Open Access

Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems

Fixed Point Theory and Applications20112011:38

https://doi.org/10.1186/1687-1812-2011-38

Received: 24 February 2011

Accepted: 18 August 2011

Published: 18 August 2011

Abstract

In this article, we introduce a new mapping generated by infinite family of nonexpansive mapping and infinite real numbers. By means of the new mapping, we prove a strong convergence theorem for finding a common element of the set of fixed point problems of infinite family of nonexpansive mappings and the set of a finite family of variational inclusion problems in Hilbert space. In the last section, we apply our main result to prove a strong convergence theorem for finding a common element of the set of fixed point problems of infinite family of strictly pseudo-contractive mappings and the set of finite family of variational inclusion problems.

Keywords

nonexpansive mappingstrict pseudo contractionstrongly positive operatorvariational inclusion problemfixed point

1 Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let A : C → H be a nonlinear mapping and let F : C × C be a bifunction. A mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y H. We denote by F(T) the set of fixed points of T (i.e. F(T) = {x H : Tx = x}). Goebel and Kirk [1] showed that F(T) is always closed convex and also nonempty provided T has a bounded trajectory.

The problem for finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see, e.g., [2, 3]).

A bounded linear operator A on H is called strongly positive with coefficient γ ̄ if there exists a constant γ ̄ > 0 with the property
A x , x γ ̄ x 2 .
A mapping A of C into H is called inverse-strongly monotone, see [4], 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.

The variational inequality problem is to find a point u C such that
v - u , A u 0 f o r a l l v C .
(1.1)

The set of solutions of (1.1) is denoted by V I(C, A). Many authors have studied methods for finding solution of variational inequality problems (see, e.g., [58]).

In 2008, Qin et al. [9] introduced the following iterative scheme:
y n = P C ( I - s n A ) x n x n + 1 = α n γ f ( W n x n ) + ( I - α n B ) W n P C ( I - r n A ) y n , n ,
(1.2)

where W n is the W-mapping generated by a finite family of nonexpansive mappings and real numbers, A : CH is relaxed (u,v) cocoercive and μ-Lipschitz continuous, and P C is a metric projection H onto C. Under suitable conditions of {s n }, {r n }{α n }, γ, they proved that {x n } converges strongly to an element of the set of variational inequality problem and the set of a common fixed point of a finite family of nonexpansive mappings.

In 2006, Marino and Xu [10] introduced the iterative scheme as follows:
x 0 H , x n + 1 = ( I - α n A ) S x n + α n γ f ( x n ) , n 0 ,
(1.3)
where S is a nonexpansive mapping, f is a contraction with the coefficient a (0, 1), A is a strongly positive bounded linear self-adjoint operator with the coefficient γ ̄ , and γ is a constant such that 0 < γ < γ a . They proved that {x n } generated by the above iterative scheme converges strongly to the unique solution of the variational inequality:
( A - γ f ) x * , x - x * 0 , x F ( S ) .
We know that a mapping B : HH is said to be monotone, if for each x, y H, we have
B x - B y , x - y 0 .

A set-valued mapping M : H → 2 H is called monotone if for all x, y H, f Mx and g My imply 〈x - y, f - g〉 ≥ 0. A monotone mapping M : H → 2 H is maximal if the graph of Graph(M) of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for (x, f) H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) Graph(M) implies f Mx.

Next, we consider the following so-called variational inclusion problem:

Find a u H such that
θ B u + M u
(1.4)

where B : HH, M : H → 2 H are two nonlinear mappings, and θ is zero vector in H (see, for instance, [1116]). The set of the solution of (1.4) is denoted by V I(H, B, M).

Let C be a nonempty closed convex subset of Banach space X. Let { T n } n = 1 be an infinite family of nonexpansive mappings of C into itself, and let λ1, λ2,..., be real numbers in [0, 1]; then we define the mapping K n : CC as follows:
U n , 0 = I (1) U n , 1 = λ 1 T 1 U n , 0 + ( 1 - λ 1 ) U n , 0 , (2) U n , 2 = λ 2 T 2 U n , 1 + ( 1 - λ 2 ) U n , 1 , (3) U n , 3 = λ 3 T 3 U n , 2 + ( 1 - λ 3 ) U n , 2 , (4) (5) U n , k = λ k T k U n , k - 1 + ( 1 - λ k ) U n , k - 1 (6) U n , k + 1 = λ k + 1 T k + 1 U n , k + ( 1 - λ k + 1 ) U n , k (7) (8) U n , n - 1 = λ n - 1 T n - 1 U n , n - 2 + ( 1 - λ n - 1 ) U n , n - 2 (9) K n = U n , n = λ n T n U n , n - 1 + ( 1 - λ n ) U n , n - 1 . (10) (11)

Such a mapping K n is called the K-mapping generated by T1, T2,..., T n and λ1, λ2,..., λ n .

Let x1 H and {x n } be the sequence generated by
x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n A ) ( γ n K n x n + ( 1 - γ n ) S x n ) ,
(1.5)

where A is a strongly positive linear-bounded self-adjoint operator with the coefficient 0 < γ < 1 , S : CC is the S - mapping generated by G1, G2,..., G N and ν1, ν2,..., ν N , where G i : HH is a mapping defined by J M i , η ( I - η B i ) x = G i x for every x H, and η (0, 2δ i ) for every i = 1, 2,..., N, f : HH is contractive mapping with coefficient θ (0, 1) and 0 < γ < γ θ , {α n }, {β n }, {γ n } are sequences in [0, 1].

In this article, by motivation of (1.3), we prove a strong convergence theorem of the proposed algorithm scheme (1.5) to an element z i = 1 F ( T i ) i = 1 N V ( H , B i , M i ) , under suitable conditions of {α n }, {β n }, {γ n }.

2 Preliminaries

In this section, we provide some useful lemmas that will be used for our main result in the next section.

Let C be a closed convex subset of a real Hilbert space H, and 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. (see [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. (see [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. (1)

    { α n } [ 0 , 1 ] , n = 1 α n = ;

     
  2. (2)

    lim sup n β n 0 o r n = 1 | α n β n | < .

     

Then limn→∞s n = 0.

Lemma 2.3. (see [19]) Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n } be a sequence of nonexpansive mappings on C. Suppose n = 1 F ( T n ) is nonempty. Let {λ n } be a sequence of positive numbers with Σ n = 1 λ n = 1 . Then a mapping S on C defined by
S ( x ) = Σ n = 1 λ n T n x n

for x C is well defined, nonexpansive and F ( S ) = n = 1 F ( T n ) hold.

Lemma 2.4. (see [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.

Lemma 2.5. (see [21]) 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 inf n→∞ β n ≤ lim supn→∞β n < 1.

Suppose xn+1= β n x n +(1 - β n )z n for all integer n ≥ 0 and lim supn→∞(||zn+1- z n || - ||xn+1- x n ||) ≤ 0. Then limn→∞||x n - z n || = 0.

In 2009, Kangtunykarn and Suantai [5] introduced the S-mapping generated by a finite family of nonexpansive mappings and real numbers as follows:

Definition 2.1. Let C be a nonempty convex subset of real Banach space. Let { T i } i = 1 N be a finite family of nonexpanxive mappings of C into itself. For each j = 1, 2,..., N, let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I where I [0, 1] and α 1 j + α 2 j + α 3 j = 1 , define the mapping S : C → C as follows:
U 0 = I (1) U 1 = α 1 1 T 1 U 0 + α 2 1 U 0 + α 3 1 I (2) U 2 = α 1 2 T 2 U 1 + α 2 2 U 1 + α 3 2 I (3) U 3 = α 1 3 T 3 U 2 + α 2 3 U 2 + α 3 3 I (4) (5) U N - 1 = α 1 N - 1 T N - 1 U N - 2 + α 2 N - 1 U N - 2 + α 3 N - 1 I (6) S = U N = α 1 N T N U N - 1 + α 2 N U N - 1 + α 3 N I . (7) (8)
(2.1)

This mapping is called the S-mapping generated by T1,..., T N and α1, α2,..., α N .

Lemma 2.6. (see [5]) Let C be a nonempty closed convex subset of strictly convex. Let { T i } i = 1 N be a finite family of nonexpanxive mappings of C into itself with i = 1 N F ( T i ) and let α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , j = 1,2,3,..., N, where I = [0, 1], α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ) for all j = 1,2,..., N-1, α 1 N ( 0 , 1 ] α 2 j , α 3 j [ 0 , 1 ) for all j = 1,2,..., N. Let S be the mapping generated by T1,..., T N and α1, α2,..., α N . Then F ( S ) = i = 1 N F ( T i ) .

Lemma 2.7. (see [5]) Let C be a nonempty closed convex subset of Banach space. Let { T i } i = 1 N be a finite family of nonexpansive mappings of C into itself and α j ( n ) = ( α 1 n , j , α 2 n , j , α 3 n , j ) , α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [0,1], α 1 n , j + α 2 n , j + α 3 n , j = 1 and α 1 j + α 2 j + α 3 j = 1 such that α i n , j α i j [ 0 , 1 ] as n → ∞ for i = 1,3 and j = 1,2,3,..., N. Moreover, for every n , let S and S n be the S-mappings generated by T1, T2,..., T N and α1, α2,..., α N and T1, T2,..., T N and α 1 ( n ) , α 2 ( n ) , . . . , α N ( n ) , respectively. Then limn→∞||S n x - Sx|| = 0 for every x C.

Definition 2.2. (see [11]) Let M : H → 2 H be a multi-valued maximal monotone mapping, then the single-valued mapping J M,λ : HH defined by
J M , λ ( u ) = ( I + λ M ) 1 ( u ) ,  u H ,

is called the resolvent operator associated with M, where λ is any positive number and I is identity mapping.

Lemma 2.8. (see [11]) u H is a solution of variational inclusion (1.4) if and only if u = JM, λ(u - λBu), λ > 0, i.e.,
V I ( H , B , M ) = F ( J M , λ ( I - λ B ) ) , λ > 0 .

Further, if λ (0, 2α], then V I(H, B, M) is closed convex subset in H.

Lemma 2.9. (see [22]) The resolvent operator J M,λ associated with M is single-valued, nonexpansive for all λ > 0 and 1-inverse-strongly monotone.

Lemma 2.10. In a strictly convex Banach space E, if
| | x | | = | | y | | = | | λ x + ( 1 - λ ) y | |

for all x, y E and λ (0, 1), then x = y.

Lemma 2.11. Let C be a nonempty closed convex subset of a strictly convex Banach space. Let { T i } i = 1 be an infinite family of nonexpanxive mappings of C into itself with i = 1 F ( T i ) and let λ1, λ2,..., be real numbers such that 0 < λ i < 1 for every i = 1, 2,..., and i = 1 λ i < . For every n , let K n be the K-mapping generated by T1, T2,..., T n and λ1, λ2,..., λ n . Then for every x C and k , limn→∞K n x exits.

Proof. Let x C. Then for k, n , we have
U n + 1 , k x - U n , k x = λ k T k U n + 1 , k - 1 x + ( 1 - λ k ) U n + 1 , k - 1 x - λ k T k U n , k - 1 x - ( 1 - λ k ) U n , k - 1 x (1) = λ k ( T k U n + 1 , k - 1 x - T k U n , k - 1 x ) + ( 1 - λ k ) ( U n + 1 , k - 1 x - U n , k - 1 x ) (2) λ k T k U n + 1 , k - 1 x - T k U n , k - 1 x + ( 1 - λ k ) U n + 1 , k - 1 x - U n , k - 1 x (3) λ k U n + 1 , k - 1 x - U n , k - 1 x + ( 1 - λ k ) U n + 1 , k - 1 x - U n , k - 1 x (4) = U n + 1 , k - 1 x - U n , k - 1 x (5) = λ k - 1 T k - 1 U n + 1 , k - 2 x + ( 1 - λ k - 1 ) U n + 1 , k - 2 x - λ k - 1 T k - 1 U n , k - 2 x - ( 1 - λ k - 1 ) U n , k - 2 x (6) λ k - 1 T k - 1 U n + 1 , k - 2 x - T k - 1 U n , k - 2 x + ( 1 - λ k - 1 ) U n + 1 , k - 2 x - U n , k - 2 x (7) U n + 1 , k - 2 x - U n , k - 2 x (8) (9) U n + 1 , 1 x - U n , 1 x (10) = λ 1 T 1 U n + 1 , 0 x + ( 1 - λ 1 ) U n + 1 , 0 x - λ 1 T 1 U n , 0 x - ( 1 - λ 1 ) U n , 0 x (11) = λ 1 T 1 x + ( 1 - λ 1 ) x - λ 1 T 1 x - ( 1 - λ 1 ) x (12) = 0 , (13) (14) 
(2.2)
which implies that Un+1,k= Un, kfor every k, n . Hence, K n = Un, n= Un+1,n. Since Kn+1x = Un+1,n+1x = λn+1Tn+1K n x + (1 - λn+1)K n x, we have
K n + 1 x - K n x = λ n + 1 ( T n + 1 K n x - K n x ) .
(2.3)
Let x * i = 1 F ( T i ) and x C. For each n , we have
K n x - x * = λ n T n U n , n - 1 x + ( 1 - λ n ) U n , n - 1 x - x * (1) λ n T n U n , n - 1 x - x * + ( 1 - λ n ) U n , n - 1 x - x * (2) U n , n - 1 x - x * (3) = λ n - 1 T n - 1 U n , n - 2 x + ( 1 - λ n - 1 ) U n , n - 2 x - x * (4) λ n - 1 T n - 1 U n , n - 2 x - x * + ( 1 - λ n - 1 ) U n , n - 2 x - x * (5) U n , n - 2 x - x * (6) . (7) . (8) . (9) U n , 1 x - x * (10) = λ 1 T 1 U n , 0 x + ( 1 - λ 1 ) U n , 0 x - x * (11) λ 1 T 1 x - x * + ( 1 - λ 1 ) x - x * (12) = x - x * , (13) (14)
(2.4)
which implies that {K n x} is bounded, and so is {T n K n x}. For mn, by (2.3) we have
K m x - K n x = K m x - K m - 1 x + K m - 1 x - K m - 2 x + K m - 2 x - + (1) - K n + 1 x + K n + 1 x - K n x (2) K m x - K m - 1 x + K m - 1 x - K m - 2 x + K m - 2 x - K m - 3 x + (3) + K n + 2 - K n + 1 x + K n + 1 x - K n x (4) = λ m T m K m - 1 x - K m - 1 x + λ m - 1 T m - 1 K m - 2 x - K m - 2 x + (5) + λ n + 1 T n + 1 K n x - K n x (6) M k = n + 1 m λ k , (7) (8) 
(2.5)

where M = supn{||Tn+1K n x - K n x||}. This implies that {K n x} is Cauchy sequence. Hence limn→∞K n x exists.

From Lemma 2.11, we can define a mapping K : CC as follows:
K x = lim n K n x , x C .

Such a mapping K is called the K-mapping generated by T1, T2,..., and λ1, λ2,.....

Remark 2.12. It is easy to see that for each n , K n is nonexpansive mappings. Let x, y C, then we have
K x - K y = lim n K n x - K n y x - y .
(2.6)
By (2.6), we have K : CC is nonexpansive mapping. Next, we will show that limn→∞supxD||K n x - Kx|| = 0 for every bounded subset D of C. To show this, let x, y C and D be a bounded subset of C. By (2.5), for mn, we have
K m x - K n x M k = n + 1 m λ k .
By letting m → ∞, for any x D, we have
K x - K n x M k = n + 1 λ k .
Since n = 1 λ n < , we have
lim n sup x D K x - K n x = 0 .

By the next lemma, we will show that F ( K ) = i = 1 F ( T i )

Lemma 2.13. Let C be a nonempty closed convex subset of a strictly convex Banach space. Let { T i } i = 1 be an infinite family of nonexpansive mappings of C into itself with i = 1 F ( T i ) , and let λ1, λ2,..., be real numbers such that 0 < λ i < 1 for every i = 1, 2,... with i = 1 λ i < . Let K n and K be the K-mapping generated by T1, T2,... T n and λ1, λ2,... λ n and T1, T2,... and λ1, λ2,..., respectively. Then F ( K ) = i = 1 F ( T i ) .

Proof. It is easy to see that i = 1 F ( T i ) F ( K ) . Next, we show that F ( K ) i = 1 F ( T i ) . Let x0 F (K) and x * i = 1 F ( T i ) . Let k be fixed. Since
K n x 0 - x * = λ n T n U n , n - 1 x 0 + ( 1 - λ n ) U n , n - 1 x 0 - x * (1)  = λ n ( T n U n , n - 1 x 0 - x * ) + ( 1 - λ n ) ( U n , n - 1 x 0 - x * ) (2)  λ n T n U n , n - 1 x 0 - x * + ( 1 - λ n ) U n , n - 1 x 0 - x * (3)  U n , n - 1 x 0 - x * (4)  = λ n - 1 ( T n - 1 U n , n - 2 x 0 - x * ) + ( 1 - λ n - 1 ) U n , n - 2 ( x 0 - x * ) (5)  λ n - 1 T n - 1 U n , n - 2 x 0 - x * + ( 1 - λ n - 1 ) U n , n - 2 x 0 - x * (6)  U n , n - 2 x 0 - x * (7)  (8)  . (9)  . (10)  U n , k x 0 - x * (11)  = λ k ( T k U n , k - 1 x 0 - x * ) + ( 1 - λ k ) ( U n , k - 1 x 0 - x * ) (12)  λ k T k U n , k - 1 x 0 - x * + ( 1 - λ k ) U n , k - 1 x 0 - x * (13)  U n , k - 1 x 0 - x * (14)  . (15)  . (16)  . (17)  U n , 1 x 0 - x * (18)  = λ 1 ( T 1 x 0 - x * ) + ( 1 - λ 1 ) ( x 0 - x * ) (19)  λ 1 T 1 x 0 - x * + ( 1 - λ 1 ) x 0 - x * (20)  x 0 - x * , (21)  (22) 
(2.7)
we have
x 0 - x * = lim n K n x 0 - x * λ 1 ( T 1 x 0 - x * ) + ( 1 - λ 1 ) ( x 0 - x * ) (1)  λ 1 T 1 x 0 - x * + ( 1 - λ 1 ) x 0 - x * (2)  x 0 - x * , (3)  (4) 
this implies that
x 0 - x * = T 1 x 0 - x * = λ 1 ( T 1 x 0 - x * ) + ( 1 - λ 1 ) ( x 0 - x * ) .
By Lemma 2.10, we have T1x0 = x0, that is x0 F (T1). It follows that Un,1x0 = x0. By (2.7), we have
K n x 0 - x * U n , 2 x 0 - x * = λ 2 ( T 2 U n , 1 x 0 - x * ) + ( 1 - λ 2 ) ( U n , 1 x 0 - x * ) (1) = λ 2 ( T 2 x 0 - x * ) + ( 1 - λ 2 ) ( x 0 - x * ) (2) λ 2 T 2 x 0 - x * + ( 1 - λ 2 ) x 0 - x * (3) x 0 - x * . (4) (5) 
It follows that
x 0 - x * = lim n K n x 0 - x * (1) λ 2 ( T 2 x 0 - x * ) + ( 1 - λ 2 ) ( x 0 - x * ) (2) λ 2 T 2 x 0 - x * + ( 1 - λ 2 ) x 0 - x * (3) x 0 - x * , (4) (5)
which implies
x 0 - x * = T 2 x 0 - x * = λ 2 ( T 2 x 0 - x * ) + ( 1 - λ 2 ) ( x 0 - x * ) .
By Lemma 2.10, we obtain that T2x0 = x0, that is x0 F (T2). It follows that Un,2x0 = x0. By using the same argument, we can conclude that T i x0 = x0 and U i x0 = x0 for i = 1, 2,..., k - 1. By (2.7), we have
K n x 0 - x * U n , k x 0 - x * (1) = λ k ( T k U n , k - 1 x 0 - x * ) + ( 1 - λ k ) ( U n , k - 1 x 0 - x * ) (2) = λ k ( T k x 0 - x * ) + ( 1 - λ k ) ( x 0 - x * ) (3) λ k T k x 0 - x * + ( 1 - λ k ) x 0 - x * (4) x 0 - x * . (5) (6)
It follows that
x 0 - x * = lim n K n x 0 - x * (1) = λ k ( T k x 0 - x * ) + ( 1 - λ k ) ( x 0 - x * ) (2) λ k T k x 0 - x * + ( 1 - λ k ) x 0 - x * (3) x 0 - x * , (4) (5)
(2.8)
which implies
x 0 - x * = T k x 0 - x * = λ k ( T k x 0 - x * ) + ( 1 - λ k ) ( x 0 - x * ) .
(2.9)

By Lemma 2.10, we have T k x0 = x0, that is x0 F (T k ). This implies that x 0 i = 1 F ( T i ) .

3 Main result

Theorem 3.1. Let H be a real Hilbert space, and let M i : H → 2 H be maximal monotone mappings for every i = 1, 2,..., N. Let B i : HH be a δ i - inverse strongly monotone mapping for every i = 1, 2,..., N and { T i } i = 1 an infinite family of nonexpansive mappings from H into itself. Let A be a strongly positive linear-bounded self-adjoint operator with the coefficient 0 < γ < 1 . Let G i : HH be defined by J M i , η ( I - η B i ) x = G i x for every x H and η (0, 2δ i ) for every i = 1, 2,..., N and let ν j = ( α 1 j , α 2 j , α 3 j ) I × I × I , j = 1, 2, 3,..., N, where I = [0, 1], α 1 j + α 2 j + α 3 j = 1 , α 1 j ( 0 , 1 ) for all j = 1,2,..., N-1, α 1 N ( 0 , 1 ] α 2 j , α 3 j [ 0 , 1 ) for all j = 1, 2,..., N.. Let S : CC be the S-mapping generated by G1, G2,..., G N and ν1, ν2,..., ν N . Let λ1, λ2,..., be real numbers such that 0 < λ i < 1 for every i = 1, 2,..., with i = 1 λ i < , and let K n be the K-mapping generated by T1, T2,..., T n and λ1, λ2,..., λ n , and let K be the K-mapping generated by T1, T2,..., and λ1, λ2,..., i.e.,
K x = lim n K n x
for every x C. Assume that F = i = 1 F ( T i ) i = 1 N V ( H , B i , M i ) . For every n , i = 1, 2,..., N, let x1 H and {x n } be the sequence generated by
x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n A ) ( γ n K n x n + ( 1 - γ n ) S x n ) ,
(3.1)

where f : HH is contractive mapping with coefficient θ (0, 1) and 0 < γ < γ θ . Let {α n }, {β n }, {γ n } be sequences in [0, 1], satisfying the following conditions:

(i) lim n α n = 0 and Σ n = 0 α n =

(ii) 0 < liminf n β n limsup n β n < 1 ,

(iii) lim n γ n = c ( 0 , 1 )

Then {x n } converges strongly to z F , which solves uniquely the following variational inequality:
( A - γ f ) z , z - x * 0 , x * F .
(3.2)

Equivalently, we have P F ( I A + γ f ) z = z .

Proof. Let z be the unique solution of (3.2). First, we will show that the mapping G i is a nonexpansive mapping for every i = 1, 2,..., N. Let x, y H, since B i is δ i - inverse strongly monotone mapping and 0 < η < 2δ i , for every i = 1, 2,..., N, we have
( I - η B i ) x - ( I - η B i ) y 2 = x - y - η ( B i x - B i y ) 2 (1) = x - y 2 - 2 η x - y , B i x - B i y + η 2 B i x - B i y 2 (2) x - y 2 - 2 δ i η B i x - B i y 2 + η 2 B i x - B i y 2 (3) = x - y 2 + η ( η - 2 δ i ) B i x - B i y 2 (4) x - y 2 . (5) (6) 
(3.3)
Thus, (I - ηB i ) is a nonexpansive mapping for every i = 1, 2,..., N . By Lemma 2.9, we have G i = J M i , η ( I - η B i ) is a nonexpansive mappings for every i = 1, 2,..., N. Let x * F ; by Lemma 2.8, we have
x * = G i x * = J M i , η ( I - η B i ) x * , i = 1 , 2 , . . . N .
(3.4)
Let e n = γ n K n x n + (1 - γ n )Sx n . Since G i is a nonexpansive mapping for every i = 1, 2,..., N, we have that S is a nonexpansive mapping. By nonexpansiveness of K n we have
e n - x * = γ n ( K n x n - x * ) + ( 1 - γ n ) ( S x n - x * ) (1) γ n K n x n - x * + ( 1 - γ n ) S x n - x * (2) γ n x n - x * + ( 1 - γ n ) x n - x * (3) x n - x * . (4) (5)
(3.5)
Without loss of generality, by conditions (i) and (ii), we have α n ≤ (1 - β n )||A||-1. Since A is a strongly positive linear-bounded self-adjoint operator, we have
A = s u p { | A x , x | : x H , x = 1 } .
(3.6)
For each x C with ||x|| = 1, we have
( ( 1 - β n ) I - α n A ) x , x = 1 - β n - α n A x , x 1 - β n - α n A 0 ,
(3.7)
then (1 - β n )I -α n A is positive. By (3.6) and (3.7), we have
( 1 β n ) I α n A = sup { ( ( 1 β n ) I α n A ) x , x : x C , x = 1 } = sup { 1 β n α n A x , x : x C , x = 1 } 1 β n α n A x , x 1 β n α n γ ¯ .
(3.8)

We shall divide our proof into six steps.

Step 1. We will show that the sequence {x n } is bounded. Let x * F , by (3.5) and (3.8), we have
x n + 1 - x * = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n A ) e n - x * (1) = α n γ f ( x n ) - α n A x * + α n A x * - β n x * + β n x * + β n x n (2) + ( ( 1 - β n ) I - α n A ) e n - x * (3) α n γ f ( x n ) - A x * + β n x n - x * + ( ( 1 - β n ) I - α n A ) ( e n - x * ) (4) α n γ f ( x n ) - A x * + β n x n - x * + ( ( 1 - β n ) I - α n γ ) x n - x * (5) α n ( γ f ( x n ) - γ f ( x * ) + γ f ( x * ) - A x * ) + ( 1 - α n γ ) x n - x * (6) α n γ θ x n - x * + α n γ f ( x * ) - A x * + ( 1 - α n γ ) x n - x * (7) = α n γ f ( x * ) - A x * + ( 1 - α n ( γ - γ θ ) ) x n - x * (8) m a x { x n - x * , γ f ( x * ) - A x * γ - γ θ } . (9) (10) 
By induction, we can prove that {x n } is bounded, and so are {e n }, {K n x n }, {Sx n } and {G i (x n )} for every i = 1, 2,..., N. Without loss of generality, we can assume that there exists a bounded set D H such that
e n , x n , S x n , K n x n , G i x n D , n a n d i = 1 , 2 , , N .
(3.9)

Step 2. We will show that l i m n x n + 1 - x n = 0 .

Define sequence {z n } by z n = 1 1 - β n ( x n + 1 - β n x n ) .

Then xn+1= β n x n + (1 - β n )z n .

Since {x n } is bounded, we have
| | z n + 1 - z n | | = | | x n + 2 - β n + 1 x n + 1 1 - β n + 1 - x n + 1 - β n x n 1 - β n | | (1) = | | α n + 1 γ f ( x n + 1 ) + ( 1 - β n + 1 ) I - α n + 1 A e n + 1 1 - β n + 1 (2) - α n γ f ( x n ) + ( 1 - β n ) I - α n A e n 1 - β n | | (3) α n + 1 | | γ f ( x n + 1 ) - A e n + 1 1 - β n + 1 | | + | | e n + 1 - e n | | (4) + α n | | γ f ( x n ) - A e n 1 - β n | | . (5) (6)
(3.10)
By definition of e n and nonexpansiveness of S, we have
e n + 1 - e n = γ n + 1 K n + 1 x n + 1 + ( 1 - γ n + 1 ) S x n + 1 - γ n K n x n - ( 1 - γ n ) S x n (1)  = γ n + 1 K n + 1 x n + 1 + ( 1 - γ n + 1 ) S x n + 1 - γ n + 1 K n x n + γ n + 1 K n x n (2)  - ( 1 - γ n + 1 ) S x n + ( 1 - γ n + 1 ) S x n - γ n K n x n - ( 1 - γ n ) S x n (3)  = γ n + 1 ( K n + 1 x n + 1 - K n x n ) + ( 1 - γ n + 1 ) ( S x n + 1 - S x n ) (4)  + ( γ n + 1 - γ n ) K n x n + ( γ n - γ n + 1 ) S x n (5)  γ n + 1 K n + 1 x n + 1 - K n x n + ( 1 - γ n + 1 ) x n + 1 - x n (6)  + γ n + 1 - γ n K n x n + γ n - γ n + 1 S x n (7)  γ n + 1 K n + 1 x n + 1 - K n x n + ( 1 - γ n + 1 ) x n + 1 - x n (8)  + 2 | γ n + 1 - γ n | M , (9) (10) 
where M = maxn{||K n x n ||, ||Sx n ||}. Substituting (3.11) into (3.10), we have
z n + 1 - z n α n + 1 | | γ f ( x n + 1 ) - A e n + 1 1 - β n + 1 | | + α n | | γ f ( x n ) - A e n 1 - β n | | + e n + 1 - e n (1) α n + 1 | | γ f ( x n + 1 ) - A e n + 1 1 - β n + 1 | | + α n | | γ f ( x n ) - A e n 1 - β n | | (2) + γ n + 1 K n + 1 x n + 1 - K n x n + ( 1 - γ n + 1 ) x n + 1 - x n (3) + 2 | γ n + 1 - γ n | M (4) α n + 1 | | γ f ( x n + 1 ) - A e n + 1 1 - β n + 1 | | + α n | | γ f ( x n ) - A e n 1 - β n | | (5) + γ n + 1 K n + 1 x n + 1 - K n + 1 x n + K n + 1 x n - K n x n (6) + ( 1 - γ n + 1 ) x n + 1 - x n + 2 | γ n + 1 - γ n | M (7) α n + 1 | | γ f ( x n + 1 ) - A e n + 1 1 - β n + 1 | | + α n | | γ f ( x n ) - A e n 1 - β n | | + x n + 1 - x n (8) + K n + 1 x n - K n x n + 2 | γ n + 1 - γ n | M . (9) (10) 
(3.12)
It implies that
z n + 1 - z n - x n + 1 - x n α n + 1 | | γ f ( x n + 1 ) - A e n + 1 1 - β n + 1 | | + α n | | γ f ( x n ) - A e n 1 - β n | | (1) + K n + 1 x n - K n x n + 2 | γ n + 1 - γ n | M . (2) (3) 
(3.13)
By (2.3), it implies that
K n + 1 x n - K n x n = λ n + 1 ( T n + 1 K n x n - K n x n ) ,
since λ n 0 as n → ∞, we have
lim n K n + 1 x n - K n x n = 0 .
(3.14)
By (3.13), (3.14) and conditions (i), (iii), we have
limsup n z n + 1 - z n - x n + 1 - x n 0 .
(3.15)
By Lemma 2.5, we have
lim n z n - x n = 0 .
(3.16)
By condition (ii) and (3.16)
lim n x n + 1 - x n = lim n ( 1 - β n ) z n - x n = 0 .
(3.17)
Step 3. We will show that
lim n e n - x n = 0 .
(3.18)
Since xn+1= α n γf + β n x n + ((1 - β n )I - α n A) e n , we have
x n + 1 - e n = α n ( γ f ( x n ) - A e n ) + β n ( x n - e n ) (1) α n γ f ( x n ) - A e n + β n x n - x n + 1 + x n + 1 - e n , (2) (3)
it implies that
( 1 - β n ) x n + 1 - e n α n γ f ( x n ) - A e n + β n x n - x n + 1 ,
and it follows that
x n + 1 - e n α n ( 1 - β n ) γ f ( x n ) - A e n + β n ( 1 - β n ) x n - x n + 1 .
By conditions (i), (ii) and (3.17), we have
lim n x n + 1 - e n = 0 .
(3.19)
Since ||e n - x n || ≤ ||e n - xn+1|| + ||xn+1- x n ||, by (3.17) and (3.19), we have
lim n e n - x n = 0 .
Step 4. Define a mapping Q : HH by
Q x = c K x + ( 1 - c ) S x , x H .
(3.20)
We will show that
lim n | | Q x n - x n | | = 0 .
(3.21)
Since
Q x n - e n = c K x n + ( 1 - c ) S x n - γ n K n x n - ( 1 - γ n ) S x n (1) c K x n - γ n K n x n + | γ n - c | S x n (2) = c K x n - γ n K x n + γ n K x n - γ n K n x n + | γ n - c | S x n (3) | c - γ n | K x n + γ n K x n - K n x n + | γ n - c | S x n (4) | c - γ n | K x n + sup> x D K x - K n x + | γ n - c | S x n . (5) (6)
By remark 2.12 and condition (iii), we have
lim n Q x n - e n = 0 .
(3.22)
Since ||Qx n - x n || ≤ ||Qx n - e n || + ||e n - x n ||, from (3.22) and (3.18), we have
lim n | | Q x n - x n | | = 0 .
Step 5. We will show that
lim sup n ( γ f A ) z , x n z 0 ,
(3.23)
where z = P F ( I - ( A - γ f ) ) z . Let { x n j } be subsequence of {x n } such that
limsup n ( γ f - A ) z , x n - z = lim j ( γ f - A ) z , x n j - z .
(3.24)
Without loss of generality, we may assume that { x n j } converses weakly to some q H. By nonexpansiveness of S and K, (3.20) and Lemma 2.3, we have that Q is nonexpansive mapping and
F ( Q ) = F ( K ) F ( S ) .
(3.25)
Since J M i , η ( I - η B i ) x = G i x for every x H and i = 1, 2,... N , by Lemma 2.8, we have
V I ( H , B i , M i ) = F ( J M i , η ( I - η B i ) ) = F ( G i ) , i = 1 , 2 , . . . N .
(3.26)
By Lemma 2.6 and Lemma 2.9, we have
F ( S ) = i = 1 N F ( G i ) = i = 1 N V I ( H , B i , M i )
(3.27)
By Lemma 2.13, we have
F ( K ) = i = 1 F ( T i ) .
(3.28)
By (3.25), (3.27), and (3.28), we have
F ( Q ) = F ( K ) F ( S ) = i = 1 F ( T i ) i = 1 N V I ( H , B i , M i ) .
(3.29)
Since x n j q as j → ∞, nonexpansiveness of Q, (3.21) and Lemma 2.4, we have
q F ( Q ) = i = 1 F ( T i ) i = 1 N V I ( H , B i , M i ) = F .
(3.30)
By (3.24) and (3.30), we have
lim sup n ( γ f A ) z , x n z = lim n ( γ f A ) z , x n j z = ( γ f A ) z , q z 0.
Step 6. Finally, we will show that x n z as n → ∞, where z = P F ( I - ( A - γ f ) ) z . Since
x n + 1 - z 2 = α n γ f ( x n ) + β n x n + ( 1 - β n ) I - α n A e n - z 2 (1) = α n ( γ f ( x n ) - A z ) + β n ( x n - z ) + ( 1 - β n ) I - α n A ( e n - z ) 2 (2) β n ( x n - z ) + ( 1 - β n ) I - α n A ( e n - z ) 2 + 2 α n γ f ( x n ) - A z , x n + 1 - z (3) = β n ( x n - z ) + ( 1 - β n ) I - α n A ( e n - z ) 2 + 2 α n γ f ( x n ) - A z , x n + 1 - z (4) β n ( x n - z ) + ( 1 - β n ) I - α n A e n - z 2 (5) + 2 α n γ f ( x n ) - γ f ( z ) , x n + 1 - z + 2 α n γ f ( z ) - A z , x n + 1 - z (6) β n ( x n - z ) + ( 1 - β n - α n γ ̄ ) e n - z 2 (7) + 2 α n γ θ x n - z x n + 1 - z + 2 α n γ f ( z ) - A z , x n + 1 - z (8) β n ( x n - z ) + ( 1 - β n - α n γ ̄ ) x n - z 2 (9) + 2 α n γ θ x n - z x n + 1 - z + 2 α n γ f ( z ) - A z , x n + 1 - z (10) ( 1 - α n γ ̄ ) x n - z 2 + α n γ θ x n - z 2 + x n + 1 - z 2 (11) + 2 α n γ f ( z ) - A z , x n + 1 - z (12) ( 1 - 2 α n γ ̄ + α n γ θ ) x n - z 2 + α n 2 γ ̄ 2 x n - z 2 + α n γ θ x n + 1 - z 2 (13) + 2 α n γ f ( z ) - A z , x n + 1 - z , (14) (15) 
it implies that
x n + 1 z 2 ( 1 2 α n γ ¯ + α n γ θ ) 1 α n γ θ x n z 2 + α n 2 γ ¯ 2 1 α n γ θ x n z 2 + 2 α n 1 α n γ θ γ f ( z ) A z , x n + 1 z = ( 1 α n γ θ + α n γ θ 2 α n γ ¯ + α n γ θ ) 1 α n γ θ x n z 2 + α n 2 γ ¯ 2 1 α n γ θ x n z 2 + 2 α n 1 α n γ θ γ f ( z ) A z , x n + 1 z = ( 1 α n γ θ 2 α n ( γ ¯ γ θ ) ) 1 α n γ θ x n z 2 + α n 2 γ ¯ 2 1 α n γ θ x n z 2 + 2 α n 1 α n γ θ γ f ( z ) A z , x n + 1 z = ( 1 2 α n ( γ ¯ γ θ ) ) 1 α n γ θ x n z 2 + α n 2 γ ¯ 2 1 α n γ θ x n z 2 + 2 α n 1 α n γ θ γ f ( z ) A z , x n + 1 z = ( 1 2 α n ( γ ¯ γ θ ) ) 1 α n γ θ x n z 2 + α n 1 α n γ θ ( α n γ ¯ 2 x n z 2 + 2 γ f ( z ) A z , x n + 1 z ) = ( 1 2 α n ( γ ¯ γ θ ) ) 1 α n γ θ x n z 2 + 2 ( γ ¯ γ θ ) 2 ( γ ¯ γ θ ) α n 1 α n γ θ ( α n γ ¯ 2 x n z 2 + 2 γ f ( z ) A z , x n + 1 z ) = ( 1 2 α n ( γ ¯ γ θ ) ) 1 α n γ θ x n z 2 + 2 α n ( γ ¯ γ θ ) 1 α n γ θ ( α n γ ¯ 2 2 ( γ ¯ γ θ ) x n z 2 + 2 2 ( γ ¯ γ θ ) γ f ( z ) A z , x n + 1 z ) ,

from condition i, step 5 and Lemma 2.2, we can conclude that x n z as n → ∞, where z = PF (I -(A - γf))z. This completes the proof.

By means of our main result, we have the following results in the framework of Hilbert space. To prove these results, we need definition and lemma as follows:

Definition 3.1. A mapping T : CC is said to be a κ-strict pseudo-contraction mapping, if there exists κ [0, 1) such that
T x - T y 2 x - y 2 + κ ( I - T ) x - ( I - T ) y 2 , x , y C .

Lemma 3.2. (see [23]) Let C be a nonempty closed convex subset of a real Hilbert space H and T : CC a κ-strict pseudo-contraction. Define S : CC by Sx = αx + (1 - α)Tx, for each x C. Then, as α [κ, 1), S is nonexpansive such that F(S) = F(T).

Corollary 3.3. Let H be a real Hilbert space and let M i : H → 2 H be maximal monotone mappings for every i = 1, 2,... N. Let B i : HH be a δ i - inverse strongly monotone mapping for every i = 1, 2,... N and { T i } i = 1 an infinite family of κ i - strictly pseudo-contractive mappings from H into itself. Define a mapping T κ i by T κ i = κ i x + (