Open Access

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Fixed Point Theory and Applications20122012:92

https://doi.org/10.1186/1687-1812-2012-92

Received: 13 September 2011

Accepted: 6 June 2012

Published: 6 June 2012

Abstract

Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the set i = 1 N Fix ( S i ) of fixed points of a finite family of nonexpansive mappings { S i } i = 1 N in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of i = 1 N Fix ( S i ) G M E P , which is the unique solution of a variational inequality.

AMS subject classifications: 49J40; 47J20; 47H09.

Keywords

Generalized mixed equilibrium problemfixed pointnonexpansive mappingvariational inequalityhybrid iterative method.

1 Introduction

The theory of equilibrium problems has played an important role in the study of a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimization, and has numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis and mechanics. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be productive and innovative. It is remarkable that the variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems [1, 2].

Let H be a real Hilbert space. Throughout this paper, we write x n x to indicate that the sequence {x n } converges weakly to x. The x n x indicates that {x n } converges strongly to x. Let C be a nonempty closed convex subset of H and Θ be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for Θ: C × CR is to find x ¯ C such that
Θ ( x ¯ , y ) 0 , y C .
(1.1)
The set of solutions of problem (1.1) is denoted by EP(Θ). Given a mapping T: CH, let Θ (x, y) = 〈Tx, y - x〉 for all x, y C. Then, z EP(Θ) if and only if 〈Tz, y - z〉 ≥ 0 for all y C. Numerous problems in physics, optimization, and economics reduce to finding a solution of problem (1.1). Equilibrium problems have been studied extensively [218]. Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP(Θ) is nonempty and derived a strong convergence theorem. Very recently, Peng and Yao [4] introduced the following generalized mixed equilibrium problem of finding x ¯ C such that
Θ ( x ¯ , y ) + φ ( y ) - φ ( x ¯ ) + A x ¯ , y - x ¯ 0 , y C ,
(1.2)

where A: HH is a nonlinear mapping, φ: CR is a function and Θ: C × CR is a bifunction. The set of solutions of problem (1.2) is denoted by GMEP.

In particular, whenever A = 0, problem (1.2) reduces to the following mixed equilibrium problem of finding x ¯ C such that
Θ ( x ¯ , y ) + φ ( y ) - φ ( x ¯ ) 0 , y C ,

which was considered by Ceng and Yao [5]. The set of solutions of this problem is denoted by MEP.

Whenever φ = 0, problem (1.2) reduces to the following generalized equilibrium problem of finding x ¯ C such that
Θ ( x ¯ , y ) + A x ¯ , y - x ¯ 0 , y C ,
(1.3)

which was introduced and studied by Takahashi and Takahashi [13]. The set of solutions of problem (1.3) is denoted by GEP. Obviously, the generalized equilibrium problem covers the equilibrium problem as a special case. It is assumed in [4] that Θ: C ×CR is a bifunction satisfying conditions (H1)-(H4) and φ: CR is a lower semicontinuous and convex function with restriction (A1) or (A2), where

(H1) Θ (x, x) = 0, x C;

(H2) Θ is monotone, i.e., Θ (x, y) + Θ (y, x) ≤ 0, x, y C;

(H3) for each y C, x Θ (x, y) is weakly upper semicontinuous;

(H4) for each x C, y Θ (x, y) is convex and lower semicontinuous;

(A1) for each x H and r > 0, there exist a bounded subset D x C and y x C such that for any z C \ D x ,
Θ ( z , y x ) + φ ( y x ) - φ ( z ) + 1 r y x - z , z - x < 0 ;

(A2) C is a bounded set.

It is worth pointing out that, related iterative methods for solving fixed point problems, variational inequalities and optimization problems can be found in [1935].

Recall that a ρ-Lipschitzian mapping T: CH is a mapping on C such that
ǁ T x - T y ǁ ρ ǁ x - y ǁ , x , y C ,
where ρ ≥ 0 is a constant. In particular, if ρ [0, 1) then T is called a contraction on C; if ρ = 1 then T is called a nonexpansive mapping on C. Denote the set of fixed points of T by Fix(T). It is well known that if C is a nonempty bounded closed convex subset of H and S: CC is nonexpansive, then Fix(S) ≠ Ø. Let P C be the metric projection of H onto C, that is, for every point x H, there exists a unique nearest point of C, denoted by P C x, such that ǀǀ x - P C x ǀǀ ≤ ǀǀ x - y ǀǀ for all y C. Recall also that a mapping A of C into H is called
  1. (i)
    monotone if
    A x - A y , x - y 0 , x , y C ;
     
  2. (ii)
    η-strongly monotone if there exists a constant η > 0 such that
    A x - A y , x - y η x - y 2 , x , y C ;
     
  3. (iii)
    δ-inverse strongly monotone if there exists a constant δ > 0 such that
    A x - A y , x - y δ A x - A y 2 , x , y C .
     
Furthermore, let A be a strongly positive bounded linear operator on H, that is, there exists a constant γ ¯ > 0 such that
A x , x γ ¯ x 2 , x H .
(1.4)

1.1 The W-mappings

The concept of W-mappings was introduced in Atsushiba and Takahashi [22]. It is very useful in establishing the convergence of iterative methods for computing a common fixed point of nonlinear mappings (see, for instance, [23, 25, 27]).

Let λn,1, λn,2..., λ n, N (0, 1], n ≥ 1. Given the nonexpansive mappings S1, S2,..., S N on H, Atsushiba and Takahashi defines, for each n ≥ 1, mappings Un,1, Un,2,..., U n, N by
U n , 1 = λ n , 1 S 1 + ( 1 - λ n , 1 ) I , U n , 2 = λ n , 2 S 2 U n , 1 + ( 1 - λ n , 2 ) I , U n , N - 1 = λ n , N - 1 S N - 1 U n , N - 2 + ( 1 - λ n , N - 1 ) I , W n : = U n , N = λ n , N S N U n , N - 1 + ( 1 - λ n , N ) I .
(1.5)

The W n is called the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Note that Nonexpansivity of S i implies the nonexpansivity of W n .

Colao et al. [14] introduced an iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space H. Moreover, they proved the strong convergence of the proposed iterative algorithm.

1.2 Theorem CMX

(See [[14], Theorem 3.1]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let { S i } i = 1 N be a finite family of nonexpansive mappings on H, A a strongly positive bounded linear operator on H with coefficient γ ¯ and f an α-contraction on H for some α (0, 1). Moreover, let {α n } be a sequence in (0, 1), { λ n , i } i = 1 N a sequence in [a, b] with 0 < ab < 1, {r n } a sequence in (0, ∞) and γ and β two real numbers such that 0 < β < 1 and 0 < γ < γ ¯ / α . Let Θ: C × CR be a bifunction satisfying assumptions (H1)-(H4) and i = 1 N Fix( S i ) E P ( Θ ) . For every n ≥ 1, let W n be the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Given x1 H arbitrarily, suppose the sequences {x n } and {u n } are generated iteratively by
Θ ( u n , y ) + 1 r n y - u n , u n - x n 0 , y C , x n + 1 = α n γ f ( x n ) + β x n + ( ( 1 - β ) I - α n A ) W n u n , n 1 ,
(1.6)
where the sequences {α n }, {r n } and the finite family of sequences { λ n , i } i = 1 N satisfy the conditions:
  1. (i)

    limn→∞α n = 0 and n = 1 α n = ;

     
  2. (ii)

    lim infn→∞r n > 0 and limn→∞r n /rn+1= 1 (or limn→∞ǀrn+1- r n ǀ = 0);

     
  3. (iii)

    limn→∞ǀλ n, i - λn- 1, iǀ = 0 for every i {1,..., N}.

     
Then both {x n } and {u n } converge strongly to x * i = 1 N Fix ( S i ) E P ( Θ ) , which is the unique fixed point of the composite mapping P i = 1 N Fix ( S i ) E P ( Θ ) ( I - A + γ f ) , i.e.,
x * = P i = 1 N Fix ( S i ) E P ( Θ ) ( I - A + γ f ) x * .
Very recently, Yao et al. [10] relaxed the β in Colao, Marino and Xu's iterative scheme (1.6) by a sequence of {β n }. They showed that if with additional condition 0 < lim infn→∞β n lim supn→∞β n < 1 holds, then the sequences {x n } and {u n } generated by (1.6) (but now with β n in the place of β) still converge strongly to x * i = 1 N Fix ( S i ) E P ( Θ ) , which is the unique fixed point of the composite mapping P i = 1 N Fix ( S i ) E P ( Θ ) ( I - A + γ f ) , i.e.,
x * = P i = 1 N Fix ( S i ) E P ( Θ ) ( I - A + γ f ) x * .

1.3 Hybrid steepest-descent method

Let F: HH be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, and let T: HH be nonexpansive such that Fix(T) ≠ Ø. Yamada [20] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem: finding x ̃ Fix ( T ) such that
F x ̃ , x - x ̃ 0 , x Fix ( T ) .
This method generates a sequence {x n } via the following iterative scheme:
x n + 1 = T x n - λ n + 1 μ F ( T x n ) , n 0 ,
(1.7)
where 0 < μ < 2η/κ2, the initial guess x0 H is arbitrary and the sequence {λ n } in (0, 1) satisfies the conditions:
λ n 0 , n = 0 λ n = and n = 0 ǀ λ n + 1 - λ n ǀ < .
A key fact in Yamada's argument is that, for small enough λ > 0, the mapping
T λ x : = T x - λ μ F ( T x ) , x H

is a contraction, due to the κ-Lipschitz continuity and η-strong monotonicity of F.

1.4 Our hybrid model

In this paper, assume Θ: C × CR is a bifunction satisfying assumptions (H1)-(H4) and φ: CR is a lower semicontinuous and convex function with restriction (A1) or (A2). Let the mapping A: HH be δ-inverse strongly monotone, and { S i } i = 1 N be a finite family of nonexpansive mappings on H such that i = 1 N Fix ( S i ) G M E P . Let F: HH be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: HH a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ2 and 0 ≤ γρ < τ, where τ = 1 - 1 - μ ( 2 η - μ κ 2 ) . By combining Yamada's hybrid steepest-descent method [20] and Colao, Marino and Xu's hybrid viscosity approximation method [14] (see also [10]), we propose the following hybrid iterative method for finding a common element of the set of solutions of generalized mixed equilibrium problem (1.2) and the set of fixed points of finitely many nonexpansive mappings { S i } i = 1 N , that is, for given x1 H arbitrarily, let {x n } and {u n } be generated iteratively by
Θ ( u n , y ) + φ ( y ) - φ ( u n ) + A x n , y - u n + 1 r n y - u n , u n - x n 0 , y C , x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n μ F ) W n u n , n 1 ,
(1.8)
where {a n }, {β n } (0, 1), {r n } (0, 2δ], { λ n , i } i = 1 N [ a , b ] with 0 < a ≤ b < 1, and W n is the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . We shall prove that under quite mild hypotheses, both sequences {x n } and {u n } converge strongly to x * i = 1 N Fix( S i ) G M E P , where x * = P i = 1 N Fix ( S i ) G M E P ( I - μ F + γ f ) x * is a unique solution of the variational inequality:
( μ F - γ f ) x * , x * - x 0 , x i = 1 N Fix ( S i ) G M E P .
(1.9)
Compared with Theorem 3.2 of Yao et al. [10], our Theorem 3.1 improves and extends their Theorem 3.2 [10] in the following aspects:
  1. (i)

    The contraction f: HH with coefficient ρ (0, 1) in [[10], Theorem 3.2] is extended to the case of general Lipschitzian mapping f on H with constant ρ ≥ 0.

     
  2. (ii)

    The strongly positive bounded linear operator A: HH with coefficient γ ̃ > 0 in [[10], Theorem 3.2] is extended to the case of general κ-Lipschitzian and η-strongly monotone operator F: HH with constants κ, η > 0.

     
  3. (iii)

    The equilibrium problem in [[10], Theorem 3.2] is extended to the case of generalized mixed equilibrium problem (1.2). Obviously, the problem (1.2) is more complicated than their problem (1.1).

     
  4. (iv)

    The hybrid viscosity approximation method in [[10], Theorem 3.2] (see also [[14], Theorem 3.1]) is extended to develop our iterative method by virtue of Yamada's hybrid steepest-descent method [20].

     

2 Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉, and norm ǀǀ · ǀǀ. Let C be a nonempty closed convex subset of H. Recall that the metric (or nearest point) projection from H onto C is the mapping P C : HC which assigns to each point x H the unique point P C x C satisfying the property
ǁ x - P C x ǁ = inf y C ǁ x - y ǁ = : d ( x , C ) .

In order to prove our main results in the next section, we need the following lemmas and propositions.

Lemma 2.1 (See [36]). Let C be a nonempty closed convex subset of a real Hilbert space H. Given x H and z C, we then have
  1. (i)

    z = P C x if and only if 〈 x - z, y - z 0, y C.

     
  2. (ii)

    z = P C x if and only if ǀǀ x - z ǀǀ2 ≤ ǀǀ x - y ǀǀ2 - ǀǀ y - z ǀǀ2, y C.

     
  3. (iii)

    P C x - P C y, x - y 〉 ≥ ǀǀ P C x - P C y ǀǀ2, x, y H.

     

Consequently, P C is nonexpansive and monotone.

Lemma 2.2 (See [5]). Let C be a nonempty closed convex subset of H. Let Θ: C×CR be a bifunction satisfying conditions (H1)-(H4) and let φ: CR be a lower semicontinuous and convex function. For r > 0 and x H, define a mapping T r ( Θ , φ ) : H C as follows:
T r ( Θ , φ ) ( x ) = { z C : Θ ( z , y ) + φ ( y ) - φ ( z ) + 1 r y - z , z - x 0 , y C }

for all x H. Assume that either (A1) or (A2) holds. Then the following assertions hold:

(i) T r ( Θ , φ ) ( x ) for each x H and T r ( Θ , φ ) is single-valued;
  1. (ii)
    T r ( Θ , φ ) is firmly nonexpansive, i.e., for any x, y H,
    T r ( Θ , φ ) x - T r ( Θ , φ ) y 2 T r ( Θ , φ ) x - T r ( Θ , φ ) y , x - y ;
     
  2. (iii)

    Fix ( T r ( Θ , φ ) ) = M E P ( Θ , φ ) ;

     
  3. (iv)

    MEP(Θ, φ)is closed and convex.

     

Remark 2.1. If φ = 0, then T r ( Θ , φ ) is rewritten as T r Θ ; if Θ = 0 additionally, then T r Θ = P C .

Lemma 2.3 (See [21]). Let {x n } and {y 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 xn+1= (1 - β n )y n +β n x n for all integers n ≥ 0 and lim supn→∞(ǀǀyn+1- y n ǀǀ - ǀǀxn+1- x n ǀǀ) 0. Then, limn→∞ǀǀy n - x n ǀǀ = 0.

Proposition 2.1 (See [[6], Proposition 2.1]). Let C, H, Θ, φ and T r ( Θ , φ ) be as in Lemma 2.2. Then the following inequality holds:
T s ( Θ , φ ) x - T t ( Θ , φ ) x 2 s - t s T s ( Θ , φ ) x - T t ( Θ , φ ) x , T s ( Θ , φ ) x - x

for all s, t > 0 and x H.

Lemma 2.4 (See [19]). Let {a n } be a sequence of nonnegative numbers satisfying the condition
a n + 1 ( 1 - δ n ) a n + δ n σ n , n 1 ,
where {δ n }, {σ n } are sequences of real numbers such that
  1. (i)
    {δ n } [0, 1] and n = 1 δ n = , or equivalently,
    n = 1 ( 1 - δ n ) : = lim n k = 1 n ( 1 - δ k ) = 0 ;
     
  2. (ii)

    lim supn→∞σ n ≤ 0, or

     

(ii)' n = 1 δ n σ n is convergent.

Then limn→∞a n = 0.

We will need the following result concerning the W-mapping W n generated by S1,..., S N and λn,1, λn,2,..., λ n, N in (1.5).

Proposition 2.2 (See [23]). Let C be a nonempty closed convex subset of a Banach space X. Let S1, S2,..., S N be a finite family of nonexpansive mappings of C into itself such that i = 1 N Fix( S i ) , and let λn,1, λn,2,..., λ n, N be real numbers such that 0 < λ n, i b < 1 for i = 1, 2,..., N. For any n ≥ 1, let W n be the W-mapping of C into itself generated by S1,..., S N and λn,1,..., λ n, N . If X is strictly convex, then Fix( W n ) = i = 1 N Fix( S i ) .

Proposition 2.3 (See [[14], Lemma 2.8]). Let C be a nonempty convex subset of a Banach space. Let { S i } i = 1 N be a finite family of nonexpansive mappings of C into itself and { λ n , i } i = 1 N be sequences in [0, 1] such that λ n, i → λ i (i = 1,..., N). Moreover for every integer n ≥ 1, let W and W n be the W-mappings generated by S1,..., S N and λ1,..., λ N and S1,..., S N and λn,1,..., λ n, N respectively. Then for every x C, it follows that
lim n ǁ W n x - W x ǁ = 0 .

The following two lemmas are the immediate consequences of the inner product on H.

Lemma 2.5. For all x, y H, there holds the inequality
x + y 2 x 2 + 2 y , x + y .
Lemma 2.6 (See [36]). For all x, y, z H and α, β, γ [0, 1] with α + β + γ = 1, there holds the equality
α x + β y + γ z 2 = α x 2 + β y 2 + γ z 2 - α β x - y 2 - β γ y - z 2 - γ α z - x 2 .

The following lemma plays a crucial role in proving strong convergence of our iterative schemes.

Lemma 2.7 (See [[19], Lemma 3.1]). Let λ be a number in (0, 1] and let μ > 0. Let F: H → H be an operator on H such that, for some constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone. Associating with a nonexpansive mapping T: HH, define the mapping T λ : HH by
T λ x : = T x - λ μ F ( T x ) , x H .
Then T λ is a contraction provided μ < 2η/κ2, that is,
ǁ T λ x - T λ y ǁ ( 1 - λ τ ) ǁ x - y ǁ , x , y H ,

where τ = 1 - 1 - μ ( 2 η - μ κ 2 ) ( 0 , 1 ] .

Remark 2.2. Put F = 1 2 I , where I is the identity operator of H. Then we have μ < 2η/κ2 = 4. Also, put μ = 2. Then it is easy to see that κ = η = 1 2 and
τ = 1 - 1 - μ ( 2 η - μ κ 2 ) = 1 - 1 - 2 ( 2 1 2 - 2 ( 1 2 ) 2 ) = 1 .

In particular, whenever λ > 0, we have Tλx: = Tx - λμF(Tx) = (1 - λ) Tx.

3 Iterative scheme and strong convergence

In this section, based on Yamada's hybrid steepest-descent method [20] and Colao, Marino and Xu's hybrid viscosity approximation method [14] (see also [10]), we introduce a hybrid iterative method for finding a common element of the set of solutions of generalized mixed equilibrium problem (1.2) and the set of fixed points of finitely many nonexpansive mappings in a real Hilbert space. Moreover, we derive the strong convergence of the proposed iterative algorithm to a common solution of problem (1.2) and the fixed point problem of finitely many nonexpansive mappings.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ: C × CR be a bifunction satisfying assumptions (H1)-(H4) and φ: CR be a lower semicontinuous and convex function with restriction (A1) or (A2). Let the mapping A: HH be δ-inverse strongly monotone, and { S i } i = 1 N be a finite family of nonexpansive mappings on H such that i = 1 N Fix ( S i ) G M E P . Let F: HH be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: HH a ρ- Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ2 and 0 ≤ γρ < τ, where τ = 1 - 1 - μ ( 2 η - μ κ 2 ) . Suppose {a n } and {β n } are two sequences in (0, 1), {r n } is a sequence in (0, 2δ] and { λ n , i } i = 1 N is a sequence in [a, b] with 0 < ab < 1. For every n ≥ 1, let W n be the W-mapping generated by S1,..., S N and λn,1, λn,2,..., λ n, N . Given x1 H arbitrarily, suppose the sequences {x n } and {u n } are generated iteratively by
Θ ( u n , y ) + φ ( y ) - φ ( u n ) + A x n , y - u n + 1 r n y - u n , u n - x n 0 , y C , x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n μ F ) W n u n , n 1 ,
(3.1)
where the sequences {a n }, {β n }, {r n } and the finite family of sequences { λ n , i } i = 1 N satisfy the conditions:
  1. (i)

    limn→∞α n = 0 and n = 1 α n = ;

     
  2. (ii)

    0 < lim infn→∞β n ≤ lim supn→∞β n < 1;

     
  3. (iii)

    0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1- r n ) = 0;

     
  4. (iv)

    limn→∞n+1, i- λ n, i ) = 0 for all i = 1, 2,..., N.

     
Then both {x n } and {u n } converge strongly to x * i = 1 N Fix ( S i ) G M E P , where x * = P i = 1 N Fix ( S i ) G M E P ( I - μ F + γ f ) x * is a unique solution of the variational inequality:
( μ F - γ f ) x * , x * - x 0 , x i = 1 N Fix ( S i ) G M E P .
(3.2)
Proof. Let Q = P i = 1 N Fix ( S i ) G M E P . Note that F: H → H is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: HH is a ρ-Lipschitzian mapping with constant ρ ≥ 0. Then, we have
( I - μ F ) x - ( I - μ F ) y 2 = x - y 2 - 2 μ x - y , F x - F y + μ 2 F x - F y 2 ( 1 - 2 μ η + μ 2 κ 2 ) x - y 2 = ( 1 - τ ) 2 x - y 2 ,
where τ = 1 - 1 - μ ( 2 η - μ κ 2 ) , and hence
ǁ Q ( I - μ F + γ f ) ( x ) - Q ( I - μ F + γ f ) ( y ) ǁ ǁ ( I - μ F + γ f ) ( x ) - ( I - μ F + γ f ) ( y ) ǁ ǁ ( I - μ F ) x - ( I - μ F ) y ǁ + γ ǁ f ( x ) - f ( y ) ǁ ( 1 - τ ) ǁ x - y ǁ + γ ρ ǁ x - y ǁ = ( 1 - ( τ - γ ρ ) ) ǁ x - y ǁ ,

for all x, y H. Since 0 ≤ γρ < τ ≤ 1, it is known that 1 - (τ - γρ) [0, 1). Therefore, Q(I - μF + γf) is a contraction of H into itself, which implies that there exists a unique element x* H such that x * = Q ( I - μ F + γ f ) x * = P i = 1 N Fix ( S i ) G M E P ( I - μ F + γ f ) x * .

From the definition of T r ( Θ , φ ) , we know that u n = T r n ( Θ , φ ) ( x n - r n A x n ) . Take p i = 1 N Fix ( S i ) G M E P arbitrarily. Since p = T r n ( Θ , φ ) ( p - r n A p ) = S i p , A is δ-inverse strongly monotone and 0 < r n ≤ 2δ, we deduce that, for any n ≥ 1,
u n - p 2 = T r n ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( p - r n A p ) 2 ( x n - r n A x n ) - ( p - r n A p ) 2 = x n - p - r n ( A x n - A p ) 2 = x n - p 2 - 2 r n x n - p , A x n - A p + r n 2 A x n - A p 2 x n - p 2 + r n ( r n - 2 δ ) A x n - A p 2 x n - p 2 .
(3.3)

First we will prove that both {x n } and {u n } are bounded.

Indeed, taking into account the control conditions (i) and (ii), we may assume, without loss of generality, that α n ≤ 1 - β n for all n ≥ 1. Now, by Proposition 2.2 we have p Fix(W n ).

Then utilizing Lemma 2.7, from (3.1) and (3.3) we obtain
ǁ x n + 1 - p ǁ = ǁ α n ( γ f ( x n ) - μ F p ) + β n ( x n - p ) + ( ( 1 - β n ) I - α n μ F ) W n u n - ( ( 1 - β n ) I - α n μ F ) W n p ǁ α n ǁ γ f ( x n ) - μ F p ǁ + β n ǁ x n - p ǁ + ǁ ( ( 1 - β n ) I - α n μ F ) W n u n - ( ( 1 - β n ) I - α n μ F ) W n p ǁ = α n ǁ γ f ( x n ) - μ F p ǁ + β n ǁ x n - p ǁ + ( 1 - β n ) ǁ ( I - α n 1 - β n μ F ) W n u n - ( I - α n 1 - β n μ F ) W n p ǁ ( 1 - β n ) ( 1 - α n τ 1 - β n ) ǁ u n - p ǁ + β n ǁ x n - p ǁ + α n ǁ γ f ( x n ) - μ F p ǁ ( 1 - β n - α n τ ) ǁ u n - p ǁ + β n ǁ x n - p ǁ + α n ǁ γ f ( x n ) - μ F p ǁ ( 1 - α n τ ) ǁ x n - p ǁ + α n γ ǁ f ( x n ) - f ( p ) ǁ + α n ǁ γ f ( p ) - μ F p ǁ ( 1 - α n τ ) ǁ x n - p ǁ + α n γ ρ ǁ x n - p ǁ + α n ǁ γ f ( p ) - μ F p ǁ = ( 1 - ( τ - γ ρ ) α n ) ǁ x n - p ǁ + α n ǁ γ f ( p ) - μ F p ǁ = ( 1 - ( τ - γ ρ ) α n ) ǁ x n - p ǁ + ( τ - γ ρ ) α n ǁ γ f ( p ) - μ F p ǁ τ - γ ρ max ǁ x n - p ǁ , ǁ γ f ( p ) - μ F p ǁ τ - γ ρ .
(3.4)
It follows from (3.4) and induction that
ǁ x n - p ǁ max { ǁ x 0 - p ǁ , ǁ γ f ( p ) - μ F p ǁ τ - γ ρ } , n 1 .

Therefore {x n } is bounded. We also obtain that {u n }, {Ax n }, {W n u n } and {f (x n )} are all bounded. We shall use M to denote the possible different constants appearing in the following reasoning.

Next, we show that ǀǀ xn+1- x n ǀǀ → 0.

Indeed, set xn+1= β n x n + (1 - β n )z n for all n ≥ 1. Then from the definition of z n we obtain
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 = α n + 1 γ f ( x n + 1 ) + ( ( 1 - β n + 1 ) I - α n + 1 μ F ) W n + 1 u n + 1 1 - β n + 1 - α n γ f ( x n ) + ( ( 1 - β n ) I - α n μ F ) W n u n 1 - β n = α n + 1 1 - β n + 1 γ f ( x n + 1 ) - α n 1 - β n γ f ( x n ) + W n + 1 u n + 1 - W n u n + α n 1 - β n μ F W n u n - α n + 1 1 - β n + 1 μ F W n + 1 u n + 1 = α n + 1 1 - β n + 1 [ γ f ( x n + 1 ) - μ F W n + 1 u n + 1 ] + α n 1 - β n [ μ F W n u n - γ f ( x n ) ] + W n + 1 u n + 1 - W n + 1 u n + W n + 1 u n - W n u n .
It follows that
ǁ z n + 1 - z n ǁ - ǁ x n + 1 - x n ǁ α n + 1 1 - β n + 1 ( γ ǁ f ( x n + 1 ) ǁ + μ ǁ F W n + 1 u n + 1 ǁ ) + α n 1 - β n ( μ ǁ F W n u n ǁ + γ ǁ f ( x n ) ǁ ) + ǁ W n + 1 u n + 1 - W n + 1 u n ǁ + ǁ W n + 1 u n - W n u n ǁ - ǁ x n + 1 - x n ǁ α n + 1 1 - β n + 1 ( γ ǁ f ( x n + 1 ) ǁ + μ ǁ F W n + 1 u n + 1 ǁ ) + α n 1 - β n ( μ ǁ F W n u n ǁ + γ ǁ f ( x n ) ǁ ) + ǁ W n + 1 u n - W n u n ǁ + ǁ u n + 1 - u n ǁ - ǁ x n + 1 - x n ǁ .
(3.5)
From (1.5), since S i and U n, i for all i = 1, 2,..., N are nonexpansive,
ǁ W n + 1 u n - W n u n ǁ = ǁ λ n + 1 , N S N U n + 1 , N - 1 u n + ( 1 - λ n + 1 , N ) u n - λ n , N S N U n , N - 1 u n - ( 1 - λ n , N ) u n ǁ ǀ λ n + 1 , N - λ n , N ǀ ǁ u n ǁ + ǁ λ n + 1 , N S N U n + 1 , N - 1 u n - λ n , N S N U n , N - 1 u n ǁ ǀ λ n + 1 , N - λ n , N ǀ ǁ u n ǁ + ǁ λ n + 1 , N ( S N U n + 1 , N - 1 u n - S N U n , N - 1 u n ) ǁ + ǀ λ n + 1 , N - λ n , N ǀ ǁ S N U n , N - 1 u n ǁ 2 M ǀ λ n + 1 , N - λ n , N ǀ + λ n + 1 , N ǁ U n + 1 , N - 1 u n - U n , N - 1 u n ǁ .
(3.6)
Again, from (1.5),
ǁ U n + 1 , N - 1 u n - U n , N - 1 u n ǁ = ǁ λ n + 1 , N - 1 S N - 1 U n + 1 , N - 2 u n + ( 1 - λ n + 1 , N - 1 ) u n - λ n , N - 1 S N - 1 U n , N - 2 u n - ( 1 - λ n , N - 1 ) u n ǁ ǀ λ n + 1 , N - 1 - λ n , N - 1 ǀ ǁ u n ǁ + ǁ λ n + 1 , N - 1 S N - 1 U n + 1 , N - 2 u n - λ n , N - 1 S N - 1 U n , N - 2 u n ǁ ǀ λ n + 1 , N - 1 - λ n , N - 1 ǀ ǁ u n ǁ + λ n + 1 , N - 1 ǁ S N - 1 U n + 1 , N - 2 u n - S N - 1 U n , N - 2 u n ǁ + ǀ λ n + 1 , N - 1 - λ n , N - 1 ǀ M 2 M ǀ λ n + 1 , N - 1 - λ n , N - 1 ǀ + λ n + 1 , N - 1 ǁ U n + 1 , N - 2 u n - U n , N - 2 u n ǁ 2 M ǀ λ n + 1 , N - 1 - λ n , N - 1 ǀ + ǁ U n + 1 , N - 2 u n - U n , N - 2 u n ǁ .
(3.7)
Therefore, we have
ǁ U n + 1 , N - 1 u n - U n , N - 1 u n ǁ 2 M ǀ λ n + 1 , N - 1 - λ n , N - 1 ǀ + 2 M ǀ λ n + 1 , N - 2 - λ n , N - 2 ǀ + ǁ U n + 1 , N - 3 u n - U n , N - 3 u n ǁ 2 M i = 2 N - 1 ǀ λ n + 1 , i - λ n , i ǀ + ǁ U n + 1 , 1 u n - U n , 1 u n ǁ = ǁ λ n + 1 , 1 S 1 u n + ( 1 - λ n + 1 , 1 ) u n - λ n , 1 S 1 u n - ( 1 - λ n , 1 ) u n ǁ + 2 M i = 2 N - 1 ǀ λ n + 1 , i - λ n , i ǀ ,
and then
ǁ U n + 1 , N - 1 u n - U n , N - 1 u n ǁ ǀ λ n + 1 , 1 - λ n , 1 ǀ ǁ u n ǁ + ǁ λ n + 1 , 1 S 1 u n - λ n , 1 S 1 u n ǁ + 2 M i = 2 N - 1 ǀ λ n + 1 , i - λ n , i ǀ 2 M i = 1 N - 1 ǀ λ n + 1 , i - λ n , i ǀ .
(3.8)
Substituting (3.8) into (3.6), we have
ǁ W n + 1 u n - W n u n ǁ 2 M ǀ λ n + 1 , N - λ n , N ǀ + 2 λ n + 1 , N M i = 1 N - 1 ǀ λ n + 1 , i - λ n , i ǀ 2 M i = 1 N ǀ λ n + 1 , i - λ n , i ǀ .
(3.9)
On the other hand, utilizing the δ-inverse strongly monotonicity of A we have
ǁ ( x n + 1 - r n + 1 A x n + 1 ) - ( x n - r n A x n ) ǁ = ǁ x n + 1 - x n - r n + 1 ( A x n + 1 - A x n ) + ( r n - r n + 1 ) A x n ǁ ǁ x n + 1 - x n - r n + 1 ( A x n + 1 - A x n ) ǁ + ǀ r n + 1 - r n ǀ ǁ A x n ǁ ǁ x n + 1 - x n ǁ + ǀ r n + 1 - r n ǀ ǁ A x n ǁ ,
(3.10)
Since u n = T r n ( Θ , φ ) ( x n - r n A x n ) and u n + 1 = T r n + 1 ( Θ , φ ) ( x n + 1 - r n + 1 A x n + 1 ) , we get
ǁ u n + 1 - u n ǁ = ǁ T r n + 1 ( Θ , φ ) ( x n + 1 - r n + 1 A x n + 1 ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ = ǁ T r n + 1 ( Θ , φ ) ( x n + 1 - r n + 1 A x n + 1 ) - T r n + 1 ( Θ , φ ) ( x n - r n A x n ) + T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ ǁ T r n + 1 ( Θ , φ ) ( x n + 1 - r n + 1 A x n + 1 ) - T r n + 1 ( Θ , φ ) ( x n - r n A x n ) ǁ + ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ ǁ ( x n + 1 - r n + 1 A x n + 1 ) - ( x n - r n A x n ) ǁ + ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ ǁ x n + 1 - x n ǁ + ǀ r n + 1 - r n ǀ ǁ A x n ǁ + ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ ,
(3.11)
Using (3.9) and (3.11) in (3.5), we get
ǁ z n + 1 - z n ǁ - ǁ x n + 1 - x n ǁ α n + 1 1 - β n + 1 ( γ ǁ f ( x n + 1 ) ǁ + μ ǁ F W n + 1 u n + 1 ǁ ) + α n 1 - β n ( μ ǁ F W n u n ǁ + γ ǁ f ( x n ) ǁ ) + 2 M i = 1 N ǀ λ n + 1 , i - λ n , i ǀ + ǁ x n + 1 - x n ǁ + ǀ r n + 1 - r n ǀ ǁ A x n ǁ + ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ - ǁ x n + 1 - x n ǁ = α n + 1 1 - β n + 1 ( γ ǁ f ( x n + 1 ) ǁ + μ ǁ F W n + 1 u n + 1 ǁ ) + α n 1 - β n ( μ ǁ F W n u n ǁ + γ ǁ f ( x n ) ǁ ) + 2 M i = 1 N ǀ λ n + 1 , i - λ n , i ǀ + ǀ r n + 1 - r n ǀ ǁ A x n ǁ + ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ .
(3.12)
Note that 0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ and limn→∞(rn+1- r n ) = 0. Then utilizing Proposition 2.1 we have
lim n ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ = 0 .
(3.13)
Consequently, it follows from (3.13) and conditions (i), (iii), (iv) that
lim  sup n ( ǁ z n + 1 - z n ǁ - ǁ x n + 1 - x n ǁ ) lim  sup n { α n + 1 1 - β n + 1 ( γ ǁ f ( x n + 1 ) ǁ + μ ǁ F W n + 1 u n + 1 ǁ ) + α n 1 - β n ( μ ǁ F W n u n ǁ + γ ǁ f ( x n ) ǁ ) + 2 M i = 1 N ǀ λ n + 1 , i - λ n , i ǀ + ǀ r n + 1 - r n ǀ ǁ A x n ǁ + ǁ T r n + 1 ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( x n - r n A x n ) ǁ }  =  0 .
Hence by Lemma 2.3 we have
lim n ǁ z n - x n ǁ = 0 .
Consequently
lim n ǁ x n + 1 - x n ǁ = lim n ( 1 - β n ) ǁ z n - x n ǁ = 0 .
(3.14)
From (3.11), (3.13), (3.14) and condition (iii) we have
lim n ǁ u n + 1 - u n ǁ = 0 .
Since xn+1= a n γ f(x n ) + β n x n + ((1 - β n )I - a n μF)W n u n , we have
ǁ x n - W n u n ǁ ǁ x n - x n + 1 ǁ + ǁ x n + 1 - W n u n ǁ ǁ x n - x n + 1 ǁ + α n ǁ γ f ( x n ) - μ F W n u n ǁ + β n ǁ x n - W n u n ǁ ,
that is
ǁ x n - W n u n ǁ 1 1 - β n ǁ x n - x n + 1 ǁ + α n 1 - β n ǁ γ f ( x n ) - μ F W n u n ǁ .
It follows that
lim n ǁ x n - W n u n ǁ = 0 .
(3.15)
On the other hand, from (3.3) and (3.4) we get
x n + 1 - p 2 [ ( 1 - β n - α n τ ) ǁ u n - p ǁ + β n ǁ x n - p ǁ + α n ǁ γ f ( x n ) - μ F p ǁ ] 2 ( 1 - β n - α n τ ) u n - p 2 + β n x n - p 2 + α n τ γ f ( x n ) - μ F p 2 ( 1 - β n - α n τ ) [ x n - p 2 + r n ( r n - 2 δ ) A x n - A p 2 ] + β n x n - p 2 + α n τ γ f ( x n ) - μ F p 2 = ( 1 - α n τ ) x n - p 2 + r n ( r n - 2 δ ) ( 1 - β n - α n τ ) A x n - A p 2 + α n τ ǁ γ f ( x n ) - μ F p ǁ 2 x n - p 2 + r n ( r n - 2 δ ) ( 1 - β n - α n τ ) A x n - A p 2 + α n τ γ f ( x n ) - μ F p 2 ,
and hence
r n ( 2 δ - r n ) ( 1 - β n - α n τ ) A x n - A p 2 x n - p 2 - x n + 1 - p 2 + α n τ γ f ( x n ) - μ F p 2 = ( ǁ x n - p ǁ + ǁ x n + 1 - p ǁ ) ǁ x n - x n + 1 ǁ + α n τ γ f ( x n ) - μ F p 2 .
Obviously, conditions (i), (ii), (iii) guarantee that α n → 0, 0 < lim infn→∞β n ≤ lim supn→∞β n < 1 and 0 < lim infn→∞r n ≤ lim supn→∞r n < 2δ. Thus from ǀǀ x n - xn+1ǀǀ → 0 we conclude that
lim n ǁ A x n - A p ǁ = 0 .
(3.16)
Note that T r ( Θ , φ ) is firmly nonexpansive. Hence we have
u n - p 2 = T r n ( Θ , φ ) ( x n - r n A x n ) - T r n ( Θ , φ ) ( p - r n A p ) 2 ( x n - r n A x n ) - ( p - r n A p ) , u n - p = 1 2 [ ( x n - r n A x n ) - ( p - r n A p ) 2 + u n - p 2 - ( x n - r n A x n ) - ( p - r n A p ) - ( u n - p ) 2 ] 1 2 [ x n - p 2 + u n - p 2 - x n - u n - r n ( A x n - A p ) 2 ] = 1 2 [ x n - p 2 + u n - p 2 - x n - u n 2 + 2 r n A x n - A p , x n - u n - r n 2 A x n - A p 2 ] ,
which implies that
u n - p 2 x n - p 2 - x n - u n 2 + 2 r n ǁ A x n - A p ǁ ǁ x n - u n ǁ .
(3.17)
Therefore, utilizing Lammas 2.5 and 2.7 we deduce from (3.17) that
x n + 1 - p 2 = α n ( γ f ( x n ) - μ F p ) + β n ( x n - W n u n ) + ( I - α n μ F ) W n u n - ( I - α n μ F ) W n p 2 ( I - α n μ F ) W n u n - ( I - α n μ F ) W n p + β n ( x n - W n u n ) 2 + 2 α n γ f ( x n ) - μ F p , x n + 1 - p ( I - α n μ F ) W n u n - ( I - α n μ F ) W n p + β n x n - W n u n 2 + 2 α n γ f ( x n ) - μ F p x n + 1 - p ( 1 - α n τ ) u n - p + β n x n - W n u n 2 + 2 α n γ f ( x n ) - μ F p x n + 1 - p u n - p + x n - W n u n 2 + 2 α n γ f ( x n ) - μ F p x n + 1 - p = u n - p 2 + x n - W n u n 2 + 2 u n - p x n - W n u n + 2 α n γ f ( x n ) - μ F p x n + 1 - p x n - p 2 - x n - u n 2 + 2 r n ǁ A x n - A p ǁ ǁ x n - u n ǁ + x n - W n u n 2 + 2 ǁ u n - p ǁ ǁ x n - W n u n ǁ + 2 α n ǁ γ f ( x n ) - μ F p ǁ ǁ x n + 1 - p ǁ .
Then we have
x n - u n 2 x n - p 2 - x n + 1 - p 2 + 2 r n ǁ A x n - A p ǁ ǁ x n - u n ǁ + x n - W n u n 2 + 2 ǁ u n - p ǁ ǁ x n - W n u n ǁ + 2 α n ǁ γ f ( x n ) - μ F p ǁ ǁ x n + 1 - p ǁ ( ǁ x n - p ǁ + ǁ x n + 1 - p ǁ ) ǁ x n - x n + 1 ǁ + 2 r n ǁ A x n - A p ǁ ǁ x n - u n ǁ + x n - W n u n 2 + 2 ǁ u n - p ǁ ǁ x n - W n u n ǁ + 2 α n ǁ γ f ( x n ) - μ F p ǁ ǁ x n + 1 - p ǁ .
So, from (3.14)-(3.16) and α n → 0, we have
lim n ǁ x n - u n ǁ = 0 .
Since
ǁ W n u n - u n ǁ ǁ W n u n - x n ǁ + ǁ x n - u n ǁ ,
we also have
lim n ǁ W n u n - u n ǁ = 0 .
Next, let us show that
limsup n ( μ F - γ f ) x * , x * - x n 0 ,
where x * = P i = 1 N Fix ( S i ) G M E P ( I - μ F + γ f ) x * is a unique solution of the variational inequality (3.2). To show this, we can choose a subsequence { u n i } of {u n } such that
lim sup n ( μ F - γ f ) x * , x * - u n = lim i ( μ F - γ f ) x * , x * - u n i .
Since { u n i } is bounded, there exists a subsequence {u ij } of { u n i } which converges weakly to w. Without loss of generality, we may assume that u n i w . From ǀǀW n u n - u n ǀǀ → 0, we obtain W n u n i w . Now we show that w GMEP. From u n = T r n ( Θ , φ ) ( x n - r n A x n ) , we know that
Θ ( u n , y ) + φ ( y ) - φ ( u n ) + A x n , y - u n + 1 r n y - u n , u n - x n 0 , y C .
From (H2) it follows that
φ ( y ) - φ ( u n ) + A x n , y - u n + 1 r n y - u n , u n - x n Θ ( y , u n ) , y C .
Replacing n by n i , we have
φ ( y ) - φ ( u n i ) + A x n i , y - u n i + y - u n i , u n i - x n i r n i Θ ( y , u n i ) , y C .
(3.18)
Put u t = ty + (1 - t)w for all t (0, 1] and y C. Then, we have u t C. So, from (3.18) we have
u t - u n i , A u t u t - u n i , A u t - φ ( u t ) + φ ( u n i ) - u t - u n i , A x n i - u t - u n i , u n i - x n i r n i + Θ ( u t , u n i ) = u t - u n i , A u t - A u n i + u t - u n i , A u n i - A x n i - φ ( u t ) + φ ( u n i ) - u t - u n i , u n i - x n i r n i + Θ ( u t , u n i ) .
Since ǁ u n i - x n i ǁ 0 , we have ǁ A u n i - A x n i ǁ 0 . Further, from the monotonicity of A, we have u t - u n i , A u t - A u n i 0 . So, from (H4), the weakly lower semicontinuity of φ , u n i - x n i r n i 0 and u n i w , we have
u t - w , A u t - φ ( u t ) + φ ( w ) + Θ ( u t , w ) ,
(3.19)
as i → ∞. From (H1), (H4) and (3.19), we also have
0 = Θ ( u t , u t ) + φ ( u t ) - φ ( u t ) t Θ ( u t , y ) + ( 1 - t ) Θ ( u t , w ) + t φ ( y ) + ( 1 - t