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

()

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

()

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

()

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,

()

converges weakly to p ∈ F(T), where p = lim _n→∞PTⁿx.

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

()

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

()

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

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,

()

Furthermore, we have that

()

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

()

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 , let {β_n} be a sequence of nonnegative real numbers with , and let {γ_n} be a sequence of real numbers with limsup _n→∞γ_n ≤ 0. Suppose that

()

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^∞,

()

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 such that

()

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

()

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 {Tⁿz∣n = 0,1, …} is bounded.

Proof. Suppose that T 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 exists z ∈ 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

()

for any n ∈ ℕ ∪ {0} and x ∈ C. By (9) we obtain that

()

Thus we have that

()

From

()

we have that

()

and hence

()

By α + β + γ + δ ≥ 0, we have that

()

From this inequality and ζ + η ≥ 0 we obtain that

()

Applying a Banach limit μ to both sides of this inequality, we obtain that

()

and hence

()

Since there exists p ∈ H by Lemma 5 such that

()

for any y ∈ H, we obtain from (24) that

()

We obtain from (9) that

()

and hence

()

Since ε + η ≥ 0, we obtain that

()

Since α + γ > 0, we obtain that

()

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

()

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,

()

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

. Then, lim _n→∞∥S_nx − TS_nx∥ = 0. In particular, if C is bounded, then

()

Proof. Let x ∈ C. Since A(T) is nonempty, we obtain that

()

for any n ∈ ℕ ∪ {0} and y ∈ A(T). Then we have that {Tⁿx} is bounded. Since

()

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

()

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

()

Replacing z by S_nx, we obtain that

()

Since {TS_nx}, {S_nx}, and {Tⁿx} are bounded, we have that

()

Since α + γ + ε + η > 0, we have that lim _n→∞∥S_nx − TS_nx∥ = 0. In particular, if C is bounded, then

()

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

()

From this inequality, we have that

()

We apply a Banach limit μ to both sides of this inequality. We have that

()

and hence

()

Thus we have

()

From

, we also have

()

From α + β + γ + δ ≥ 0 we obtain that

()

and hence

()

Since x_n⇀z, we have that

()

Using (9), we have that

()

and hence

()

Since α + β > 0 and ζ + η ≥ 0, we have that

()

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,

()

is weakly convergent to an attractive point p of T, where p = lim _n→∞PTⁿx.

Proof. Let x ∈ C. Since A(T) is nonempty, we obtain that

()

for any n ∈ ℕ ∪ {0} and y ∈ A(T). Thus {Tⁿx} is bounded. Since

()

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

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

()

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

()

for any z ∈ H and for any u ∈ A(T); see [25]. Therefore

()

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

()

That is, {∥Tⁿx − PTⁿx∥∣n = 0,1, …} is nonincreasing. Therefore we obtain

()

Summing up these inequalities with respect to k = 0,1, …, n − 1 and dividing by n, we obtain

()

Since

converges weakly to p and {PTⁿx∣n = 0,1, …} converges strongly to q, we obtain that

()

Putting y = p, we obtain

()

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

()

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,

()

converges weakly to p ∈ A(T), where p = lim _n→∞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. 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

()

Then {x_n} converges weakly to v ∈ A(T), where v = lim _n→∞Px_n.

Proof. Let z ∈ A(T). We have that

()

for all n ∈ ℕ. Hence

exists. Then {x_n} is bounded. We also have from (8) that

()

Thus we have

()

Since

exists and liminf _n→∞α_n(1 − α_n) > 0, we have that

()

Since {x_n} is bounded, there exists a subsequence

of {x_n} such that

. By Lemma 11 and (70), we obtain that v ∈ A(T). Let

and

be two subsequences of {x_n} such that

and

. To complete the proof, we show v₁ = v₂. We know that v₁, v₂ ∈ A(T), and hence

and

exist. Put

()

Note that for n = 1,2, …,

()

From

and

, we have

()

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

()

for all u ∈ A(T) and n ∈ ℕ. Since x_n⇀v and Px_n → p, we obtain

()

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

()

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

()

for all n = 1,2, …, where 0 ≤ α_n ≤ 1, α_n → 0 and

. 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

()

for any x, y ∈ C. Since A(T) ≠ ∅, we have that for all q ∈ A(T) and n = 1,2, 3, …,

()

Then we have

()

Hence, by induction, we obtain

()

for all n ∈ ℕ. This implies that {x_n} and {z_n} are bounded. Since ∥Tⁿx_n − q∥ ≤ ∥x_n − q∥, we have also that {Tⁿx_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

()

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

()

Since {z_n} is bounded, there exists a subsequence

of {z_n} such that

. Replacing n by n_i, we have that

()

Since {x_n} and {Tⁿx_n} are bounded, we have that

()

as i → ∞. Using (9), we have that

()

and hence

()

Since α + γ > 0 and ε + η ≥ 0, we have that

()

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

()

We may assume without loss of generality that there exists a subsequence

of {x_n+1} such that

()

and

. From ∥x_n+1 − z_n∥ → 0, we have

. From the above argument, we have v ∈ A(T). Since P is the metric projection of H onto A(T), we have

()

This implies

()

Since x_n+1 − Pu = (1 − α_n)(z_n − Pu) + α_n(u − Pu), from (7) and (80) we have

()

Putting

, β_n = 0, and γ_n = 2〈u − Pu, x_n+1 − Pu〉 in Lemma 4, we have from

and (93) that

()

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

()

for all n = 1,2, …, where 0 ≤ α_n ≤ 1, α_n → 0 and

. 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.

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

Abstract

1. Introduction

2. Preliminaries

3. Attractive Point Theorems

4. Nonlinear Ergodic Theorems

5. Weak Convergence Theorems of Mann’s Type

6. Strong Convergence Theorem

Acknowledgments

References

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

Abstract

1. Introduction

2. Preliminaries

3. Attractive Point Theorems

4. Nonlinear Ergodic Theorems

5. Weak Convergence Theorems of Mann’s Type

6. Strong Convergence Theorem

Acknowledgments

References

References

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