Open Access

A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems

Journal of Inequalities and Applications20112011:51

https://doi.org/10.1186/1029-242X-2011-51

Received: 21 February 2011

Accepted: 8 September 2011

Published: 8 September 2011

Abstract

In this article, we introduce a new general composite iterative scheme for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality problem for an inverse-strongly monotone mapping in Hilbert spaces. It is shown that the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets under suitable control conditions, which solves a certain optimization problem. The results of this article substantially improve, develop, and complement the previous well-known results in this area.

2010 Mathematics Subject Classifications: 49J30; 49J40; 47H09; 47H10; 47J20; 47J25; 47J05; 49M05.

Keywords

generalized mixed equilibrium problemfixed pointnonexpansive mapping; inverse-strongly monotone mappingvariational inequality; optimization problemmetric projectionstrongly positive bounded linear operator

1 Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||. Let C be a nonempty closed convex subset of H and S : CC be a self-mapping on C. Let us denote by F(S) the set of fixed points of S and by P C the metric projection of H onto C.

Let B : CH be a nonlinear mapping and φ : C be a function, and Θ be a bifunction of C × C into , where is the set of real numbers.

Then, we consider the following generalized mixed equilibrium problem of finding x C such that
Θ ( x , y ) + B x , y - x + φ ( y ) - φ ( x ) 0 , y C ,
(1.1)

which was recently introduced by Peng and Yao [1]. The set of solutions of the problem (1.1) is denoted by GMEP(Θ, φ, B). Here, some special cases of the problem (1.1) are stated as follows:

If B = 0, then the problem (1.1) reduces the following mixed equilibrium problem of finding x C such that
Θ ( x , y ) + φ ( y ) - φ ( x ) 0 , y C ,
(1.2)

which was studied by Ceng and Yao [2] (see also [3]). The set of solutions of the problem (1.2) is denoted by MEP (Θ,φ).

If φ = 0 and B = 0, then the problem (1.1) reduces the following equilibrium problem of finding x C such that
Θ ( x , y ) 0 , y C .
(1.3)

The set of solutions of the problem (1.3) is denoted by EP(Θ).

If φ = 0 and Θ(x, y) = 0 for all x, y C, then the problem (1.1) reduces the following variational inequality problem of finding x C such that
B x , y - x 0 , y C .
(1.4)

The set of solutions of the problem (1.4) is denoted by V I(C, B).

The problem (1.1) is very general in the sense that it includes, as special cases, fixed point problems, optimization problems, variational inequality problems, minmax problems, Nash equilibrium problems in noncooperative games, and others; see [2, 46].

Recently, in order to study the problem (1.3) coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem (1.3) and the set of fixed points of a countable family of nonexpansive mappings; see [716] and the references therein.

In 2008, Su et al. [17] gave an iterative scheme for the problem (1.3), the problem (1.4) for an inverse-strongly monotone mapping, and fixed point problems of non-expansive mappings. In 2009, Yao et al. [18] considered an iterative scheme for the problem (1.2), the problem (1.4) for a Lipschitz and relaxed-cocoercive mapping and fixed point problems of nonexpansive mappings, and in 2008, Peng and Yao [1] studied an iterative scheme for the problem (1.1), the problem (1.4) for a monotone, and Lipschitz continuous mapping and fixed point problems of nonexpansive mappings.

In particular, in 2010, Jung [9] introduced the following new composite iterative scheme for finding a common element of the set of solutions of the problem (1.3) and the set of fixed points of a nonexpansive mapping: x1 C and
Θ ( u n , y ) + 1 r n y - u n , u n - x n 0 , y C , y n = α n f ( x n ) + ( 1 - α n ) T u n , x n + 1 = ( 1 - β n ) y n + β n T y n , n 1 ,
(1.5)

where T is a nonexpansive mapping, f is a contraction with constant k (0, 1), {α n }, {β n } [0, 1], and {r n } (0, ∞). He showed that the sequences {x n } and {u n } generated by (1.5) converge strongly to a point in F(T ) ∩ EP (Θ) under suitable conditions.

On the other hand, the following optimization problem has been studied extensively by many authors:
min x Ω μ 2 A x , x + 1 2 x - u 2 - h ( x ) ,

where Ω = n = 1 C n , C 1 , C 2 , are infinitely many closed convex subsets of H such that n = 1 C n , u H, μ ≥ 0 is a real number, A is a strongly positive bounded linear operator on H (i.e., there is a constant γ ̄ > 0 such that A x , x γ ̄ x 2 , x H) and h is a potential function for γ f (i.e., h'(x) = γ f(x) for all x H). For this kind of optimization problems, see, for example, Bauschke and Borwein [19], Combettes [20], Deutsch and Yamada [21], Jung [22], and Xu [23] when Ω = i = 1 N C i ; and h(x) = 〈x, b〉 for a given point b in H.

In 2009, Yao et al. [3] considered the following iterative scheme for the problem (1.2) and optimization problems:
Θ ( y n , y ) + φ ( y ) - φ ( y n ) + 1 r K ( y n ) - K ( x n ) , y - y n 0 , y H , x n + 1 = α n ( u + γ f ( x n ) ) + β n x n + ( ( 1 - β n ) I - α n ( I + μ A ) ) W n y n , n 1 ,
(1.6)
where u H; {α n } and {β n } are two sequences in (0,1), μ > 0, r > 0, γ > 0; K'(x) is the Fréchet derivative of a functional K : H at x; and W n is the so-called W-mapping related to a sequence {T n } of nonexpansive mappings. They showed that under appropriate conditions, the sequences {x n } and {y n } generated by (1.6) converge strongly to a solution of the optimization problem:
min x n = 1 F ( T n ) M E P ( Θ , φ ) μ 2 A x , x + 1 2 | | x - u | | 2 - h ( x ) .

In 2010, using the method of Yao et al. [3], Jaiboon and Kumam [24] also introduced a general iterative method for finding a common element of the set of solutions of the problem (1.2), the set of fixed points of a sequence {T n } of nonexpansive mappings, and the set of solutions of the problem (1.4) for a α-inverse-strongly monotone mapping. We point out that in the main results of [3, 24], the condition of the sequentially continuity from the weak topology to the strong topology for the derivative K' of the function K : C is very strong. Even if K ( x ) = x 2 2 , then K' (x) = x is not sequentially continuous from the weak topology to the strong topology.

In this article, inspired and motivated by above mentioned results, we introduce a new iterative method for finding a common element of the set of solutions of a generalized mixed equilibrium problem (1.1), the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of the variational inequality problem (1.4) for an inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem. The results of this article can be viewed as an improvement and complement of the recent results in this direction.

2 Preliminaries and lemmas

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. In the following, we write x n x to indicate that the sequence {x n } converges weakly to x. x n x implies that {x n } converges strongly to x.

First, we know that a mapping f : CC is a contraction on C if there exists a constant k (0, 1) such that ||f(x) - f(y)|| ≤ k||x - y||, x, y C. A mapping T : CC is called nonexpansive if ||Tx - Ty|| ||x - y||, x, y C.

In a real Hilbert space H, we have
λ x + ( 1 - λ ) y 2 = λ x 2 + ( 1 - λ ) y 2 - λ ( 1 - λ ) x - y 2
for all x, y H and λ . For every point x H, there exists the unique nearest point in C, denoted by P C (x), such that
| | x - P C ( x ) | | | | x - y | |
for all y C. P C is called the metric projection of H onto C. It is well known that P C is nonexpansive and P C satisfies
x - y , P C ( x ) - P C ( y ) P C ( x ) - P C ( y ) 2
(2.1)
for every x, y H. Moreover, P C (x) is characterized by the properties:
x - y 2 x - P C ( x ) 2 + y - P C ( x ) 2
and
u = P C ( x ) x - u , u - y 0 f o r a l l x H , y C .
In the context of the variational inequality problem for a nonlinear mapping F, this implies that
u V I ( C , F ) u = P C ( u - λ F u ) , f o r a n y λ > 0 .
(2.2)
It is also well known that H satisfies the Opial condition, that is, for any sequence {x n } with x n x, the inequality
liminf n x n - x < liminf n x n - y

holds for every y H with yx.

A mapping F of C into H is called α-inverse-strongly monotone if there exists a constant α > 0 such that
x - y , F x - F y α F x - F y 2 , x , y C .
We know that if F = I - T, where T is a nonexpansive mapping of C into itself and I is the identity mapping of H, then F is 1 2 -inverse-strongly monotone and V I (C, F ) = F(T ). A mapping F of C into H is called strongly monotone if there exists a positive real number η such that
x - y , F x - F y η x - y 2 , x , y C .
In such a case, we say F is η-strongly monotone. If F is η-strongly monotone and κ-Lipschitz continuous, that is, ||Fx - Fy|| ≤ κ||x - y|| for all x, y C, then F is η κ 2 -inverse-strongly monotone. If F is an α-inverse-strongly monotone mapping of C into H, then it is obvious that F is 1 α -Lipschitz continuous. We also have that for all x, y C and λ > 0,
( I - λ F ) x - ( I - λ F ) y 2 = ( x - y ) - λ ( F x - F y ) 2 (1)  = x - y 2 - 2 λ x - y , F x - F y + λ 2 F x - F y 2 (2)  x - y 2 + λ ( λ - 2 α ) F x - F y 2 . (3)  (4) 

Hence, if λ ≤ 2α, then I - λF is a nonexpansive mapping of C into H. The following result for the existence of solutions of the variational inequality problem for inverse-strongly monotone mappings was given in Takahashi and Toyoda [25].

Proposition Let C be a bounded closed convex subset of a real Hilbert space, and F be an α-inverse-strongly monotone mapping of C into H. Then, V I(C, F) is nonempty.

A set-valued mapping Q : H → 2 H is called monotone if for all x, y H, f Qx and g Qy imply 〈x - y, f - g 〉 ≥ 0. A monotone mapping Q : H → 2 H is maximal if the graph G(Q) of Q is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping Q is maximal if and only if for (x, f ) H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) G(Q) implies f Qx. Let F be an inverse-strongly monotone mapping of C into H, and let N C v be the normal cone to C at v, that is, N C v = {w H : 〈v - u, w〉 ≥ 0, for all u C}, and define
Q v = F v + N C v , v C v C .

Then, Q is maximal monotone and 0 Qv if and only if v V I(C, F ); see [26, 27].

For solving the equilibrium problem for a bifunction Θ : C × C, let us assume that Θ and φ satisfy the following conditions:

(A1) Θ(x, x) = 0 for all x C;

(A2) Θ is monotone, that is, Θ(x, y) + Θ (y, x) ≤ 0 for all x, y C;

(A3) for each x, y, z C,
lim t 0 Θ ( t z + ( 1 - t ) x , y ) Θ ( x , y ) ;

(A4) for each x C, y α Θ (x, y) is convex and lower semicontinuous;

(A5) For each y C, x α Θ (x, y) is weakly upper semicontinuous;

(B1) For each x H and r > 0, there exists 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 ;

(B2) C is a bounded set;

The following lemmas were given in [1, 4].

Lemma 2.1 ([4]) Let C be a nonempty closed convex subset of H, and Θ be a bifunction of C × C into satisfying (A1)-(A4). Let r > 0 and x H. Then, there exists z C such that
Θ ( z , y ) + 1 r y - z , z - x 0 y C .
Lemma 2.2 ([1]) Let C be a nonempty closed convex subset of H. Let Θ be a bifunction form C × C to satisfying (A1)-(A5) and φ : C be a proper lower semicontinuous and convex function. For r > 0 and x H, define a mapping S r : HC as follows:
S r ( x ) = { z C : Θ ( z , y ) + φ ( y ) - φ ( z ) + 1 r y - z , z - x 0 , y C }
for all z H. Assume that either (B1) or (B2) holds. Then, the following hold:
  1. (1)

    For each x H, S r (x) ≠ ;

     
  2. (2)

    S r is single-valued;

     
  3. (3)
    S r is firmly nonexpansive, that is, for any x, y H,
    | | S r x - S r y | | 2 S r x - S r y , x - y ;
     
  4. (4)

    F(S r ) = MEP (Θ, φ);

     
  5. (5)

    MEP (Θ, φ) is closed and convex.

     

We also need the following lemmas for the proof of our main results.

Lemma 2.3 ([23]) Let {s n } be a sequence of non-negative real numbers satisfying
s n + 1 ( 1 - λ n ) s n + β n , n 1 ,
where {λ n } and {β n } satisfy the following conditions:
  1. (i)

    {λ n } [0, 1] and n = 1 λ n = or, equivalently, n = 1 ( 1 - λ n ) = 0 ,

     
  2. (ii)

    lim sup n β n λ n 0 or n = 1 | β n | < ,

     

Then, limn→∞s n = 0.

Lemma 2.4 In a Hilbert space, there holds the inequality
| | x + y | | 2 | | x | | 2 + 2 y , x + y , x , y H .
Lemma 2.5 (Aoyama et al. [28]) Let C be a nonempty closed convex subset of H and {T n } be a sequence of nonexpansive mappings of C into itself. Suppose that
n = 1 sup { | | T n + 1 z - T n z | | : z C } < .

Then, for each y C, {T n y} converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by Ty = limn→∞T n y for all y C. Then, limn→∞sup{||Tz - T n z|| : z C} = 0.

The following lemma can be found in [3](see also Lemma 2.1 in [22]).

Lemma 2.6 Let C be a nonempty closed convex subset of a real Hilbert space H and g : C {} be a proper lower semicontinuous differentiable convex function. If x* is a solution to the minimization problem
g ( x * ) = inf x C g ( x ) ,
then
g ( x ) , x - x * 0 , x C .
In particular, if x* solves the optimization problem
min x C μ 2 A x , x + 1 2 | | x - u | | 2 - h ( x ) ,
then
u + ( γ f - ( I + μ A ) ) x * , x - x * 0 , x C ,

where h is a potential function for γ f.

3 Main results

In this section, we introduce a new composite iterative scheme for finding a common point of the set of solutions of the problem (1.1), the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of the problem (1.4) for an inverse-strongly monotone mapping.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H such that C ± C C. Let Θ be a bifunction from C × C to satisfying (A1)-(A5) and φ : C be a lower semicontinuous and convex function. Let F, B be two α, β-inverse-strongly monotone mappings of C into H, respectively. Let {T n } be a sequence of nonexpansive mappings of C into itself such that Ω 1 : = n = 1 F ( T n ) V I ( C , F ) G M E P ( Θ , φ , B ) . Let μ > 0 and γ > 0 be real numbers. Let f be a contraction of C into itself with constant k (0, 1) and A be a strongly positive bounded linear operator on C with constant γ ̄ ( 0 , 1 ) such that 0 < γ < ( 1 + μ ) γ ̄ k . Assume that either (B1) or (B2) holds. Let u C, and let {x n },{y n }, and {u n } be sequences generated by x1 C and
Θ ( u n , y ) + B x n , y - u n + φ ( y ) - φ ( u n ) + 1 r n y - u n , u n - x n 0 , y C , y n = α n ( u + γ f ( x n ) ) + ( I - α n ( I + μ A ) ) T n P C ( u n - λ n F u n ) , x n + 1 = ( 1 - β n ) y n + β n T n P C ( y n - λ n F y n ) , n 1 , ( IS )

where {α n }, {β n } [0, 1], λ n [a, b] (0, 2α) and r n [c, d] (0, 2β). Let {α n }, {λ n } and {β n } satisfy the following conditions:

(C1) α n → 0 (n → ∞); n = 1 α n = ;

(C2) β n [0, a) for all n ≥ 0 and for some a (0, 1);

(C3) n = 1 | α n + 1 - α n | < , n = 1 | β n + 1 - β n | < , n = 1 | λ n + 1 - λ n | < , n = 1 | r n + 1 - r n | < .

Suppose that n = 1 sup { | | T n + 1 z - T n z | | : z D } < for any bounded subset D of C. Let T be a mapping of C into itself defined by Tz = limn→∞T n z for all z C and suppose that F ( T ) = n = 1 F ( T n ) . Then {x n } and {u n } converge strongly to q Ω1, which is a solution of the optimization problem:
min x Ω 1 μ 2 A x , x + 1 2 | | x - u | | 2 - h ( x ) , ( O P 1 )

where h is a potential function for γ f.

Proof First, from α n → 0 (n → ∞) in the condition (C1), we assume, without loss of generality, that α n ≤ (1 + μ||A||)-1 and 2 ( ( 1 + μ ) γ ̄ - γ k ) α n < 1 for n ≥ 1. We know that if A is bounded linear self-adjoint operator on H, then
| | A | | = sup { | A u , u | : u H , | | u | | = 1 } .
Observe that
( I - α n ( I + μ A ) ) u , u = 1 - α n - α n μ A u , u (1) 1 - α n - α n μ | | A | | (2) 0 , (3) (4)
which is to say I - α n (I + μA) is positive. It follows that
| | I - α n ( I + μ A ) | | = sup { ( I - α n ( I + μ A ) ) u , u : u H , | | u | | = 1 } (1) = sup { 1 - α n - α n μ A u , u : u H , | | u | | = 1 } (2) 1 - α n ( 1 + μ γ ̄ ) (3) < 1 - α n ( 1 + μ ) γ ̄ . (4) (5)

Let us divide the proof into several steps. From now on, we put z n = P C (u n -λ n Fu n ) and w n = P C (y n - λ n Fy n ).

Step 1: We show that {x n } is bounded. To this end, let p Ω 1 : = n = 1 F ( T n ) V I ( C , F ) G M E P ( Θ , φ , B ) and { S r n } be a sequence of mappings defined as in Lemma 2.2. Then p = T n p = S r n ( p - r n B p ) and p = P C (p - λ n Fp) from (2.2). From z n = P C (u n - λ n Fu n ) and the fact that P C and I - λ n F are nonexpansive, it follows that
| | z n - p | | | | ( I - λ n F ) u n - ( I - λ n F ) p | | | | u n - p | | .
Also, by u n = S r n ( x n - r n B x n ) C and the β-inverse-strongly monotonicity of B, we have with r n (0, 2β),
| | u n - p | | 2 | | x n - r n B x n - ( p - r n B p ) | | 2 (1) | | x n - p | | 2 - 2 r n x n - p , B x n - B p + r n 2 | | B x n - B p | | 2 (2) | | x n - p | | 2 + r n ( r n - 2 β ) β | | B x n - B p | | 2 (3) | | x n - p | | 2 , (4) (5)
that is, ||u n - p|| ≤ ||x n - p||, and so
| | z n - p | | | | x n - p | | .
(3.1)
Similarly, we have
| | w n - p | | | | y n - p | | .
(3.2)
Now, set Ā = ( I + μ A ) . Then, from (IS) and (3.1), we obtain
| | y n - p | | ( 1 - ( 1 + μ γ ̄ ) α n ) | | z n - p | | + α n | | u | | + α n γ | | f ( x n ) - f ( p ) | | + α n | | γ f ( p ) - Ā p | | ( 1 - ( 1 + μ γ ̄ ) α n ) | | z n - p | | + α n | | u | | + α n γ k | | x n - p | | + α n | | γ f ( p ) - Ā p | | ( 1 - ( ( 1 + μ ) γ ̄ - γ k ) α n ) | | x n - p | | + α n ( | | γ f ( p ) - Ā p | | + | | u | | ) = ( 1 - ( ( 1 + μ ) γ ̄ - γ k ) α n ) | | x n - p | | + ( ( 1 + μ ) γ ̄ - γ k ) α n | | γ f ( p ) - Ā p | | + | | u | | ( 1 + μ ) γ ̄ - γ k .
(3.3)
From (3.2) and (3.3), it follows that
| | x n + 1 - p | | ( 1 - β n ) | | y n - p | | + β n | | w n - p | | (1) ( 1 - β n ) | | y n - p | | + β n | | y n - p | | (2) = | | y n - p | | (3) max | | x n - p | | , | | γ f ( p ) - Ā p | | + | | u | | ( 1 + μ ) γ ̄ - γ k . (4) (5)
(3.4)
By induction, it follows from (3.4) that
| | x n - p | | max | | x 1 - p | | , | | γ f ( p ) - Ā p | | + | | u | | ( 1 + μ ) γ ̄ - γ k , n 1 .
Therefore, {x n } is bounded. Hence {u n }, {y n }, {z n }, {w n }, {f(x n )}, {Fu n }, {Fy n }, and { Ā T n z n } are bounded. Moreover, since ||T n z n - p|| ≤ ||x n - p|| and ||T n w n - p|| ≤ ||y n - p||, {T n z n } and {T n w n } are also bounded, and since α n → 0 in the condition (C1), we have
| | y n - T n z n | | = α n | | ( u + γ f ( x n ) ) - Ā T n z n | | 0 ( a s n ) .
(3.5)
Step 2: We show that limn→∞||xn+1- x n || = 0. Indeed, since I - λ n F and P C are nonexpansive, we have
| | z n - z n - 1 | | = | | P C ( u n - λ n F u n ) - P C ( u n - 1 - λ n - 1 F u n - 1 ) | | (1) | | ( I - λ n F ) u n - ( I - λ n F ) u n - 1 + ( λ n - λ n - 1 ) F u n - 1 | | (2) | | u n - u n - 1 | | + | λ n - λ n - 1 | | | F u n - 1 | | . (3) (4)
(3.6)
Similarly, we get
| | w n - w n - 1 | | | | y n - y n - 1 | | + | λ n - λ n - 1 | | | F y n - 1 | | .
(3.7)
On the other hand, from u n - 1 = S r n - 1 ( x n - 1 - r n B x n - 1 ) and u n = S r n ( x n - r n B x n ) , it follows that
Θ ( u n - 1 , y ) + B x n - 1 , y - u n - 1 (1)  + φ ( y ) - φ ( u n - 1 ) + 1 r n - 1 y - u n - 1 , u n - 1 - x n - 1 0 , y C , (2)  (3) 
(3.8)
and
Θ ( u n , y ) + B x n , y - u n (1) + φ ( y ) - φ ( u n ) + 1 r n y - u n , u n - x n 0 , y C . (2) (3)
(3.9)
Substituting y = u n into (3.8) and y = un - 1into (3.9), we obtain
Θ ( u n - 1 , u n ) + B x n - 1 , u n - u n - 1 + φ ( u n ) - φ ( u n - 1 ) + 1 r n - 1 u n - u n - 1 , u n - 1 - x n - 1 0
and
Θ ( u n , u n - 1 ) + B x n , u n - 1 - u n + φ ( u n - 1 ) - φ ( u n ) + 1 r n u n - 1 - u n , u n - x n 0 .
From (A2), we have
u n - u n - 1 , B x n - 1 - B x n + u n - 1 - x n - 1 r n - 1 - u n - x n r n 0 ,
and then
u n - u n - 1 , r n - 1 ( B x n - 1 - B x n ) + u n - 1 - x n - 1 - r n - 1 r n ( u n - x n ) 0 .
Hence, it follows that
u n u n 1 , ( I r n l B ) x n ( I r n 1 B ) x n 1 + u n 1 u n + u n x n r n 1 r n ( u n x n ) 0.
(3.10)
Without loss of generality, let us assume that there exists a real number c such that r n > c > 0 for all n ≥ 1. Then, by (3.10) and the fact that (I - rn-1B) is nonexpansive, we have
| | u n - u n - 1 | | 2 u n - u n - 1 , ( I - r n - 1 B ) x n - ( I - r n - 1 B ) x n - 1 + ( 1 - r n - 1 r n ) ( u n - x n ) | | u n - u n - 1 | | | | ( I - r n - 1 B ) x n - ( I - r n - 1 B ) x n - 1 | | + 1 - r n - 1 r n | | u n - x n | | | | u n - u n - 1 | | | | x n - x n - 1 | | + 1 - r n - 1 r n | | u n - x n | | ,
which implies that
| | u n - u n - 1 | | | | x n - x n - 1 | | + 1 r n | r n - r n - 1 | | | u n - x n | | (1) | | x n - x n - 1 | | + M 1 c | r n - r n - 1 | , (2) (3)
(3.11)
where M1 = sup {||u n - x n || : n ≥ 1}. Substituting (3.11) into (3.6), we have
| | z n - z n - 1 | | | | x n - x n - 1 | | + M 1 c | r n - r n - 1 | + | λ n - λ n - 1 | | | F u n - 1 | | .
(3.12)
Simple calculations show that
y n - y n - 1 = ( α n - α n - 1 ) ( u + γ f ( x n - 1 ) - Ā T n - 1 z n - 1 ) + α n γ ( f ( x n ) - f ( x n - 1 ) ) (1) + ( I - α n Ā ) ( T n z n - T n z n - 1 ) + ( I - α n Ā ) ( T n z n - 1 - T n - 1 z n - 1 ) . (2) (3) 
Hence, by (3.11) and (3.12), we obtain
| | y n - y n - 1 | | | α n - α n - 1 | ( | | u | | + γ | | f ( x n - 1 ) | | + | | Ā | | | | T n - 1 z n - 1 | | ) (1) + α n γ k | | x n - x n - 1 | | + ( 1 - ( 1 + μ ) γ ̄ α n ) | | z n - z n - 1 | | (2) + ( 1 - ( 1 + μ ) γ ̄ α n ) | | T n z n - 1 - T n - 1 z n - 1 | | (3) | α n - α n - 1 | ( | | u | | + γ | | f ( x n - 1 ) | | + | | Ā | | | | T n - 1 z n - 1 | | ) (4) + α n γ k | | x n - x n - 1 | | + ( 1 - ( 1 + μ ) γ ̄ α n ) | | x n - x n - 1 | | (5) + M 1 c | r n - r n - 1 | + | λ n - λ n - 1 | | | F u n - 1 | | + sup z D 1 | | T n z - T n - 1 z | | , (6) (7) 
(3.13)
where D1 is a bounded subset of C containing {z n }. Also observe that
x n + 1 - x n = ( 1 - β n ) ( y n - y n - 1 ) + ( β n - β n - 1 ) ( T n - 1 w n - 1 - y n - 1 ) (1) + β n ( T n w n - T n w n - 1 ) + β n ( T n w n - 1 - T n - 1 w n - 1 ) . (2) (3)
(3.14)
By (3.7), (3.13) and (3.14), we have
| | x n + 1 x n | | ( 1 β n ) | | y n y n 1 | | + | β n β n 1 | ( | | T n 1 x n 1 | | + | | y n 1 | | ) + β n | | w n w n 1 | | + β n | | T n w n 1 T n 1 w n 1 | | ( 1 β n ) | | y n y n 1 | | + β n | | y n y n 1 | | + β n | λ n λ n 1 | | | F y n 1 | | + | β n β n 1 | ( | | T n 1 w n 1 | | + | | y n 1 | | ) + β n | | T n w n 1 T n 1 w n 1 | | | | y n y n 1 | | + | λ n λ n 1 | | | F y n 1 | | + | β n β n 1 | ( | | T n 1 w n 1 | | + | | y n 1 | | ) + | | T n w n 1 T n 1 w n 1 | | ( 1 ( ( 1 + μ ) γ ¯ γ k ) α n ) | | x n x n 1 | | + | ( α n α n 1 | ( | | u | | + γ | | f ( x n 1 ) | | + | | A ¯ | | | | T n 1 z n 1 | | ) + | λ n λ n 1 | ( | | F y n 1 | | + | | F u n 1 | | ) + M 1 c | r n r n 1 | + | β n β n 1 | ( | | T n 1 w n 1 | | + | | y n 1 | | ) + 2 sup z D 2 { | | T n z T n 1 z | | ( 1 ( ( 1 + μ ) γ ¯ γ k ) α n ) | | x n x n 1 | | + M 1 c | r n r n 1 | + M 2 | α n α n 1 | + M 3 | λ n λ n 1 | + M 4 | β n β n 1 | + 2 sup z D 2 | | T n z T n 1 z | | ,
(3.15)
where D2 is a bounded subset of C containing {z n } and {w n }, M 2 = sup { | | u | | + γ | | f ( x n ) | | + | | Ā | | | | T n z n | | : n 1 } , M3 = sup{||Fy n || + ||Fu n || : n ≥ 1}, and M4 = sup{||T n w n || + ||y n || : n ≥ 1}. From the conditions (C1) and (C3) and the condition n = 1 sup { | | T n + 1 z - T z | | : z D } < for any bounded subset D of C, it is easy to see that
lim n ( ( 1 + μ ) γ ̄ - γ k ) α n = 0 , n = 1 ( ( 1 + μ ) γ ̄ - γ k ) α n = ,
and
n = 2 M 1 c | r n - r n - 1 | + M 2 | α n - α n - 1 | + M 3 | λ n - λ n - 1 | + M 4 | β n - β n - 1 | + 2 sup z D 2 | | T n z - T n - 1 z | | < .
Applying Lemma 2.3 to (3.15), we obtain
lim n | | x n + 1 - x n | | = 0 .
Moreover, from (3.11), it follows that
lim n | | u n + 1 - u n | | = 0 .
From (3.12) and (3.13), we also have
lim n | | z n + 1 - z n | | = 0 a n d lim n | | y n + 1 - y n | | = 0 .
Step 3: We show that limn-∞||x n - u n || = 0. To this end, let p Ω1. Since S r n is firmly nonexpansive and u n = S r n ( x n - r n B x n ) , we have
| | u n p | | 2 S r n ( x n r n B x n ) S r n ( p r n B p ) , x n r n B x n ( p r n B p ) = 1 2 { | | u n p | | 2 + | | x n r n B x n ( p r n B p ) | | 2 } 1 2 { | | x n r n B x n ( p r n B p ) ( u n p ) | | 2 } 1 2 { | | u n p | | 2 + | | x n p | | 2 | | x n u n r n ( B x n B p ) | | 2 } 1 2 { | | u n p | | 2 + | | x n p | | 2 | | x n u n | | 2 + 2 r n B x n B p , x n u n r n 2 | | B x n B p | | 2 } .
Hence,
| | u n - p | | 2 | | x n - p | | 2 - | | x n - u n | | 2 + 2 r n B x n - B p , x n - u n - r n 2 | | B x n - B p | | 2 | | x n - p | | 2 - | | x n - u n | | 2 + 2 r n B x n - B p , x n - u n | | x n - p | | 2 - | | x n - u n | | 2 + 2 r n | | B x n - B p | | | | x n - u n | | .
(3.16)
On the other hand, since z n = P C (u n - λ n Fu n ), we get
| | z n - p | | 2 | | ( I - λ n F ) u n - ( I - λ n F ) p | | 2 (1) | | u n - p | | 2 . (2) (3)
(3.17)
From (3.16), (3.17), and the convexity of || ||2, we obtain
| | y n - p | | 2 α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | T n z n - p | | 2 (1)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | z n - p | | 2 (2)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 (3)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (4)  + ( 1 - α n ) { | | x n - p | | 2 - | | x n - u n | | 2 + 2 r n | | x n - u n | | | | B x n - B p | | } (5)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - p | | 2 (6)  - ( 1 - α n ) | | x n - u n | | 2 + 2 ( 1 - α n ) r n | | x n - u n | | | | B x n - B p | | . (7)  (8) 
(3.18)
On the another hand, we note that
| | u n - p | | 2 | | ( x n - r n B x n ) - ( p - r n B p ) | | 2 (1) | | x n - p | | 2 - 2 r n x n - p , B x n - B p + r n 2 | | B x n - B p | | 2 (2) | | x n - p | | 2 - 2 r n β | | B x n - B p | | 2 + r n 2 | | B x n - B p | | 2 . (3) (4)
(3.19)
Using the convexity of || ||2, (3.2), (3.18), and (3.19), we have
| | x n + 1 - p | | 2 | | y n - p | | 2 (1)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 (2)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (3)  + ( 1 - α n ) { | | x n - p | | 2 - 2 r n β | | B x n - B p | | 2 + r n 2 | | B x n - B p | | 2 } (4)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - p | | 2 (5)  + ( 1 - α n ) r n ( r n - 2 β ) | | B x n - B p | | 2 . (6)  (7) 
(3.20)
Hence, we have
( 1 - α n ) c ( 2 β - d ) | | B x n - B p | | 2 ( 1 - α n ) r n ( 2 β - r n ) | | B x n - B p | | 2 α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( | | x n - p | | 2 - | | x n + 1 - p | | 2 ) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - x n + 1 | | ( | | x n - p | | + | | x n + 1 - p | | ) .
From the condition (C1), {r n } [c, d] (0, 2β) and Step 2, it follows that
lim n | | B x n - B p | | 2 = 0 .
(3.21)
Also, by (3.16) and (3.20), we have
| | x n + 1 - p | | 2 | | y n - p | | 2 (1)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | u n - p | | 2 (2)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (3)  + ( 1 - α n ) { | | x n - p | | 2 - | | x n - u n | | 2 + 2 r n | | x n - u n | | | | B x n - B p | | } (4)  α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - p | | 2 (5)  - ( 1 - α n ) | | x n - u n | | 2 + 2 r n ( 1 - α n ) | | x n - u n | | | | B x n - B p | | , (6)  (7) 
and so
( 1 - α n ) | | x n - u n | | 2 α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - p | | 2 (1)  - | | x n + 1 - p | | 2 + 2 r n ( 1 - α n ) | | x n - u n | | | | B x n - B p | | (2)  α n M 5 + | | x n - x n + 1 | | ( | | x n - p | | + | | x n + 1 - p | | ) (3)  + 2 r n ( 1 - α n ) | | x n - u n | | | | B x n - B p | | , (4)  (5) 
where M 5 = sup { | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 : n 1 } . Since ||xn+1- x n || → 0, α n → 0, and ||Bx n - Bp|| → 0 as n → ∞, we obtain
lim n | | x n - u n | | = 0 .
(3.22)
Moreover, since lim infn→∞r n > 0, we also have
lim n | | x n - u n | | r n = lim n 1 r n | | x n - u n | | = 0 .
(3.23)
Step 4: We show that limn→∞||x n - T n z n || = 0 and limn-∞||x n - y n || = 0. Indeed, since z n = P C (u n - λ n Fu n ) and w n = P C (y n - λ n Fy n ), we obtain with the condition (C2)
| | x n + 1 - y n | | β n ( | | T n z n - T n w n | | + | | y n - T n z n | | ) (1)  a ( | | z n - w n | | + | | y n - T n z n | | ) (2)  a ( | | u n - y n | | + | | y n - T n z n | | ) (3)  a ( | | u n - x n | | + | | x n - x n + 1 | | + | | x n + 1 - y n | | + | | y n - T n z n | | ) , (4)  (5) 
which implies that
| | x n + 1 - y n | | a 1 - a ( | | u n - x n | | + | | x n - x n + 1 | | + | | y n - T n z n | | ) .
Thus, from (3.5), Step 2, and Step 3, we have
| | x n + 1 - y n | | 0 a s n .
Also we have
| | x n - y n | | | | x n - x n + 1 | | + | | x n + 1 - y n | | 0 a s n .
Since limn→∞||y n - T n z n || = 0 by (3.5) in Step 1, we obtain
| | x n - T n z n | | | | x n - y n | | + | | y n - T n z n | | 0 a s n .
Step 5: We show that limn→∞||T n z n - z n || = 0. Let p Ω1. Using the convexity of || ||2, we compute
| | y n - p | | 2 α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | T n z n - p | | 2 (1) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( 1 - α n ) | | z n - p | | 2 (2) = α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (3) + ( 1 - α n ) | | P C ( u n - λ n F u n ) - P C ( p - λ n F p ) | | 2 (4) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (5) + ( 1 - α n ) | | ( u n - p ) - λ n ( F u n - F p ) | | 2 (6) = α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (7) + ( 1 - α n ) { | | u n - p | | 2 - 2 λ n u n - p , F u n - F p + λ n 2 | | F u n - F p | | 2 } (8) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (9) + ( 1 - α n ) { | | u n - p | | 2 - 2 λ n α | | F u n - F p | | 2 + λ n 2 | | F u n - F p | | 2 } (10) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (11) + | | x n - p | | 2 + ( 1 - α n ) λ n ( λ n - 2 α ) | | F u n - F p | | 2 . (12) (13) 
(3.24)
Using (3.24), we obtain
| | x n + 1 - p | | 2 | | y n - p | | 2 (1) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 (2) + | | x n - p | | 2 + ( 1 - α n ) λ n ( λ n - 2 α ) | | F u n - F p | | 2 . (3) (4)
Hence, we have
( 1 - α n ) a ( 2 α - b ) | | F u n - F p | | 2 ( 1 - α n ) λ n ( 2 α - λ n ) | | F u n - F p | | 2 α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + ( | | x n - p | | - | | x n + 1 - p | | ) ( | | x n - p | | + | | x n + 1 - p | | ) α n M 5 + | | x n - x n + 1 | | ( | | x n - p | | + | | x n + 1 - p | | ) .
where M 5 = sup { | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 : n 1 } . From the condition (C1), λ n [a, b] (0, 2α), and Step 2, it follows that
lim n | | F u n - F p | | = 0 .
(3.25)
On the other hand, using z n = P C (u n - λ n Fu n ) and (2.1), we observe that
| | z n p | | 2 ( u n λ n F u n ) ( p λ n F p ) , z n p 1 2 { | | u n p | | 2 + | | z n p | | 2 | | ( u n z n ) λ n ( F u n F p ) | | 2 } 1 2 { | | x n p | | 2 + | | z n p | | 2 | | u n z n | | 2 + 2 λ n u n z n , F u n F p λ n 2 | | F u n F p | | 2 } ,
and so
| | z n - p | | 2 | | x n - p | | 2 - | | u n - z n | | 2 (1) + 2 λ n u n - z n , F u n - F p - λ n 2 | | F u n - F p | | 2 (2) (3)
(3.26)
Thus, from (3.26), we have
| | y n p | | 2 α n | | u + γ f ( x n ) + ( I A ¯ ) T n z n p | | 2 + ( 1 α n ) | | T n z n p | | 2 α n | | u + γ f ( x n ) + ( I A ¯ ) T n z n p | | 2 + ( 1 α n ) | | z n p | | 2 α n | | u + γ f ( x n ) + ( I A ¯ ) T n z n p | | 2 + ( 1 α n ) { | | x n p | | 2 | | u n z n | | 2 + 2 λ n u n z n , F u n F p λ n 2 | | F u n F p | | 2 } α n | | u + γ f ( x n ) + ( I A ¯ ) T n z n p | | 2 + | | x n p | | 2 ( 1 α n ) | | u n z n | | 2 + 2 λ n ( 1 α n ) u n z n , F u n F p λ n 2 | | F u n F p | | 2 .
Hence, we obtain
| | x n + 1 - p | | 2 | | y n - p | | 2 (1) α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - p | | 2 (2) - ( 1 - α n ) | | u n - z n | | 2 + 2 λ n ( 1 - α n ) u n - z n , F u n - F p (3) - λ n 2 | | F u n - F p | | 2 , (4) (5)
(3.27)
which implies that
( 1 - α n ) | | u n - z n | | 2 α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n - p | | 2 - | | x n + 1 - p | | 2 + 2 λ n ( 1 - α n ) | | u n - z n | | | | F u n - F p | | α n | | u + γ f ( x n ) + ( I - Ā ) T n z n - p | | 2 + | | x n + 1 - x n | | ( | | x n - p | | + | | x n + 1 - p | | ) + 2 λ n ( 1 - α n ) | | u n - z n | | | | F u n - F p | | .
From ||xn+1- x n || → 0, α n → 0 and ||Fu n - Fp|| → 0 as n → ∞, it follows that
lim n | | u n - z n | | = 0 .
(3.28)
Since ||T n z n - z n ||||T n z n - x n || + ||x n - u n || + ||u n - z n ||, from Step 3, Step 4, and (3.28), we conclude that
lim n | | T n z n - z n | | = 0 .
(3.29)
We notice that by the assumption on T, (3.29) and Lemma 2.5,
lim n | | T z n - z n | | lim n ( | | T z n - T n z n | | + | | T n z n - z n | | ) (1)  lim n sup { | | T y - T n y | | : y C } + lim n | | T n z n - z n | | = 0 . (2)  (3) 
(3.30)
Step 6: We show that
limsup n u + ( γ f - ( I + μ A ) ) q , y n - q = limsup n u + ( γ f - Ā ) q , y n - q 0 ,
where q is a solution of the optimization problem (OP1). To this end, first we prove that
limsup n u + ( γ f - Ā ) q , T z n - q 0 .
Since {z n } is bounded, we can choose a subsequence { z n i } of {z n } such that
limsup n u + ( γ f - Ā ) q , T z n - q = lim i u + ( γ f - Ā ) q , T z n i - q .
(3.31)

Without loss of generality, we may assume that { z n i } converges weakly to z C. From ||Tz n - z n || → 0 in (3.30), it follows that T z n i z .

Now, we will show that z Ω1. First, we show that z n = 1 F ( T n ) = F ( T ) . Assume that z F(T). Since z n i z and Tzz, by the Opial condition, we obtain
lim inf i | | z n i z | | < lim inf i | | z n i T z | | lim inf