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

Authors: Sy-Ming Guu; Wataru Takahashi
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

LetH 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 mappingT: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
(1)2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2,3∥Tx-Ty∥2≤∥x-y∥2+∥Tx-y∥2+∥Ty-x∥2
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
(2)∥Fx-Fy∥2≤〈x-y,Fx-Fy〉
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
(3)α∥Tx-Ty∥2+(1-α)∥x-Ty∥2≤β∥Tx-y∥2+(1-β)∥x-y∥2
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]).
LetH 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,
(4)Snx=1n∑k=0n-1‍Tkx
converges weakly to p∈F(T), where p=limn→∞PTnx.We see that the setC 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.
LetH 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 {vn} and {bn} be sequences defined by
(5)v1∈C,vn+1=Tvn,bn=1n∑k=1n‍vk
for all n∈ℕ. If {vn} is bounded, then the followings hold:  (1)
A(T) is nonempty, closed, and convex;(2)
{bn} converges weakly to u0∈A(T), where u0=limn→∞PA(T)vn and PA(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
(6)α∥Tx-Ty∥2+β∥x-Ty∥2+γ∥Tx-y∥2+δ∥x-y∥2+ε∥x-Tx∥2+ζ∥y-Ty∥2+η∥(x-Tx)-(y-Ty)∥2≤0
for any x,y∈C; see also Kawasaki and Takahashi [17].A mappingT: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 byℕ 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 {xn} to x∈H by xn→x and xn⇀x, respectively. Let A be a nonempty subset of H. We denote by co¯A 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,
(7)∥y∥2-∥x∥2≤2〈y-x,y〉,(8)∥αx+(1-α)y∥2=α∥x∥2+(1-α)∥y∥2-α(1-α)∥x-y∥2.Furthermore, we have that(9)2〈x-y,z-w〉=∥x-w∥2+∥y-z∥2-∥x-z∥2-∥y-w∥2
for any x,y,z,w∈H.LetD 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∥=infy∈D∥x-y∥. We denote such a correspondence by z=PDx. The mapping PD is called the metric projection of H onto D. It is known that PD is nonexpansive and
(10)〈x-PDx,PDx-u〉≥0
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.
LetD be a nonempty closed convex subset of H. Let P be the metric projection from H onto D. Let {un} be a sequence in H. If ∥un+1-u∥≤∥un-u∥ for any u∈D and n∈ℕ, then {Pun} 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{sn} be a sequence of nonnegative real numbers, let {αn} be a sequence of [0,1] with ∑n=1∞‍αn=∞, let {βn} be a sequence of nonnegative real numbers with ∑n=1∞‍βn<∞, and let {γn} be a sequence of real numbers with limsupn→∞γn≤0. Suppose that
(11)sn+1≤(1-αn)sn+αnγn+βn
for all n=1,2,…. Then limn→∞sn=0.Letl∞ 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 μn(xn) 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 μn(xn+1)=μn(xn). 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∞,
(12)liminfn→∞xn≤μn(xn)≤limsupn→∞xn.
In particular, if f=(x1,x2,x3,…)∈l∞ and xn→a∈ℝ, then we have μ(f)=μn(xn)=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.
LetH be a real Hilbert space, let {xn} be a bounded sequence in H, and let μ be a mean on l∞. Then there exists a unique point z0∈co¯{xn∣n∈ℕ} such that
(13)μn(〈xn,y〉)=〈z0,y〉
for any y∈H.The following result obtained by Takahashi and Takeuchi [12] is important in this paper.Lemma 6.
LetH 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.
LetH 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

LetH 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
(14)α∥Tx-Ty∥2+β∥x-Ty∥2+γ∥Tx-y∥2+δ∥x-y∥2+ε∥x-Tx∥2+ζ∥y-Ty∥2+η∥(x-Tx)-(y-Ty)∥2≤0
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.
LetH 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.
ThenT has an attractive point if and only if there exists z∈C such that {Tnz∣n=0,1,…} is bounded.Proof.
Suppose thatT has an attractive point z. Then ∥Tn+1x-z∥≤∥Tnx-z∥ for all x∈C and n∈ℕ. Therefore {Tnz∣n=0,1,…} is bounded.
Conversely suppose that there existsz∈C such that {Tnz∣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
(15)α∥Tx-Tn+1z∥2+β∥x-Tn+1z∥2+γ∥Tx-Tnz∥2+δ∥x-Tnz∥2+ε∥x-Tx∥2+ζ∥Tnz-Tn+1z∥2+η∥(x-Tx)-(Tnz-Tn+1z)∥2≤0
for any n∈ℕ∪{0} and x∈C. By (9) we obtain that
(16)∥(x-Tx)-(Tnz-Tn+1z)∥2=∥x-Tx∥2+∥Tnz-Tn+1z∥2-2〈x-Tx,Tnz-Tn+1z〉=∥x-Tx∥2+∥Tnz-Tn+1z∥2+∥x-Tnz∥2+∥Tx-Tn+1z∥2-∥x-Tn+1z∥2-∥Tx-Tnz∥2.
Thus we have that
(17)(α+η)∥Tx-Tn+1z∥2+(β-η)∥x-Tn+1z∥2+(γ-η)∥Tx-Tnz∥2+(δ+η)∥x-Tnz∥2+(ε+η)∥x-Tx∥2+(ζ+η)∥Tnz-Tn+1z∥2≤0.
From
(18)(γ-η)∥Tx-Tnz∥2=(α+γ)(∥x-Tx∥2+∥x-Tnz∥2-2〈x-Tx,x-Tnz〉)-(α+η)∥Tx-Tnz∥2,
we have that
(19)(α+η)∥Tx-Tn+1z∥2+(β-η)∥x-Tn+1z∥2+(α+γ)(∥x-Tx∥2+∥x-Tnz∥2-2〈x-Tx,x-Tnz〉)-(α+η)∥Tx-Tnz∥2+(δ+η)∥x-Tnz∥2+(ε+η)∥x-Tx∥2+(ζ+η)∥Tnz-Tn+1z∥2≤0,
and hence
(20)(α+η)(∥Tx-Tn+1z∥2-∥Tx-Tnz∥2)+(β-η)∥x-Tn+1z∥2-2(α+γ)〈x-Tx,x-Tnz〉+(α+γ+δ+η)∥x-Tnz∥2+(α+γ+ε+η)∥x-Tx∥2+(ζ+η)∥Tnz-Tn+1z∥2≤0.
By α+β+γ+δ≥0, we have that
(21)-(β-η)=-(β+δ)+δ+η≤α+γ+δ+η.
From this inequality and ζ+η≥0 we obtain that
(22)(α+η)(∥Tx-Tn+1z∥2-∥Tx-Tnz∥2)+(β-η)(∥x-Tn+1z∥2-∥x-Tnz∥2)-2(α+γ)〈x-Tx,x-Tnz〉+(α+γ+ε+η)∥x-Tx∥2≤0.
Applying a Banach limit μ to both sides of this inequality, we obtain that
(23)(α+η)(μn∥Tx-Tn+1z∥2-μn∥Tx-Tnz∥2)+(β-η)(μn∥x-Tn+1z∥2-μn∥x-Tnz∥2)-2(α+γ)μn〈x-Tx,x-Tnz〉+(α+γ+ε+η)μn∥x-Tx∥2≤0,
and hence
(24)-2(α+γ)μn〈x-Tx,x-Tnz〉+(α+γ+ε+η)∥x-Tx∥2≤0.
Since there exists p∈H by Lemma 5 such that
(25)μn〈y,Tnz〉=〈y,p〉
for any y∈H, we obtain from (24) that
(26)-2(α+γ)〈x-Tx,x-p〉+(α+γ+ε+η)∥x-Tx∥2≤0.
We obtain from (9) that
(27)-(α+γ)(∥x-p∥2+∥Tx-x∥2-∥Tx-p∥2)+(α+γ+ε+η)∥x-Tx∥2≤0,
and hence
(28)-(α+γ)(∥x-p∥2-∥Tx-p∥2)+(ε+η)∥x-Tx∥2≤0.
Since ε+η≥0, we obtain that
(29)-(α+γ)(∥x-p∥2-∥Tx-p∥2)≤0.
Since α+γ>0, we obtain that
(30)∥Tx-p∥2≤∥x-p∥2.
This implies that p∈H is an attractive point.
In the case ofα+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 Theorem8, we can show the following attractive point theorem for generalized hybrid mappings in a Hilbert space.Theorem 9 (Takahashi and Takeuchi [12]).
LetH 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
(31)α∥Tx-Ty∥2+(1-α)∥x-Ty∥2≤β∥Tx-y∥2+(1-β)∥x-y∥2
for all x,y∈C. Then T has an attractive point if and only if there exists z∈C such that {Tnz∣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,
(32)α+(1-α)-β-(1-β)=0≥0,α+(1-α)=1>0,0+0≥0,0+0≥0.
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.
LetC 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 Snx=(1/n)∑k=1n‍Tkx. Then, limn→∞∥Snx-TSnx∥=0. In particular, if C is bounded, then
(33)limn→∞supx∈C∥Snx-TSnx∥=0.Proof.
Letx∈C. Since A(T) is nonempty, we obtain that
(34)∥Tn+1x-y∥≤∥Tnx-y∥
for any n∈ℕ∪{0} and y∈A(T). Then we have that {Tnx} is bounded. Since
(35)∥Snx-y∥≤1n∑k=0n-1‍∥Tkx-y∥≤∥x-y∥
for any n∈ℕ∪{0} and y∈A(T), {Snx∣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
(36)(α+η)(∥Tz-Tk+1x∥2-∥Tz-Tkx∥2)+(β-η)(∥z-Tk+1x∥2-∥z-Tkx∥2)-2(α+γ)〈z-Tz,z-Tkx〉+(α+γ+ε+η)∥z-Tz∥2≤0
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
(37)α+ηn(∥Tz-Tnx∥2-∥Tz-x∥2)+β-ηn(∥z-Tnx∥2-∥z-x∥2)-2(α+γ)〈z-Tz,z-Snx〉+(α+γ+ε+η)∥z-Tz∥2≤0.
Replacing z by Snx, we obtain that
(38)α+ηn(∥TSnx-Tnx∥2-∥TSnx-x∥2)+β-ηn(∥Snx-Tnx∥2-∥Snx-x∥2)+(α+γ+ε+η)∥Snx-TSnx∥2≤0.
Since {TSnx}, {Snx}, and {Tnx} are bounded, we have that
(39)(α+γ+ε+η)limsupn→∞∥Snx-TSnx∥≤0.
Since α+γ+ε+η>0, we have that limn→∞∥Snx-TSnx∥=0. In particular, if C is bounded, then
(40)limsupn→∞supx∈C∥Snx-TSnx∥≤0,
and hence limn→∞supx∈C∥Snx-TSnx∥=0.
Similarly, we can obtain the desired result for the case ofα+β+γ+δ≥0, α+β+ζ+η>0, and ε+η≥0. This completes the proof.Lemma 11.
LetH 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 xn⇀z and xn-Txn→0, then z∈A(T).Proof.
LetT:C→H be an (α,β,γ,δ,ε,ζ,η)-widely more generalized hybrid mapping, and suppose that xn⇀z and xn-Txn→0. Replacing x by xn in (14), we have that
(41)α∥Txn-Ty∥2+β∥xn-Ty∥2+γ∥Txn-y∥2+δ∥xn-y∥2+ε∥xn-Txn∥2+ζ∥y-Ty∥2+η∥(xn-Txn)-(y-Ty)∥2≤0.
From this inequality, we have that
(42)α(∥Txn-xn∥2+∥xn-Ty∥2+2〈Txn-xn,xn-Ty〉)+β∥xn-Ty∥2+γ(∥Txn-xn∥2+∥xn-y∥2+2〈Txn-xn,xn-y〉∥Txn-xn∥2)+δ∥xn-y∥2+ε∥xn-Txn∥2+ζ∥y-Ty∥2+η∥(xn-Txn)-(y-Ty)∥2≤0.
We apply a Banach limit μ to both sides of this inequality. We have that
(43)αμn(∥Txn-xn∥2+∥xn-Ty∥2+2〈Txn-xn,xn-Ty〉)+βμn∥xn-Ty∥2+γμn(∥Txn-xn∥2+∥xn-y∥2+2〈Txn-xn,xn-y〉∥Txn-xn∥2)+δμn∥xn-y∥2+εμn∥xn-Txn∥2+ζ∥y-Ty∥2+ημn∥(xn-Txn)-(y-Ty)∥2≤0,
and hence
(44)αμn∥xn-Ty∥2+βμn∥xn-Ty∥2+γμn∥xn-y∥2+δμn∥xn-y∥2+ζ∥y-Ty∥2+ημn∥(xn-Txn)-(y-Ty)∥2≤0.
Thus we have
(45)(α+β)μn∥xn-Ty∥2+(γ+δ)μn∥xn-y∥2+(ζ+η)∥y-Ty∥2≤0.
From ∥xn-Ty∥2=∥xn-y∥2+∥y-Ty∥2+2〈xn-y,y-Ty〉, we also have
(46)(α+β)(μn∥xn-y∥2+∥y-Ty∥2+2μn〈xn-y,y-Ty〉)+(γ+δ)μn∥xn-y∥2+(ζ+η)∥y-Ty∥2≤0.
From α+β+γ+δ≥0 we obtain that
(47)(α+β)∥y-Ty∥2+2(α+β)μn〈xn-y,y-Ty〉+(ζ+η)∥y-Ty∥2≤0,
and hence
(48)(α+β+ζ+η)∥y-Ty∥2+2(α+β)μn〈xn-y,y-Ty〉≤0.
Since xn⇀z, we have that
(49)(α+β+ζ+η)∥y-Ty∥2+2(α+β)〈z-y,y-Ty〉≤0.
Using (9), we have that
(50)(α+β+ζ+η)∥y-Ty∥2+(α+β)(∥z-Ty∥2-∥z-y∥2-∥y-Ty∥2)≤0,
and hence
(51)(ζ+η)∥y-Ty∥2+(α+β)(∥z-Ty∥2-∥z-y∥2)≤0.
Since α+β>0 and ζ+η≥0, we have that
(52)∥z-Ty∥2-∥z-y∥2≤0
for all y∈C. This implies that z∈A(T).
Similarly, we can obtain the desired result for the case ofα+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.
LetH 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,
(53)Snx=1n∑k=0n-1‍Tkx
is weakly convergent to an attractive point p of  T, where p=limn→∞PTnx.Proof.
Letx∈C. Since A(T) is nonempty, we obtain that
(54)∥Tn+1x-y∥≤∥Tnx-y∥
for any n∈ℕ∪{0} and y∈A(T). Thus {Tnx} is bounded. Since
(55)∥Snx-y∥≤1n∑k=0n-1‍∥Tkx-y∥≤∥x-y∥
for any n∈ℕ∪{0} and y∈A(T), {Snx∣n=0,1,…} is also bounded. Therefore there exists a strictly increasing sequence {ni} and p∈H such that {Snix∣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
(56)limn→∞∥Snx-TSnx∥=0.
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 {PTnx∣n=0,1,…} converges strongly to q. To complete the proof, we show that q=p. Note that the metric projection P satisfies
(57)〈z-Pz,Pz-u〉≥0
for any z∈H and for any u∈A(T); see [25]. Therefore
(58)〈Tkx-PTkx,PTkx-y〉≥0
for any k∈ℕ∪{0} and y∈A(T). Since P is the metric projection from H onto A(T) and PTn-1x∈A(T), we obtain that
(59)∥Tnx-PTnx∥≤∥Tnx-PTn-1x∥≤∥Tn-1x-PTn-1x∥.
That is, {∥Tnx-PTnx∥∣n=0,1,…} is nonincreasing. Therefore we obtain
(60)〈Tkx-PTkx,y-q〉≤〈Tkx-PTkx,PTkx-q〉≤∥Tkx-PTkx∥·∥PTkx-q∥≤∥x-Px∥·∥PTkx-q∥.
Summing up these inequalities with respect to k=0,1,…,n-1 and dividing by n, we obtain
(61)〈Snx-1n∑k=0n-1‍PTkx,y-q〉≤∥x-Px∥n∑k=0n-1‍∥PTkx-q∥.
Since {Snix∣i=0,1,…} converges weakly to p and {PTnx∣n=0,1,…} converges strongly to q, we obtain that
(62)〈p-q,y-q〉≤0.
Putting y=p, we obtain
(63)∥p-q∥2≤0,
and hence q=p. This completes the proof.
Similarly, we can obtain the desired result for the case ofα+β+γ+δ≥0, α+β>0, ζ+η>0, and ε+η≥0.As the proof of Theorem9, we can prove Takahashi and Takeuchi’s mean convergence theorem for generalized hybrid mappings in a Hilbert space.Theorem 13.
LetH 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
(64)α∥Tx-Ty∥2+(1-α)∥x-Ty∥2≤β∥Tx-y∥2+(1-β)∥x-y∥2
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,
(65)Snx=1n∑k=0n-1‍Tkx
converges weakly to p∈A(T), where p=limn→∞PTnx.


## 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.
LetH 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.
LetP be the metric projection of H onto A(T). Let {αn} be a sequence of real numbers such that 0≤αn≤1 and liminfn→∞αn(1-αn)>0. Suppose that {xn} is the sequence generated by x1=x∈C and
(66)xn+1=αnxn+(1-αn)Txn,n∈ℕ.
Then {xn} converges weakly to v∈A(T), where v=limn→∞Pxn.Proof.
Letz∈A(T). We have that
(67)∥xn+1-z∥2=∥αnxn+(1-αn)Txn-z∥2≤αn∥xn-z∥2+(1-αn)∥Txn-z∥2≤αn∥xn-z∥2+(1-αn)∥xn-z∥2=∥xn-z∥2
for all n∈ℕ. Hence limn→∞∥xn-z∥2 exists. Then {xn} is bounded. We also have from (8) that
(68)∥xn+1-z∥2=∥αnxn+(1-αn)Txn-z∥2=αn∥xn-z∥2+(1-αn)∥Txn-z∥2-αn(1-αn)∥Txn-xn∥2≤αn∥xn-z∥2+(1-αn)∥xn-z∥2-αn(1-αn)∥Txn-xn∥2=∥xn-z∥2-αn(1-αn)∥Txn-xn∥2.
Thus we have(69)αn(1-αn)∥Txn-xn∥2≤∥xn-z∥2-∥xn+1-z∥2.
Since limn→∞∥xn-z∥2 exists and liminfn→∞αn(1-αn)>0, we have that
(70)∥Txn-xn∥⟶0.
Since {xn} is bounded, there exists a subsequence {xni} of {xn} such that xni⇀v. By Lemma 11 and (70), we obtain that v∈A(T). Let {xni} and {xnj} be two subsequences of {xn} such that xni⇀v1 and xnj⇀v2. To complete the proof, we show v1=v2. We know that v1,v2∈A(T), and hence limn→∞∥xn-v1∥2 and limn→∞∥xn-v2∥2 exist. Put
(71)a=limn→∞(∥xn-v1∥2-∥xn-v2∥2).
Note that for n=1,2,…,
(72)∥xn-v1∥2-∥xn-v2∥2=2〈xn,v2-v1〉+∥v1∥2-∥v2∥2.
From xni⇀v1 and xnj⇀v2, we have
(73)a=2〈v1,v2-v1〉+∥v1∥2-∥v2∥2,(74)a=2〈v2,v2-v1〉+∥v1∥2-∥v2∥2.
Combining (73) and (74), we obtain 0=2〈v2-v1,v2-v1〉. Thus we obtain v2=v1. This implies that {xn} converges weakly to an element v∈A(T). Since ∥xn+1-z∥≤∥xn-z∥ for all z∈A(T) and n∈ℕ, we obtain from Lemma 3 that {Pxn} converges strongly to an element p∈A(T). On the other hand, we have from the property of P that
(75)〈xn-Pxn,Pxn-u〉≥0
for all u∈A(T) and n∈ℕ. Since xn⇀v and Pxn→p, we obtain
(76)〈v-p,p-u〉≥0
for all u∈A(T). Putting u=v, we obtain p=v. This means v=limn→∞Pxn. This completes the proof.Using Theorem14, 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]).
LetH 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 {αn} be a sequence of real numbers such that 0≤αn≤1 and liminfn→∞αn(1-αn)>0. Suppose {xn} is the sequence generated by x1=x∈C and
(77)xn+1=αnxn+(1-αn)Txn,n∈ℕ.
Then the sequence {xn} converges weakly to an element v∈F(T).Proof.
As in the proof of Theorem9, a generalized hybrid mapping is a widely more generalized hybrid mapping. Since {xn}⊂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 {xn} 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.
LetC 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.
Letu∈C, and define sequences {xn} and {zn} in C as follows: x1=x∈C and
(78)xn+1=αnu+(1-αn)zn,zn=1n∑k=0n-1‍Tkxn
for all n=1,2,…, where 0≤αn≤1, αn→0 and ∑n=1∞‍αn=∞. If A(T) is nonempty, then {xn} and {zn} converge strongly to Pu, where P is the metric projection of H onto A(T).Proof.
SinceT:C→C is a widely more generalized hybrid mapping, there exist α,β,γ,δ,ε,ζ,η∈ℝ such that
(79)α∥Tx-Ty∥2+β∥x-Ty∥2+γ∥Tx-y∥2+δ∥x-y∥2+ε∥x-Tx∥2+ζ∥y-Ty∥2+η∥(x-Tx)-(y-Ty)∥2≤0
for any x,y∈C. Since A(T)≠∅, we have that for all q∈A(T) and n=1,2,3,…,
(80)∥zn-q∥=∥1n∑k=0n-1‍Tkxn-q∥≤1n∑k=0n-1‍∥Tkxn-q∥≤1n∑k=0n-1‍∥xn-q∥=∥xn-q∥.
Then we have
(81)∥xn+1-q∥=∥αnu+(1-αn)zn-q∥∥xn+1-q∥≤αn∥u-q∥+(1-αn)∥zn-q∥∥xn+1-q∥≤αn∥u-q∥+(1-αn)∥xn-q∥.
Hence, by induction, we obtain
(82)∥xn-q∥≤max{∥u-q∥,∥x-q∥}
for all n∈ℕ. This implies that {xn} and {zn} are bounded. Since ∥Tnxn-q∥≤∥xn-q∥, we have also that {Tnxn} is bounded.
Letn∈ℕ. Using α+β+γ+δ≥0 and ζ+η≥0, as in the proof of Theorem 8 we have that
(83)(α+η)(∥Tz-Tk+1xn∥2-∥Tz-Tkxn∥2)+(β-η)(∥z-Tk+1xn∥2-∥z-Tkxn∥2)-2(α+γ)〈z-Tz,z-Tkxn〉+(α+γ+ε+η)∥z-Tz∥2≤0
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
(84)α+ηn(∥Tz-Tnxn∥2-∥Tz-xn∥2)+β-ηn(∥z-Tnxn∥2-∥z-xn∥2)-2(α+γ)〈z-Tz,z-zn〉+(α+γ+ε+η)∥z-Tz∥2≤0.
Since {zn} is bounded, there exists a subsequence {zni} of {zn} such that zni⇀w∈H. Replacing n by ni, we have that
(85)α+ηni(∥Tz-Tnixni∥2-∥Tz-xni∥2)+β-ηni(∥z-Tnixni∥2-∥z-xni∥2)-2(α+γ)〈z-Tz,z-zni〉+(α+γ+ε+η)∥z-Tz∥2≤0.
Since {xn} and {Tnxn} are bounded, we have that
(86)-2(α+γ)〈z-Tz,z-w〉+(α+γ+ε+η)∥z-Tz∥2≤0
as i→∞. Using (9), we have that
(87)(α+γ+ε+η)∥z-Tz∥2+(α+γ)(∥w-Tz∥2-∥w-z∥2-∥z-Tz∥2)≤0,
and hence
(88)(ε+η)∥z-Tz∥2+(α+γ)(∥w-Tz∥2-∥w-z∥2)≤0.
Since α+γ>0 and ε+η≥0, we have that
(89)∥w-Tz∥2-∥w-z∥2≤0
for all z∈C. This implies that w∈A(T).
On the other hand, sincexn+1-zn=αn(u-zn), {zn} is bounded, and αn→0, we have limn→∞∥xn+1-zn∥=0. Let us show
(90)limsupn→∞〈u-Pu,xn+1-Pu〉≤0.
We may assume without loss of generality that there exists a subsequence {xni+1} of {xn+1} such that
(91)limsupn→∞〈u-Pu,xn+1-Pu〉=limi→∞〈u-Pu,xni+1-Pu〉
and xni+1⇀v. From ∥xn+1-zn∥→0, we have zni⇀v. From the above argument, we have v∈A(T). Since P is the metric projection of H onto A(T), we have
(92)limi→∞〈u-Pu,xni+1-Pu〉=〈u-Pu,v-Pu〉≤0.
This implies
(93)limsupn→∞〈u-Pu,xn+1-Pu〉≤0.
Since xn+1-Pu=(1-αn)(zn-Pu)+αn(u-Pu), from (7) and (80) we have
(94)∥xn+1-Pu∥2=∥(1-αn)(zn-Pu)+αn(u-Pu)∥2∥xn+1-Pu∥2≤(1-αn)2∥zn-Pu∥2+2αn〈u-Pu,xn+1-Pu〉∥xn+1-Pu∥2≤(1-αn)∥xn-Pu∥2+2αn〈u-Pu,xn+1-Pu〉.
Putting sn=∥xn-Pu∥2, βn=0, and γn=2〈u-Pu,xn+1-Pu〉 in Lemma 4, we have from ∑n=1∞‍αn=∞ and (93) that
(95)limn→∞∥xn-Pu∥=0.
By limn→∞∥xn+1-zn∥=0, we also obtain zn→Pu as n→∞.
Similarly, we can obtain the desired result for the case ofα+β+γ+δ≥0, α+β>0, ζ+η≥0, and ε+η≥0.Using Theorem16, we can show the following result obtained by Kurokawa and Takahashi [30].Theorem 17 (Hojo and Takahashi [22]).
LetC 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 {xn} and {zn} in C as follows: x1=x∈C and
(96)xn+1=αnu+(1-αn)zn,zn=1n∑k=0n-1‍Tkxn
for all n=1,2,…, where 0≤αn≤1, αn→0 and ∑n=1∞‍αn=∞. If F(T) is nonempty, then {xn} and {zn} converge strongly to Pu∈F(T), where P is the metric projection of H onto A(T).Proof.
As in the proof of Theorem9, a generalized hybrid mapping is a widely more generalized hybrid mapping. Since {xn},{zn}⊂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 {xn} and {zn} converge strongly to an element Pu∈F(T).



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*Source: 904164-2013-06-27.xml*