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# Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems

## Abstract

We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some well-known results in the literature.

Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25.

## 1. Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.

Let A: CH be a nonlinear mapping. The classical variational inequality problem is to fined x C such that

$〈 A x , y - x 〉 ≥ 0 , ∀ y ∈ C .$
(1)

The set of solution of (1) is denoted by VI(C, A), i.e.,

$VI ( C , A ) = { x ∈ C : 〈 A x , y - x 〉 ≥ 0 , ∀ y ∈ C } .$
(2)

Recall that the following definitions:

1. (1)

A is called monotone if

$〈 A x - A y , x - y 〉 ≥ 0 , ∀ x , y ∈ C .$
2. (2)

A is called α-strongly monotone if there exists a positive constant α such that

$〈 A x - A y , x - y 〉 ≥ α x - y 2 , ∀ x , y ∈ C .$
3. (3)

A is called μ-Lipschitzian if there exist a positive constant μ such that

$A x - A y ≤ μ x - y , ∀ x , y ∈ C .$
4. (4)

A is called α-inverse strongly monotone, if there exists a positive real number α > 0

such that

$〈 A x - A y , x - y 〉 ≥ α A x - A y 2 , ∀ x , y ∈ C .$

It is obvious that any α-inverse strongly monotone mapping B is $1 α$-Lipschitzian.

1. (5)

A mapping T : CC is called nonexpansive if Tx - Ty x - y for all x, y C. Next, we denote by Fix(T) the set of fixed point of T.

2. (6)

A mapping f : CC is said to be contraction if there exists a coefficient α (0, 1) such that

$f ( x ) - f ( y ) ≤α x - y ,∀x,y∈C.$
3. (7)

A set-valued mapping U : H → 2His called monotone if for all x, y H, f Ux and g Uy imply 〈x - y, f - g〉 ≥ 0.

4. (8)

A monotone mapping U : H → 2His maximal if the graph G(U) of U is not properly contained in the graph of any other monotone mapping.

It is known that a monotone mapping U is maximal if and only if for (x, f) H × H, 〈x - y, f - g〉 ≤ 0 for every (y, g) G(U) implies that f Ux. Let B be a monotone mapping of C into H and let N C x be the normal cone to C at x C, that is, N C x = {y H : 〈x - z, y〉 ≤ 0, z C} and define

$U x = B x + N C x , x ∈ C , ∅ x ∉ C .$

Then U is the maximal monotone and 0 Ux if and only if x VI(C, B); see [1].

Let F be a bi-function of C×C into , where is the set of real numbers. The equilibrium problem for F : C × C is to determine its equilibrium points, i.e the set

$EP ( F ) = { x ∈ C : F ( x , y ) ≥ 0 , ∀ y ∈ C } .$

Let $J= { F i } i ∈ I$ be a family of bi-functions from C × C into . The system of equilibrium problems for $J= { F i } i ∈ I$ is to determine common equilibrium points for $J= { F i } i ∈ I$, i.e the set

$EP ( J ) = { x ∈ C : F i ( x , y ) ≥ 0 , ∀ y ∈ C , ∀ i ∈ I } .$
(3)

Numerous problems in physics, optimization, and economics reduce into finding some element of EP(F). Some method have been proposed to solve the equilibrium problem; see, for instance [25]. The formulation (3), extend this formalism to systems of such problems, covering in particular various forms of feasibility problems [6, 7].

Given any r > 0 the operator $J r F :H→C$ 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 [3] that under suitable hypotheses on F (to be stated precisely in Sect. 2), $J r F :H→C$ is single- valued and firmly nonexpansive and

satisfies

$Fix ( J r F ) = EP ( F ) , ∀ r > 0 .$

Using this result, in 2007, Yao et al. [8], proposed the following explicit scheme with respect to W-mappings for an infinite family of nonexpansive mappings:

$x n + 1 = α n f ( x n ) + β n x n + γ n W n J r n F x n$
(4)

They proved that if the sequences {α n }, {β n }, {γ n } and {r n } of parameters satisfy appropriate conditions, then, the sequences {x n } and ${ J r n F x n }$ both converge strongly to the unique $x * ∈ ∩ i = 1 ∞ Fix ( T i ) ∩EP ( F )$, where $x * ∈ P ∩ i = 1 ∞ Fix ( T i ) ∩ E P ( F ) f ( x * )$. Their results extend and improve the corresponding results announced by Combettes and Hirstoaga [3] and Takahashi and Takahashi [5].

Very recently, Jitpeera et al. [9], introduced the iterative scheme based on viscosity and Cesàro mean

$ϕ ( u n , y ) + φ ( y ) - φ ( u n ) + 1 r n 〈 y - u n , u n - x n 〉 ≥ 0 , ∀ y ∈ C , y n = δ n u n + ( 1 - δ n ) P C ( u n - λ n B u n ) , x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n A ) 1 n + 1 ∑ i = 0 n T i y n , ∀ n ≥ 0 ,$

where B : CH is β-inverse strongly monotone, φ: C {∞} is a proper lower semi-continuous and convex function, Ti: CC is a nonexpansive mapping for all i = 1, 2, ..., n, {α n }, {β n }, {δ n } (0, 1), {λ n } (0, 2β) and {r n } (0, ∞) satisfy the following conditions

1. (i)

lim n →∞ α n = 0, $∑ n = 1 ∞ α n =∞$,

2. (ii)

lim n →∞ δ n = 0

3. (iii)

0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1.

4. (iv)

{λ n } [a, b] (0, 2β) and lim inf n →∞ | λ n +1 - λ n |= 0,

5. (v)

lim inf n →∞ r n > 0 and lim inf n →∞ | r n +1 - r n |= 0.

They show that if $θ= ∩ i = 1 n Fix ( T i ) ∩VI ( C , B ) ∩MEP ( ϕ , φ )$ is nonempty, then the sequence {x n } converges strongly to the z = P θ (I - A + γf )z which is the unique solution of the variational inequality

$〈 ( γ f - A ) z , x - z 〉 ≤0∀y∈θ.$

In this paper, motivated and inspired by Yao et al. [8, 1015], Lau et al. [16], Jitpeera et al. [9], Kangtunyakarn [17] and Kim [18], Atsushiba and Takahashi [19], Saeidi [20], Piri [2123] and Piri and Badali [24], we introduce the following iterative scheme for finding a common element of the set of solutions for a system of equilibrium problems $EP ( J )$ for a family $J= { F k : k = 1 , 2 , 3 , … , M }$ of equilibrium bi-functions, systems of variational inequalities, the set of common fixed points for an infinite family ψ = {T i , i = 1, 2, ...} of nonexpansive mappings and a left amenable semigroup φ = {T t : t S} of nonexpansive mappings, with respect to W-mappings and a left regular sequence {μ n } of means defined on an appropriate space of bounded real-valued functions of the semigroup

$z n = J r M , n F M … J r 2 , n F 2 J r 1 , n F 1 x n , y n = η n P C ( z n - ζ n A z n ) + ( 1 - η n ) P C ( z n - δ n B z n ) , x n + 1 = α n f ( T μ n W n y n ) + β n x n + γ n T μ n W n y n , n ≥ 1 ,$
(5)

where A: CH be β-inverse monotone map and B : CH be δ-inverse monotone map. We prove that under mild assumptions on parameters like that in Yao et al. [8], the sequences {x n } and ${ J r k , n F k x n } k = 1 M$ converge strongly to $x * ∈F= ∩ i = 1 ∞ Fix ( T i ) ∩Fix ( φ ) ∩EP ( J ) ∩VI ( C , A ) ∩VI ( C , B )$, where $x * = P F f ( x * )$.

Compared to the similar works, our results have the merit of studying the solutions of systems of equilibrium problems, systems of variational inequalities and fixed point problems of amenable semigroup of nonexpansive mappings. Consequence for nonnegative integer numbers is also presented.

## 2. Preliminaries

Let S be a semigroup and let B(S) be the space of all bounded real valued functions defined on S with supremum norm. For s S and f B(S), we define elements l s f and r s f in B(S) by

$( l s f ) ( t ) =f ( s t ) , ( r s f ) ( t ) =f ( t s ) ,∀t∈S.$

Let X be a subspace of B(S) containing 1 and let X* be its topological dual. An element μ of X* is said to be a mean on X if μ = μ(1) = 1. We often write μ t (f(t)) instead of μ(f) for μ X* and f X. Let X be left invariant (respectively right invariant), i.e., l s (X) X (respectively r s (X) X) for each s S. A mean μ on X is said to be left invariant (respectively right invariant) if μ(l s f) = μ(f) (respectively μ(r s f) = μ(f)) for each s S and f X. X is said to be left (respectively right) amenable if X has a left (respectively right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup, see [25]. A net {μ α } of means on X is said to be strongly left regular if

$lim α l s * μ α - μ α =0,$

for each s S, where $l s *$ is the adjoint operator of l s .

Let S be a semigroup and let C be a nonempty closed and convex subset of a reflexive Banach space E. A family φ = {T t : t S} of mapping from C into itself is said to be a nonexpansive semigroup on C if T t is nonexpansive and T ts = T t T s for each t, s S. By Fix(φ) we denote the set of common fixed points of φ, i.e.

$Fix ( φ ) = ⋂ t ∈ S { x ∈ C : T t x = x } .$

Lemma 2.1. [25] Let S be a semigroup and C be a nonempty closed convex subset of a reflexive Banach space E. Let φ = {T t : t S} be a nonexpansive semigroup on H such that {T t x : t S} is bounded for some x C, let X be a subspace of B(S) such that 1 X and the mapping t → 〈T t x, y*〉 is an element of X for each x C and y* E*, and μ is a mean on X. If we write T μ x instead of ∫ T t xdμ(t), then the followings hold.

1. (i)

T μ is nonexpansive mapping from C into C.

2. (ii)

T μ x = x for each x Fix(φ).

3. (iii)

$T μ x ∈ c o ¯ { T t x : t ∈ S }$ for each x C.

Let C be a nonempty subset of a Hilbert space H and T : CH a mapping. Then T is said to be demiclosed at v H if, for any sequence {x n } in C, the following implication holds:

$x n ⇀u∈C,T x n →vimplyTu=v,$

where → (respectively ) denotes strong (respectively weak) convergence.

Lemma 2.2. [26] Let C be a nonempty closed convex subset of a Hilbert space H and suppose that T : CH is nonexpansive. then, the mapping I - T is demiclosed at zero.

Lemma 2.3. [27] For a given x H, y C,

$y= P C x⇔ 〈 y - x , z - y 〉 ≥0,∀z∈C.$

It is well known that P C is a firmly nonexpansive mapping of H onto C and satisfies

$P C x - P C y 2 ≤ 〈 P C x - P C y , x - y 〉 ,∀x,y∈H.$
(6)

Moreover, P C is characterized by the following properties: P C x C and for all x H, y C,

$〈 x - P C x , y - P C x 〉 ≤0.$
(7)

It is easy to see that (7) is equivalent to the following inequality

$x - y 2 ≥ x - P C x 2 + y - P C x 2 .$
(8)

Using Lemma 2.3, one can see that the variational inequality (1) is equivalent to a fixed point problem. It is easy to see that the following is true:

$u∈VI ( C , A ) ⇔u= P C ( u - λ A u ) ,λ>0.$
(9)

Lemma 2.4. [28] Let {x n } and {y n } be bounded sequences in a Banach space E and let {α n } be a sequence in [0, 1] with $0< lim inf n → ∞ α n ≤ lim sup n → ∞ α n <1$. Suppose x n +1 = α n x n +(1-α n )y n for all integers n ≥ 0 and

$lim sup n → ∞ ( y n + 1 - y n - x n + 1 - x n ) ≤1.$

Then, $lim n → ∞ y n - x n =0$.

Let F : C × C be a bi-function. Given any r > 0, the operator $J r F :H→C$ 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]. The equilibrium problem for F is to determine its equilibrium points, i.e., the set

$EP ( F ) = { x ∈ C : F ( x , y ) ≥ 0 , ∀ y ∈ C } .$

Let $J= { F i } i ∈ I$ be a family of bi-functions from C × C into . The system of equilibrium problems for is to determine common equilibrium points for $J= { F i } i ∈ I$. i.e, the set

$EP ( J ) = { x ∈ C : F i ( x , y ) ≥ 0 , ∀ y ∈ C , ∀ i ∈ I } .$

Lemma 2.5. [3] Let C be a nonempty closed convex subset of H and F : C × C satisfy

(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 all x, y, z C, limt→0F(tz + (1 - t)x, y) ≤ F (x, y),

(A4) for all x C, yF(x, y) is convex and lower semi-continuous.

Given r > 0, define the operator $J r F :H→C$, the resolvent of F, by

$J r F ( x ) = { z ∈ C : F ( z , y ) + 1 r 〈 y - z , z - x 〉 ≥ 0 , ∀ y ∈ C } .$

Then,

1. (1)

$J r F$ is single valued,

2. (2)

$J r F$ is firmly nonexpansive, i.e, $J r F x - J r F y 2 ≤ 〈 J r F x - J r F y , x - y 〉$ for all x, y H,

3. (3)

$Fix ( J r F ) =EP ( F )$,

4. (4)

EP(F) is closed and convex.

Let T1, T2, ... be an infinite family of mappings of C into itself and let λ1, λ2, ... be a real numbers such that 0 ≤ λ i < 1 for every i . For any n , define a mapping W n of C into C as follows:

$U n , n + 1 = I , U n , n = λ n T n U n , n + 1 + ( 1 - λ n ) I , U n , n - 1 = λ n - 1 T n - 1 U n , n + ( 1 - λ n - 1 ) I , ⋮ U n , k = λ k T k U n , k + 1 + ( 1 - λ k ) I , U n , k - 1 = λ k - 1 T k - 1 U n , k + ( 1 - λ k - 1 ) I , ⋮ U n , 2 = λ 2 T 2 U n , 3 + ( 1 - λ 2 ) I , W n = U n , 1 = λ 1 T 1 U n , 2 + ( 1 - λ 1 ) I .$
(10)

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

Lemma 2.6. [29] Let C be a nonempty closed convex subset of a Hilbert space H, {T i : CC} be an infinite family of nonexpansive mappings with $∩ i = 1 ∞ Fix ( T i ) ≠∅$, {λ i } be a real sequence such that 0 < λ i b < 1, i ≥ 1. Then

1. (1)

W n is nonexpansive and $Fix ( W n ) = ∩ i = 1 n Fix ( T i )$ for each n ≥ 1,

2. (2)

for each x C and for each positive integer j, the limit lim n →∞ U n , j exists.

3. (3)

The mapping W : CC defined by

$Wx:= lim n → ∞ W n x= lim n → ∞ U n , 1 x,∀x∈C,$

is a nonexpansive mapping satisfying $Fix ( W ) = ∩ i = 1 ∞ Fix ( T i )$ and it is called the W-mapping generated by T1, T2, ... and λ1, λ2, ....

Lemma 2.7. [30] Let C be a nonempty closed convex subset of a Hilbert space H, {T i : CC} be a countable family of nonexpansive mappings with $∩ i = 1 ∞ Fix ( T i ) ≠∅$,{λ i } be a real sequence such that 0 < λ i b < 1, i ≥ 1. If D is any bounded subset of C, then

$lim n → ∞ sup x ∈ D W x - W n x =0.$

Lemma 2.8. [31] Let {a n } be a sequence of nonnegative real numbers such that

$a n + 1 ≤ ( 1 - b n ) a n + b n c n ,n≥0,$

where {b n } and {c n } are sequences of real numbers satisfying the following conditions:

1. (i)

{b n } [0, 1], $∑ n = 0 ∞ b n =∞$,

2. (ii)

either $lim sup n → ∞ c n ≤0$ or $∑ n = 0 ∞ | b n c n |<∞$.

Then, $lim n → ∞ a n =0$.

Lemma 2.9. [32] Let (E, 〈., .〉) be an inner product space. Then for all x, y, z E and α, β, γ, [0, 1] such that α + β + γ = 1, we have

$α x + β y + γ z 2 = α x 2 + β y 2 + γ z 2 - α β x - y 2 - α γ x - z 2 - β γ y - z 2 .$

Notation Throughout the rest of this paper the open ball of radius r centered at 0 is denoted by B r . For a subset A of H we denote by $c o ¯ A$ the closed convex hull of A. For ε > 0 and a mapping T : DH, we let F ϵ (T; D) be the set of ϵ-approximate fixed points of T, i.e., F ϵ (T ; D) = {x D : x - Tx ϵ}. Weak convergence is denoted by and strong convergence is denoted by →.

## 3. Strong convergence

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, A: CH a β-inverse strongly monotone, B : CH a γ-inverse strongly monotone, S a semigroup and φ = {T t : t S} be a nonexpansive semigroup from C into C such that $Fix ( φ ) = ∩ t ∈ S Fix ( T t ) ≠ ∅$. Let X be a left invariant subspace of B(S) such that 1 X, and the function t → 〈T t x, yis an element of X for each x C and y H, {μ n } a left regular sequence of means on X such that lim n →∞ μ n +1 - μ n = 0. Let $J= { F k : k = 1 , 2 , … , M }$ be a finite family of bi-functions from C × C into which satisfy (A1)-(A4) and ${ T i } i = 1 ∞$ an infinite family of nonexpansive mappings of C into C such that $T i ( F i x ( φ ) ∩ EP ( J ) ) ⊂Fix(φ)$ for each i and $F= ∩ i = 1 ∞ Fix ( T i ) ∩Fix ( φ ) ∩EP ( J ) ∩VI ( C , A ) ∩VI ( C , B ) ≠∅$. Let {α n }, {β n }, {γ n } and {η n } be a sequences in (0, 1). Let {ζ n } a sequence in (0, 2β), {δ n } a sequence in (0, 2γ), ${ r k , n } k = 1 M$ be sequences in (0, ∞) and {λ n } a sequence of real numbers such that 0 < λ n b < 1. Assume that,

(B1) lim n →∞ η n = η (0, 1), lim n →∞ α n = 0 and $∑ n = 1 ∞ α n =∞$,

(B2) 0 < lim inf n →∞ β n ≤ lim sup n →∞ β n < 1,

(B3) α n + β n + γ n = 1,

(B4) lim n →∞ | ζ n +1 - ζ n |= lim n →∞ | δ n +1 - δ n |= 0,

(B5) lim inf n →∞ r k , n > 0 and lim n →∞ (r k , n +1 - r k , n ) = 0 for k {1, 2, · · ·, M}.

Let f be a contraction of C into itself with coefficient α (0, 1) and given x1 C arbitrarily. If the sequences {x n }, {y n } and {z n } are generated iteratively by x1 C and

$z n = J r M , n F M … J r 2 , n F 2 J r 1 , n F 1 x n , y n = η n P C ( z n - ζ n A z n ) + ( 1 - η n ) P C ( z n - δ n B z n ) , x n + 1 = α n f ( T μ n W n y n ) + β n x n + γ n T μ n W n y n , n ≥ 1 ,$
(11)

then, the sequences {x n }, {y n } and ${ J r k , n F k x n } k = 1 M$ converge strongly to $x * ∈F$, which is the unique solution of the system of variational inequalities:

$〈 f ( x * ) - x * , x * - y 〉 ≥ 0 , ∀ y ∈ F , 〈 B x * , y - x * 〉 ≥ 0 ∀ y ∈ C , 〈 A x * , y - x * 〉 ≥ 0 ∀ y ∈ C .$

Proof. Since A is a β-inverse strongly monotone map, for any x, y C, we have

$( I - ζ n A ) x - ( I - ζ n A ) y 2 = ( x - y ) - ζ n ( A x - A y ) 2 = | x - y 2 - 2 ζ n 〈 x - y , A x - A y 〉 + ζ n 2 A x - A y 2 ≤ x - y 2 - 2 ζ n β A x - A y 2 + ζ n 2 A x - A y 2 = x - y 2 + ζ n ( ζ n - 2 β ) A x - A y 2 ≤ x - y 2$

It follows that

$( I - ζ n A ) x - ( I - ζ n A ) y ≤ x - y .$
(12)

Since B is a β-inverse strongly monotone map, repeating the same argument as above, we can deduce that

$( I - δ n B ) x - ( I - δ n B ) y ≤ x - y .$
(13)

Let $p∈F$, in the context of the variational inequality problem the characterization of projection (9) implies that p = P C (p - ζ n Ap) and p = P C (p - δ n Bp). Using (12) and (13), we get

(14)

By taking v n = P C (z n - ζ n Az n ), w n = P C (z n - δ n Bz n ) and $J n k = J r k , n F k ⋯ J r 2 , n F 2 J r 1 , n F 1$ for k {1, 2, ..., M} and $J n 0 =I$ for all n , we shall equivalently write scheme (11) as follows:

$z n = J n M x n , y n = η n v n + ( 1 - η n ) w n , x n + 1 = α n f ( T μ n W n y n ) + β n x n + γ n T μ n W n y n , n ≥ 1 .$

We shall divide the proof into several steps.

Step 1. The sequence {x n } is bounded.

Proof of Step 1. Let $p∈F$. Since for each k {1, 2, ..., M}, $J r k , n F k$ is nonexpansive we have

$J n k x n - p = J n k x n - J n k p ≤ x n - p , ∀ k ∈ { 1 , 2 , … , M } .$
(15)

Thus, by Lemmas 2.1, 2.5 and (14), we have

$x n + 1 - p ≤ α n f ( T μ n W n y n ) - p + β n x n - p + γ n T μ n W n J n M y n - p ≤ α n [ f ( T μ n W n y n ) - f ( p ) + f ( p ) - p ] + β n x n - p + γ n y n - p ≤ α n α x n - p + α n f ( p ) - p + ( β n + γ n ) x n - p = [ 1 - α n ( 1 - α ) ] x n - p + α n f ( p ) - p ≤ max x n - p , 1 1 - α f ( p ) - p .$

By induction,

$x n - p ≤max x 1 - p , 1 1 - α f ( p ) - p ,n≥1.$

Step 2. Let {u n } be a bounded sequence in H. Then

$lim n → ∞ J n + 1 k u n - J n k u n = 0 ,$
(16)

for every k {1, 2, ..., M}.

Proof of Step 2. This assertion is proved in [27, Step 2].

Step 3. Let {u n } be a bounded sequence in H. Then

This assertion is proved in [21, Step 3].

Step 4. limn→∞ x n +1 - x n = 0.

Proof of Step 4. Setting x n +1 = β n x n + (1 - β n )t n for all n ≥ 1, we have

$t n + 1 - t n = 1 1 - β n + 1 [ x n + 2 - β n + 1 x n + 1 ] - 1 1 - β n [ x n + 1 - β n x n ] = α n + 1 1 - β n + 1 [ f ( T μ n + 1 W n + 1 y n + 1 ) - f ( T μ n W n y n ) ] + α n + 1 1 - β n + 1 - α n 1 - β n f ( T μ n W n y n ) + γ n + 1 1 - β n + 1 [ T μ n + 1 W n + 1 y n + 1 - T μ n W n y n ] + γ n + 1 1 - β n + 1 - γ n 1 - β n T μ n W n y n .$

Therefore, we have

$t n + 1 - t n ≤ α n + 1 1 - β n + 1 [ α T μ n + 1 W n + 1 y n + 1 - T μ n W n y n + T μ n + 1 W n + 1 y n + 1 - T μ n W n y n ] + α n + 1 1 - β n + 1 - α n 1 - β n [ f ( T μ n W n y n ) + T μ n W n y n ] + T μ n + 1 W n + 1 y n + 1 - T μ n W n y n .$

On the other hand

$T μ n + 1 W n + 1 y n + 1 - T μ n W n y n ≤ T μ n + 1 W n + 1 y n + 1 - T μ n + 1 W n + 1 y n + T μ n + 1 W n + 1 y n - T μ n + 1 W n y n + T μ n + 1 W n y n - T μ n W n y n ≤ y n + 1 - y n + W n + 1 y n - W n y n + T μ n + 1 W n y n - T μ n W n y n$

Observing that $z n = J n M x n$, $z n + 1 = J n + 1 M x n + 1$ and $J n M x n = J r M , n F M J n M - 1 x n$ we get

$1 r M , n 〈 y - z n , z n - J n M - 1 x n 〉 + F M ( z n , y ) ≥ 0 , ∀ y ∈ C ,$
(17)

and

$1 r M , n + 1 〈 y - z n + 1 , z n + 1 - J n + 1 M - 1 x n + 1 〉 + F M ( z n + 1 , y ) ≥ 0 , ∀ y ∈ C ,$
(18)

Take y = z n +1 in (17) and y = z n in (18), by using (A2), it follows that

$z n + 1 - z n , z n - J n M - 1 x n - r M , n r M , n + 1 ( z n + 1 - J n + 1 M - 1 x n + 1 ) ≥ 0 ,$

and hence

Thus, we have

$z n + 1 - z n ≤ J n + 1 M - 1 x n + 1 - J n M - 1 x n + 1 - r M , n r M , n + 1 z n + 1 - J n + 1 M - 1 x n + 1 ≤ J n + 1 M - 1 x n + 1 - J n + 1 M - 1 x n + J n + 1 M - 1 x n - J n M - 1 x n + 1 - r M , n r M , n + 1 z n + 1 - J n + 1 M - 1 x n + 1 ≤ x n + 1 - x n + J n + 1 M - 1 x n - J n M - 1 x n + 1 - r M , n r M , n + 1 z n + 1 - J n + 1 M - 1 x n + 1 .$

Since v n = P C (z n - ζ n Az n ) and w n = P C (z n - δ n Bz n ), it follows from the definition of {y n } that

Therefore,

$t n + 1 - t n - x n + 1 - x n ≤ α n + 1 1 - β n + 1 [ y n + 1 - y n + T μ n + 1 W n + 1 y n + 1 - T μ n W n y n ] + α n + 1 1 - β n + 1 - α n 1 - β n [ f ( T μ n W n y n ) + T μ n W n y n ] + J n + 1 M - 1 x n - J n M - 1 x n + 1 - r M , n r M , n + 1 z n + 1 - J n + 1 M - 1 x n + 1 + η n + 1 ζ n + 1 - ζ n A z n + η n + 1 - η n ( v n + w n ) + δ n + 1 - δ n B z n + W n + 1 y n - W n y n + T μ n + 1 W n y n - T μ n W n y n .$

This together with conditions (B1), (B4), Steps 2 and 3 imply that

$lim sup n → ∞ ( t n + 1 - t n - x n + 1 - x n ) ≤0.$

Hence by Lemma 2.4, we obtain lim n →∞ t n - x n = 0. Consequently,

$lim n → ∞ x n + 1 - x n = ( 1 - β n ) t n - x n =0.$

Step 5. $lim n → ∞ J n k + 1 x n - J n k x n =0$, k {0, 1, 2, ..., M - 1}.

Proof of Step 5. Let $p∈F$ and k {1, 2, ..., M - 1}. Since $J r k + 1 , n F k + 1$ is firmly nonexpansive, we obtain

$J n k + 1 x n - p 2 = J r k + 1 , n F k + 1 J n k x n - J r k + 1 , n F k + 1 p 2 = J r k + 1 , n F k + 1 J n k x n - p , J n k x n - p = 1 2 J r k + 1 , n F k + 1 J n k x n - p 2 + J n k x n - p 2 - J r k + 1 , n F k + 1 J n k x n - J n k x n 2 .$

It follows that

$J n k + 1 x n - p 2 ≤ x n - p 2 - J n k + 1 x n - J n k x n 2 .$
(19)

Using Lemma 2.9, (14) and (19), we obtain

$x n + 1 - p 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n T μ n W n y n - p 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n y n - p 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n z n - p 2 = α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n J n M x n - p 2 = α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n J r M , n F M … J r k + 2 , n F k + 2 J n k + 1 y n - J r M , n F M … J r k + 2 , n F k + 2 p 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n J n k + 1 y n - p 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n J n k + 1 x n - p 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + γ n [ x n - p 2 - J n k + 1 x n - J n k x n ] .$

Then, we have

$γ n J n k + 1 x n - J n k x n 2 ≤ α n f ( T μ n W n y n ) - p 2 + β n x n - p 2 + ( 1 - α n - β n ) x n - p 2 - x n + 1 - p 2 = α n [ f ( T μ n W n y n ) - p 2 - x n - p 2 ] + x n - p 2 - x n + 1 - p 2 ≤ α n [ f ( T μ n W n y n ) - p 2 - x n - p 2 ] + x n - x n + 1 [ x n - p + x n + 1 - p ] .$

It is easily seen that lim inf n →∞ γ n > 0. So we have

$lim n → ∞ J n k + 1 x n - J n k x n =0.$

Step 6. $lim n → ∞ x n - T μ n W n J n M y n =0$.

Proof of Step 6. Observe that

hence

$x n - T μ n W n J n M y n ≤ 1 1 - β n x n + 1 - x n + α n 1 - β n [ f ( T μ n W n y n ) + T μ n W n J n M y n ] .$

It follows from conditions (B1), (B2) and Step 4, that

$lim n → ∞ x n - T μ n W n J n M y n =0.$

Step 7. limn→∞ x n - T t x n = 0, for all t S.

Proof of Step 7. Let $p∈F$ and set $M 0 =max { x 1 - p , 1 1 - α f ( p ) - p }$ and D = {y H : y - p M0}, we remark that D is bounded closed convex set, {y n } D and it is invariant under ${ J r k , n F k : k = 1 , 2 , . . . , M , ∀ n ∈ ℕ }$, φ and W n for all n . We will show that

$lim sup n → ∞ sup y ∈ D T μ n y - T t T μ n y =0,∀t∈S$
(20)

Let ϵ > 0. By [33, Theorem 1.2], there exists δ > 0 such that

$c o ¯ F δ ( T t ; D ) + B δ ⊂ F ε ( T t ; D ) ,∀t∈S.$
(21)

Also by [33, Corollary 1.1], there exists a natural number N such that

$1 N + 1 ∑ i = 0 N T t i s y - T t 1 N + 1 ∑ i = 0 N T t i s y ≤ δ ,$
(22)

for all t, s S and y D. Let t S. Since {μ n } is strongly left regular, there exists N0 such that $μ n - l t i * μ n ≤ δ ( M 0 + | | p | | )$ for nN0 and i = 1, 2, ..., N. Then, we have

$sup y ∈ D ‖ T μ n y − ∫ 1 N + 1 ∑ i = 0 N T t i s y d μ n ( s ) ‖ = sup y ∈ D sup | | z | | = 1 | 〈 T μ n y , z 〉 − 〈 ∫ 1 N + 1 ∑ i = 0 N T t i s y d μ n ( s ) , z 〉 | = sup y ∈ D sup | | z | | = 1 | 1 N + 1 ∑ i = 0 N ( μ n ) s 〈 T s y , z 〉 − 1 N + 1 ∑ i = 0 N ( μ n ) s 〈 T t i s y , z 〉 | ≤ 1 N + 1 ∑ i = 0 N sup y ∈ D sup | | z | | = 1 | ( μ n ) s 〈 T s y , z 〉 − ( l t i ∗ μ n ) s 〈 T s y , z 〉 | ≤ max i = 1 , 2 , … , N | | μ n − l t i ∗ μ n | | ( M 0 + | | p | | ) ≤ δ , ∀ n ≥ N 0 .$
(23)

By Lemma 2.1 we have

$∫ 1 N + 1 ∑ i = 0 N T t i S y d μ n ( s ) ∈ c o ¯ 1 N + 1 ∑ i = 0 N T t i ( T s y ) : s ∈ S .$
(24)

It follows from (21), (22), (23) and (24) that

$T μ n y ∈ c o ¯ 1 N + 1 ∑ i = 0 N T t i S y : s ∈ S + B δ ⊂ c o ¯ F δ ( T t ; D ) + B δ ⊂ F ε ( T t ; D ) ,$

for all y D and nN0. Therefore,

$lim sup n → ∞ sup y ∈ D T t ( T μ n y ) - T μ n y ≤ε.$

Since ϵ > 0 is arbitrary, we get (20).

Let t S and ϵ > 0. Then, there exists δ > 0, which satisfies (21). From condition (B1), (20) and Step 6, there exists N1 such that $α n < δ 4 M 0$, $T μ n y∈ F δ ( T t , D )$ for all y D and $x n - T μ n W n y n < δ 2$ for all nN1. We note that

$α n f ( T μ n W n y n ) - T μ n W n y n ≤ α n [ f ( T μ n W n y n ) - f ( p ) + f ( p ) - p + p - T μ n W n y n ] ≤ α n [ α y n - p + f ( p ) - p + p - y n ] ≤ 2 M 0 α n ≤ δ 2 ,$

for all nN1. Therefore, we have

$x n + 1 = T μ n W n y n + α n ( f ( T μ n W n y n ) - T μ n W n y n ) + β n ( x n - T μ n W n y n ) ∈ F δ ( T t ; D ) + B δ 2 + B δ 2 ⊂ F δ ( T t ; D ) +$