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Fixed point theorems for mappings with condition (B)

  • Lai-Jiu Lin
  • Chih-Sheng Chuang
  • Zenn-Tsun Yu
Open Access
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
2 T x - T y 2 T x - y 2 + T y - x 2 Open image in new window
for all x, yC. Indeed, T : CC is a nonspreading mapping if and only if
T x - T y 2 x - y 2 + 2 x - T x , y - T y Open image in new window

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
2 T x - T y 2 x - y 2 + T x - y 2 Open image in new window
for all x, yC. A mapping T : CC is called a TY-2 [3] mapping if
3 T x - T y 2 2 T x - y 2 + T y - x 2 Open image in new window

for all x, yC.

In 2010, Takahashi [4] introduced the hybrid mappings. A mapping T : CC is hybrid [4] if
T x - T y 2 x - y 2 + x - T x , y - T y Open image in new window
for each x, yC. Indeed, T : CC is a hybrid mapping if and only if
3 T x - T y 2 x - y 2 + T x - y 2 + T y - x 2 Open image in new window

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
T x - T y 2 x - y 2 + 2 λ x - T x , y - T y Open image in new window

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
α T x - T y 2 + ( 1 - α ) T y - x 2 β T x - y 2 + ( 1 - β ) x - y 2 Open image in new window

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
T x - T y 2 α T x - y 2 + α T y - x 2 + ( 1 - 2 α ) x - y 2 Open image in new window

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,
1 2 x - T x x - y T x - T y x - y . Open image in new window

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 ( 1 2 , 1 2 ) Open image in new window-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,
1 2 x - T x x - y T x - T y 2 + x - T y 2 T x - y 2 + x - y 2 . Open image in new window
Remark 1.1.
  1. (i)

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

     
  2. (ii)

    Every ( 1 2 , 1 2 ) Open image in new window-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
1 2 x - T x x - y ( T x ) 2 + x 2 - 2 x T x 4 x 2 + 4 y 2 - 8 x y ( T x ) 2 - 2 x T x 3 x 2 + 4 y 2 - 8 x y T x ( T x - 2 x ) ( 3 x - 2 y ) ( x - 2 y ) , Open image in new window
and
| T x - T y | 2 + | x - T y | 2 | T x - y | 2 + | x - y | 2 ( T x ) 2 + ( T y ) 2 - 2 TxTy + x 2 + ( T y ) 2 - 2 x T y ( T x ) 2 + y 2 - 2 yTx + x 2 + y 2 - 2 xy 2 ( T y ) 2 - 2 T y ( T x + x ) 2 y 2 - 2 y ( T x + x ) 2 ( T y ) 2 - 2 y 2 2 ( T y - y ) ( T x + x ) ( T y - y ) ( T y + y ) ( T y - y ) ( T x + x ) ( T y - y ) [ ( T y + y ) - ( T x + x ) ] 0 . Open image in new window
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 ( 1 2 , 1 2 ) Open image in new window-generalized hybrid mapping, and T satisfies condition (B).

     
  2. (2)
    T is not a nonspreading mapping. Indeed, if x = 1 and y = -1, then
    2 T x - T y 2 = 8 > 0 = T x - y 2 + T y - x 2 . Open image in new window
     
  3. (3)
    T is not a TY-1 mapping. Indeed, if x = 1 and y = -1, then
    2 T x - T y 2 = 8 > 4 = 4 + 0 = x - y 2 + T x - y 2 . Open image in new window
     
  4. (4)
    T is not a TY-2 mapping. Indeed, if x = 1 and y = -1, then
    3 T x - T y 2 = 12 > 0 = 2 T x - y 2 + T y - x 2 . Open image in new window
     
  5. (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 . Open image in new window
     
  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,
    T x - T y 2 = 4 > 4 - 8 α = α T x - y 2 + α T y - x 2 + ( 1 - 2 α ) x - y 2 . Open image in new window
     
For α < 0, let x = y = 1,
T x - T y 2 = 0 > 8 α = α T x - y 2 + α T y - x 2 + ( 1 - 2 α ) x - y 2 . Open image in new window
  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
T ( x ) : = x if x [ - 1 , 0 ] , - x if x ( 0 , 1 ] , Open image in new window
for each xC. First, we consider the following conditions:
  1. (a)

    For x ∈ [-1, 0] and 1 2 x - T x x - y Open image in new window, 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 1 2 x - T x x - y Open image in new window, 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 ( 1 2 , 1 2 ) Open image in new window-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
2 T x - T y 2 = 2 x - y 2 2 x + y 2 = T x - y 2 + T y - x 2 ; Open image in new window
(1)2 If x ≤ 0 and y ≤ 0, then
2 T x - T y 2 = 2 x - y 2 = T x - y 2 + T y - x 2 ; Open image in new window
(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,
T x - y 2 + T y - x 2 - 2 T x - T y 2 = - 4 x y 0 . Open image in new window
  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
x n + 1 = α n x n + ( 1 - α n ) T x n , n , Open image in new window

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
x 1 C chosen arbitrary , y n : = ( 1 - β n ) x n + β n T x n , x n + 1 : = ( 1 - α n ) x n + α n T y n , Open image in new window

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
x 1 E chosen arbitrary , y n : = ( 1 - β n ) x n + β n T x n + u n , x n + 1 : = ( 1 - α n ) x n + α n T y n + v n , Open image in new window

where {α n } and {β n } are sequences in [0,1], and {u n } and {v n } are sequences in E with n = 1 u n < Open image in new window and n = 1 v n < Open image in new window.

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
x 1 C chosen arbitrary , y n : = a n x n + b n T x n + c n u n , x n + 1 : = a n x n + b n T y n + c n v n , Open image in new window

where {a n }, {b n }, {c n }, { a n } Open image in new window, { b n } Open image in new window, { c n } Open image in new window are sequences in [0,1] with a n + b n + c n + = 1 and a n + b n + c n = 1 Open image in new window, 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
x n C such that G ( x n , y ) + 1 r n y - x n , x n - u n 0 y C ; u n + 1 : = a n x n + b n T x n + θ n ω n . Open image in new window

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
λ x + ( 1 - λ ) y 2 = λ x 2 + ( 1 - λ ) y 2 - λ ( 1 - λ ) x - y 2 . Open image in new window
Hence, we also have
2 x - y , u - v = x - v 2 + y - u 2 - x - u 2 - y - v 2 Open image in new window
for all x, y, u, vH. Furthermore, we know that
α x + β y + γ z 2 = α x 2 + β y 2 + γ z 2 - α β x - y 2 - α γ x - z 2 - β γ y - z 2 Open image in new window

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,
lim inf n x n μ n x n lim sup n x n . Open image in new window

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 g ( z 0 ) = min z C g ( z ) Open image in new window.

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 n = 1 λ n < Open image in new window and n = 1 δ n < Open image in new window. 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 n = 1 t n < Open image in new window, then lim n s n Open image in new window exists.

The equilibrium problem is to find zC such that
G ( z , y ) 0 for each y C . Open image in new window
(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, lim t↓ 0 G ( tz + ( 1 - t ) x , y ) G ( x , y ) Open image in new window;

(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
G ( z , y ) + 1 r y - z , z - x 0 f o r a l l y C . Open image in new window
Furthermore, if
T r ( x ) : = { z C : G ( z , y ) + 1 r y - z , z - x 0 f o r a l l y C } , Open image in new window

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 1 2 x - T x x - y Open image in new window or 1 2 T x - T 2 x T x - y Open image in new window holds;

     
  4. (iv)
    either
    T x - T y 2 + x - T y 2 T x - y 2 + x - y 2 Open image in new window
     
or
T 2 x - T y 2 + T x - T y 2 T 2 x - y 2 + T x - y 2 Open image in new window
holds;
  1. (v)

    lim n T n x - T n + 2 x = 0 Open image in new window.

     
Proof Since 1 2 x - T x x - T x Open image in new window, it is easy to see (i) and (ii) are satisfied. (iii) Suppose that
1 2 x - T x > x - y and 1 2 T x - T 2 x > T x - y Open image in new window
holds. So,
x - T x x - y + y - T x < 1 2 x - T x + 1 2 T x - T 2 x 1 2 x - T x + 1 2 x - T x = x - T x . Open image in new window

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
T n + 1 x - T n + 2 x 2 + T n x - T n + 2 x 2 T n x - T n + 1 x 2 . Open image in new window
Then {||T n x - T n+1 x||} is a decreasing sequence, and lim n T n x - T n + 1 x Open image in new window exists. Furthermore, we have:
lim n T n x - T n + 2 x 2 lim n T n x - T n + 1 x 2 - lim n T n + 1 x - T n + 2 x 2 = 0 . Open image in new window

So, lim n T n x - T n + 2 x = 0 Open image in new window.

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,
T x - T y , y - T y x - y , T y - y + T 2 x - x T y - y . Open image in new window
Proof By Proposition 3.1(iv), for each x, yC, either
T x - T y 2 + x - T y 2 T x - y 2 + x - y 2 Open image in new window
or
T 2 x - T y 2 + T x - T y 2 T 2 x - y 2 + T x - y 2 Open image in new window
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 + 2 x - y , y - T y + T y - y 2 T x - T y 2 + 2 T x - T y , T y - y + T y - y 2 + x - y 2 x - y , y - T y T x - T y , T y - y T x - T y , y - T y x - y , T y - y . Open image in new window
In the second case, we have
T 2 x - T y 2 + T x - T y 2 T x - y 2 + T 2 x - y 2 T 2 x - y 2 + 2 T 2 x - y , y - T y + y - T y 2 + T x - T y 2 T x - T y 2 + 2 T x - T y , T y - y + y - T y 2 + T 2 x - y 2 T x - T y , y - T y T 2 x - y , T y - y x - y , T y - y + T 2 x - x T y - y . Open image in new window

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 lim n x n - T x n = 0 Open image in new window. Then Tx= x.

Proof By Remark 3.1, we get:
T x n - T x , x - T x x n - x , T x - x + x n - T x n x - T x Open image in new window

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 μ n x n - z 2 = min y C μ n x n - y 2 Open image in new window. By Proposition 3.2, for each n ∈ ℕ,
x n + 1 - Tz , z - Tz x n - z , Tz - z + x n - x n + 2 z - Tz 1 2 x n + 1 - Tz 2 + 1 2 Tz - z 2 - 1 2 x n + 1 - z 2 1 2 x n - z 2 + 1 2 z - Tz 2 - 1 2 x n - Tz 2 + x n - x n + 2 z - Tz μ n x n - Tz 2 μ n x n - z 2 + μ n x n - x n + 2 z - Tz . Open image in new window

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, S n x = 1 n k = 0 n - 1 T k x Open image in new window 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, where v = lim n P F ( T ) T n x Open image in new windowand 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:
T T k x - Tv , v - Tv T k x - v , Tv - v + T 2 T k x - T k x Tv - v T k + 1 x - Tv , v - Tv T k x - v , Tv - v + T k + 2 x - T k x Tv - v k = 0 n - 2 T k + 1 x - Tv , v - Tv k = 0 n - 2 T k x - Tv , Tv - v + k = 0 n - 2 T k + 2 x - T k x Tv - v n S n x - x - ( n - 1 ) Tv , v - Tv ( n - 1 ) S n - 1 x - ( n - 1 ) Tv , Tv - v + k = 0 n - 2 T k + 2 x - T k x Tv - v n n - 1 S n x - x n - 1 - Tv , v - Tv S n - 1 x - Tv , Tv - v + 1 n - 1 k = 0 n - 2 T k + 2 x - T k x Tv - v . Open image in new window
By Proposition 3.1(v), lim n T k + 2 x - T k x = 0 Open image in new window. This implies that
lim n 1 n - 1 k = 0 n - 2 T k + 2 x - T k x = 0 . Open image in new window
Since S n xv, we have:
v - Tv , v - Tv v - Tv , Tv - v . Open image in new window

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, lim n T n x - u Open image in new window exists and this implies that {T n x} is a bounded sequence. By Lemma 2.4, there exists zF(T) such that lim n P F ( T ) T n x = z Open image in new window. Clearly, zF(T). Besides, we have:
S n x - u 1 n k = 0 n - 1 T k x - u x - u . Open image in new window
So, {S n x} is a bounded sequence. Then there exist a subsequence { S n i x } Open image in new window of {S n x} and vC such that S n i x v Open image in new window. By the above proof, we have:
n n - 1 S n x - x n - 1 - Tv , v - Tv S n - 1 x - Tv , Tv - v + 1 n - 1 k = 0 n - 2 T k + 2 x - T k x Tv - v . Open image in new window
This implies that
n i n i - 1 S n i x - x n i - 1 - T v , v - T v S n i - 1 x - x n i - 1 - T v , T v - v + 1 n i - 1 k = 0 n i - 2 T k + 2 x - T k x T v - v . Open image in new window

Since S n i x v Open image in new window, {T n x} is a bounded sequence, and lim n 1 n - 1 k = 0 n - 2 T k + 2 x - T k x = 0 Open image in new window, 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
T k x - P F ( T ) T k x , u - z T k x - P F ( T ) T k x , P F ( T ) T k x - z T k x - P F ( T ) T k x P F ( T ) T k x - z T k x - z P F ( T ) T k x - z x - z P F ( T ) T k x - z . Open image in new window
Adding these inequalities from k = 0 to k = n - 1 and dividing by n, we have
S n x - 1 n k = 0 n - 1 P F ( T ) T k x , u - z x - z n k = 0 n - 1 P F ( T ) T k x - z . Open image in new window

Since S n k x v Open image in new window and P F(T) Tkx → z, we get 〈v - z, u - z〉 ≤ 0. Since u is any point of F(T), we know that v = z = lim n P F ( T ) T n x Open image in new window.

Furthermore, if { S n i x } Open image in new window is a subsequence of {S n x} and S n i x q Open image in new window, then q = v by following the same argument as the above proof. Therefore, S n x υ = lim n P T n x Open image in new window, 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
a n + c n + θ n = b n + d n + λ n = 1 , n . Open image in new window
Let {u n } and {v n } be bounded sequences in C. Let {x n } and {y n } be defined by
x 1 C c h o s e n a r b i t r a r y , y n : = a n x n + c n T 1 x n + θ n u n , x n + 1 : = b n x n + d n T 2 y n + λ n v n . Open image in new window
Assume that:
  1. (i)

    lim inf n a n c n > 0 Open image in new window and lim inf n b n d n > 0 Open image in new window ;

     
  2. (ii)

    n = 1 θ n < Open image in new window and n = 1 λ n < Open image in new window.

     

Then x n z and y n z, where z = lim n P Ω x n Open image in new window.

Proof Take any w ∈ Ω and let w be fixed. Then for each n ∈ ℕ, we have:
y n - w 2 = a n x n + c n T 1 x n + θ n u n - w 2 = a n x n - w 2 + c n T 1 x n - w 2 + θ n u n - w 2 - a n c n x n - T 1 x n 2 - a n θ n x n - w 2 - c n θ n T 1 x n - u n 2 a n x n - w 2 + c n x n - w 2 + θ n u n - w 2 - a n c n x n - T 1 x n 2 - a n θ n x n - w 2 - c n θ n T 1 x n - u n 2 x n - w 2 + θ n u n - w 2 , Open image in new window
and
x n + 1 - w 2 = b n x n + d n T 2 y n + λ n v n - w 2 = b n x n - w 2 + d n T 2 y n - w 2 + λ n v n - w 2 - b n d n x n - T 2 y n 2 - b n λ n x n - v n 2 - d n λ n T 2 y n - v n 2 b n x n - w 2 + d n y n - w 2 + λ n v n - w 2 - b n d n x n - T 2 y n 2 - b n λ n x n - v n 2 - d n λ n T 2 y n - v n 2 b n x n - w 2 + d n ( x n - w 2 + θ n u n - w 2 ) + λ n v n - w 2 - b n d n x n - T 2 y n 2 - b n λ n x n - v n 2 - d n λ n T 2 y n - v n 2 x n - w 2 + d n θ n u n - w 2 + λ n v n - w 2 - b n d n x n - T 2 y n 2 - b n λ n x n - v n 2 - d n λ n T 2 y n - v n 2 x n - w 2 + d n θ n u n - w 2 + λ n v n - w 2 . Open image in new window
By Lemma 2.5, lim n x n - w Open image in new window exists. So, {x n } is a bounded sequence. Now, we set lim n x n - w = t Open image in new window. Besides,
b n d n x n - T 2 y n 2 + b n λ n x n - v n 2 + d n λ n T 2 y n - v n 2 x n - w 2 + d n θ n u n - w 2 + λ n v n - w 2 - x n + 1 - w 2 . Open image in new window
This implies that
lim n b n d n x n - T 2 y n 2 = 0 . Open image in new window
By assumption, lim n x n - T 2 y n = 0 Open image in new window. Furthermore, we have:
x n + 1 - w 2 b n x n - w 2 + d n y n - w 2 + λ n v n - w 2 . Open image in new window
This implies that
b n d n ( x n - w 2 + θ n u n - w 2 - y n - w 2 ) d n ( x n - w 2 + θ n u n - w 2 - y n - w 2 ) ( 1 - b n ) x n - w 2 + d n θ n u n - w 2 - d n y n - w 2 x n - w 2 - x n + 1 - w 2 + λ n v n - w 2 + d n θ n u n - w 2 . Open image in new window
Hence, lim n b n d n ( x n - w 2 + θ n u n - w 2 - y n - w 2 ) = 0 Open image in new window. By assumption,
lim n ( x n - w 2 + θ n u n - w 2 - y n - w 2 ) = 0 . Open image in new window
Since lim n θ n u n - w 2 = 0 Open image in new window,
lim n ( x n - w 2 - y n - w 2 ) = 0 . Open image in new window
Hence, lim n y n - w = lim n x n - w = t Open image in new window. Similar to the above proof, we also get
lim n x n - T 1 x n = 0 . Open image in new window
Besides,
y n - x n = a n x n + c n T 1 x n + θ n u n - x n c n T 1 x n - x n + θ n x n - u n T 1 x n - x n + θ n x n - u n . Open image in new window

This implies that lim n y n - x n = 0 Open image in new window and lim n y n - T 2 y n = 0 Open image in new window. Since {x n } is a bounded sequence, there exists a subsequence { x n k } Open image in new window of {x n } such that x n k z Open image in new window. By Theorem 3.1, z = T1z.

If x n j Open image in new window is a subsequence of {x n } and x n j q Open image in new window, then T1q = q. Suppose that qz. Then we have:
lim inf k x n k - z < lim inf k x n k - q = lim n x n - q = lim j x n j - q < lim inf j x n j - z = lim n x n - z = lim inf k x n k - z . Open image in new window

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 lim n x n - y n = 0 Open image in new window, 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 lim n P Ω x n = v Open image in new window. By Lemma 2.3, PΩz = v. Since z ∈ Ω, z = v = lim n P Ω x n Open image in new window, 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
x 1 C c h o s e n a r b i t r a r y , y n : = a n x n + ( 1 - a n ) T 1 x n , x n + 1 : = b n x n + ( 1 - b n ) T 2 y n . Open image in new window

Assume that lim inf n a n ( 1 - a n ) > 0 Open image in new windowand lim inf n b n ( 1 - b n ) > 0 Open image in new window. Then x n z and y n z, where z = lim n P Ω x n Open image in new window.

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
x 1 C c h o s e n a r b i t r a r y , y n : = a n x n + ( 1 - a n ) T x n , x n + 1 : = b n x n + ( 1 - b n ) T y n . Open image in new window

Assume that lim inf n a n ( 1 - a n ) > 0 Open image in new windowand lim inf n b n ( 1 - b n ) > 0 Open image in new window. Then x n z and y n z, where z = lim n P F ( T ) x n Open image in new window.

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
x 1 C c h o s e n a r b i t r a r y , x n + 1 : = b n x n + ( 1 - b n ) T x n . Open image in new window

Assume that lim inf n b n ( 1 - b n ) > 0 Open image in new window. Then x n z, where z = lim n P F ( T ) x n Open image in new window.

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
x n C such that G ( x n , y ) + 1 r n y - x n , x n - u n 0 y C ; u n + 1 : = a n x n + b n T x n + θ n ω n . Open image in new window

Assume that: lim inf n a n b n > 0 Open image in new window, and n = 1 θ n < Open image in new window. Then x n z, where z = lim n P ( E P ) F ( T ) x n Open image in new window.

Proof Take any w ∈ Ω and let w be fixed. Putting x n = T r n u n Open image in new window for each n ∈ ℕ. Then we have:
x n + 1 - w 2 = T r n + 1 u n + 1 - w 2 u n + 1 - w 2 a n x n + b n T x n + θ n ω n - w 2 a n x n - w 2 + b n T x n - w 2 + θ n ω n - w 2 - a n b n x n - T x n 2 a n x n - w 2 + b n x n - w 2 + θ n ω n - w 2 - a n b n x n - T x n 2 x n - w 2 + θ n ω n - w 2 - a n b n x n - T x n 2 . Open image in new window
By Lemma 2.5, lim n x n - w Open image in new window exists. So, {x n } is bounded. Furthermore, we have:
  1. (a)

    lim n a n b n x n - T x n 2 = 0 Open image in new window;

     
  2. (b)

    lim n x n - T x n = 0 Open image in new window;

     
  3. (c)

    u n + 1 - x n = b n T x n - b n x n + θ n w n - θ n x n T x n - x n + θ n w n - x n Open image in new window;

     
  4. (d)

    lim n u n + 1 - x n = 0 Open image in new window;

     
  5. (e)

    lim n u n - w = lim n x n - w Open image in new window.

     
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
x n + 1 - w 2 = T r n + 1 u n + 1 - w 2 = T r n + 1 u n + 1 - T r n + 1 w 2 T r n + 1 u n + 1 - T r n + 1 w , u n + 1 - w x n + 1 - w , u n + 1 - w = 1 2 x n + 1 - w 2 + 1 2 u n + 1 - w 2 - 1 2 x n + 1 - u n + 1 2 . Open image in new window
This implies that
x n + 1 - u n + 1 2 u n + 1 - w 2 - x n + 1 - w 2 . Open image in new window
By (e), lim n x n - u n = 0 Open image in new window. Next, we want to show that z ∈ (EP). Since x n = T r n u n Open image in new window,
G ( x n , y ) + 1 r n y - x n , x n - u n 0 y C . Open image in new window
By (A2),
1 r n y - x n , x n - u n G ( y , x n ) y C . Open image in new window
By (A4), (i), and lim n x n - u n = 0 Open image in new window, we get
0 lim n G ( y , x n ) G ( y , z ) y C . Open image in new window

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 lim n P ( E P ) F ( T ) x n = v Open image in new window. 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
x n C s u c h t h a t G ( x n , y ) + 1 r n y - x n , x n - u n 0 y C ; u n + 1 : = a n x n + ( 1 - a n ) T x n . Open image in new window

Assume that: {r n } ⊆ [a, ∞) for some a > 0 and lim inf n a n ( 1 - a n ) > 0 Open image in new window. Then x n z, where z = lim n P ( E P ) F ( T ) x n Open image in new window.

Notes

References

  1. 1.
    Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch Math 2008, 91: 166–177. 10.1007/s00013-008-2545-8MathSciNetCrossRefGoogle Scholar
  2. 2.
    Iemoto S, Takahashi W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal 2009, 71: e2082-e2089. 10.1016/j.na.2009.03.064MathSciNetCrossRefGoogle Scholar
  3. 3.
    Takahashi W, Yao JC: Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces. Taiwanese J Math 2011, 15: 457–472.MathSciNetGoogle Scholar
  4. 4.
    Takahashi W: Fixed point theorems for new nonlinear mappings in a Hilbert spaces. J Nonlinear Convex Anal 2010, 11: 79–88.MathSciNetGoogle Scholar
  5. 5.
    Aoyama K, Iemoto S, Kohsaka F, Takahashi W: Fixed point and ergodic theorems for λ -hybrid mappings in Hilbert spaces. J Nonlinear Convex Anal 2010, 11: 335–343.MathSciNetGoogle Scholar
  6. 6.
    Kocourek P, Takahashi W, Yao JC: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwanese J Math 2010, 14: 2497–2511.MathSciNetGoogle Scholar
  7. 7.
    Aoyama K, Kohsaka F: Fixed point theorem for α -nonexpansive mappings in Banach spaces. Nonlinear Anal 2011, 74: 4387–4391. 10.1016/j.na.2011.03.057MathSciNetCrossRefGoogle Scholar
  8. 8.
    Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J Math Anal Appl 2008, 340: 1088–1095. 10.1016/j.jmaa.2007.09.023MathSciNetCrossRefGoogle Scholar
  9. 9.
    Itoh S, Takahashi W: The common fixed point theory of single-valued mappings and multi-valued mappings. Pacific J Math 1978, 79: 493–508.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mann WR: Mean value methods in iteration. Proc Am Math Soc 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3CrossRefGoogle Scholar
  11. 11.
    Ishikawa S: Fixed points by a new iteration method. Proc Am Math Soc 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu L: Ishikawa and Mann iteration processes with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl 1995, 194: 114–125. 10.1006/jmaa.1995.1289MathSciNetCrossRefGoogle Scholar
  13. 13.
    Xu Y: Ishikawa and Mann iteration processes with errors for nonlinear strongly accretive operator equations. J Math Anal Appl 1998, 224: 91–101. 10.1006/jmaa.1998.5987MathSciNetCrossRefGoogle Scholar
  14. 14.
    Phuengrattana W: Approximating fixed points of Suzuki-generalized nonexpansive mappings. Nonlinear Anal Hybrid Syst 2011, 5: 583–590. 10.1016/j.nahs.2010.12.006MathSciNetCrossRefGoogle Scholar
  15. 15.
    Takahashi W: Introduction to Nonlinear and Convex Analysis. Yokohoma Publishers, Yokohoma; 2009.Google Scholar
  16. 16.
    Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput Math Appl 2000, 40: 559–567. 10.1016/S0898-1221(00)00179-6MathSciNetCrossRefGoogle Scholar
  17. 17.
    Takahashi W: Nonlinear Functional Analysis-Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.Google Scholar
  18. 18.
    Huang S: Hybrid extragradient methods for asymptotically strict pseudo-contractions in the intermediate sense and variational inequality problems. Optimization 2011, 60: 739–754. 10.1080/02331934.2010.527340MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J Math Anal Appl 1993, 178: 301–308. 10.1006/jmaa.1993.1309MathSciNetCrossRefGoogle Scholar
  20. 20.
    Blum E, Oettli W: From optimization and variational inequalities. Math Student 1994, 63: 123–146.MathSciNetGoogle Scholar

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© Lin et al.; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan
  2. 2.Department of Electronic EngineeringNan Kai University of TechnologyNantourTaiwan

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