since there exists 𝑝∈𝐻 by Lemma 5 such that 𝜇_𝑛𝑦,𝑇^𝑛𝑧=𝑦,𝑝 (25) for any 𝑦∈𝐻, we obtain from (24) that −2𝛼+𝛾𝑥−𝑇𝑥,𝑥−𝑝+𝛼+𝛾+𝜀+𝜂‖𝑥−𝑇𝑥‖²≤0. (26) We obtain from (9) that −𝛼+𝛾𝑥−𝑝²+‖𝑇𝑥−𝑥‖²−𝑇𝑥−𝑝²+𝛼+𝛾+𝜀+𝜂‖𝑥−𝑇𝑥‖²≤0, (27) and hence −𝛼+𝛾𝑥−𝑝²−𝑇𝑥−𝑝²+𝜀+𝜂‖𝑥−𝑇𝑥‖²≤0. (28) Since 𝜀+𝜂≥0, we obtain that −𝛼+𝛾𝑥−𝑝²−𝑇𝑥−𝑝²≤0. (29) Since 𝛼+𝛾>0, we obtain that 𝑇𝑥−𝑝²≤𝑥−𝑝². (30) This implies that 𝑝∈𝐻 is an attractive point.
In the case of 𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0, 𝜁+𝜂≥0, and 𝜀+𝜂≥0, we can obtain the result by replacing the variables 𝑥 and 𝑦. This completes the proof.

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 𝐻 be a Hilbert space, let 𝐶 be a nonempty subset of 𝐻, and let 𝑇 be a generalized hybrid mapping from 𝐶 into 𝐶; that is, there exist real numbers 𝛼 and 𝛽 such that 𝛼𝑇𝑥−𝑇𝑦²+(1−𝛼)𝑥−𝑇𝑦²≤𝛽𝑇𝑥−𝑦²+1−𝛽𝑥−𝑦² (31) for all 𝑥,𝑦∈𝐶. Then 𝑇 has an attractive point if and only if there exists 𝑧∈𝐶 such that {𝑇^𝑛𝑧∣𝑛=0,1,…} is bounded.

Proof. An (𝛼,𝛽)-generalized hybrid mapping 𝑇 is an (𝛼,1−𝛼,−𝛽,−(1−𝛽),0,0,0)-widely more generalized hybrid mapping. Furthermore, the mapping satisfies the condition (2) in Theorem 8, that is, 𝛼+(1−𝛼)−𝛽−1−𝛽=0≥0,𝛼+(1−𝛼)=1>0,0+0≥0,0+0≥0. (32) Then we have the desired result from Theorem 8.

4. 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.

Lemma 10. Let 𝐶 be a nonempty subset of a real Hilbert space 𝐻. Let 𝑇 be an (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping from 𝐶 into itself such that 𝐴(𝑇)≠∅. Suppose that it satisfies either of the following conditions: (1)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾+𝜀+𝜂>0 and 𝜁+𝜂≥0; (2)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽+𝜁+𝜂>0 and 𝜀+𝜂≥0. For any 𝑥∈𝐶, define 𝑆_𝑛𝑥=(1/𝑛)∑^𝑛_𝑘=1𝑇^𝑘𝑥. Then, lim_𝑛→∞‖𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥‖=0. In particular, if 𝐶 is bounded, then lim_𝑛→∞sup_𝑥∈𝐶𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥=0. (33)

Proof. Let 𝑥∈𝐶. Since 𝐴(𝑇) is nonempty, we obtain that 𝑇^𝑛+1𝑥−𝑦≤𝑇^𝑛𝑥−𝑦 (34) for any 𝑛∈ℕ∪{0} and 𝑦∈𝐴(𝑇). Then we have that {𝑇^𝑛𝑥} is bounded. Since 𝑆_𝑛≤1𝑥−𝑦𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥−𝑦≤𝑥−𝑦 (35) for any 𝑛∈ℕ∪{0} and 𝑦∈𝐴(𝑇), {𝑆_𝑛𝑥∣𝑛=0,1,…} is also bounded. Using 𝛼+𝛽+𝛾+𝛿≥0 and 𝜁+𝜂≥0, as in the proof of Theorem 8 we have that 𝛼+𝜂𝑇𝑧−𝑇^𝑘+1𝑥²−𝑇𝑧−𝑇^𝑘𝑥²+𝛽−𝜂𝑧−𝑇^𝑘+1𝑥²−𝑧−𝑇^𝑘𝑥²−2𝛼+𝛾𝑧−𝑇𝑧,𝑧−𝑇^𝑘𝑥+𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²≤0 (36) for any 𝑘∈ℕ∪{0} and 𝑧∈𝐶. Summing up these inequalities with respect to 𝑘=0,1,…,𝑛−1 and dividing by 𝑛, we obtain that 𝛼+𝜂𝑛𝑇𝑧−𝑇^𝑛𝑥²−‖𝑇𝑧−𝑥‖²+𝛽−𝜂𝑛𝑧−𝑇^𝑛𝑥²−‖𝑧−𝑥‖²−2𝛼+𝛾𝑧−𝑇𝑧,𝑧−𝑆_𝑛𝑥+𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²≤0. (37) Replacing 𝑧 by 𝑆_𝑛𝑥, we obtain that 𝛼+𝜂𝑛𝑇𝑆_𝑛𝑥−𝑇^𝑛𝑥²−𝑇𝑆_𝑛𝑥−𝑥²+𝛽−𝜂𝑛𝑆_𝑛𝑥−𝑇^𝑛𝑥²−𝑆_𝑛𝑥−𝑥²+𝛼+𝛾+𝜀+𝜂𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥²≤0. (38) Since {𝑇𝑆_𝑛𝑥}, {𝑆_𝑛𝑥}, and {𝑇^𝑛𝑥} are bounded, we have that 𝛼+𝛾+𝜀+𝜂limsup_𝑛→∞𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥≤0. (39) Since 𝛼+𝛾+𝜀+𝜂>0, we have that lim_𝑛→∞‖𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥‖=0. In particular, if 𝐶 is bounded, then limsup_𝑛→∞sup_𝑥∈𝐶𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥≤0, (40) and hence lim_𝑛→∞sup_𝑥∈𝐶‖𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥‖=0.
Similarly, we can obtain the desired result for the case of 𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽+𝜁+𝜂>0, and 𝜀+𝜂≥0. This completes the proof.

Lemma 11. Let 𝐻 be a Hilbert space, and let 𝐶 be a nonempty subset of 𝐻. Let 𝑇∶𝐶→𝐻 be an (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping. Suppose that it satisfies either of the following conditions: (1)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0 and 𝜀+𝜂≥0; (2)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0 and 𝜁+𝜂≥0. If 𝑥_𝑛⇀𝑧 and 𝑥_𝑛−𝑇𝑥_𝑛→0, then 𝑧∈𝐴(𝑇).

Proof. Let 𝑇∶𝐶→𝐻 be an (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping, and suppose that 𝑥_𝑛⇀𝑧 and 𝑥_𝑛−𝑇𝑥_𝑛→0. Replacing 𝑥 by 𝑥_𝑛 in (14), we have that 𝛼𝑇𝑥_𝑛−𝑇𝑦²+𝛽𝑥_𝑛−𝑇𝑦²+𝛾𝑇𝑥_𝑛−𝑦²+𝛿𝑥_𝑛−𝑦²+𝜀𝑥_𝑛−𝑇𝑥_𝑛²+𝜁𝑦−𝑇𝑦²+𝜂𝑥_𝑛−𝑇𝑥_𝑛−𝑦−𝑇𝑦²≤0. (41) From this inequality, we have that 𝛼𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑇𝑦²+2𝑇𝑥_𝑛−𝑥_𝑛,𝑥_𝑛𝑥−𝑇𝑦+𝛽_𝑛−𝑇𝑦²+𝛾𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑦²+2𝑇𝑥_𝑛−𝑥_𝑛,𝑥_𝑛𝑥−𝑦+𝛿_𝑛−𝑦²+𝜀𝑥_𝑛−𝑇𝑥_𝑛²+𝜁𝑦−𝑇𝑦²+𝜂𝑥_𝑛−𝑇𝑥_𝑛−𝑦−𝑇𝑦²≤0. (42) We apply a Banach limit 𝜇 to both sides of this inequality. We have that 𝛼𝜇_𝑛𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑇𝑦²+2𝑇𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑇𝑦+𝛽𝜇_𝑛𝑥_𝑛−𝑇𝑦²+𝛾𝜇_𝑛𝑇𝑥_𝑛−𝑥_𝑛²+𝑥_𝑛−𝑦²+2𝑇𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑦+𝛿𝜇_𝑛𝑥_𝑛−𝑦²+𝜀𝜇_𝑛𝑥_𝑛−𝑇𝑥_𝑛²+𝜁𝑦−𝑇𝑦²+𝜂𝜇_𝑛𝑥_𝑛−𝑇𝑥_𝑛−𝑦−𝑇𝑦²≤0, (43) and hence 𝛼𝜇_𝑛𝑥_𝑛−𝑇𝑦²+𝛽𝜇_𝑛𝑥_𝑛−𝑇𝑦²+𝛾𝜇_𝑛𝑥_𝑛−𝑦²+𝛿𝜇_𝑛𝑥_𝑛−𝑦²+𝜁𝑦−𝑇𝑦²+𝜂𝜇_𝑛𝑥_𝑛−𝑇𝑥_𝑛−𝑦−𝑇𝑦²≤0. (44) Thus we have 𝛼+𝛽𝜇_𝑛𝑥_𝑛−𝑇𝑦²+𝛾+𝛿𝜇_𝑛𝑥_𝑛−𝑦²+𝜁+𝜂𝑦−𝑇𝑦²≤0. (45) From ‖𝑥_𝑛−𝑇𝑦‖²=‖𝑥_𝑛−𝑦‖²+‖𝑦−𝑇𝑦‖²+2⟨𝑥_𝑛−𝑦,𝑦−𝑇𝑦⟩, we also have 𝛼+𝛽𝜇_𝑛𝑥_𝑛−𝑦²+𝑦−𝑇𝑦²+2𝜇_𝑛𝑥_𝑛+𝜇−𝑦,𝑦−𝑇𝑦𝛾+𝛿_𝑛𝑥_𝑛−𝑦²+𝜁+𝜂𝑦−𝑇𝑦²≤0. (46) From 𝛼+𝛽+𝛾+𝛿≥0 we obtain that 𝛼+𝛽𝑦−𝑇𝑦²+2𝛼+𝛽𝜇_𝑛⟨𝑥_𝑛+−𝑦,𝑦−𝑇𝑦⟩𝜁+𝜂𝑦−𝑇𝑦²≤0, (47) and hence 𝛼+𝛽+𝜁+𝜂𝑦−𝑇𝑦²+2𝛼+𝛽𝜇_𝑛𝑥_𝑛−𝑦,𝑦−𝑇𝑦≤0. (48) Since 𝑥_𝑛⇀𝑧, we have that 𝛼+𝛽+𝜁+𝜂𝑦−𝑇𝑦²+2𝛼+𝛽𝑧−𝑦,𝑦−𝑇𝑦≤0. (49) Using (9), we have that 𝛼+𝛽+𝜁+𝜂𝑦−𝑇𝑦²+𝛼+𝛽𝑧−𝑇𝑦²−𝑧−𝑦²−𝑦−𝑇𝑦²≤0, (50) and hence 𝜁+𝜂𝑦−𝑇𝑦²+𝛼+𝛽𝑧−𝑇𝑦²−𝑧−𝑦²≤0. (51) Since 𝛼+𝛽>0 and 𝜁+𝜂≥0, we have that 𝑧−𝑇𝑦²−𝑧−𝑦²≤0 (52) for all 𝑦∈𝐶. This implies that 𝑧∈𝐴(𝑇).
Similarly, we can obtain the desired result for the case of 𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0, and 𝜀+𝜂≥0. This completes the proof.

Now we have the following nonlinear ergodic theorem for widely more generalized hybrid mappings in a Hilbert space.

Theorem 12. Let 𝐻 be a real Hilbert space, let 𝐶 be a nonempty subset of 𝐻, let 𝑇 be an (𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂)-widely more generalized hybrid mapping from 𝐶 into itself such that 𝐴(𝑇)≠∅, and let 𝑃 be the metric projection of 𝐻 onto 𝐴(𝑇). Suppose that 𝑇 satisfies either of the conditions: (1)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0, 𝜀+𝜂≥0 and 𝜁+𝜂≥0; (2)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0, 𝜁+𝜂≥0 and 𝜀+𝜂≥0. Then for any 𝑥∈𝐶, 𝑆_𝑛1𝑥=𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥 (53) is weakly convergent to an attractive point 𝑝 of  𝑇, where 𝑝=lim_𝑛→∞𝑃𝑇^𝑛𝑥.

Proof. Let 𝑥∈𝐶. Since 𝐴(𝑇) is nonempty, we obtain that 𝑇^𝑛+1𝑥−𝑦≤𝑇^𝑛𝑥−𝑦 (54) for any 𝑛∈ℕ∪{0} and 𝑦∈𝐴(𝑇). Thus {𝑇^𝑛𝑥} is bounded. Since 𝑆_𝑛≤1𝑥−𝑦𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥−𝑦≤𝑥−𝑦 (55) for any 𝑛∈ℕ∪{0} and 𝑦∈𝐴(𝑇), {𝑆_𝑛𝑥∣𝑛=0,1,…} is also bounded. Therefore there exists a strictly increasing sequence {𝑛_𝑖} and 𝑝∈𝐻 such that {𝑆_𝑛𝑖𝑥∣𝑖=0,1,…} converges weakly to 𝑝. Using 𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0, 𝜀+𝜂≥0, and 𝜁+𝜂≥0, we have from Lemma 10 that lim_𝑛→∞𝑆_𝑛𝑥−𝑇𝑆_𝑛𝑥=0. (56) We have from Lemma 11 that 𝑝∈𝐴(𝑇). Since 𝐴(𝑇) is closed and convex from Lemma 6, the metric projection 𝑃 from 𝐻 onto 𝐴(𝑇) is well defined. By Lemma 3, there exists 𝑞∈𝐴(𝑇) such that {𝑃𝑇^𝑛𝑥∣𝑛=0,1,…} converges strongly to 𝑞. To complete the proof, we show that 𝑞=𝑝. Note that the metric projection 𝑃 satisfies ⟨𝑧−𝑃𝑧,𝑃𝑧−𝑢⟩≥0 (57) for any 𝑧∈𝐻 and for any 𝑢∈𝐴(𝑇); see [25]. Therefore 𝑇^𝑘𝑥−𝑃𝑇^𝑘𝑥,𝑃𝑇^𝑘𝑥−𝑦≥0 (58) for any 𝑘∈ℕ∪{0} and 𝑦∈𝐴(𝑇). Since 𝑃 is the metric projection from 𝐻 onto 𝐴(𝑇) and 𝑃𝑇^𝑛−1𝑥∈𝐴(𝑇), we obtain that 𝑇^𝑛𝑥−𝑃𝑇^𝑛𝑥≤𝑇^𝑛𝑥−𝑃𝑇^𝑛−1𝑥≤𝑇^𝑛−1𝑥−𝑃𝑇^𝑛−1𝑥. (59) That is, {‖𝑇^𝑛𝑥−𝑃𝑇^𝑛𝑥‖∣𝑛=0,1,…} is nonincreasing. Therefore we obtain 𝑇^𝑘𝑥−𝑃𝑇^𝑘𝑥,𝑦−𝑞≤𝑇^𝑘𝑥−𝑃𝑇^𝑘𝑥,𝑃𝑇^𝑘≤𝑇𝑥−𝑞_𝑘𝑥−𝑃𝑇^𝑘𝑥⋅𝑃𝑇^𝑘≤‖‖⋅𝑥−𝑞𝑥−𝑃𝑥𝑃𝑇^𝑘𝑥−𝑞. (60) Summing up these inequalities with respect to 𝑘=0,1,…,𝑛−1 and dividing by 𝑛, we obtain 𝑆_𝑛1𝑥−𝑛^𝑛−1_𝑘=0𝑃𝑇^𝑘≤‖‖𝑥,𝑦−𝑞𝑥−𝑃𝑥𝑛^𝑛−1_𝑘=0𝑃𝑇^𝑘𝑥−𝑞. (61) Since {𝑆_𝑛𝑖𝑥∣𝑖=0,1,…} converges weakly to 𝑝 and {𝑃𝑇^𝑛𝑥∣𝑛=0,1,…} converges strongly to 𝑞, we obtain that 𝑝−𝑞,𝑦−𝑞≤0. (62) Putting 𝑦=𝑝, we obtain 𝑝−𝑞²≤0, (63) and hence 𝑞=𝑝. This completes the proof.
Similarly, we can obtain the desired result for the case of 𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0, 𝜁+𝜂>0, and 𝜀+𝜂≥0.

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 𝐻 be a Hilbert space, let 𝐶 be a nonempty subset of 𝐻, and let 𝑇 be a generalized hybrid mapping from 𝐶 into itself; that is, there exist 𝛼,𝛽∈ℝ such that 𝛼𝑇𝑥−𝑇𝑦²+(1−𝛼)𝑥−𝑇𝑦²≤𝛽𝑇𝑥−𝑦²+1−𝛽𝑥−𝑦² (64) for all 𝑥,𝑦∈𝐶. Suppose that 𝐴(𝑇)≠∅, and let 𝑃 be the metric projection from 𝐻 onto 𝐴(𝑇). Then for any 𝑥∈𝐶, 𝑆_𝑛1𝑥=𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥 (65) converges weakly to 𝑝∈𝐴(𝑇), where 𝑝=lim_𝑛→∞𝑃𝑇^𝑛𝑥.

5. 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].

Theorem 14. Let 𝐻 be a Hilbert space, and let 𝐶 be a convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a widely more generalized hybrid mapping with 𝐴(𝑇)≠∅ such that it satisfies either of the conditions: (1)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0 and 𝜀+𝜂≥0; (2)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0 and 𝜁+𝜂≥0.
Let 𝑃 be the metric projection of 𝐻 onto 𝐴(𝑇). Let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛≤1 and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. Suppose that {𝑥_𝑛} is the sequence generated by 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛∈ℕ. (66) Then {𝑥_𝑛} converges weakly to 𝑣∈𝐴(𝑇), where 𝑣=lim_𝑛→∞𝑃𝑥_𝑛.

Proof. Let 𝑧∈𝐴(𝑇). We have that 𝑥_𝑛+1−𝑧²=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛−𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑇𝑥_𝑛−𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑥_𝑛−𝑧²=𝑥_𝑛−𝑧² (67) for all 𝑛∈ℕ. Hence lim_𝑛→∞‖𝑥_𝑛−𝑧‖² exists. Then {𝑥_𝑛} is bounded. We also have from (8) that 𝑥_𝑛+1−𝑧²=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛−𝑧²=𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑇𝑥_𝑛−𝑧²−𝛼_𝑛1−𝛼_𝑛𝑇𝑥_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑥_𝑛−𝑧²+1−𝛼_𝑛𝑥_𝑛−𝑧²−𝛼_𝑛1−𝛼_𝑛𝑇𝑥_𝑛−𝑥_𝑛²=𝑥_𝑛−𝑧²−𝛼_𝑛1−𝛼_𝑛𝑇𝑥_𝑛−𝑥_𝑛². (68)
Thus we have 𝛼_𝑛1−𝛼_𝑛𝑇𝑥_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑧²−𝑥_𝑛+1−𝑧². (69) Since lim_𝑛→∞‖𝑥_𝑛−𝑧‖² exists and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, we have that 𝑇𝑥_𝑛−𝑥_𝑛⟶0. (70) Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑣. By Lemma 11 and (70), we obtain that 𝑣∈𝐴(𝑇). Let {𝑥_𝑛𝑖} and {𝑥_𝑛𝑗} be two subsequences of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑣₁ and 𝑥_𝑛𝑗⇀𝑣₂. To complete the proof, we show 𝑣₁=𝑣₂. We know that 𝑣₁,𝑣₂∈𝐴(𝑇), and hence lim_𝑛→∞‖𝑥_𝑛−𝑣₁‖² and lim_𝑛→∞‖𝑥_𝑛−𝑣₂‖² exist. Put 𝑎=lim_𝑛→∞𝑥_𝑛−𝑣₁²−𝑥_𝑛−𝑣₂². (71) Note that for 𝑛=1,2,…, 𝑥_𝑛−𝑣₁²−𝑥_𝑛−𝑣₂²=2𝑥_𝑛,𝑣₂−𝑣₁+𝑣₁²−𝑣₂². (72) From 𝑥_𝑛𝑖⇀𝑣₁ and 𝑥_𝑛𝑗⇀𝑣₂, we have 𝑎=2𝑣₁,𝑣₂−𝑣₁+𝑣₁²−𝑣₂²,𝑎=2𝑣₂,𝑣₂−𝑣₁+𝑣₁²−𝑣₂². (73)(74) Combining (73) and (74), we obtain 0=2⟨𝑣₂−𝑣₁,𝑣₂−𝑣₁⟩. Thus we obtain 𝑣₂=𝑣₁. This implies that {𝑥_𝑛} converges weakly to an element 𝑣∈𝐴(𝑇). Since ‖𝑥_𝑛+1−𝑧‖≤‖𝑥_𝑛−𝑧‖ for all 𝑧∈𝐴(𝑇) and 𝑛∈ℕ, we obtain from Lemma 3 that {𝑃𝑥_𝑛} converges strongly to an element 𝑝∈𝐴(𝑇). On the other hand, we have from the property of 𝑃 that 𝑥_𝑛−𝑃𝑥_𝑛,𝑃𝑥_𝑛−𝑢≥0 (75) for all 𝑢∈𝐴(𝑇) and 𝑛∈ℕ. Since 𝑥_𝑛⇀𝑣 and 𝑃𝑥_𝑛→𝑝, we obtain 𝑣−𝑝,𝑝−𝑢≥0 (76) for all 𝑢∈𝐴(𝑇). Putting 𝑢=𝑣, we obtain 𝑝=𝑣. This means 𝑣=lim_𝑛→∞𝑃𝑥_𝑛. This completes the proof.

Using Theorem 14, we can show the following weak convergence theorem of Mann’s type for generalized hybrid mappings in a Hilbert space.

Theorem 15 (Kocourek et al. [9]). Let 𝐻 be a Hilbert space, and let 𝐶 be a closed convex subset of 𝐻. Let 𝑇∶𝐶→𝐶 be a generalized hybrid mapping with 𝐹(𝑇)≠∅. Let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛≤1 and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. Suppose {𝑥_𝑛} is the sequence generated by 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛∈ℕ. (77) Then the sequence {𝑥_𝑛} converges weakly to an element 𝑣∈𝐹(𝑇).

Proof. As in the proof of Theorem 9, a generalized hybrid mapping is a widely more generalized hybrid mapping. Since {𝑥_𝑛}⊂𝐶 and 𝐶 is closed and convex, we have from Theorem 14 that 𝑣∈𝐴(𝑇)∩𝐶. A generalized hybrid mapping with 𝐹(𝑇)≠∅ is quasi-nonexpansive, we have from Lemma 7 that 𝐴(𝑇)∩𝐶=𝐹(𝑇). Thus {𝑥_𝑛} converges weakly to an element 𝑣∈𝐹(𝑇).

6. Strong Convergence Theorem

In this section, using an idea of mean convergence by Shimizu and Takahashi [21, 29], we prove the following strong convergence theorem for widely more generalized hybrid mappings in a Hilbert space.

Theorem 16. Let 𝐶 be a nonempty convex subset of a real Hilbert space 𝐻. Let 𝑇 be a widely more generalized hybrid mapping of 𝐶 into itself with 𝐴(𝑇)≠∅ such that it satisfies either of the following conditions: (1)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛾>0, 𝜀+𝜂≥0 and 𝜁+𝜂≥0; (2)𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0, 𝜁+𝜂≥0 and 𝜀+𝜂≥0.
Let 𝑢∈𝐶, and define sequences {𝑥_𝑛} and {𝑧_𝑛} in 𝐶 as follows: 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=1𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥_𝑛 (78) for all 𝑛=1,2,…, where 0≤𝛼_𝑛≤1, 𝛼_𝑛→0 and ∑^∞_𝑛=1𝛼_𝑛=∞. If 𝐴(𝑇) is nonempty, then {𝑥_𝑛} and {𝑧_𝑛} converge strongly to 𝑃𝑢, where 𝑃 is the metric projection of 𝐻 onto 𝐴(𝑇).

Proof. Since 𝑇∶𝐶→𝐶 is a widely more generalized hybrid mapping, there exist 𝛼,𝛽,𝛾,𝛿,𝜀,𝜁,𝜂∈ℝ such that 𝛼𝑇𝑥−𝑇𝑦²+𝛽𝑥−𝑇𝑦²+𝛾𝑇𝑥−𝑦²+𝛿𝑥−𝑦²+𝜀‖𝑥−𝑇𝑥‖²+𝜁𝑦−𝑇𝑦²+𝜂(𝑥−𝑇𝑥)−𝑦−𝑇𝑦²≤0 (79) for any 𝑥,𝑦∈𝐶. Since 𝐴(𝑇)≠∅, we have that for all 𝑞∈𝐴(𝑇) and 𝑛=1,2,3,…, 𝑧_𝑛=1−𝑞𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥_𝑛≤1−𝑞𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥_𝑛≤1−𝑞𝑛^𝑛−1_𝑘=0𝑥_𝑛−𝑞=𝑥_𝑛−𝑞. (80) Then we have 𝑥_𝑛+1−𝑞=𝛼_𝑛𝑢+1−𝛼_𝑛𝑧_𝑛−𝑞≤𝛼_𝑛𝑢−𝑞+1−𝛼_𝑛𝑧_𝑛−𝑞≤𝛼_𝑛𝑢−𝑞+1−𝛼_𝑛𝑥_𝑛−𝑞. (81) Hence, by induction, we obtain 𝑥_𝑛−𝑞≤max𝑢−𝑞,𝑥−𝑞 (82) for all 𝑛∈ℕ. This implies that {𝑥_𝑛} and {𝑧_𝑛} are bounded. Since ‖𝑇^𝑛𝑥_𝑛−𝑞‖≤‖𝑥_𝑛−𝑞‖, we have also that {𝑇^𝑛𝑥_𝑛} is bounded.
Let 𝑛∈ℕ. Using 𝛼+𝛽+𝛾+𝛿≥0 and 𝜁+𝜂≥0, as in the proof of Theorem 8 we have that 𝛼+𝜂𝑇𝑧−𝑇^𝑘+1𝑥_𝑛²−𝑇𝑧−𝑇^𝑘𝑥_𝑛²+𝛽−𝜂𝑧−𝑇^𝑘+1𝑥_𝑛²−𝑧−𝑇^𝑘𝑥_𝑛²−2𝛼+𝛾𝑧−𝑇𝑧,𝑧−𝑇^𝑘𝑥_𝑛+𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²≤0 (83) for any 𝑘∈ℕ∪{0} and 𝑧∈𝐶. Summing up these inequalities with respect to 𝑘=0,1,…,𝑛−1 and dividing by 𝑛, we obtain that 𝛼+𝜂𝑛𝑇𝑧−𝑇^𝑛𝑥_𝑛²−𝑇𝑧−𝑥_𝑛²+𝛽−𝜂𝑛𝑧−𝑇^𝑛𝑥_𝑛²−𝑧−𝑥_𝑛²−2𝛼+𝛾𝑧−𝑇𝑧,𝑧−𝑧_𝑛+𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²≤0. (84) Since {𝑧_𝑛} is bounded, there exists a subsequence {𝑧_𝑛𝑖} of {𝑧_𝑛} such that 𝑧_𝑛𝑖⇀𝑤∈𝐻. Replacing 𝑛 by 𝑛_𝑖, we have that 𝛼+𝜂𝑛_𝑖𝑇𝑧−𝑇^𝑛𝑖𝑥_𝑛𝑖²−𝑇𝑧−𝑥_𝑛𝑖²+𝛽−𝜂𝑛_𝑖𝑧−𝑇^𝑛𝑖𝑥_𝑛𝑖²−𝑧−𝑥_𝑛𝑖²−2𝛼+𝛾𝑧−𝑇𝑧,𝑧−𝑧_𝑛𝑖+𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²≤0. (85) Since {𝑥_𝑛} and {𝑇^𝑛𝑥_𝑛} are bounded, we have that −2𝛼+𝛾⟨𝑧−𝑇𝑧,𝑧−𝑤⟩+𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²≤0 (86) as 𝑖→∞. Using (9), we have that 𝛼+𝛾+𝜀+𝜂‖𝑧−𝑇𝑧‖²+𝛼+𝛾‖𝑤−𝑇𝑧‖²−‖𝑤−𝑧‖²−‖𝑧−𝑇𝑧‖²≤0, (87) and hence 𝜀+𝜂‖𝑧−𝑇𝑧‖²+𝛼+𝛾‖𝑤−𝑇𝑧‖²−‖𝑤−𝑧‖²≤0. (88) Since 𝛼+𝛾>0 and 𝜀+𝜂≥0, we have that ‖𝑤−𝑇𝑧‖²−‖𝑤−𝑧‖²≤0 (89) for all 𝑧∈𝐶. This implies that 𝑤∈𝐴(𝑇).
On the other hand, since 𝑥_𝑛+1−𝑧_𝑛=𝛼_𝑛(𝑢−𝑧_𝑛), {𝑧_𝑛} is bounded, and 𝛼_𝑛→0, we have lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0. Let us show limsup_𝑛→∞𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢≤0. (90) We may assume without loss of generality that there exists a subsequence {𝑥_𝑛𝑖+1} of {𝑥_𝑛+1} such that limsup_𝑛→∞𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢=lim_𝑖→∞𝑢−𝑃𝑢,𝑥_𝑛𝑖+1−𝑃𝑢 (91) and 𝑥_𝑛𝑖+1⇀𝑣. From ‖𝑥_𝑛+1−𝑧_𝑛‖→0, we have 𝑧_𝑛𝑖⇀𝑣. From the above argument, we have 𝑣∈𝐴(𝑇). Since 𝑃 is the metric projection of 𝐻 onto 𝐴(𝑇), we have lim_𝑖→∞𝑢−𝑃𝑢,𝑥_𝑛𝑖+1−𝑃𝑢=⟨𝑢−𝑃𝑢,𝑣−𝑃𝑢⟩≤0. (92) This implies limsup_𝑛→∞𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢≤0. (93) Since 𝑥_𝑛+1−𝑃𝑢=(1−𝛼_𝑛)(𝑧_𝑛−𝑃𝑢)+𝛼_𝑛(𝑢−𝑃𝑢), from (7) and (80) we have 𝑥_𝑛+1−𝑃𝑢²=1−𝛼_𝑛𝑧_𝑛−𝑃𝑢+𝛼_𝑛(𝑢−𝑃𝑢)²≤1−𝛼_𝑛²𝑧_𝑛−𝑃𝑢²+2𝛼_𝑛𝑢−𝑃𝑢,𝑥_𝑛+1≤−𝑃𝑢1−𝛼_𝑛𝑥_𝑛−𝑃𝑢²+2𝛼_𝑛𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢. (94) Putting 𝑠_𝑛=‖𝑥_𝑛−𝑃𝑢‖², 𝛽_𝑛=0, and 𝛾_𝑛=2⟨𝑢−𝑃𝑢,𝑥_𝑛+1−𝑃𝑢⟩ in Lemma 4, we have from ∑^∞_𝑛=1𝛼_𝑛=∞ and (93) that lim_𝑛→∞𝑥_𝑛−𝑃𝑢=0. (95) By lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0, we also obtain 𝑧_𝑛→𝑃𝑢 as 𝑛→∞.
Similarly, we can obtain the desired result for the case of 𝛼+𝛽+𝛾+𝛿≥0, 𝛼+𝛽>0, 𝜁+𝜂≥0, and 𝜀+𝜂≥0.

Using Theorem 16, we can show the following result obtained by Kurokawa and Takahashi [30].

Theorem 17 (Hojo and Takahashi [22]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇 be a generalized hybrid mapping of 𝐶 into itself. Let 𝑢∈𝐶 and define two sequences {𝑥_𝑛} and {𝑧_𝑛} in 𝐶 as follows: 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=1𝑛^𝑛−1_𝑘=0𝑇^𝑘𝑥_𝑛 (96) for all 𝑛=1,2,…, where 0≤𝛼_𝑛≤1, 𝛼_𝑛→0 and ∑^∞_𝑛=1𝛼_𝑛=∞. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} and {𝑧_𝑛} converge strongly to 𝑃𝑢∈𝐹(𝑇), where 𝑃 is the metric projection of 𝐻 onto 𝐴(𝑇).

Proof. As in the proof of Theorem 9, a generalized hybrid mapping is a widely more generalized hybrid mapping. Since {𝑥_𝑛},{𝑧_𝑛}⊂𝐶 and 𝐶 is closed and convex, we have from Theorem 16 that 𝑃𝑢∈𝐴(𝑇)∩𝐶. A generalized hybrid mapping with 𝐹(𝑇)≠∅ is quasi-nonexpansive, we have from Lemma 7 that 𝐴(𝑇)∩𝐶=𝐹(𝑇). Thus {𝑥_𝑛} and {𝑧_𝑛} converge strongly to an element 𝑃𝑢∈𝐹(𝑇).

Acknowledgments

Research for Sy-Ming Guu is partially supported by NSC 100-2221-E-182-072-MY2. Wataru Takahashi is partially supported by Grant-in-Aid for Scientific Research no. 23540188 from Japan Society for the Promotion of Science.