# Fixed point theorems for mappings with condition (B)

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Research

## Abstract

In this article, a new type of mappings that satisfies condition (B) is introduced. We study Pazy's type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings with condition (B).

### Keywords

fixed point equilibrium problem Banach limit generalized hybrid mapping projection

## 1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : CH be a mapping, and let F(T) denote the set of fixed points of T. A mapping T : CH is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, yC. A mapping T : CH is said to be quasi-nonexpansive mapping if F(T) ≠ ∅ and ||Tx - Ty|| ≤ ||x - y|| for all xC and yF(T).

In 2008, Kohsaka and Takahashi [1] introduced nonspreading mapping, and obtained a fixed point theorem for a single nonspreading mapping, and a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. A mapping T : CC is called nonspreading [1] if
for all x, yC. Indeed, T : CC is a nonspreading mapping if and only if

for all x, yC[2].

Recently, Takahashi and Yao [3] introduced two nonlinear mappings in Hilbert spaces. A mapping T : CC is called a TY-1 mapping [3] if
for all x, yC. A mapping T : CC is called a TY-2 [3] mapping if

for all x, yC.

In 2010, Takahashi [4] introduced the hybrid mappings. A mapping T : CC is hybrid [4] if
for each x, yC. Indeed, T : CC is a hybrid mapping if and only if

for all x, yC[4].

In 2010, Aoyoma et al. [5] introduced λ-hybrid mappings in a Hilbert space. Note that the class of λ-hybrid mappings contain the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings. Let λ be a real number. A mapping T : CC is called λ-hybrid [5] if

for all x, yC.

In 2010, Kocourek et al. [6] introduced (α, β)-generalized hybrid mappings, and studied fixed point theorems and weak convergence theorems for such nonlinear mappings in Hilbert spaces. Let α, β ∈ ℝ. A mapping T : CH is (α, β)-generalized hybrid [6] if

for all x, yC.

In 2011, Aoyama and Kohsaka [7] introduced α-nonexpansive mapping on Banach spaces. Let C be a nonempty closed convex subset of a Banach space E, and let α be a real number such that α < 1. A mapping T : CE is said to be α-nonexpansive if

for all x, yC.

Furthermore, we observed that Suzuki [8] introduced a new class of nonlinear mappings which satisfy condition (C) in Banach spaces. Let C be a nonempty subset of a Banach space E. Then, T : CE is said to satisfy condition (C) if for all x, yC,

In fact, every nonexpansive mapping satisfies condition (C), but the converse may be false [8, Example 1]. Besides, if T : CE satisfies condition (C) and F(T) ≠ ∅, then T is a quasi-nonexpansive mapping. However, the converse may be false [8, Example 2].

Motivated by the above studies, we introduced Takahashi's -generalized hybrid mappings with Suzuki's sense on Hilbert spaces.

Definition 1.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CH be a mapping. Then, we say T satisfies condition (B) if for all x, yC,
Remark 1.1.
1. (i)

In fact, if T is the identity mapping, then T satisfies condition (B).

2. (ii)

Every -generalized hybrid mapping satisfies condition (B). But the converse may be false.

3. (iii)

If T : CC satisfies condition (B) and F(T) ≠ ∅, then T is a quasi-nonexpansive mapping, and this implies that F(T) is a closed convex subset of C [9].

Remark 1.2. Let H = ℝ, let C be nonempty closed convex subset of H, and let T : CH be a function. In fact, we have
and
Example 1.1. Let H = C = ℝ, and let T : CH be defined by Tx : = -x for each xC. Hence, we have the following conditions:
1. (1)

T is -generalized hybrid mapping, and T satisfies condition (B).

2. (2)
T is not a nonspreading mapping. Indeed, if x = 1 and y = -1, then

3. (3)
T is not a TY-1 mapping. Indeed, if x = 1 and y = -1, then

4. (4)
T is not a TY-2 mapping. Indeed, if x = 1 and y = -1, then

5. (5)
T is not a hybrid mapping. Indeed, if x = 1 and y = -1, then

6. (6)
Now, we want to show that if α ≠ 0, then T is not a α-nonexpansive mapping. For α > 0, let x = 1 and y = -1,

For α < 0, let x = y = 1,
1. (7)

Similar to (6), if α + β ≠ 1, then T is not a (α, β)-generalized hybrid mapping.

Example 1.2. Let H = ℝ, C = [-1, 1], and let T : CC be defined by
for each xC. First, we consider the following conditions:
1. (a)

For x ∈ [-1, 0] and , we know that

(a)1 if y ∈ [-1, 0], then Ty = y and (Ty - y)[(Ty + y) - (Tx + x)] = 0;

(a)2 if y ∈ [0,1], then Ty = -y and (Ty - y)[(Ty + y) - (Tx + x)] = 4xy ≤ 0.
1. (b)

For x ∈ (0, 1] and , we know that

(b)1 if yx, then xy - x, Tx = -x, and Ty = -y. So, (Ty - y)[(Ty + y) - (Tx + x)] = 0;

(b)2 if y <x, then xx - y and this implies that y ≤ 0. So, (Ty - y)[(Ty + y) - (Tx + x)] = 0.

By these conditions and Remark 1.2, we know that T satisfies condition (B). In fact, T is -generalized hybrid mapping. Furthermore, we know that the following conditions:
1. (1)

T is a nonspreading mapping. Indeed, we know that the following conditions hold.

(1)1 If x > 0 and y > 0, then
(1)2 If x ≤ 0 and y ≤ 0, then
(1)3 If x > 0 and y ≤ 0, then ||Tx - Ty||2 = ||Tx - y||2 = || x+y||2, and ||Ty - x||2 = ||x - y||2. Hence,
1. (2)

Similar to the above, we know that T is a TY-1 mapping, a TY-2 mapping, a hybrid mapping, (α, β)-generalized hybrid mapping, and T is a α-nonexpansive mapping.

On the other hand, the following iteration process is known as Mann's type iteration process [10] which is defined as

where the initial guess x0 is taken in C arbitrarily and {α n } is a sequence in [0,1].

In 1974, Ishikawa [11] gave an iteration process which is defined recursively by

where {α n } and {β n } are sequences in [0,1].

In 1995, Liu [12] introduced the following modification of the iteration method and he called Ishikawa iteration method with errors: for a normed space E, and T : EE a given mapping, the Ishikawa iteration method with errors is the following sequence

where {α n } and {β n } are sequences in [0,1], and {u n } and {v n } are sequences in E with and .

In 1998, Xu [13] introduced an Ishikawa iteration method with errors which appears to be more satisfactory than the one introduced by Liu [12]. For a nonempty convex subset C of E and T : CC a given mapping, the Ishikawa iteration method with errors is generated by

where {a n }, {b n }, {c n }, , , are sequences in [0,1] with a n + b n + c n + = 1 and , and {u n } and {v n } are bounded sequences in C.

Motivated by the above studies, we consider an Ishikawa iteration method with errors for mapping with condition (B).

We also consider the following iteration for mappings with condition (B). Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C × C → ℝ be a function. Let T : CH be a mapping. Let {a n }, {b n }, and {θ n } be sequences in [0,1] with a n + b n + θ n = 1. Let {ω n } be a bounded sequence in C. Let {r n } be a sequence of positive real numbers. Let {x n } be defined by u1H

Furthermore, we observed that Phuengrattana [14] studied approximating fixed points of for a nonlinear mapping T with condition (C) by the Ishikawa iteration method on uniform convex Banach space with Opial property. Here, we also consider the Ishikawa iteration method for a mapping T with condition (C) and improve some conditions of Phuengrattana's result.

In this article, a new type of mappings that satisfies condition (B) is introduced. We study Pazy's type fixed point theorems, demiclosed principles, and ergodic theorem for mappings with condition (B). Next, we consider the weak convergence theorems for equilibrium problems and the fixed points of mappings with condition (B).

## 2 Preliminaries

Throughout this article, let ℕ be the set of positive integers and let ℝ be the set of real numbers. Let H be a (real) Hilbert space with inner product 〈·, ·〉 and norm || · ||, respectively. We denote the strongly convergence and the weak convergence of {x n } to xH by x n x and x n x, respectively. From [15], for each x, yH and λ ∈ [0,1], we have
Hence, we also have
for all x, y, u, vH. Furthermore, we know that

for each x, y, zH and α, β, γ ∈ [0,1] with α + β + γ = 1 [16].

Let ℓ be the Banach space of bounded sequences with the 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, . . .). For x = (x1, x2, x3, . . .), A Banach limit on ℓ is an invariant mean, that is, μ n x n = μ n x n+1 for any n ∈ ℕ. 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. For details, we can refer [17].

Lemma 2.1. [17]Let C be a nonempty closed convex subset of a Hilbert space H, {x n } be a bounded sequence in H, and μ be a Banach limit. Let g : C → ℝ be defined by g(z): = μ n ||x n - z||2for all zC. Then there exists a unique z0C such that .

Lemma 2.2. [17]Let C be a nonempty closed convex subset of a Hilbert space H. Let P C be the metric projection from H onto C. Then for each xH, we havex - P C x, P C x - y〉 ≥ 0 for all yC.

Lemma 2.3. [17]Let D be a nonempty closed convex subset of a real Hilbert space H. Let P D be the metric projection from H onto D, and let {x n }n∈ℕbe a sequence in H. If x n x0and P D x n y0, then P D x0= y0.

Lemma 2.4. [18]Let D be a nonempty closed convex subset of a real Hilbert space H. Let P D be the metric projection from H onto D. Let {x n }n∈ℕbe a sequence in H with ||x n+1 - u||2 ≤ (1 + λ n ) ||x n - u||2+ δ n for all uD and n ∈ ℕ, where {λ n } and {δ n } are sequences of nonnegative real numbers such that and . Then{P D x n } converges strongly to an element of D.

Lemma 2.5. [19]Let {s n } and {t n } be two nonnegative sequences satisfying sn+1s n + t n for each n ∈ ℕ. If , then exists.

The equilibrium problem is to find zC such that
(2.1)

The solution set of equilibrium problem (2.1) is denoted by (EP). For solving the equilibrium problem, let us assume that the bifunction G : C × C → ℝ satisfies the following conditions:

(A1) G(x, x) = 0 for each xC;

(A2) G is monotone, i.e., G(x, y) + G(y, x) ≤ 0 for any x, yC;

(A3) for each x, y, zC, ;

(A4) for each xC, the scalar function yG(x, y) is convex and lower semicontinuous.

Lemma 2.6. [20]Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C × C → ℝ be a bifunction which satisfies conditions (A1)-(A4). Let r > 0 and ×H. Then there exists zC such that
Furthermore, if

then we have:

(i) T r is single-valued;

(ii) T r is firmly nonexpansive, that is, ||T r x - T r y||2 ≤ 〈T r x - T r y, x - yfor each x, yH;

(iii) (EP) is a closed convex subset of C;

(iv) (EP) = F(T r ).

## 3 Fixed point theorems

Proposition 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : CC be a mapping with condition (B). Then for each x, yC, we have:
1. (i)

||Tx - T 2 x||2 + ||x - T 2 x|| ≤ ||x - Tx ||2 ;

2. (ii)

||Tx - T 2 x|| ≤ ||x - Tx || and ||x - T 2 x|| ≤ ||x - Tx ||;

3. (iii)

either or holds;

4. (iv)
either

or
holds;
1. (v)

.

Proof Since , it is easy to see (i) and (ii) are satisfied. (iii) Suppose that
holds. So,

This is a contradiction. Therefore, we obtain the desired result. Next, it is easy to get (iv) by (iii).

(v): By (i), we know that
Then {||T n x - T n+1 x||} is a decreasing sequence, and exists. Furthermore, we have:

So, .

Proposition 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : CC be a mapping with condition (B). Then for each x, yC,
Proof By Proposition 3.1(iv), for each x, yC, either
or
holds. In the first case, we have
In the second case, we have

Therefore, the proof is completed.

Remark 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : CC be a mapping with condition (B). Then for each x, yC, we have:
1. (a)

||Tx - Ty||2 + ||x - Ty||2 ≤ ||x - y||2 + ||Tx - y||2 + ||T 2 x - x|| · ||Ty - y||.

2. (b)

Tx - Ty, y - Ty〉 ≤ 〈x - y, Ty - y〉 + ||Tx - x|| · ||Ty - y||.

Proof By Proposition 3.2, it is easy to prove Remark 3.1.

The following theorem shows that demiclosed principle is true for mappings with condition (B).

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CC be a mapping with condition (B). Let {x n } be a sequence in C with x n x and . Then Tx= x.

Proof By Remark 3.1, we get:

for each n ∈ ℕ. By assumptions, 〈x - Tx , x - Tx 〉 ≤ 0. So, Tx = x.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CC be a mapping with condition (B). Then {T n x} is a bounded sequence for some xC if and only if F(T) ≠ ∅.

Proof For each n ∈ ℕ, let x n := T n x. Clearly, {x n } is a bounded sequence. By Lemma 2.1, there is a unique zC such that . By Proposition 3.2, for each n ∈ ℕ,

By Proposition 3.1(v), μ n ||x n - Tz||2μ n ||x n - z||2. This implies that Tz = z and F(T) ≠ ∅. Conversely, it is easy to see.

Corollary 3.1. Let C be a nonempty bounded closed convex subset of a real Hilbert space H, and let T : CC be a mapping with condition (B). Then F(T) ≠ ∅.

The following theorem shows that Ballion's type Ergodic's theorem is also true for the mapping with condition (B).

Theorem 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CC be a mapping with condition (B). Then the following conditions are equivalent:
1. (i)

for each xC, converges weakly to an element of C;

2. (ii)

F(T) ≠ ∅.

In fact, if F(T) ≠ ∅, then for each xC, we know that S n xv, whereand P F(T) is the metric projection from H onto F(T).

Proof (i)⇒ (ii): Take any xC and let x be fixed. Then there exists vC such that S n xv. By Proposition 3.2, for each k ∈ ℕ, we have:
By Proposition 3.1(v), . This implies that
Since S n xv, we have:

So, Tv = v.

(ii)⇒ (i): Take any xC and uF(T), and let x and u be fixed. Since T satisfies condition (B), ||T n x - u|| ≤ ||Tn-1x - u|| for each n ∈ ℕ. Hence, exists and this implies that {T n x} is a bounded sequence. By Lemma 2.4, there exists zF(T) such that . Clearly, zF(T). Besides, we have:
So, {S n x} is a bounded sequence. Then there exist a subsequence of {S n x} and vC such that . By the above proof, we have:
This implies that

Since , {T n x} is a bounded sequence, and , it is easy to see that Tv = v. So, vF(T).

By Lemma 2.2, for each k ∈ ℕ, 〈T k x - P F(T) T k x, P F(T) T k x - u〉 ≥ 0. This implies that
Adding these inequalities from k = 0 to k = n - 1 and dividing by n, we have

Since and P F(T) Tkx → z, we get 〈v - z, u - z〉 ≤ 0. Since u is any point of F(T), we know that .

Furthermore, if is a subsequence of {S n x} and , then q = v by following the same argument as the above proof. Therefore, , and the proof is completed.

## 4 Weak convergence theorems with errors

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T1, T2 : CC be two mappings with condition (B) and Ω: = F(T1) ∩ F(T2) ≠ ∅. Let {a n }, {b n }, {c n }, {d n }, {θ n }, and {λ n } be sequences in [0,1] with
Let {u n } and {v n } be bounded sequences in C. Let {x n } and {y n } be defined by
Assume that:
1. (i)

and ;

2. (ii)

and .

Then x n z and y n z, where.

Proof Take any w ∈ Ω and let w be fixed. Then for each n ∈ ℕ, we have:
and
By Lemma 2.5, exists. So, {x n } is a bounded sequence. Now, we set . Besides,
This implies that
By assumption, . Furthermore, we have:
This implies that
Hence, . By assumption,
Since ,
Hence, . Similar to the above proof, we also get
Besides,

This implies that and . Since {x n } is a bounded sequence, there exists a subsequence of {x n } such that . By Theorem 3.1, z = T1z.

If is a subsequence of {x n } and , then T1q = q. Suppose that qz. Then we have:

And this leads to a contradiction. Then every weakly convergent subsequence of x n has the same limit. So, x n zF(T1). Since x n z and , y n z. By Theorem 3.1, zF(T2). Hence, z ∈ Ω.

Next, by Lemma 2.4, PΩx n converges. Then there exists v ∈ Ω such that . By Lemma 2.3, PΩz = v. Since z ∈ Ω, , and the proof is completed.

In Theorem 4.1, if θ n = λ n = 0 for each n ∈ ℕ, then we have the following result.

Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T1, T2 : CC be two mappings with condition (B) and Ω: = F(T1) ∩ F(T2) ≠ ∅. Let {a n } and (b n ) be two sequences in [0,1]. Let {x n } be defined by

Assume thatand. Then x n z and y n z, where.

Furthermore, we also have the following corollaries from Theorem 4.2.

Corollary 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CC be a mapping with condition (B) and F(T) ≠ ∅. Let {a n } and {b n } be two sequences in [0,1]. Let {x n } be defined by

Assume thatand. Then x n z and y n z, where.

Corollary 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : CC be a mapping with condition (B) and F(T) ≠ ∅. Let {b n } be a sequence in [0,1]. Let {x n } be defined by

Assume that. Then x n z, where.

Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C × C → ℝ be a function satisfying (A1)-(A4). Let T : CC be a mapping with condition (B) and Ω: = F(T) ∩ (EP) ≠ ∅. Let {a n }, {b n }, and {θ n } be sequences in [0,1] with a n + b n + θ n = 1. Let {ω n } be a bounded sequence in C. Let {r n } ⊆ [a, ∞) for some a > 0. Let {x n } be defined by u1H

Assume that:, and. Then x n z, where.

Proof Take any w ∈ Ω and let w be fixed. Putting for each n ∈ ℕ. Then we have:
By Lemma 2.5, exists. So, {x n } is bounded. Furthermore, we have:
1. (a)

;

2. (b)

;

3. (c)

;

4. (d)

;

5. (e)

.

Following the same argument as the proof of Theorem 4.2, there exists zC such that x n z and Tz = z. Besides, we also have
This implies that
By (e), . Next, we want to show that z ∈ (EP). Since ,
By (A2),
By (A4), (i), and , we get

By (A2), G(z, y) ≥ 0 for all yC. So, z ∈ (EP) ∩ F(T) = Ω. By Lemma 2.4, there exists v ∈ (EP) ∩ F(T) such that . By Lemma 2.3, z = P(EP)∩F(T)z = v, and the proof is completed.

In Theorem 4.3, if θ n = 0 for each n ∈ ℕ, then we have the following result.

Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let G : C × C → ℝ be a function satisfying (A1)-(A4). Let T : CC be a mapping with condition (B) and Ω: = F(T) ∩ (EP) ≠ ∅. Let {a n } be a sequence in [0,1]. Let {x n } be defined by u1H

Assume that: {r n } ⊆ [a, ∞) for some a > 0 and . Then x n z, where .

## Notes

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