Open Access

Convergence theorem of κ-strictly pseudo-contractive mapping and a modifıcation of generalized equilibrium problems

Fixed Point Theory and Applications20122012:89

https://doi.org/10.1186/1687-1812-2012-89

Received: 15 January 2012

Accepted: 23 May 2012

Published: 23 May 2012

Abstract

The purpose of this article, we first introduce strong convergence theorem of κ-strictly pseudo-contractive mapping without assumption of the mapping S = κI + (1 - κ)T. Then, we prove strong convergence of proposed iterative scheme for finding a common element of the set of fixed points of κ-strictly pseudo-contractive mapping and the set of solution of a modification of generalized equilibrium problem. Moreover, by using our main result and a new lemma in the last section we obtain strong convergence theorem for finding a common element of the set of fixed points of κ-strictly pseudo-contractive mapping and two sets of solutions of variational inequalities.

Keywords

nonexpansive mappinga strictly pseudo-contractive mapping generalized equilibrium problem inverse-strongly monotone variational inequality problem

1 Introduction

Throughout this article, we assume that H is a real Hilbert space and C is a nonempty subset of H. A mapping T of C into itself is nonlinear mapping. A point x is called a fixed point of T if Tx = x. We use F(T) to denote the set of fixed point of T. Recalled the following definitions;

Definition 1.1. The mapping T is said to be nonexpansive if
| | T x - T y | | | | x - y | | , x , y H
Definition 1.2. The mapping T is said to be strictly pseudo-contractive [1] with the coefficient κ [0, 1) if
| | T x - T y | | 2 | | x - y | | 2 + κ | | ( I - T ) x - ( I - T ) y | | 2 x , y H .
(1.1)

For such case, T is also said to be a κ-strictly pseudo contractive mapping.

The class of κ-strictly pseudo-contractive mapping strictly includes the class of nonexpansive mapping.

Let A : CH. The variational inequality problem is to find a point u C such that
A u , v - u 0
(1.2)

for all v C.

The variational inequality has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences (see, e.g. [25]).

A mapping A of C into H is called α-inverse strongly monotone; see [6], if there exists a positive real number α such that
x - y , A x - A y α | | A x - A y | | 2

for all x, y C.

Let F:C×C→ be a bifunction. The equilibrium problem for F is to determine its equilibrium points, i.e. the set
E P ( F ) = { x C : F ( x , y ) 0 , y C } .
(1.3)
From (1.2) and (1.3), we have the following generalized equilibrium problem, i.e.
Find  z C such that  F ( z , y ) + A z , y - z 0 , y C .
(1.4)
The set of such z C is denoted by EP (F, A), i.e.,
E P ( F , A ) = { z C : F ( z , y ) + A z , y - z 0 , y C }

In the case of A ≡ 0, EP (F, A) is denoted by EP(F). In the case of F ≡ 0, EP(F, A) is also denoted by VI(C, A).

Numerous problems in physics, optimization and economics reduce to find a solution of EP(F) (see, for example [79]). Recently, many authors considered the iterative scheme for finding a common element of the set of solution of equilibrium problem and the set of solutions of fixed point problem (see, for example [1014]). In 2005, Combettes and Hirstoaga [8] introduced an iterative scheme for finding the best approximation to the initial data when EP(F) is nonempty and they also proved the strong convergence theorem.

In 2007, Takahashi and Takahashi [11] introduced viscosity approximation method in framework of a real Hilbert space H. They defined the iterative sequence {x n } and {u n } as follows:
x 1 H , arbitrarily ; F ( u n , y ) + 1 r n y - u n , u n - x n 0 , y C , x n + 1 = α n f ( x n ) + ( 1 - α n ) T u n , n ,
(1.5)

where f : HH is a contraction mapping with constant α (0, 1) and {α n } [0,1], {r n } (0, ∞). They proved under some suitable conditions on the sequence {α n }, {r n } and bifunction F that {x n }, {u n } strongly converge to z F(T) ∩ EP(F), where z = PF(T) ∩ EP(F)f(z).

Recently, in 2008, Takahashia and Takahashi [14] introduced a general iterative method for finding a common element of EP (F, A) and F(T). They defined {x n } in the following way:
u , x 1 C , arbitrarily ; F ( z n , y ) + A x n , y - z n + 1 λ n y - z n , z n - x n 0 , y C , x n + 1 = β n x n + ( 1 - β n ) T ( a n u + ( 1 - a n ) z n ) , n ,
(1.6)

where A be an α-inverse strongly monotone mapping of C into H with positive real number α and {a n } [0, 1], {β n } [0, 1], {λ n } [0, 2α], and proved strong convergence of the scheme (1.6) to z i = 1 N F ( T i ) E P ( F , A ) , where z = P i = 1 N F ( T i ) E P u in the framework of a Hilbert space, under some suitable conditions on {a n }, {β n }, {λ n } and bifunction F.

In 2009, Inchan [15] proved the following theorem:

Theorem 1.1. Let H be a Hilbert space, C be a nonempty closed convex subset of H such that C ± C C, and let T : CH be a κ-strictly pseudo-contractive mapping with a fixed point for some 0 ≤ κ < 1. Let A be a strongly positive bounded linear operator on C with coefficient γ ̄ and f: CC be a contraction with the contractive constant (0 < α < 1) such that 0 < γ < γ ̄ α . Let{x n } be the sequence generated by
x 1 C , x n + 1 = α n γ f ( x n ) + β n x n + ( 1 - β n ) I - α n A P C S x n ,
where S : CH is a mapping defined by
S x = κ x + ( 1 - κ ) T x
(1.7)
If the control sequence {α n }, {β n } (0, 1) satisfying
( i ) lim n α n = 0 a n d lim n β n = 0 , ( i i ) n = 1 α n = , ( i i i ) n = 1 | α n + 1 - α n | < , n = 1 | β n + 1 - β n | < .
Then {x n } converges strongly to a fixed point q of T, which solves the following solution of variational inequality;
( A - γ f ) q , q - x 0 , x F ( T ) .

In 2010, Jung [16] proved the following theorem:

Theorem 1.2. Let H be a Hilbert space, C be a nonempty closed convex subset of H such that C ± C C, and let T : CH be a κ-strictly pseudo-contractive mapping with F(T) ≠ for some 0 ≤ κ < 1. Let A be a strongly positive bounded linear operator on C with coefficient γ ̄ and f: CC be a contraction with the contractive coefficient 0 < α < 1 such that 0 < γ < γ ̄ α . Let{α n } and {β n } (0, 1) be sequences which satisfy the following conditions:
( C 1 ) lim n α n = 0 , ( C 2 ) n = 0 α n = , ( B ) 0 < lim inf n β n lim sup n β n < a f o r s o m e a c o n s t a n t a ( 0 , 1 ) .
Let {x n } be a sequence in C generated by
x 0 = x C , y n = β n x n + ( 1 - β n ) P C S x n x n + 1 = α n γ f ( x n ) + ( I - α n A ) y n , n 0 ,
where S : CH is a:mapping defined by
S x = κ x + ( 1 - κ ) T x
(1.8)
Then {x n } converges strongly to a fixed point q of T, which solves the following solution of variational inequality;
( A - γ f ) q , q - x 0 , x F ( T ) .

Question A. How can we prove strong convergence theorem of κ-strictly pseudo-contractive mapping without assumption of the mapping S = κI + (1 - κ)T in Theorems 1.1 and 1.2?

Let A, B : CH be two mappings. By modification of (1.2), we have
V I ( C , a A + ( 1 - a ) B ) = x C : y - x , ( a A + ( 1 - a ) B ) x 0 , y C , a ( 0 , 1 ) .
(1.9)
From (1.4) and (1.9), we have
E P ( F , ( a A + ( 1 a ) B ) ) = { z C : F ( z , y ) + ( a A + ( 1 a ) B ) z , y z 0 , y C and a ( 0 , 1 ) } .

In this article, we prove strong convergence theorem to answer question A and to approximate a common element of the set of fixed points of κ-strictly pseudo-contractive mapping and the set of solution of a modification of generalized equilibrium problem. Moreover, by using our main result and a new lemma in the last section we obtain strong convergence theorem for finding a common element of the set of fixed points of κ-strictly pseudo-contractive mapping and two sets of solutions of variational inequalities.

2 Preliminaries

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H, let P C be the metric projection of H onto C i.e., for x H, P C x satisfies the property
| | x - P C x | | = min y C | | x - y | | .

The following characterizes the projection P C .

Lemma 2.1. [17] Given x H and y C. Then P C x = y if and only if there holds the inequality
x - y , y - z 0 z C .
Lemma 2.2. [18] Let {s n } be a sequence of nonnegative real number satisfying
s n + 1 = ( 1 - α n ) s n + α n β n , n 0
where {α n }, {β n } satisfy the conditions
( 1 ) { α n } [ 0 , 1 ] , n = 1 α n = ; ( 2 ) lim sup n β n 0 o r n = 1 | α n β n | < .

Then limn→∞s n = 0.

Lemma 2.3. [17] Let H be a Hibert space, let C be a nonempty closed convex subset of H and let A be a mapping of C into H. Let u C. Then for λ > 0,
u = P C ( I - λ A ) u u V I ( C , A ) ,

where P C is the metric projection of H onto C.

Lemma 2.4. [19] Let {x n } and {z n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0,1] with 0 < lim infn→∞β n ≤ lim supn→∞β n < 1. Suppose
x n + 1 = β n x n + ( 1 - β n ) z n
for all n ≥ 0 and
lim sup n ( | | z n + 1 - z n | | - | | x n + 1 - x n | | ) 0 .

Then limn→∞||x n - z n || = 0.

Lemma 2.5. [20] Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : CC be a nonexpansive mapping. Then, I - S is demi-closed at zero.

For solving the equilibrium problem for a bifunction F:C×C→ , let us assume that F satisfies the following conditions:

(A 1) F (x, x) = 0 x C;

(A 2) F is monotone, i.e. F(x, y) + F(y, x) ≤ 0, x, y C;

(A 3) x, y, z C,

limt→0+F(tz + (1 - t)x, y) ≤ F(x, y);

(A 4) x C, y α F(x, y) is convex and lower semicontinuous.

The following lemma appears implicitly in [7].

Lemma 2.6. [7] Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into satisfying (A 1)-(A 4). Let r > 0 and x H. Then, there exists z C such that
F ( z , y ) + 1 r y - z , z - x 0 ,

for all x C.

Lemma 2.7. [8] Assume that F:C×C→ satisfies (A 1)-(A 4). For r > 0 and x H, define a mapping T r : HC as follows:
T r ( x ) = z C : F ( z , y ) + 1 r y - z , z - x 0 , y C .

for all z H. Then, the following hold:

(1) T r is single-valued;

(2) T r is firmly nonexpansive i.e.

||T r (x) - T r (y)||2 ≤ 〈T r (x) - T r (y), x - yx, y H;

(3) F(T r ) = EP(F);

(4) EP(F) is closed and convex.

Remark 2.8. If C is nonempty closed convex subset of H and T : CC is κ-strictly pseudocontractive mapping with F(T) ≠ . Then F(T) = VI(C, (I - T)). To show this, put A = I - T. Let z VI(C, (I - T)) and z* F(T). Since z VI(C, (I - T)), 〈y - z, (I - T)z〉 ≥ 0, y C. Since T : CC is κ-strictly pseudocontractive mapping, we have
| | T z - T z * | | 2 = | | ( I - A ) z - ( I - A ) z * | | 2 = | | z - z * - ( A z - A z * ) | | 2 = | | z - z * | | 2 - 2 z - z * , A z - A z * + | | A z - A z * | | 2 = | | z - z * | | 2 - 2 z - z * , ( I - T ) z + | | ( I - T ) z | | 2 | | z - z * | | 2 + κ | | ( I - T ) z | | 2 .
It implies that
( 1 - κ ) | | ( I - T ) z | | 2 2 z - z * , ( I - T ) z 0 .

Then, we have z = Tz, therefore z F(T). Hence VI(C, (I - T)) F(T). It is easy to see that F(T) VI(C, (I - T)).

Remark 2.9. A = I - T is 1 - κ 2 - inverse strongly monotone mapping. To show this, let x, y C, we have
| | T x - T y | | 2 = | | ( I - A ) x - ( I - A ) y | | 2 = | | x - y - ( A x - A y ) | | 2 = | | x - y | | 2 - 2 x - y , A x - A y + | | A x - A y | | 2 | | x - y | | 2 + κ | | ( I - T ) x - ( I - T ) y | | 2 . = | | x - y | | 2 + κ | | A x - A y | | 2 .
Then, we have
x - y , A x - A y 1 - κ 2 | | A x - A y | | 2 .

3 Main result

Theorem 3.1. Let C be a closed convex subset of Hilbert space H and let F:C×C→ be a bifunction satisfying (A1)-(A4), let A, B : CH be α and β-inverse strongly monotone, respectively. Let T : CC be κ-strictly pseudo contractive mapping with F = F ( T ) E P F , a A + ( 1 - a ) B for all a (0, 1). Let {x n } and {u n } be the sequences generated by x1, u C and
F ( u n , y ) + ( a A + ( 1 - a ) B ) x n , y - u n + 1 r n y - u n , u n - x n 0 , y C , x n + 1 = α n u + β n x n + γ n P C ( I - λ ( I - T ) ) u n , n 1 ,
(3.1)
where {α n }, {β n }, {γ n } [0, 1], λ (0, 1 - κ), α n + β n + γ n = 1, n and {r n } [0, 2γ], γ = min{α, β} satisfy;
( i ) n = 1 α n = , lim n α n = 0 ; ( i i ) 0 < c β n d < 1 , 0 < e r n f < 2 γ ; ( i i i ) lim n | r n + 1 - r n | = 0 .

Then {x n } converges strongly to z 0 = P F u .

Proof. We divide the proof into seven steps.

Step 1. For every a (0, 1), we prove that aA + (1 - a)B is γ-inverse strongly monotone mapping. Put D = aA + (1 - a)B. For x, y C, we have
D x - D y , x - y = a A x + ( 1 - a ) B x - a A y - ( 1 - a ) B y , x - y = a ( A x - A y ) + ( 1 - a ) ( B x - B y ) , x - y = a A x - A y , x - y + ( 1 - a ) B x - B y , x - y a α | | A x - A y | | 2 + ( 1 - a ) β | | B x - B y | | 2 γ ( a | | A x - A y | | 2 + ( 1 - a ) | | B x - B y | | 2 ) γ | | a ( A x - A y ) + ( 1 - a ) ( B x - B y ) | | 2 = γ | | a A x + ( 1 - a ) B x - a A y - ( 1 - a ) B y | | 2 = γ | | D x - D y | | 2
(3.2)
Step 2. We show that I - r n D is a nonexpansive mapping for every n and so is P C (I - λ(I - T)). For every n, let x, y C. From step 1, we have
| | ( I - r n D ) x - ( I - r n D ) y | | 2 = | | x - y - r n ( D x - D y ) | | 2 = | | x - y | | 2 - 2 r n x - y , D x - D y + r n 2 | | D x - D y | | 2 | | x - y | | 2 - 2 r n γ | | D x - D y | | 2 + r n 2 | | D x - D y | | 2 = | | x - y | | 2 + r n ( r n - 2 γ ) | | D x - D y | | 2 | | x - y | | 2 .
(3.3)

Then I - r n D is a nonexpansive mapping.

Putting E = I - T, from Remark 2.9, we have E is η-inverse strong monotone mapping, where η = 1 - κ 2 . By using the same method as (3.3), we have I - λE is nonexpansive mapping. Then, we have P C (I - λ(I - T)) is a nonexpansive mapping.

Step 3. We prove that the sequence {x n } is bounded. From F and (3.1), we have u n = T r n ( I - r n D ) x n , n. Let z F . From Remark 2.8 and Lemma 2.3, we have z = P C (I - λE)z, where E = I - T. Since z EP(F, D), we have F(z, y) + 〈y - z, Dz〉 ≥ 0 y C, so we have
F ( z , y ) + 1 r n y - z , z - z + r n D z 0 , n and y C .
From Lemma 2.7, we have z = T r n ( I - r n D ) z , n. By nonexpansiveness of T r n ( I - r n D ) , we have
| | x n + 1 - z | | = | | α n ( u - z ) + β n ( x n - z ) + γ n ( P C ( I - λ E ) u n - z ) | | α n | | u - z | | + β n | | x n - z | | + γ n | | P C ( I - λ E ) u n - z | | α n | | u - z | | + β n | | x n - z | | + γ n | | T r n ( I - r n D ) x n - z | | α n | | u - z | | + ( 1 - α n ) | | x n - z | | max { | | x n - z | | , | | u - z | | } .

By induction we can prove that {x n } is bounded and so are {u n }, {P C (I - λE)u n }.

Step 4. We will show that
lim n | | x n + 1 - x n | | = 0 .
(3.4)
Let p n = x n + 1 - β n x n 1 - β n , we have
x n + 1 = ( 1 - β n ) p n + β n x n .
(3.5)
From (3.5), we have
| | p n + 1 - p n | | = x n + 2 - β n + 1 x n + 1 1 - β n + 1 - x n + 1 - β n x n 1 - β n = α n + 1 u + γ n + 1 P C ( I - λ E ) u n + 1 1 - β n + 1 - α n u + γ n P C ( I - λ E ) u n 1 - β n = α n + 1 1 - β n + 1 - α n 1 - β n u + γ n + 1 1 - β n + 1 P C ( I - λ E ) u n + 1 - P C ( I - λ E ) u n + γ n + 1 1 - β n + 1 - γ n 1 - β n P C ( I - λ E ) u n α n + 1 1 - β n + 1 - α n 1 - β n | | u | | + γ n + 1 1 - β n + 1 | | u n + 1 - u n | | + γ n + 1 1 - β n + 1 - γ n 1 - β n | | P C ( I - λ E ) u n | | = α n + 1 1 - β n + 1 - α n 1 - β n | | u | | + γ n + 1 1 - β n + 1 | | u n + 1 - u n | | + 1 - β n + 1 - α n + 1 1 - β n + 1 - 1 - β n - α n 1 - β n | | P C ( I - λ E ) u n | | = α n + 1 1 - β n + 1 - α n 1 - β n | | u | | + γ n + 1 1 - β n + 1 | | u n + 1 - u n | | + α n + 1 1 - β n + 1 - α n 1 - β n | | P C ( I - λ E ) u n | | = α n + 1 1 - β n + 1 - α n 1 - β n ( | | u | | + | | P C ( I - λ E ) u n | | ) + γ n + 1 1 - β n + 1 | | u n + 1 - u n | | .
(3.6)
Putting v n = x n - r n Dx n , we have u n = T r n ( x n - r n D x n ) = T r n v n . From definition of u n , we have
F ( u n , y ) + 1 r n y - u n , u n - v n 0 , y C ,
(3.7)
and
F ( u n + 1 , y ) + 1 r n + 1 y - u n + 1 , u n + 1 - v n + 1 0 , y C .
(3.8)
Putting y = un+1in (3.7) and y = u n in (3.8), we have
F ( u n , u n + 1 ) + 1 r n u n + 1 - u n , u n - v n 0 ,
(3.9)
and
F ( u n + 1 , u n ) + 1 r n + 1 u n - u n + 1 , u n + 1 - v n + 1 0 .
(3.10)
Summing up (3.9) and (3.10) and using (A 2), we have
0 1 r n u n + 1 - u n , u n - v n + 1 r n + 1 u n - u n + 1 , u n + 1 - v n + 1 = u n + 1 - u n , u n - v n r n + u n - u n + 1 , u n + 1 - v n + 1 r n + 1 = u n + 1 - u n , u n - v n r n - u n + 1 - v n + 1 r n + 1 .
It implies that
0 u n + 1 - u n , u n - v n - r n r n + 1 ( u n + 1 - v n + 1 ) = u n + 1 - u n , u n - u n + 1 + u n + 1 - v n - r n r n + 1 ( u n + 1 - v n + 1 ) .
It implies that
| | u n + 1 - u n | | 2 u n + 1 - u n , u n + 1 - v n - r n r n + 1 ( u n + 1 - v n + 1 ) = u n + 1 - u n , u n + 1 - v n + 1 + v n + 1 - v n - r n r n + 1 ( u n + 1 - v n + 1 ) = u n + 1 - u n , v n + 1 - v n + 1 - r n r n + 1 ( u n + 1 - v n + 1 ) | | u n + 1 - u n | | | | v n + 1 - v n | | + 1 r n + 1 | r n + 1 - r n | | | u n + 1 - v n + 1 | | .
It follows that
| | u n + 1 - u n | | | | v n + 1 - v n | | + 1 e | r n + 1 - r n | | | u n + 1 - v n + 1 | | .
(3.11)
Since v n = x n - r n Dx n , we have
| | v n + 1 - v n | | = | | x n + 1 - r n + 1 D x n + 1 - x n + r n D x n | | = | | ( I - r n + 1 D ) x n + 1 - ( I - r n + 1 D ) x n + ( I - r n + 1 D ) x n - ( I - r n D ) x n | | | | ( I - r n + 1 D ) x n + 1 - ( I - r n + 1 D ) x n | | + | | ( r n - r n + 1 ) D x n | | | | x n + 1 - x n | | + | r n - r n + 1 | | | D x n | | .
(3.12)
Substitute (3.12) into (3.11), we have
| | u n + 1 - u n | | | | v n + 1 - v n | | + 1 e | r n + 1 - r n | | | u n + 1 - v n + 1 | | | | x n + 1 - x n | | + | r n - r n + 1 | | | D x n | | + 1 e | r n + 1 - r n | | | u n + 1 - v n + 1 | | | | x n + 1 - x n | | + | r n - r n + 1 | L + 1 e | r n + 1 - r n | L ,
(3.13)
where L=maxn{||Dx n ||, ||u n n ||}. Substitute (3.13) into (3.6), we have
| | p n + 1 - p n | | α n + 1 1 - β n + 1 - α n 1 - β n ( | | u | | + | | P C ( I - λ E ) u n | | ) + γ n + 1 1 - β n + 1 | | u n + 1 - u n | | α n + 1 1 - β n + 1 - α n 1 - β n ( | | u | | + | | P C ( I - λ E ) u n | | ) + | | x n + 1 - x n | | + | r n - r n + 1 | L + 1 e | r n + 1 - r n | L ,
(3.14)
From conditions (i), (iii) and (3.14), we have
lim sup n | | p n + 1 - p n | | - | | x n + 1 - x n | | 0 .
(3.15)
From Lemma 2.4, (3.15) and (3.5), we have
lim n | | p n - x n | | = 0 .
(3.16)
From (3.5), we have
x n + 1 - x n = ( 1 - β n ) ( p n - x n ) .
(3.17)
From (3.16), (3.17) and condition (ii), we have
lim n | | x n + 1 - x n | | = 0 .
(3.18)
Since
x n + 1 - x n = α n ( u - x n ) + γ n ( P C ( I - λ ( I - T ) ) u n - x n ) ,
from conditions (i), (ii) and (3.18), we have
lim n | | P C ( I - λ E ) u n - x n | | = 0 ,
(3.19)

where E = I - T .

Step 5. We will show that
lim n | | u n - x n | | = 0 .
(3.20)
Since u n = T r n ( x n - r n D x n ) , we have
u n - z 2 = T r n ( x n - r n D x n ) - T r n ( I - r n D ) z 2 ( I - r n D ) x n - ( I - r n D ) z , u n - z = 1 2 ( I - r n D ) x n - ( I - r n D ) z 2 + u n - z 2 - ( I - r n D ) x n - ( I - r n D ) z - u n + z 2 1 2 x n - z 2 + u n - z 2 - ( x n - u n ) - r n ( D x n - D z ) 2 1 2 x n - z 2 + u n - z 2 - x n - u n 2 - r n 2 D x n - D z 2 + 2 r n x n - u n , D x n - D z ,
it implies that
| | u n - z | | 2 | | x n - z | | 2 - | | x n - u n | | 2 - r n 2 | | D x n - D z | | 2 + 2 r n x n - u n , D x n - D z .
(3.21)
By nonexpansiveness of T r n and using the same method as (3.3), we have
| | u n - z | | 2 = | | T r n ( I - r n D ) x n - T r n ( I - r n D ) z | | 2 | | ( I - r n D ) x n - ( I - r n D ) z | | 2 | | x n - z | | 2 + r n ( r n - 2 γ ) | | D x n - D z | | 2 = | | x n - z | | 2 - r n ( 2 γ - r n ) | | D x n - D z | | 2 .
(3.22)
By nonexpansiveness of P C (I - λE) and (3.22), we have
| | x n + 1 z | | 2 = | | α n ( u z ) + β n ( x n z ) + γ n ( P C ( I λ E ) u n z ) | | 2 α n | | u z | | 2 + β n | | x n z | | 2 + γ n | | u n z | | 2 α n | | u z | | 2 + β n | | x n z | | 2 + γ n ( | | x n z | | 2 r n ( 2 γ r n ) | | D x n D z | | 2 ) α n | | u z | | 2 + | | x n z | | 2 r n γ n ( 2 γ r n ) | | D x n D z | | 2 ,
(3.23)
it implies that
r n γ n ( 2 γ r n ) | | D x n D z | | 2 α n | | u z | | 2 + | | x n z | | 2 | | x n + 1 z | | 2 α n | | u z | | 2 + ( | | x n z | | + | | x n + 1 z | | ) | | x n + 1 x n | | .
(3.24)
From (3.18), (3.24), conditions (i) and (ii), we have
lim n | | D x n - D z | | = 0
(3.25)
From (3.23) and (3.21). we have
| | x n + 1 - z | | 2 | | α n ( u - z ) + β n ( x n - z ) + γ n ( P C ( I - λ E ) u n - z ) | | 2 α n | | u - z | | 2 + β n | | x n - z | | 2 + γ n | | u n - z | | 2 α n | | u - z | | 2 + β n | | x n - z | | 2 + γ n | | x n - z | | 2 - | | x n - u n | | 2 - r n 2 | | D x n - D z | | 2 + 2 r n x n - u n , D x n - D z α n | | u - z | | 2 + β n | | x n - z | | 2 + γ n | | x n - z | | 2 - γ n | | x n - u n | | 2 + 2 r n γ n | | x n - u n | | | | D x n - D z | | α n | | u - z | | 2 + | | x n - z | | 2 - γ n | | x n - u n | | 2 + 2 r n γ n | | x n - u n | | | | D x n - D z | | ,
which implies that
γ n | | x n - u n | | 2 α n | | u - z | | 2 + | | x n - z | | 2 - | | x n + 1 - z | | 2 + 2 r n γ n | | x n - u n | | | | D x n - D z | | α n | | u - z | | 2 + ( | | x n - z | | + | | x n + 1 - z | | ) | | x n + 1 - x n | | + 2 r n γ n | | x n - u n | | | | D x n - D z | | ,
from condition (i), (3.25) and (3.18), we have
lim n | | x n - u n | | = 0 .
Step 6. We prove that
lim sup n u - z 0 , x n - z 0 0 ,
(3.26)
where z 0 = P F u . To show this equality, take a subsequence { x n k } of {x n } such that
lim sup n u - z 0 , x n - z 0 = lim k u - z 0 , x n k - z 0 ,
(3.27)
Without loss of generality, we may assume that x n k ω as k → ∞ where ω C. We first show ω EP(F, D), where D = aA + (1 - a)B, a [0,1]. From (3.20), we have u n k ω as k → ∞. Since u n = T r n ( x n - r n D x n ) , we obtain
F ( u n , y ) + D x n , y - u n + 1 r n y - u n , u n - x n 0 , y C .
From (A 2), we have D x n , y - u n + 1 r n y - u n , u n - x n F ( y , u n ) . Then
D x n k , y - u n k + 1 r n k y - u n k , u n k - x n k F ( y , u n k ) , y C .
(3.28)
Put z t = ty + (1 - t)ω for all t (0, 1] and y C. Then, we have z t C. So, from (3.28) we have
z t - u n k , D z t z t - u n k , D z t - z t - u n k , D x n k - z t - u n k , u n k - x n k r n k + F ( z t , u n k ) = z t - u n k , D z t - D u n k + z t - u n k , D u n k - D x n k - z t - u n k , u n k - x n k r n k + F ( z t , u n k ) .
Since | | u n k - x n k | | 0 , we have | | D u n k - D x n k | | 0 . Further, from monotonicity of D, we have z t - u n k , D z t - D u n k 0 . So, from (A4) we have
z t - ω , D z t F ( z t , ω ) as  k .
(3.29)
From (A 1), (A 4) and (3.29), we also have
0 = F ( z t , z t ) t F ( z t , y ) + ( 1 - t ) F ( z t , ω ) t F ( z t , y ) + ( 1 - t ) z t - ω , D z t = t F ( z t , y ) + ( 1 - t ) t y - ω , D z t ,
hence
0 F ( z t , y ) + ( 1 - t ) y - ω , D z t .
Letting t → 0, we have
0 F ( ω , y ) + y - ω , D ω y C .
(3.30)
Therefore ω EP(F, D), where D = aA + (1 - a)B, a [0,1]. Since
| | P C ( I - λ E ) u n - u n | | | | | | P C ( I - λ E ) u n - x n | | | | + | | x n - u n | | ,
where E = I - T from (3.19) and (3.20), we have
lim n 0 | | P C ( I - λ E ) u n - u n | | = 0 .
(3.31)
Since u n k ω as kω, (3.31) and Lemma 2.5, we have ω F(P C (I - λE)). From Lemma 2.3 and Remark 2.8, we have ω F(T). Therefore ω F . Since x n k ω as k → ∞ and ω F , we have
lim sup n u - z 0 , x n - z 0 = lim n u - z 0 , x n k - z 0 = u - z 0 , ω - z 0 0 .
Step 7. Finally, we show that {x n } converses strongly to z 0 = P F u . From definition of x n , we have
| | x n + 1 - z 0 | | 2 = | | α n ( u - z 0 ) + β n ( x n - z 0 ) + γ n ( P C ( I - λ ( I - T ) ) u n - z 0 ) | | 2 | | β n ( x n - z 0 ) + γ n ( P C ( I - λ ( I - T ) ) u n - z 0 ) | | 2 + 2 α n u - z 0 , x n + 1 - z 0 β n | | x n - z 0 | | 2 + γ n | | P C ( I - λ ( I - T ) ) u n - z 0 | | 2 + 2 α n u - z 0 , x n + 1 - z 0 β n | | x n - z 0 | | 2 + γ n | | T r n ( I - r n D ) x n - z 0 | | 2 + 2 α n u - z 0 , x n + 1 - z 0 ( 1 - α n ) | | x n - z 0 | | 2 + 2 α n u - z 0 , x n + 1 - z 0

From (3.26) and Lemma 2.2, we have {x n } converses strongly to z 0 = P F u . This completes the prove. □

4 Applications

To prove strong convergence theorem in this section, we needed the following lemma.

Lemma 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H and let A, B : CH be α and β-inverse strongly monotone mappings, respectively, with α, β > 0 and VI(C, A) ∩ VI(C, B) ≠ . Then
V I ( C , a A + ( 1 - a ) B ) = V I ( C , A ) V I ( C , B ) , a ( 0 , 1 ) .
(4.1)

Furthermore if 0 < γ < 2η, where η = min{α, β}, we have I - γ(aA + (1 - a)B) is a nonexpansive mapping.

Proof. It is easy to see that VI(C, A) ∩ VI(C, B) VI(C, aA + (1 - a)B). Next, we will show that VI(C, aA + (1 - a)B) VI(C, A) ∩ VI(C, B). Let x0 VI(C, aA + (1 - a)B) and x* VI(C, A) ∩ VI(C, B). Then, we have
y - x * , A x * 0 , y C ,
and
y - x * , B x * 0 , y C .
For every a (0, 1), we have
y - x * , a A x * 0 , y C ,
(4.2)
and
y - x * , ( 1 - a ) B x * 0 , y C .
(4.3)
By monotonicity of A, B and x*, x0 C, we have
x * - x 0 , a A x 0 = x * - x 0 , a A x 0 + ( 1 - a ) B x 0 - ( 1 - a ) B x 0 = x * - x 0 , a A x 0 + ( 1 - a ) B x 0 - x * - x 0 , ( 1 - a ) B x 0 ( 1 - a ) x 0 - x * , B x 0 = ( 1 - a ) x 0 - x * , B x 0 - B x * + x 0 - x * , B x * 0 .
(4.4)
It implies that
x * - x 0 , A x 0 0 .
(4.5)
By monotonicity of A, x* VI(C, A) and (4.5), we have
0 x * - x 0 , A x 0 = x * - x 0 , A x 0 - A x * + A x * = x * - x 0 , A x 0 - A x * + x * - x 0 , A x * - α | | A x * - A x 0 | | 2 + x * - x 0 , A x * - α | | A x * - A x 0 | | 2 ,
it implies that
A x * = A x 0 .
(4.6)
For every y C, from (4.5), (4.6) and x* VI(C, A), we have
y - x 0 , A x 0 = y - x * , A x 0 + x * - x 0 , A x 0 y - x * , A x * 0 .
Then, we have
x 0 V I ( C , A ) .
(4.7)
From (4.4), we have
( 1 - a ) x * - x 0 , B x 0 a x 0 - x * , A x 0 = a x 0 - x * , A x 0 - A x * + x 0 - x * , A x * 0 .
(4.8)
It implies that
x * - x 0 , B x 0 0 .
(4.9)
By monotonicity of B, x* VI(C, B) and (4.9), we have
0 x * - x 0 , B x 0 = x * - x 0 , B x 0 - B x * + B x * = x * - x 0 , B x 0 - B x * + x * - x 0 , B x * - β | | B x * - B x 0 | | 2 + x * - x 0 , B x * - β | | B x * - B x 0 | | 2 ,
it implies that
B x * = B x 0 .
(4.10)
For every y C, from (4.9), (4.10) and x* VI(C, B), we have
y - x 0 , B x 0 = y - x * , B x 0 + x * - x 0 , B x 0 y - x * , B x * 0 .
Then, we have
x 0 V I ( C , B ) .
(4.11)
By (4.7) and (4.11), we have x0 VI(C, A) ∩ VI(C, B). Hence, we have
V I ( C , a A + ( 1 - a ) B ) V I ( C , A ) V I ( C , B ) .
Next, we will show that I - γ(aA + (1 - a)B) is a nonexpansive mapping. To show this let x, y C, then we have
( I γ ( a A + ( 1 a ) B ) ) x ( I γ ( a A + ( 1 a ) B ) ) y 2 = x y γ ( ( a A + ( 1 a ) B ) x ( a A + ( 1 a ) B ) y ) 2 = x y γ ( a ( A x A y ) + ( 1 a ) ( B x B y ) ) 2 = | | x y | | 2 2 γ a ( A x A y ) + ( 1 a ) ( B x B y ) , x y + γ 2 | | a ( A x A y ) + ( 1 a ) ( B x B y ) | | 2 | | x y | | 2 2 γ a A x A y , x y 2 γ ( 1 a ) B x B y , x y + a γ 2 | | A x A y | | 2 + ( 1 a ) γ 2 | | B x B y | | 2 | | x y | | 2 2 γ a α | | A x A y | | 2 2 γ ( 1 a ) β | | B x B y | | 2 + a γ 2 | | A x A y | | 2 + ( 1 a ) γ 2 | | B x B y | | 2 = | | x y | | 2 + a γ ( γ 2 α ) | | A x A y | | 2 + ( 1 a ) γ ( γ 2 β ) | | B x B y | | 2 | | x y | | 2 .
(4.12)

Theorem 4.2. Let C be a closed convex subset of Hilbert space H and let A, B : CH be α and β-inverse strongly monotone, respectively. Let T be κ-strictly pseudo contractive mapping with F = F ( T ) V I C , A V I C , B . Let {x n } be the sequence generated by x1, u C and
x n + 1 = α n u + β n x n + γ n P C ( I - λ ( I - T ) ) P C ( I - r n ( a A + ( 1 - a ) B ) ) x n , n 1 ,
(4.13)
where {α n }, {β n }, {γ n } [0, 1], a (0, 1), λ (0, 1 - κ), α n + β n + γ n = 1, n and {r n } [0, 2γ], γ = min{α, β} satisfy;
( i ) n = 1 α n = , lim n α n = 0 ; ( i i ) 0 < c β n d < 1 , 0 < e r n f < 2 γ ; ( i i i ) lim n | r n + 1 - r n | = 0 .

Then {x n } converges strongly to z 0 = P F u .

Proof. From 3.1 putting F ≡ 0 in Theorem 3.1, we have
y - u n , u n - ( I - r n D ) x n 0 , y C ,
where D = aA + (1 - a)B, a [0,1] It implies that
u n = P C ( I - r n D ) x n .

Then, we have (4.13). From Theorem 3.1 and Lemma 4.1, we can conclude the desired conclusion. □

Theorem 4.3. Let C be a closed convex subset of Hilbert space H and let F:C×C be a bifunction satisfying (A1)-(A4), let A : CH be α-inverse strongly monotone. Let T : CC be κ-strictly pseudo contractive mapping with F = F ( T ) E P ( F , A ) . Let {x n } and {u n } be the sequences generated by x1, u C and
F ( u n , y ) + A x n , y - u n + 1 r n y - u n , u n - x n 0 , y C , x n + 1 = α n u + β n x n + γ n P C ( I - λ ( I - T ) ) u n , n 1 ,
(4.14)
where {α n }, {β n }, {γ n } [0, 1], λ (0, 1 - κ), α n + β n + γ n = 1, n and {r n } [0, 2γ], γ = min{α, β} satisfy;
( i ) n = 1 α n = , lim n α n = 0 ; ( i i ) 0 < c β n d < 1 , 0 < e r n f < 2 γ ; ( i i i ) lim n | r n + 1 - r n | = 0 .

Then {x n } converges strongly to z 0 = P F u .

Proof. From Theorem 3.1, putting AB, we can conclude the desired conclusion. □

Declarations

Acknowledgements

This research was supported by the Research Administration Division of King Mongkut's Institute of Technology Ladkrabang.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang

References

  1. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleGoogle Scholar
  2. Chang SS, Joseph Lee HW, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleGoogle Scholar
  3. Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradientmethod for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2006, 128: 191–201. 10.1007/s10957-005-7564-zMathSciNetView ArticleGoogle Scholar
  4. Yao JC, Chadli O: Pseudomonotone complementarity problems and variational in-equalities. Edited by: Crouzeix JP, Haddjissas N, Schaible, S. Handbook of generalized convexity and monotonicity, Springer, Netherlands; 2005:501–558.Google Scholar
  5. Yao Y, Yao JC: On modified iterative method for nonexpansive mappings and mono-tone mappings. Appl Math Comput 2007, 186(2):1551–1558. 10.1016/j.amc.2006.08.062MathSciNetView ArticleGoogle Scholar
  6. Iiduka H, Takahashi W: Weak convergence theorem by Ces'aro means for nonexpansive mappings and inverse-strongly monotone mappings. J Nonlinear Convex Anal 2006, 7105–113.Google Scholar
  7. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math Stud 1994, 63: 123–145.MathSciNetGoogle Scholar
  8. Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 2005, 6: 117–136.MathSciNetGoogle Scholar
  9. Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J Nonlinear Convex Anal 2008, 9: 37–43.MathSciNetGoogle Scholar
  10. Kangtunyakarn A, Suantai S: Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings. Nonlinear Anal Hybrid Syst 2009, 3: 296–309. 10.1016/j.nahs.2009.01.012MathSciNetView ArticleGoogle Scholar
  11. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleGoogle Scholar
  12. Kangtunyakarn A: Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions. Fixed Point Theory Appl 2011, 2011(23):1–16.View ArticleGoogle Scholar
  13. Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. J Optim Theory Appl 2007, 133: 359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleGoogle Scholar
  14. Takahashia S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleGoogle Scholar
  15. Inchan I: Strong convergence theorems for a new iterative method of κ -strictly pseudo-contractive mappings in Hilbert spaces. Comput Math Appl 2009, 58: 1397–1407. 10.1016/j.camwa.2009.07.034MathSciNetView ArticleGoogle Scholar
  16. Jung JS: Strong convergence of iterative methods for κ -strictly pseudo-contractive map-pings in Hilbert spaces. Appl Math Comput 2010, 215: 3746–3753. 10.1016/j.amc.2009.11.015MathSciNetView ArticleGoogle Scholar
  17. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.Google Scholar
  18. Xu HK: An iterative approach to quadratic optimization. J Optim Theory Appl 2003, 116(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
  19. Suzuki T: Strong convergence of Krasnoselskii and Manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl 2005, 305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleGoogle Scholar
  20. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc Sympos Pure Math 1976, 18: 78–81.Google Scholar

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© Kangtunyakarn; licensee Springer. 2012

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