Open Access

Convergence of iterative sequences for fixed points of an infinite family of nonexpansive mappings based on a hybrid steepest descent methods

  • Nawitcha Onjai-uea1, 2,
  • Chaichana Jaiboon2, 3Email author,
  • Poom Kumam1, 2 and
  • Usa Wannasingha Humphries1, 2
Journal of Inequalities and Applications20122012:101

https://doi.org/10.1186/1029-242X-2012-101

Received: 29 November 2011

Accepted: 26 April 2012

Published: 26 April 2012

Abstract

The propose of this article is to consider the strong convergence of an iterative sequences for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequalities for inverse strongly monotone mappings, and the set of solutions of system of equilibrium problems in Hilbert spaces by using a hybrid steepest descent methods. Our results improve and generalize many known corresponding results.

AMS (2000) Subject Classification: 46C05; 47H09; 47H10.

Keywords

nonexpansive mappings system equilibrium problem variational inequality problem hybrid steepest descent method

1. Introduction

Let H be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ||·||, respectively. Let C be a nonempty closed convex subset of H and let F: C × C be a bifunction, where is the set of real numbers. The equilibrium problem for F: C × C is to find x* C such that
F ( x * , y ) 0 , y C .
(1.1)

The set of solutions of (1.1) is denoted by EP(F).

Let {F i , i = 1, 2,..., N} be a finite family of bifunctions from C × C into , where is the set of real numbers. The system of equilibrium problems for {F1, F2,..., F N } is to find a common element x* C such that
F 1 ( x * , y ) 0 , y C , F 2 ( x * , y ) 0 , y C , F N ( x * , y ) 0 , y C .
(1.2)
We denote the set of solutions of (1.2) by i = 1 N S E P ( F i ) , where SEP(F i ) is the set of solutions to the equilibrium problems, that is,
F i ( x * , y ) 0 , y C .
(1.3)

If N = 1, then the problem (1.2) is reduced to the equilibrium problems.

If N = 1 and F(x*,y) = 〈Bx*, y - x*〉, then the problem (1.2) is reduced to the variational inequality problems of finding x* C such that
B x * , y - x * 0 , y C .
(1.4)

The set of solutions of (1.4) is denoted by VI(C, B).

Many problems in applied sciences, such as monotone inclusion problems, saddle point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases. Some methods have been proposed to solve VI(C, B), EP(F), and SEP(F i ); see, for example [122] and references therein. The above formulations (1.2) extends this formulism to such problems, covering in particular various forms of feasibility problems [23, 24].

Let P C be the metric projection of H onto the closed convex subset C. Let S: CC be a nonexpansive mapping, that is, ||Sx - Sy|| ≤ ||x - y|| for all x, y C. The set of fixed points of S is denoted by F(S) = {x C: Sx = x}. If C H is nonempty, bounded, closed and convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty; see, for example, [25, 26]. A mapping f: CC is a contraction on C if there exists a constant η (0,1) such that ||f(x) - f(y)|| ≤ η||x - y|| for all x, y C.

Definition 1.1. Let B: CH be nonlinear mappings. Then B is called
  1. (1)

    monotone if 〈Bx - By, x - y〉 ≥ 0, x, y C,

     
  2. (2)
    ξ-inverse-strongly monotone if there exists a constant ξ > 0 such that
    B x - B y , x - y ξ B x - B y 2 , x , y C .
     
  3. (3)
    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 called maximal monotone, if it is monotone and if for any (x, f) H × H
    x - y , f - g 0 , ( y , g ) G r a p h ( Q )
     

(the graph of mapping Q) implies that f Qx.

A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping defined on a real Hilbert space H:
min x F 1 2 A x , x - x , b ,

where F is the fixed point set of a nonexpansive mapping S defined on H and b is a given point in H.

A linear bounded operator A is strong positive if there exists a constant γ ̄ > 0 with the property
A x , x γ ̄ x 2 , x H .
Marino and Xu [27] introduced a new iterative scheme by the viscosity approximation method:
x n + 1 = ε n γ f ( x n ) + ( 1 - ε n A ) S x n .
(1.5)
They proved that the sequences {x n } generated by (1.5) converges strongly to the unique solution of the variational inequality:
γ f z - A z , x - z 0 , x F ( S ) ,
which is the optimality condition for the minimization problem:
min x F ( S ) 1 2 A x , x - h ( x ) ,
(1.6)

where h is a potential function for γf.

In order to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequalities for a ξ-inverse-strongly monotone mapping, Takahashi and Toyoda [28] introduced the following iterative scheme:
x 0 C chosen arbitrary , x n + 1 = γ n x n + ( 1 - γ n ) S P C ( x n - α n B x n ) , n 0 ,
(1.7)

where B is a ξ-inverse-strongly monotone mapping, {γ n } is a sequence in (0, 1), and {α n } is a sequence in (0,2ξ). They showed that if F(S) VI(C,B) is nonempty, then the sequence {x n } generated by (1.7) converges weakly to some z F(S) VI(C, B).

In order to find a common element of F(S) VI(C, B), let S: HH be a nonexpansive mapping, Yamada [29] introduced the following iterative scheme called the hybrid steepest descent method:
x n + 1 = S x n - α n μ B S x n , n 1 ,
(1.8)

where x1 = x H, {α n } (0,1), let B: HH be a strongly monotone and Lipschitz continuous mapping and μ is a positive real number. He proved that the sequence {x n } generated by (1.8) converges strongly to the unique solution of the F(S) VI(C, B).

Let C be a nonempty closed convex subset of H. Given r > 0 the operators J r F : H C defined by
J r F ( x ) = z C : F ( z , y ) + 1 r y - z , z - x 0 , y C ,
is called the resolvent of F (see [3]). It is shown in [3] that under suitable hypotheses on F (to be stated precisely in Section 2), J r F : H C is single-valued and firmly nonexpansive and satisfied
F ( J r F ) = E P ( F ) , r > 0 .
Using the result, in 2009, Colao et al. [10] introduced and considered an implicit iterative scheme for finding a common element of the set of solutions of the system equilibrium problems and the set of common fixed points of an infinite family of nonexpansive mappings on C. Starting with an arbitrary initial x0 C and defining a sequence {z n } recursively by
x n = ε n γ f ( x n ) + ( 1 - ε n A ) W n J r M , n F M J r M - 1 , n F M - 1 J r M - 2 , n F M - 2 J r 2 , n F 2 J r 1 , n F 1 x n ,
(1.9)

where {ϵ n } be a sequences in (0,1). It is proved [10] that under certain appropriate conditions imposed on {ϵ n } and {r n }, the sequence {x n } generated by (1.9) converges strongly to z n = 1 F ( T n ) ( k = 1 M S E P ( F k ) ) , where z is the unique solution of the variational inequality and which is the optimality condition for the minimization problem.

In 2010, Colao and Marino [30] introduced the following explicit viscosity scheme with respect to W-mappings for an infinite family of nonexpansive mappings
x n + 1 = ε n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - ε n A ) W n J r n F x n .
(1.10)
They prove that sequence {x n } and { J r n F } converge strongly to z n = 1 F ( T n ) E P ( F ) , where z is an equilibrium point for F and is the unique solution of the variational inequality:
γ f z - A z , x - z 0 , x n = 1 F ( T n ) E P ( F )
or, equivalently, the unique solution of the minimization problem
min x n = 1 F ( T n ) E P ( F ) 1 2 A x , x - h ( x ) ,

where h is a potential function for γf. Recently, Chantarangsi et al. [11] introduced some iterative processes based on the viscosity hybrid steepest descent method for finding a common solutions of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem in a real Hilbert space.

In this article, motivated by above results, we introduce an iterative scheme for finding a common element of the set of solutions of system of equilibrium problems, the set of fixed points of an infinite family of nonexpansive mapping, and the set of solutions of variational inequality problems for inverse strongly monotone mapping in a real Hilbert space by using a new hybrid steepest descent methods. The results shown in this article improve and extend the recent ones announced by many others.

2. Preliminaries

Let H be a real Hilbert space, when {x n } is a sequence in H, we denote strong convergence of {x n } to x H by x n x and weak convergence by x n x. Let C be nonempty closed convex subset of H. The nearest point projection P C : HC defined from H onto C is the function which assigns to each x H its nearest point denoted by P C x in C. Thus, P C x is the unique point in C such that ||x - P C x|| ≤ ||x - y||, y C. It easy to see that P C is nonexpansive and
x * V I ( C , B ) x * = P C ( x * - λ B x * ) , λ > 0 .
(2.1)

Lemma 2.1. [26] Let H be a Hilbert space, let C be a nonempty closed convex subset of H. Let ξ > 0 and let A: CH be ξ-inverse strongly monotone. If 0 < ϱ ≤ 2ξ, then I - ϱB is a nonexpansive mapping of C into H.

Lemma 2.2. [26] Let H be a real Hilbert spaces, there hold the following identities:
  1. (i)

    for each x H and x* C, x* = P C x x - x*, y - x*〉 ≤ 0 for all y C;

     
  2. (ii)

    P C : HC is nonexpansive, that is, ||P C x - P C y|| ≤ ||x - y|| for all x, y H;

     
  3. (iii)

    P C is firmly nonexpansive, that is, ||P C x - P C y||2 ≤ 〈P C x - P C y, x - yfor all x,y H;

     
  4. (iv)

    ||tx + (1 - t)y||2 = t||x||2 + (1 - t)||y||2 - t(1 - t)||x - y||2, t [0,1], for all x,y H;

     
  5. (v)

    ||x + y||2 ≤ ||x||2 + 2〈y,x + y〉.

     
Lemma 2.3. [31] Each Hilbert space H satisfies Opial's condition, that is, for any sequence {x n } H with x n x, the inequality
lim inf n x n - x < lim inf n x n - y ,

hold for each y H with yx.

Lemma 2.4. [27] Let C be a nonempty closed convex subset of H and let f be a contraction of H into itself with η (0,1), and A be a strongly positive linear bounded operator on H with coefficient γ ̄ > 0 . Then, for 0 < γ < γ ̄ η ,
x - y , ( A - γ f ) x - ( A - γ f ) y ( γ ̄ - η γ ) x - y 2 , x , y H .

That is, A - γf is a strongly monotone with coefficient γ ̄ - γ η .

Lemma 2.5. [27] Assume A be a strongly positive linear bounded operator on H with coefficient γ ̄ > 0 and 0 < ρ ≤ ||A||-1. Then I - ρ A 1 - ρ γ ̄ .

Throughout this article, we assume that a bifunction F : C × C satisfies the following conditions:

(A1) F(x, x) = 0 for all x C;

(A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ 0 for all x, y C;

(A3) for each x, y, z C, limt↓0F(tz + (1 - t)x, y) ≤ F(x, y);

(A4) for each x C, y F(x, y) is convex and lower semicontinuous.

Then, we have the following lemmas.

Lemma 2.6. [1] Let C be a nonempty closed convex subset of H and let F be a bifunction of C × C into satisfying (A1)-(A4). Let r > 0 and x H. Then, there exists z C such that
F ( z , y ) + 1 r y - z , z - x 0 , y C .
Lemma 2.7. [3] Assume that F : C × C satisfies (A1)-(A4). For r > 0 and x H, define a mapping J r F : H C as follows:
J r F ( x ) = z C : F ( z , y ) + 1 r y - z , z - x 0 , y C
for all z H. Then, the following hold:
  1. (1)

    J r F is single-valued;

     
  2. (2)
    J r F is firmly nonexpansive, that is, for any x,y H,
    J r F x - J r F y 2 J r F x - J r F y , x - y ;
     
  3. (3)

    F ( J r F ) = E P ( F ) ;

     
  4. (4)

    EP(F) is closed and convex.

     

Lemma 2.8. [32] Let {x n } and {l 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 )l n + β n x n for all integers n ≥ 0 and lim supn→∞(||ln+1- l n || - ||xn+1- x n ||) ≤ 0. Then, limn→∞||l n - x n || = 0.

Lemma 2.9. [33] Assume {a n } is a sequence of nonnegative real numbers such that
a n + 1 ( 1 - b n ) a n + c n , n 0 ,
where {b n } is a sequence in (0,1) and {c n } is a sequence in such that
  1. (1)

    n = 1 b n = ,

     
  2. (2)

    lim sup n c n b n 0 or n = 1 c n < ,

     

Then, limn→∞a n = 0.

3. Main results

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T n n = 1 be a family of infinitely of nonexpansive mappings of C into itself and let μ n n = 1 be a sequence of nonnegative numbers in [0,1]. For any n ≥ 1, define a mapping W n : CC as follows:
U n , n + 1 = I , U n , n = μ n T n U n , n + 1 + ( 1 - μ n ) I , U n , n - 1 = μ n - 1 T n - 1 U n , n + ( 1 - μ n - 1 ) I , U n , k = μ k T k U n , k + 1 + ( 1 - μ k ) I , U n , k - 1 = μ k - 1 T k - 1 U n , k + ( 1 - μ k - 1 ) I , U n , 2 = μ 2 T 2 U n , 3 + ( 1 - μ 2 ) I , W n = U n , 1 = μ 1 T 1 U n , 2 + ( 1 - μ 1 ) I ,
(3.1)

such a mappings W n is nonexpansive from C to C and it is called the W-mapping generated by T1,T2,...,T n and μ1, μ2, ..., μ n (see [34]).

Lemma 3.1. [34, 35] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1,T2,..., be an infinite family of nonexpansive mappings of C into itself such that n = 1 F ( T n ) , let μ1, μ2, ... be real numbers such that 0 ≤ μ n b < 1 for every n ≥ 1. Then,
  1. (1)

    for every x C and k , the limit limn→∞U n,k x exists;

     
  2. (2)
    the mapping W of C into itself as follows:
    W x = lim n W n x = lim n U n , 1 x , x C
    (3.2)
     
is a nonexpansive mapping satisfying F ( W ) = n = 1 F ( T n ) , which it is called the W-mapping generated by T1, T2, ... and μ1, μ2, ...;
  1. (3)

    F ( W n ) = n = 1 F ( T n ) , for each n ≥ 1;

     
  2. (4)

    If E is any bounded subset of C, then lim n sup x E W x - W n x = 0 . .

     
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H, let F k , k {1, 2, 3,..., M} be a bifunction from C × C to satisfying (A1)-(A4), let {T n } be an infinite family of nonexpansive mappings of C into itself and let B be ξ-inverse strongly monotone such that
Θ : = n = 1 F ( T n ) ( k = 1 M S E P ( F k ) ) V I ( C , B ) .
Let f be a contraction of H into itself with η (0,1) and let A be a strongly positive linear bounded operator on H with coefficient γ ̄ > 0 and 0 < γ < γ ̄ η . Let {x n }, {y n } and {u n } be sequences generated by
x 1 = x C c h o s e n a r b i t r a r y , y n = ( 1 - δ n ) x n + δ n P C ( x n - α n B x n ) , u n = J r M , n F M J r M - 1 , n F M - 1 J r M - 2 , n F M - 2 J r 2 , n F 2 J r 1 , n F 1 y n , x n + 1 = ε n γ f ( u n ) + β n x n + ( ( 1 - β n ) I - ε n A ) P C ( W n u n - λ n B W n u n ) , n 1 ,
(3.3)

where {W n } is the sequence generated by (3.1) and {ϵ n }, {β n } are two sequences in (0,1) and {r k,n }, k {1,2,3,..., M} are a real sequence in (0, ∞) satisfy the following conditions:

(C1) limn→∞ϵ n = 0 and n = 1 ε n = ,

(C2) 0 < lim infn→∞β n ≤ lim supn→∞β n < 1,

(C3) {α n }, {λ n } [e, g] (0, 2ξ), limn→∞α n = 0 and limn→∞λ n = 0,

(C4) {δ n } [0, b], for some b (0,1) and limn→∞|δn+1- δ n | = 0,

(C5) lim infn→∞r k,n > 0 and limn→∞|rk,n+1- r k,n | = 0 for each k {1, 2, 3,..., M}.

Then, {x n } and {u n } converge strongly to a point z Θ, which is the unique solution of the variational inequality
( A - γ f ) z , x - z 0 , x Θ .
(3.4)

Equivalently, we have z = P Θ (I - A + γf)(z).

Proof. From the restrictions on control sequence, without loss of generality, that ϵ n ≤ (1 - β n )||A||-1 for all n ≥ 1. From Lemma 2.5, we know that if 0 ≤ ρ ≤ ||A||-1, then I - ρ A 1 - ρ γ ̄ . We will assume that I - A 1 - γ ̄ . Since A is a strongly positive bounded linear operator on H, we have
A = sup A x , x : x H , x = 1 .
Observe that
( ( 1 - β n ) I - ε n A ) x , x = 1 - β n - ε n A x , x 1 - β n - ε n A 0 ,
this show that (1 - β n )I - ϵ n A is positive. It follows that
( 1 - β n ) I - ε n A = sup ( ( 1 - β n ) I - ε n A ) x , x : x H , x = 1 = sup 1 - β n - ε n A x , x : x H , x = 1 1 - β n - ε n γ ̄ .

We divide the proof of Theorem 3.2 into seven steps.

Step 1. We show that the mapping PΘ(γf + (I - A)) has a unique fixed point.

Since f be a contraction of C into itself with coefficient η (0,1). Then, we have
P Θ ( γ f + ( I - A ) ) ( x ) - P Θ ( γ f + ( I - A ) ) ( y ) ( γ f + ( I - A ) ) ( x ) - ( γ f + ( I - A ) ) ( y ) γ f ( x ) - f ( y ) + I - A x - y γ η x - y + ( 1 - γ ̄ ) x - y = ( 1 - ( γ ̄ - η γ ) ) x - y , x , y C .
Since 0 < 1 - ( γ ̄ - η γ ) < 1 , it follows that PΘ (γf + (I - A)) is a contraction of C into itself. Therefore, by the Banach Contraction Mapping Principle, has a unique fixed point, say z C, that is,
z = P Θ ( γ f + ( I - A ) ) ( z ) .

Step 2. We show that W n - λ n BW n is nonexpansive.

For all x, y C, let W n is the sequence defined by (3.1) and λ n (0, 2ξ), we obtain W n - λ n BW n is a nonexpansive. Indeed,
( W n - λ n B W n ) x - ( W n - λ n B W n ) y 2 = ( W n x - W n y ) - λ n ( B W n x - B W n y ) 2 = W n x - W n y 2 - 2 λ n W n x - W n y , B W n x - B W n y + λ n 2 B W n x - B W n y 2 x - y 2 - 2 λ n ξ B W n x - B W n y + λ n 2 B W n x - B W n y 2 = x - y 2 - λ n ( λ n - 2 ξ ) B W n x - B W n y 2 x - y 2 ,
(3.5)

which implies that W n - λ n BW n is a nonexpansive.

Step 3. We show that the sequence {x n } is bounded.

In fact, let x ̃ Θ , then
x ̃ = P C ( x ̃ - α n B x ̃ ) .
Setting v n = P C (x n - α n Bx n ) and I - α n B is a nonexpansive mapping (Lemma 2.1), we obtain
v n - x ̃ = P C ( x n - α n B x n ) - P C ( x ̃ - α n B x ̃ ) ( x n - α n B x n ) - ( x ̃ - α n B x ̃ ) = ( I - α n B ) x n - ( I - α n B ) x ̃ x n - x ̃
(3.6)
and
y n - x ̃ ( 1 - δ n ) x n - x ̃ + δ n v n - x ̃ ( 1 - δ n ) x n - x ̃ + δ n x n - x ̃ = x n - x ̃ .
(3.7)
Let n k = J r k , n F k J r k - 1 , n F k - 1 J r k - 2 , n F k - 2 J r 2 , n F 2 J r 1 , n F 1 for k {1, 2, 3,..., M} and n 0 = I for all n. Because J r k , n F k is nonexpansive for each k = 1, 2, 3,..., M, x ̃ = n k x ̃ and (3.7), we note that u n = n M y n . It follows that
u n - x ̃ = n M y n - n M x ̃ y n - x ̃ x n - x ̃ .
(3.8)
Let e n = P C (W n u n - λ n BW n u n ), we can prove that
e n - x ̃ = P C ( W n u n - λ n B W n u n ) - P C ( W n x ̃ - λ n B W n x ̃ ) ( W n u n - λ n B W n u n ) - ( W n x ̃ - λ n B W n x ̃ ) = ( W n - λ n B W n ) u n - ( W n - λ n B W n ) x ̃ u n - x ̃ x n - x ̃ ,
(3.9)
which yields that
x n + 1 - x ̃ = ε n ( γ f ( u n ) - A x ̃ ) + β n ( x n - x ̃ ) + ( ( 1 - β n ) I - ε n A ) ( e n - x ̃ ) ε n γ f ( u n ) - A x ̃ + β n x n - x ̃ + ( 1 - β n ) I - ε n A e n - x ̃ ε n γ f ( u n ) - f ( x ̃ ) + ε n γ f ( x ̃ ) - A x ̃ + β n x n - x ̃ + ( 1 - β n - ε n γ ̄ ) e n - x ̃ ε n γ η u n - x ̃ + ε n γ f ( x ̃ ) - A x ̃ + β n x n - x ̃ + ( 1 - β n - ε n γ ̄ ) x n - x ̃ ε n γ η x n - x ̃ + ε n γ f ( x ̃ ) - A x ̃ + β n x n - x ̃ + ( 1 - β n - ε n γ ̄ ) x n - x ̃ = ( 1 - ( γ ̄ - γ η ) ε n ) x n - x ̃ + ( γ ̄ - γ η ) ε n ( γ ̄ - γ η ) γ f ( x ̃ ) - A x ̃ .
By induction, we have
x n - x ̃ max x 1 - x ̃ , γ f ( x ̃ ) - A x ̃ γ ̄ - γ η , n .
(3.10)

This implies that {x n } is bounded, and hence so are {u n }, {e n }, {y n }, {BW n u n }, {Bx n }, {Ae n }, {v n - x n }, and {f(u n )}.

Step 4. We show that lim n x n + 1 - x n = 0 .

We claim that if ω n be a bounded sequence in C, then
lim n n k ω n - n + 1 k ω n = 0 ,
(3.11)
for every k {1, 2, 3,..., M}. From Step 2 of the proof of Theorem 3.1 in [10], we have that for k {1,2,3,...,M},
lim n J r k , n + 1 F k ω n - J r k , n F k ω n = 0 .
(3.12)
Note that for every k {1,2,3,...,M}, we obtain
n k = J r k , n F k J r k - 1 , n F k - 1 J r k - 2 , n F k - 2 J r 2 , n F 2 J r 1 , n F 1 = J r k , n F k n k - 1 .
Thus,
n k ω n - n + 1 k ω n = J r k , n F k n k - 1 ω n - J r k , n + 1 F k n + 1 k - 1 ω n J r k , n F k n k - 1 ω n - J r k , n + 1 F k n k - 1 ω n + J r k , n + 1 F k n k - 1 ω n - J r k , n + 1 F k n + 1 k - 1 ω n J r k , n F k n k - 1 ω n - J r k , n + 1 F k n k - 1 ω n + n k - 1 ω n - n + 1 k - 1 ω n J r k , n F k n k - 1 ω n - J r k , n + 1 F k n k - 1 ω n + J r k - 1 , n F k - 1 n k - 2 ω n - J r k - 1 , n + 1 F k - 1 n k - 2 ω n + n k - 2 ω n - n + 1 k - 2 ω n J r k , n F k n k - 1 ω n - J r k , n + 1 F k n k - 1 ω n + J r k - 1 , n F k - 1 n k - 2 ω n - J r k - 1 , n + 1 F k - 1 n k - 2 ω n + + J r 2 , n F 2 n 1 ω n - J r 2 , n + 1 F 2 n 1 ω n + J r 1 , n F 1 ω n - J r 1 , n + 1 F 1 ω n .
(3.13)

Now, apply (3.12) to conclude (3.11).

Since T n and U n,n are nonexpansive, we have
W n + 1 x n - W n x n = μ 1 T 1 U n + 1 , 2 x n - μ 1 T 1 U n , 2 x n μ 1 U n + 1 , 2 x n - U n , 2 x n = μ 1 μ 2 T 2 U n + 1 , 3 x n - μ 2 T 2 U n , 3 x n μ 1 μ 2 U n + 1 , 3 x n - U n , 3 x n μ 1 μ 2 μ n U n + 1 , n + 1 x n - U n , n + 1 x n M 1 i = 1 n μ i ,
(3.14)
where M1 ≥ 0 is an appropriate constant such that ||Un+1,n+1x n - Un,n+1x n || ≤ M1 for all n ≥ 0. From I - α n B is nonexpansive, we have
v n + 1 - v n = P C ( x n + 1 - α n + 1 B x n + 1 ) - P C ( x n - α n B x n ) ( x n + 1 - α n + 1 B x n + 1 ) - ( x n - α n B x n ) ( x n + 1 - α n + 1 B x n + 1 ) - ( x n - α n + 1 B x n ) + α n + 1 - α n B x n x n + 1 - x n + α n + 1 - α n B x n .
(3.15)
From (3.3) and (3.15), we have
y n + 1 - y n = ( 1 - δ n + 1 ) ( x n + 1 - x n ) + δ n + 1 ( v n + 1 - v n ) + ( δ n + 1 - δ n ) ( v n - x n ) ( 1 - δ n + 1 ) x n + 1 - x n + δ n + 1 v n + 1 - v n + δ n + 1 - δ n v n - x n ( 1 - δ n + 1 ) x n + 1 - x n + δ n + 1 x n + 1 - x n + α n + 1 - α n B x n + δ n - δ n + 1 x n - v n = x n + 1 - x n + δ n + 1 α n + 1 - α n B x n + δ n - δ n + 1 x n - v n .
(3.16)
Now, we compute ||un+1- u n || and ||en+1- e n ||. Consider the following computation:
u n + 1 - u n = n + 1 M y n + 1 - n M y n n + 1 M y n + 1 - n + 1 M y n + n + 1 M y n - n M y n y n + 1 - y n + n + 1 M y n - n M y n x n + 1 - x n + δ n + 1 α n + 1 - α n B x n + δ n - δ n + 1 x n - v n + n + 1 M y n - n M y n
(3.17)
and
e n + 1 - e n = p C ( W n + 1 u n + 1 - λ n + 1 B W n + 1 u n + 1 ) - P C ( W n u n - λ n B W n u n ) ( W n + 1 u n + 1 - λ n + 1 B W n + 1 u n + 1 ) - ( W n u n - λ n B W n u n ) = ( W n + 1 u n + 1 - λ n + 1 B W n + 1 u n + 1 ) - ( W n + 1 u n - λ n + 1 B W n + 1 u n ) + ( W n + 1 u n - λ n + 1 B W n + 1 u n ) - ( W n u n - λ n B W n u n ) ( W n + 1 u n + 1 - λ n + 1 B W n + 1 u n + 1 ) - ( W n + 1 u n - λ n + 1 B W n + 1 u n ) + W n + 1 u n - W n u n + λ n B W n u n - λ n + 1 B W n + 1 u n u n + 1 - u n + M 1 i = 1 n μ i + λ n B W n u n + λ n + 1 B W n + 1 u n x n + 1 - x n + δ n + 1 α n + 1 - α n B x n + δ n - δ n + 1 x n - v n + n + 1 M y n - n M y n + M 1 i = 1 n μ i + λ n B W n u n + λ n + 1 B W n + 1 u n .
(3.18)
Setting
l n = x n + 1 - β n x n 1 - β n = ε n γ f ( u n ) + ( ( 1 - β n ) I - ε n A ) e n 1 - β n ,
we have xn+1= (1 - β n )l n + β n x n , n ≥ 1. It follows that
l n + 1 - l n = ε n + 1 γ f ( u n + 1 ) + ( ( 1 - β n + 1 ) I - ε n + 1 A ) e n + 1 1 - β n + 1 - ε n γ f ( u n ) + ( ( 1 - β n ) I - ε n A ) e n 1 - β n = ε n + 1 1 - β n + 1 γ f ( u n + 1 ) - A e n + 1 + ε n 1 - β n A e n - γ f ( u n ) + ( e n + 1 - e n ) .
(3.19)
It follows from (3.18) and (3.19) that
l n + 1 - l n - x n + 1 - x n ε n + 1 1 - β n + 1 γ f ( u n + 1 ) - A e n + 1 + ε n 1 - β n A e n - γ f ( u n ) + δ n + 1 α n + 1 - α n B x n + δ n - δ n + 1 x n - v n + n + 1 M y n - n M y n + M 1 i = 1 n μ i + λ n B W n u n + λ n + 1 B W n + 1 u n ε n + 1 1 - β n + 1 γ f ( u n + 1 ) + A e n + 1 + ε n 1 - β n A e n + γ f ( u n ) + δ n + 1 α n + 1 - α n B x n + δ n - δ n + 1 x n - v n + n + 1 M y n - n M y n + M 1 i = 1 n μ i + λ n B W n u n + λ n + 1 B W n + 1 u n .
(3.20)
This together with conditions (C1)-(C4) and (3.11) imply that
lim  sup n ( l n + 1 - l n - x n + 1 - x n ) 0 .
By Lemma 2.8, we obtain
lim n l n - x n = 0 .
Consequently,
lim n x n + 1 - x n = lim n ( 1 - β n ) l n - x n = 0 .
(3.21)
Applying (3.11), (3.21) and conditions (C3), (C4) to (3.15) and (3.17), we obtain that
lim n u n + 1 - u n = lim n v n + 1 - v n = 0 .
(3.22)

Step 5. We show that lim n W n e n - e n = 0 .

For any x ̃ Θ and (3.5), we obtain
v n - x ̃ 2 = P C ( x n - α n B x n ) - P C ( x ̃ - α n B x ̃ ) 2 ( x n - α n B x n ) - ( x ̃ - α n B x ̃ ) 2 x n - x ̃ 2 + ( α n 2 - 2 α n ξ ) B x n - B x ̃ 2 .
(3.23)
By Lemma 2.2(iv) and (3.23), we have
y n - x ̃ 2 ( 1 - δ n ) x n - x ̃ 2 + δ n v n - x ̃ 2 ( 1 - δ n ) x n - x ̃ 2 + δ n { x n - x ̃ 2 + ( α n 2 - 2 α n ξ ) B x n - B x ̃ 2 } = x n - x ̃ 2 + ( α n 2 - 2 α n ξ ) δ n B x n - B x ̃ . 2
(3.24)
So, from (3.8) and (3.24), we derive
e n - x ̃ 2 u n - x ̃ 2 y n - x ̃ 2 x n - x ̃ 2 + ( α n 2 - 2 α n ξ ) δ n B x n - B x ̃ . 2
(3.25)
From (3.3), we have
x n + 1 x ˜ 2 = ( ( 1 β n ) I ε n A ) ( e n x ˜ ) + β n ( x n x ˜ ) + ε n ( γ f ( u n ) A x ˜ ) 2 = ( ( 1 β n ) I ε n A ) ( e n x ˜ ) + β n ( x n x ˜ ) 2 + ε n 2 γ f ( u n ) A x ˜ 2 + 2 β n ε n x n x ˜ , γ f ( u n ) A x ˜ + 2 ε n ( ( 1 β n ) I ε n A ) ( e n x ˜ ) , γ f ( u n ) A x ˜ ( ( 1 β n ε n γ ¯ ) e n x ˜ + β n x n x ˜ ) 2 + ε n L n ( 1 β n ε n γ ¯ ) 2 e n x ˜ 2 + β n 2 x n x ˜ 2 + 2 ( 1 β n ε n γ ¯ ) β n e n x ˜ x n x ˜ + ε n L n [ ( 1 ε n γ ¯ ) 2 2 ( 1 ε n γ ¯ ) β n + β n 2 ] e n x ˜ 2 + ( 1 β n ε n γ ¯ ) β n { e n x ˜ 2 + x n x ˜ 2 } + β n 2 x n x ˜ 2 + ε n L n = ( 1 ε n γ ¯ ) ( 1 β n ε n γ ¯ ) e n x ˜ 2 + ( 1 ε n γ ¯ ) β n x n x ˜ 2 + ε n L n ( 1 ε n γ ¯ ) ( 1 β n ε n γ ¯ ) { x n x ˜ 2 + ( α n 2 2 α n ξ ) δ n B x n B x ˜ 2 } + ( 1 ε n γ ¯ ) β n x n x ˜ 2 + ε n L n = ( 1 ε n γ ¯ ) 2 x n x ˜ 2 + ( 1 ε n γ ¯ ) ( 1 β n ε n γ ¯ ) ( α n 2 2 α n ξ ) δ n B x n B x ˜ 2 + ε n L n x n x ˜ 2 + ( 1 ε n γ ¯ ) ( 1 β n ε n γ ¯ ) ( α n 2 2 α n ξ ) δ n B x n B x ˜ 2 + ε n L n .
(3.26)
It follows that
( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) ( 2 g ξ - e 2 ) b B x n - B x ̃ 2 ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) ( 2 α n ξ - α n 2 ) δ n B x n - B x ̃ 2 x n - x ̃ 2 - x n + 1 - x ̃ 2 + ε n L n x n - x n + 1 ( x n - x ̃ + x n + 1 - x ̃ ) + ε n L n ,
where
L n = ε n γ f ( u n ) - A x ̃ 2 + 2 β n x n - x ̃ , γ f ( u n ) - A x ̃ + 2 ( ( 1 - β n ) I - ε n A ) ( e n - x ̃ ) , γ f ( u n ) - A x ̃ .
By conditions (C1), (C2) and (3.21), we obtain
lim n B x n - B x ̃ = 0 .
(3.27)
Since P C is firmly nonexpansive mapping, we have
v n - x ̃ 2 = P C ( x n - α n B x n ) - P C ( x ̃ - α n B x ̃ ) 2 ( x n - α n B x n ) - ( x ̃ - α n B x ̃ ) , v n - x ̃ = 1 2 ( x n - α n B x n ) - ( x ̃ - α n B x ̃ ) 2 + v n - x ̃ 2 - ( x n - α n B x n ) - ( x ̃ - α n B x ̃ ) - ( v n - x ̃ ) 2 1 2 x n - x ̃ 2 + v n - x ̃ 2 - ( x n - v n ) - α n ( B x n - B x ̃ ) 2 1 2 x n - x ̃ 2 + v n - x ̃ 2 - x n - v n 2 - α n 2 B x n - B x ̃ 2 + 2 α n x n - v n B x n - B x ̃ .
Hence, we have
v n - x ̃ 2 x n - x ̃ 2 - x n - v n 2 + 2 α n x n - v n B x n - B x ̃
and so
y n - x ̃ 2 ( 1 - δ n ) x n - x ̃ 2 + δ n v n - x ̃ 2 ( 1 - δ n ) x n - x ̃ 2 + δ n { x n - x ̃ 2 - x n - v n 2 + 2 α n x n - v n B x n - B x ̃ } = x n - x ̃ 2 - δ n x n - v n 2 + 2 δ n α n x n - v n B x n - B x ̃ .
(3.28)
Using (3.26) and (3.28), we also have
x n + 1 - x ̃ 2 ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) e n - x ̃ 2 + ( 1 - ε n γ ̄ ) β n x n - x ̃ 2 + ε n L n ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) u n - x ̃ 2 + ( 1 - ε n γ ̄ ) β n x n - x ̃ 2 + ε n L n ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) y n - x ̃ 2 + ( 1 - ε n γ ̄ ) β n x n - x ̃ 2 + ε n L n ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) x n - x ̃ 2 - δ n x n - v n 2 + 2 δ n α n x n - v n B x n - B x ̃ + ( 1 - ε n γ ̄ ) β n x n - x ̃ 2 + ε n L n x n - x ̃ 2 - ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) δ n x n - v n 2 + 2 ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) δ n α n x n - v n B x n - B x ̃ + ε n L n .
It follow that
( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) δ n x n - v n 2 x n - x n + 1 ( x n - x ̃ + x n + 1 - x ̃ ) + 2 ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) δ n α n x n - v n B x n - B x ̃ + ε n L n .
From conditions (C1), C(4), (3.21) and (3.27), we obtain
lim n x n - u n = 0 .
(3.29)
Observe also that if e n = P C (W n u n - λ n BW n u n ), then
e n - x ̃ 2 = P C ( W n u n - λ n B W n u n ) - P C ( x ̃ - λ n B x ̃ ) 2 ( W n u n - λ n B W n u n ) - ( x ̃ - λ n B x ̃ ) 2 = ( W n u n - λ n B W n u n ) - ( W n x ̃ - λ n B W n x ̃ ) 2 u n - x ̃ 2 + ( λ n 2 - 2 λ n ξ ) B W n u n - B x ̃ 2 x n - x ̃ 2 + ( λ n 2 - 2 λ n ξ ) B W n u n - B x ̃ 2 .
(3.30)
Substituting (3.30) in (3.26), we have
x n + 1 - x ̃ 2 ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) e n - x ̃ 2 + ( 1 - ε n γ ̄ ) β n x n - x ̃ 2 + ε n L n ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) x n - x ̃ 2 + ( λ n 2 - 2 λ n ξ ) B W n u n - B x ̃ 2 + ( 1 - ε n γ ̄ ) β n x n - x ̃ 2 + ε n L n x n - x ̃ 2 + ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) ( λ n 2 - 2 λ n ξ ) B W n u n - B x ̃ 2 + ε n L n .
It follows that
( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) ( 2 g ξ - e 2 ) B W n u n - B x ̃ 2 ( 1 - ε n γ ̄ ) ( 1 - β n - ε n γ ̄ ) ( 2 λ n ξ - λ n 2 ) B W n u n - B x ̃ 2 x n - x n + 1 ( x n - x ̃ + x n + 1 - x ̃ ) + ε n L n
Since ||xn+1- x n || → 0 (n → ∞) and conditions (C1) and (C2), we obtain
lim n B W n u n - B x ̃ = 0 .
(3.31)
Since P C is firmly nonexpansive (Lemma 2.2 (iii)), we have
e n - x ̃ 2 = P C ( W n u n - λ