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Fixed point theorems for mappings with condition B Fixed Point Theory and Applications 2011, 2011:92 doi:10.1186/1687-1812-2011-92 Lai-Jiu Lin maljlin@cc.ncue.edu.twChih Sheng Chuang csc

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Fixed point theorems for mappings with condition (B)

Fixed Point Theory and Applications 2011, 2011:92 doi:10.1186/1687-1812-2011-92

Lai-Jiu Lin (maljlin@cc.ncue.edu.tw)Chih Sheng Chuang (cschuang1977@gmail.com)

Zenn Tsun Yu (t106@nkut.edu.tw)

ISSN 1687-1812

Article type Research

Submission date 19 July 2011

Acceptance date 2 December 2011

Publication date 2 December 2011

Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/92

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Fixed point theorems for mappings with

condition (B)

Lai-Jiu Lin∗1, Chih-Sheng Chuang1 and Zenn-Tsun Yu2

1Department of Mathematics, National Changhua University of Education,

Abstract

In this article, a new type of mappings that satisfies condition (B) is

in-troduced 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

hy-brid mapping; projection

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C → H

be a mapping, and let F (T ) denote the set of fixed points of T A mapping

T : C → H is said to be nonexpansive if ||T x − T y|| ≤ ||x − y|| for all x, y ∈ C.

A mapping T : C → H is said to be quasi-nonexpansive mapping if F (T ) 6= ∅ and

||T x − y|| ≤ ||x − y|| for all x ∈ C and y ∈ F (T ).

In 2008, Kohsaka and Takahashi [1] introduced nonspreading mapping, andobtained a fixed point theorem for a single nonspreading mapping, and a commonfixed point theorem for a commutative family of nonspreading mappings in Banach

spaces A mapping T : C → C is called nonspreading [1] if

2||T x − T y||2 ≤ ||T x − y||2+ ||T y − x||2for all x, y ∈ C Indeed, T : C → C is a nonspreading mapping if and only if

||T x − T y||2 ≤ ||x − y||2+ 2hx − T x, y − T yi for all x, y ∈ C [2].

Recently, Takahashi and Yao [3] introduced two nonlinear mappings in Hilbert

spaces A mapping T : C → C is called a T J-1 mapping [3] if

3||T x − T y||2 ≤ ||x − y||2+ ||T x − y||2+ ||T y − x||2

for all x, y ∈ C [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 map- pings, nonspreading mappings, and hybrid mappings Let λ be a real number A mapping T : C → C is called λ-hybrid [5] if

||T x − T y||2 ≤ ||x − y||2+ 2λhx − T x, y − T yi for all x, y ∈ C.

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 : C → H is (α,

Ba-α-nonexpansive if

||T x − T y||2 ≤ α||T x − y||2+ α||T y − x||2 + (1 − 2α)||x − y||2

for all x, y ∈ C.

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 : C → E is said to satisfy condition (C) if for all x, y ∈ C,

1

2||x − T x|| ≤ ||x − y|| ⇒ ||T x − T y|| ≤ ||x − y||.

In fact, every nonexpansive mapping satisfies condition (C), but the converse may

be false [8, Example 1] Besides, if T : C → E satisfies condition (C) and F (T ) 6= ∅,

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 (1

2,1

2)-generalizedhybrid 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 : C → H be a mapping Then, we say T satisfies condition (B) if for

2)-generalized hybrid mapping satisfies condition (B) But the

converse may be false

(iii) If T : C → C satisfies condition (B) and F (T ) 6= ∅, then T is a nonexpansive mapping, and this implies that F (T ) is a closed convex subset

quasi-of C [9].

Remark 1.2 Let H = R, let C be nonempty closed convex subset of H, and let

T : C → H be a function In fact, we have

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

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

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

3||T x − T y||2 = 12 > 4 = ||x − y||2+ ||T x − y||2+ ||T y − x||2.

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

||T x − T y||2 = 4 > 4 − 8α = α||T x − y||2+ α||T y − x||2+ (1 − 2α)||x − y||2.

For α < 0, let x = y = 1,

||T x − T y||2 = 0 > 8α = α||T x − y||2+ α||T y − x||2+ (1 − 2α)||x − y||2.

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

Example 1.2 Let H = R, C = [−1, 1], and let T : C → C be defined by

for each x ∈ C First, we consider the following conditions:

(a) For x ∈ [−1, 0] and 1

2||x − T x|| ≤ ||x − y||, we know that

(a)1 if y ∈ [−1, 0], then T y = y and (T y − y)[(T y + y) − (T x + x)] = 0;

(a)2 if y ∈ (0, 1], then T y = −y and (T y − y)[(T y + y) − (T x + x)] = 4xy ≤ 0.

(b) For x ∈ (0, 1] and 1

2||x − T x|| ≤ ||x − y||, we know that

(b)1 if y ≥ x, then x ≤ y − x, T x = −x, and T y = −y So, (T y − y)[(T y +

2||T x − T y||2 = 2||x − y||2 = ||T x − y||2+ ||T y − x||2;

(1)3 If x > 0 and y ≤ 0, then ||T x − T y||2 = ||T x − y||2 = ||x + y||2, and

||T y − x||2 = ||x − y||2 Hence,

||T x − y||2+ ||T y − x||2− 2||T x − T y||2 = −4xy ≥ 0.

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

map-nonexpansive mapping

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

x n+1 = α n x n + (1 − α n )T x n , n ∈ N,

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 : E → E a given mapping, the Ishikawa iteration method with errors is the

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

E with P∞ n=1 ||u n || < ∞ and P∞ n=1 ||v n || < ∞.

In 1998, Xu [13] introduced an Ishikawa iteration method with errors which pears to be more satisfactory than the one introduced by Liu [12] For a nonempty

ap-convex subset C of E and T : C → C a given mapping, the Ishikawa iteration

method with errors is generated by

n = 1, and {u n } and {v n } are bounded sequences in C.

Motivated by the above studies, we consider an Ishikawa iteration method witherrors 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 : C → C 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 u1 ∈ H

Furthermore, we observed that Phuengrattana [14] studied approximating fixed

points of for a nonlinear mapping T with condition (C) by the Ishikawa

itera-tion method on uniform convex Banach space with Opal 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 ergodictheorem for mappings with condition (B) Next, we consider the weak convergencetheorems for equilibrium problems and the fixed points of mappings with condition(B)

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 h·, ·i and norm

|| · ||, respectively We denote the strongly convergence and the weak convergence

of {x n } to x ∈ H by x n → x and x n * x, respectively From [15], for each x, y ∈ H

and λ ∈ [0, 1], we have

||λx + (1 − λ)y||2 = λ||x||2 + (1 − λ)||y||2− λ(1 − λ)||x − y||2.

Hence, we also have

2hx − y, u − vi = ||x − v||2+ ||y − u||2− ||x − u||2− ||y − v||2

for all x, y, u, v ∈ H Furthermore, we know that

||αx+βy +γz||2 = α||x||2+β||y||2+γ||z||2−αβ||x−y||2−αγ||x−z||2−βγ||y −z||2

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

Let ` ∞ be the Banach space of bounded sequences with the supremum norm

Let µ be an element of (` ∞)∗ (the dual space of ` ∞ ) Then, we denote by µ(f ) the value of µ at f = (x1, x2, x3, ) ∈ ` ∞ Sometimes, we denote by µ n x n the

value µ(f ) A linear functional µ on ` ∞ is called a mean if µ(e) = ||µ|| = 1, where

e = (1, 1, 1, ) For x = (x1, x2, x3, ), A Banach limit on ` ∞ 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 ` ∞, then

In particular, if f = (x1, x2, x3, ) ∈ ` ∞ 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||2 for all z ∈ C Then there exists a unique z0 ∈ C such that g(z0) = min

z∈C g(z).

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 x ∈ H, we have

hx − P C x, P C x − yi ≥ 0 for all y ∈ C.

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∈N be a sequence

in H If x n * x0 and 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∈N be a sequence in

H with ||x n+1 − u||2 ≤ (1 + λ n )||x n − u||2 + δ n for all u ∈ D and n ∈ N, where {λ n } and {δ n } are sequences of nonnegative real numbers such that

∞

X

n=1

λ n < ∞ and

∞

X

n=1

δ n < ∞ 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 s n+1 ≤

the following conditions:

(A1) G(x, x) = 0 for each x ∈ C;

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

(A3) for each x, y, z ∈ C, lim

t↓0 G(tz + (1 − t)x, y) ≤ G(x, y);

(A4) for each x ∈ C, the scalar function y → G(x, y) is convex and lower

semicon-tinuous

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 x ∈ C Then there exists z ∈ C such that

Proposition 3.1 Let C be a nonempty closed convex subset of a real Hilbert space

H Let T : C → C be a mapping with condition (B) Then for each x, y ∈ C, we have:

Proposition 3.2 Let C be a nonempty closed convex subset of a real Hilbert space

H Let T : C → C be a mapping with condition (B) Then for each x, y ∈ C,

hT x − T y, y − T yi ≤ hx − y, T y − yi + ||T2x − x|| · ||T y − y||.

Proof By Proposition 3.1(iv), for each x, y ∈ C, either

||T x − T y||2+ ||x − T y||2 ≤ ||T x − y||2+ ||x − y||2

or

||T2x − T y||2+ ||T x − T y||2 ≤ ||T2x − y||2 + ||T x − y||2

holds In the first case, we have

||T x − T y||2+ ||x − T y||2 ≤ ||T x − y||2+ ||x − y||2

⇒ ||T x − T y||2 + ||x − y||2 + 2hx − y, y − T yi + ||T y − y||2 ≤ ||T x − T y||2 +

2hT x − T y, T y − yi + ||T y − y||2+ ||x − y||2

⇒ hx − y, y − T yi ≤ hT x − T y, T y − yi

⇒ hT x − T y, y − T yi ≤ hx − y, T y − yi.

In the second case, we have

||T2x − T y||2+ ||T x − T y||2 ≤ ||T x − y||2+ ||T2x − y||2

⇒ ||T2x − y||2+ 2hT2x − y, y − T yi + ||y − T y||2+ ||T x − T y||2 ≤ ||T x − T y||2+

2hT x − T y, T y − yi + ||y − T y||2+ ||T2x − y||2

⇒ hT x − T y, y − T yi ≤ hT2x − y, T y − yi

≤ hx − y, T y − yi + ||T2x − x|| · ||T y − y||.

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

(a) ||T x − T y||2+ ||x − T y||2 ≤ ||x − y||2+ ||T x − y||2+ ||T2x − x|| · ||T y − y||.

(b) hT x − T y, y − T yi ≤ hx − y, T y − yi + ||T x − x|| · ||T y − y||.

The following theorem shows that demiclosed principle is true for mappingswith condition (B)

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space

H, and let T : C → C be a mapping with condition (B) Let {x n } be a sequence

in C with x n * x and lim

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space

H, and let T : C → C be a mapping with condition (B) Then {T n x} is a bounded sequence for some x ∈ C if and only if F (T ) 6= ∅.

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 z ∈ C such that µ n ||x n − z||2 = min

By Proposition 3.1(v), µ n ||x n − T z||2 ≤ µ n ||x n − z||2 This implies that T z = z and

Corollary 3.1 Let C be a nonempty bounded closed convex subset of a real Hilbert space H, and let T : C → C be a mapping with condition (B) Then F (T ) 6= ∅.

The following theorem shows that Ballion’s type Ergodic’s theorem is also truefor 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 : C → C be a mapping with condition (B) Then the following conditions are equivalent:

(i) for each x ∈ C, S n x = 1

n→∞ P F (T ) T n x and P F (T ) is the metric projection from H onto F (T ).

Proof (i)⇒ (ii): Take any x ∈ C and let x be fixed Then there exists υ ∈ C such

that S n x * υ By Proposition 3.2, for each k ∈ N, we have:

exists and this implies that {T n x} is a bounded sequence By Lemma 2.4, there

exists z ∈ F (T ) such that lim

n→∞ P F (T ) T n x = z Clearly, z ∈ F (T ) Besides, we have:

So, {S n x} is a bounded sequence Then there exist a subsequence {S n i x} of {S n x}

and υ ∈ C such that S n i x * υ By the above proof, we have:

This implies that

0, it is easy to see that T υ = υ So, υ ∈ F (T ).

By Lemma 2.2, for each k ∈ N, hT k x − P F (T ) T k x, P F (T ) T k x − ui ≥ 0 This

Since S n k x * υ and P F (T ) T k x → z, we get hυ − z, u − zi ≤ 0 Since u is any point

of F (T ), we know that υ = z = lim

n→∞ P F (T ) T n x.

Furthermore, if {S n i x} is a subsequence of {S n x} and S n i x * q, then q = υ

by following the same argument as in the above proof Therefore, S n x * υ =

lim

Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space

H, and let T1, T2 : C → C be two mappings with condition (B) and Ω := F (T1) ∩