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: C → H be a nonlinear mapping. The classical variational inequality problem is to fined x ∈ C such that

$$〈Ax,y-x〉\ge 0,\forall y\in C.$$

(1)

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

$$\text{VI}\left(C,A\right)=\left\{x\in C:〈Ax,y-x〉\ge 0,\forall y\in C\right\}.$$

(2)

Recall that the following definitions:

• (1) A is called monotone if

  $$〈Ax-Ay,x-y〉\ge 0,\forall x,y\in C.$$

• (2) A is called α-strongly monotone if there exists a positive constant α such that

  $$〈Ax-Ay,x-y〉\ge \alpha {∥x-y∥}^2,\forall x,y\in C.$$

• (3) A is called μ-Lipschitzian if there exist a positive constant μ such that

  $$∥Ax-Ay∥\le \mu ∥x-y∥,\forall x,y\in C.$$

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

such that

$$〈Ax-Ay,x-y〉\ge \alpha {∥Ax-Ay∥}^2,\forall x,y\in C.$$

It is obvious that any α-inverse strongly monotone mapping B is $\frac{1}{\alpha }$-Lipschitzian.

• (5) A mapping T : C → C 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.

• (6) A mapping f : C → C is said to be contraction if there exists a coefficient α ∈ (0, 1) such that

  $$∥f\left(x\right)-f\left(y\right)∥\le \alpha ∥x-y∥,\forall x,y\in C.$$

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

• (8) A monotone mapping U : H → 2^His 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

$$Ux=\left\{\begin{array}{cc}Bx+N_{C}x,\hfill & x\in C,\hfill \\ \varnothing \hfill & x\notin C.\hfill \end{array}\right.$$

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 ℝ, where ℝ is the set of real numbers. The equilibrium problem for F : C × C → ℝ is to determine its equilibrium points, i.e the set

$$\text{EP}\left(F\right)=\left\{x\in C:F\left(x,y\right)\ge 0,\forall y\in C\right\}.$$

Let $\mathcal{J}={\left\{F_i\right\}}_{i\in I}$ be a family of bi-functions from C × C into ℝ. The system of equilibrium problems for $\mathcal{J}={\left\{F_i\right\}}_{i\in I}$ is to determine common equilibrium points for $\mathcal{J}={\left\{F_i\right\}}_{i\in I}$, i.e the set

$$\text{EP}\left(\mathcal{J}\right)=\left\{x\in C:F_i\left(x,y\right)\ge 0,\forall y\in C,\forall i\in I\right\}.$$

(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 [2–5]. 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 $J_r^F:H\to C$ defined by

$$J_r^F\left(x\right)=\left\{z\in C:F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0,\forall y\in C\right\},$$

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), $J_r^F:H\to C$ is single- valued and firmly nonexpansive and

satisfies

$$\text{Fix}\left(J_r^F\right)=\text{EP}\left(F\right),\forall r>0.$$

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:

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+{\gamma }_n{W}_n{J}_{r_n}^F{x}_n$$

(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 $\left\{J_{r_n}^F{x}_n\right\}$ both converge strongly to the unique $x^{*}\in {\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{EP}\left(F\right)$, where $x^{*}\in P_{{\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{E}{\text{P}}_{\left(F\right)}}f\left(x^{*}\right)$. 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

$$\left\{\begin{array}{c}\varphi \left(u_n,y\right)+\phi \left(y\right)-\phi \left(u_n\right)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C,\hfill \\ y_n={\delta }_n{u}_n+\left(1-{\delta }_n\right)P_C\left(u_n-{\lambda }_{n}B{u}_n\right),\hfill \\ x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\alpha }_{n}A\right)\frac{1}{n+1}{\sum }_{i=0}^n{T}^i{y}_n,\forall n\ge 0,\hfill \end{array}\right.$$

where B : C → H is β-inverse strongly monotone, φ: C → ℝ ∪ {∞} is a proper lower semi-continuous and convex function, Tⁱ: C → C 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

• (i) lim _{n →∞} α _n = 0, ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$,

• (ii) lim _{n →∞} δ _n = 0

• (iii) 0 < lim inf _{n →∞} β _n ≤ lim sup _{n →∞} β _n < 1.

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

• (v) lim inf _{n →∞} r _n > 0 and lim inf _{n →∞} | r _{n +1} - r _n |= 0.

They show that if $\theta ={\cap }_{i=1}^n\text{Fix}\left(T^i\right)\cap \text{VI}\left(C,B\right)\cap \text{MEP}\left(\varphi ,\phi \right)$ 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

$$〈\left(\gamma f-A\right)z,x-z〉\le 0\forall y\in \theta .$$

In this paper, motivated and inspired by Yao et al. [8, 10–15], Lau et al. [16], Jitpeera et al. [9], Kangtunyakarn [17] and Kim [18], Atsushiba and Takahashi [19], Saeidi [20], Piri [21–23] 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 $\text{EP}\left(\mathcal{J}\right)$ for a family $\mathcal{J}=\left\{F_k:k=1,2,3,\dots ,M\right\}$ 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

$$\left\{\begin{array}{c}{z}_n=J_{r_M,n}^{F_M}\dots J_{r_2,n}^{F_2}{J}_{r_1,n}^{F_1}{x}_n,\hfill \\ y_n={\eta }_n{P}_C\left(z_n-{\zeta }_{n}A{z}_n\right)+\left(1-{\eta }_n\right)P_C\left(z_n-{\delta }_{n}B{z}_n\right),\hfill \\ x_{n+1}={\alpha }_{n}f\left(T_{{\mu }_n}{W}_n{y}_n\right)+{\beta }_n{x}_n+{\gamma }_n{T}_{{\mu }_n}{W}_n{y}_n,n\ge 1,\hfill \end{array}\right.$$

(5)

where A: C → H be β-inverse monotone map and B : C → H 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 ${\left\{J_{r_{k,n}}^{F_k}{x}_n\right\}}_{k=1}^M$ converge strongly to $x^{*}\in \mathcal{F}={\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{Fix}\left(\phi \right)\cap \text{EP}\left(\mathcal{J}\right)\cap \text{VI}\left(C,A\right)\cap \text{VI}\left(C,B\right)$, where $x^{*}=P_{\mathcal{F}}f\left(x^{*}\right)$.

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

$$\left(l_{s}f\right)\left(t\right)=f\left(st\right),\left(r_{s}f\right)\left(t\right)=f\left(ts\right),\forall t\in S.$$

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

$$\underset{\alpha }{\text{lim}}∥l_s^{*}{\mu }_{\alpha }-{\mu }_{\alpha }∥=0,$$

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.

$$\text{Fix}\left(\phi \right)=\bigcap _{t\in S}\left\{x\in C:T_{t}x=x\right\}.$$

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.

• (i) T _μ is nonexpansive mapping from C into C.

• (ii) T _μ x = x for each x ∈ Fix(φ).

• (iii) $T_{\mu }x\in \overline{co}\left\{T_{t}x:t\in S\right\}$ for each x ∈ C.

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

$$x_n\rightharpoonup u\in C,T{x}_n\to v\text{imply}Tu=v,$$

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 : C → H is nonexpansive. then, the mapping I - T is demiclosed at zero.

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

$$y=P_{C}x\iff 〈y-x,z-y〉\ge 0,\forall z\in C.$$

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

$${∥P_{C}x-P_{C}y∥}^2\le 〈P_{C}x-P_{C}y,x-y〉,\forall x,y\in H.$$

(6)

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

$$〈x-P_{C}x,y-P_{C}x〉\le 0.$$

(7)

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

$${∥x-y∥}^2\ge {∥x-P_{C}x∥}^2+{∥y-P_{C}x∥}^2.$$

(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:

$$u\in \text{VI}\left(C,A\right)\iff u=P_C\left(u-\lambda Au\right),\lambda >0.$$

(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 $0<\underset{n\to \infty }{\text{lim}\text{inf}}{\alpha }_n\le \underset{n\to \infty }{\text{lim}\text{sup}}{\alpha }_n<1$. Suppose x _{n +1} = α _n x _n +(1-α _n )y _n for all integers n ≥ 0 and

$$\underset{n\to \infty }{\text{lim}\text{sup}}\left(∥y_{n+1}-y_n∥-∥x_{n+1}-x_n∥\right)\le 1.$$

Then, $\underset{n\to \infty }{\text{lim}}∥y_n-x_n∥=0$.

Let F : C × C → ℝ be a bi-function. Given any r > 0, the operator $J_r^F:H\to C$ defined by

$$J_r^{F}x=\left\{z\in C:F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0,\forall y\in C\right\}$$

is called the resolvent of F, see [3]. The equilibrium problem for F is to determine its equilibrium points, i.e., the set

$$\text{EP}\left(F\right)=\left\{x\in C:F\left(x,y\right)\ge 0,\forall y\in C\right\}.$$

Let $\mathcal{J}={\left\{F_i\right\}}_{i\in I}$ be a family of bi-functions from C × C into ℝ. The system of equilibrium problems for is to determine common equilibrium points for $\mathcal{J}={\left\{F_i\right\}}_{i\in I}$. i.e, the set

$$\text{EP}\left(\mathcal{J}\right)=\left\{x\in C:F_i\left(x,y\right)\ge 0,\forall y\in C,\forall i\in I\right\}.$$

Lemma 2.5. [3] Let C be a nonempty closed convex subset of H and F : C × C → ℝ 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, lim_t→0F(tz + (1 - t)x, y) ≤ F (x, y),

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

Given r > 0, define the operator $J_r^F:H\to C$, the resolvent of F, by

$$J_r^F\left(x\right)=\left\{z\in C:F\left(z,y\right)+\frac{1}{r}〈y-z,z-x〉\ge 0,\forall y\in C\right\}.$$

Then,

• (1) $J_r^F$ is single valued,

• (2) $J_r^F$ is firmly nonexpansive, i.e, ${∥J_r^{F}x-J_r^{F}y∥}^2\le 〈J_r^{F}x-J_r^{F}y,x-y〉$ for all x, y ∈ H,

• (3) $\text{Fix}\left(J_r^F\right)=\text{EP}\left(F\right)$,

• (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 ∈ ℕ. For any n ∈ ℕ, define a mapping W _n of C into C as follows:

$$\begin{array}{c}{U}_{n,n+1}=I,\hfill \\ U_{n,n}={\lambda }_n{T}_n{U}_{n,n+1}+\left(1-{\lambda }_n\right)I,\hfill \\ U_{n,n-1}={\lambda }_{n-1}{T}_{n-1}{U}_{n,n}+\left(1-{\lambda }_{n-1}\right)I,\hfill \\ ⋮\hfill \\ U_{n,k}={\lambda }_k{T}_k{U}_{n,k+1}+\left(1-{\lambda }_k\right)I,\hfill \\ U_{n,k-1}={\lambda }_{k-1}{T}_{k-1}{U}_{n,k}+\left(1-{\lambda }_{k-1}\right)I,\hfill \\ ⋮\hfill \\ U_{n,2}={\lambda }_2{T}_2{U}_{n,3}+\left(1-{\lambda }_2\right)I,\hfill \\ W_n=U_{n,1}={\lambda }_1{T}_1{U}_{n,2}+\left(1-{\lambda }_1\right)I.\hfill \end{array}$$

(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 : C → C} be an infinite family of nonexpansive mappings with ${\cap }_{i=1}^{\infty }Fix\left(T_i\right)\ne \varnothing$, {λ _i } be a real sequence such that 0 < λ _i ≤ b < 1, ∀i ≥ 1. Then

• (1) W _n is nonexpansive and $Fix\left(W_n\right)={\cap }_{i=1}^n\text{Fix}\left(T_i\right)$ for each n ≥ 1,

• (2) for each x ∈ C and for each positive integer j, the limit lim _{n →∞} U _{n , j} exists.

• (3) The mapping W : C → C defined by

  $$Wx:=\underset{n\to \infty }{\text{lim}}{W}_{n}x=\underset{n\to \infty }{\text{lim}}{U}_{n,1}x,\forall x\in C,$$

is a nonexpansive mapping satisfying $Fix\left(W\right)={\cap }_{i=1}^{\infty }Fix\left(T_i\right)$ 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 : C → C} be a countable family of nonexpansive mappings with ${\cap }_{i=1}^{\infty }Fix\left(T_i\right)\ne \varnothing$,{λ _i } be a real sequence such that 0 < λ _i ≤ b < 1, ∀i ≥ 1. If D is any bounded subset of C, then

$$\underset{n\to \infty }{\text{lim}}\underset{x\in D}{\text{sup}}∥Wx-W_{n}x∥=0.$$

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

$$a_{n+1}\le \left(1-b_n\right)a_n+b_n{c}_n,n\ge 0,$$

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

• (i) {b _n } ⊂ [0, 1], $\sum _{n=0}^{\infty }{b}_n=\infty$,

• (ii) either $\underset{n\to \infty }{\text{lim}\text{sup}}{c}_n\le 0$ or $\sum _{n=0}^{\infty }\left|b_n{c}_n\right|<\infty$.

Then, $\underset{n\to \infty }{\text{lim}}{a}_n=0$.

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

$$\begin{array}{ll}\hfill {∥\alpha x+\beta y+\gamma z∥}^2=& \alpha {∥x∥}^2+\beta {∥y∥}^2+\gamma {∥z∥}^2\\ -\alpha \beta {∥x-y∥}^2-\alpha \gamma {∥x-z∥}^2-\beta \gamma {∥y-z∥}^2.\end{array}$$

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 $\overline{co}A$ the closed convex hull of A. For ε > 0 and a mapping T : D → H, 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: C → H a β-inverse strongly monotone, B : C → H a γ-inverse strongly monotone, S a semigroup and φ = {T _t : t ∈ S} be a nonexpansive semigroup from C into C such that $\text{Fix}\left(\phi \right)={\cap }_{t\in S}\text{Fix}\left(T_t\right)\ne \varnothing$. Let X be a left invariant subspace of B(S) such that 1 ∈ X, and the function t → 〈T _t x, y〉 is 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 $\mathcal{J}=\left\{F_k:k=1,2,\dots ,M\right\}$ be a finite family of bi-functions from C × C into ℝ which satisfy (A₁)-(A₄) and ${\left\{T_i\right\}}_{i=1}^{\infty }$ an infinite family of nonexpansive mappings of C into C such that $T_i\left(Fix\left(\phi \right)\cap \text{EP}\left(\mathcal{J}\right)\right)\subset \text{Fix(}\phi \text{)}$ for each i ∈ ℕ and $\mathcal{F}={\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{Fix}\left(\phi \right)\cap \text{EP}\left(\mathcal{J}\right)\cap \text{VI}\left(C,A\right)\cap \text{VI}\left(C,B\right)\ne \varnothing$. 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), ${\left\{r_{k,n}\right\}}_{k=1}^M$ 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 ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$,

(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

$$\left\{\begin{array}{c}{z}_n=J_{r_M,n}^{F_M}\dots J_{r_2,n}^{F_2}{J}_{r_1,n}^{F_1}{x}_n,\hfill \\ y_n={\eta }_n{P}_C\left(z_n-{\zeta }_{n}A{z}_n\right)+\left(1-{\eta }_n\right)P_C\left(z_n-{\delta }_{n}B{z}_n\right),\hfill \\ x_{n+1}={\alpha }_{n}f\left(T_{{\mu }_n}{W}_n{y}_n\right)+{\beta }_n{x}_n+{\gamma }_n{T}_{{\mu }_n}{W}_n{y}_n,n\ge 1,\hfill \end{array}\right.$$

(11)

then, the sequences {x _n }, {y _n } and ${\left\{J_{r_{k,n}}^{F_k}{x}_n\right\}}_{k=1}^M$ converge strongly to $x^{*}\in \mathcal{F}$, which is the unique solution of the system of variational inequalities:

$$\left\{\begin{array}{c}〈f\left(x^{*}\right)-x^{*},x^{*}-y〉\ge 0,\forall y\in \mathcal{F},\hfill \\ 〈B{x}^{*},y-x^{*}〉\ge 0\forall y\in C,\hfill \\ 〈A{x}^{*},y-x^{*}〉\ge 0\forall y\in C.\hfill \end{array}\right.$$

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

$$\begin{array}{c}{∥\left(I-{\zeta }_{n}A\right)x-\left(I-{\zeta }_{n}A\right)y∥}^2\hfill \\ ={∥\left(x-y\right)-{\zeta }_n\left(Ax-Ay\right)∥}^2\hfill \\ ={∥|x-y∥}^2-2{\zeta }_n〈x-y,Ax-Ay〉+{\zeta }_n^2{∥Ax-Ay∥}^2\hfill \\ \le {∥x-y∥}^2-2{\zeta }_n\beta {∥Ax-Ay∥}^2+{\zeta }_n^2{∥Ax-Ay∥}^2\hfill \\ ={∥x-y∥}^2+{\zeta }_n\left({\zeta }_n-2\beta \right){∥Ax-Ay∥}^2\hfill \\ \le {∥x-y∥}^2\hfill \end{array}$$

It follows that

$$∥\left(I-{\zeta }_{n}A\right)x-\left(I-{\zeta }_{n}A\right)y∥\le ∥x-y∥.$$

(12)

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

$$∥\left(I-{\delta }_{n}B\right)x-\left(I-{\delta }_{n}B\right)y∥\le ∥x-y∥.$$

(13)

Let $p\in \mathcal{F}$, 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

$$\begin{array}{ll}\hfill ∥y_n-p∥& =∥{\eta }_n{P}_C\left(z_n-{\zeta }_{n}A{z}_n\right)+\left(1-{\eta }_n\right)P_C\left(z_n-{\delta }_{n}B{z}_n\right)-p∥\\ =∥{\eta }_n\left[P_C\left(z_n-{\zeta }_{n}A{z}_n\right)-P_C\left(p-{\zeta }_{n}Ap\right)\right]\\ +\left(1-{\eta }_n\right)\left[P_C\left(z_n-{\delta }_{n}B{z}_n\right)-P_C\left(p-{\delta }_{n}Bp\right)\right]∥\\ \le {\eta }_n∥P_C\left(z_n-{\zeta }_{n}A{z}_n\right)-P_C\left(p-{\zeta }_{n}Ap\right)∥\\ +\left(1-{\eta }_n\right)∥P_C\left(z_n-{\delta }_{n}B{z}_n\right)-P_C\left(p-{\delta }_{n}Bp\right)∥\\ \le {\eta }_n∥z_n-p∥+\left(1-{\eta }_n\right)∥z_n-p∥=∥z_n-p∥.\end{array}$$

(14)

By taking v _n = P _C (z _n - ζ _n Az _n ), w _n = P _C (z _n - δ _n Bz _n ) and ${\mathcal{J}}_n^k=J_{r_{k,n}}^{F_k}\cdots J_{r_{2,n}}^{F_2}{J}_{r_{1,n}}^{F_1}$ for k ∈ {1, 2, ..., M} and ${\mathcal{J}}_n^0=I$ for all n ∈ ℕ, we shall equivalently write scheme (11) as follows:

$$\begin{array}{c}{z}_n={\mathcal{J}}_n^M{x}_n,\hfill \\ y_n={\eta }_n{v}_n+\left(1-{\eta }_n\right)w_n,\hfill \\ x_{n+1}={\alpha }_{n}f\left(T_{{\mu }_n}{W}_n{y}_n\right)+{\beta }_n{x}_n+{\gamma }_n{T}_{{\mu }_n}{W}_n{y}_n,n\ge 1.\hfill \end{array}$$

We shall divide the proof into several steps.

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

Proof of Step 1. Let $p\in \mathcal{F}$. Since for each k ∈ {1, 2, ..., M}, $J_{r_{k,n}}^{F_k}$ is nonexpansive we have

$$∥{\mathcal{J}}_n^k{x}_n-p∥=∥{\mathcal{J}}_n^k{x}_n-{\mathcal{J}}_n^{k}p∥\le ∥x_n-p∥,\forall k\in \left\{1,2,\dots ,M\right\}.$$

(15)

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

$$\begin{array}{c}∥x_{n+1}-p∥\hfill \\ \le {\alpha }_n∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥+{\beta }_n∥x_n-p∥+{\gamma }_n∥T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n-p∥\hfill \\ \le {\alpha }_n\left[∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-f\left(p\right)∥+∥f\left(p\right)-p∥\right]+{\beta }_n∥x_n-p∥+{\gamma }_n∥y_n-p∥\hfill \\ \le {\alpha }_n\alpha ∥x_n-p∥+{\alpha }_n∥f\left(p\right)-p∥+\left({\beta }_n+{\gamma }_n\right)∥x_n-p∥\hfill \\ =\left[1-{\alpha }_n\left(1-\alpha \right)\right]∥x_n-p∥+{\alpha }_n∥f\left(p\right)-p∥\hfill \\ \le \text{max}\left\{∥x_n-p∥,\frac{1}{1-\alpha }∥f\left(p\right)-p∥\right\}.\hfill \end{array}$$

By induction,

$$∥x_n-p∥\le \text{max}\left\{∥x_1-p∥,\frac{1}{1-\alpha }∥f\left(p\right)-p∥\right\},n\ge 1.$$

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

$$\underset{n\to \infty }{\text{lim}}∥{\mathcal{J}}_{n+1}^k{u}_n-{\mathcal{J}}_n^k{u}_n∥=0,$$

(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

$$\underset{n\to \infty }{\text{lim}}∥W_{n+1}{u}_n-W_n{u}_n∥=0\text{and li}{\text{m}}_{n\to \infty }∥T_{{\mu }_{n+1}}{u}_n-T_{{\mu }_n}{u}_n∥=0.$$

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

Step 4. lim_n→∞∥ 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

$$\begin{array}{c}{t}_{n+1}-t_n\hfill \\ =\frac{1}{1-{\beta }_{n+1}}\left[x_{n+2}-{\beta }_{n+1}{x}_{n+1}\right]-\frac{1}{1-{\beta }_n}\left[x_{n+1}-{\beta }_n{x}_n\right]\hfill \\ =\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left[f\left(T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}\right)-f\left(T_{{\mu }_n}{W}_n{y}_n\right)\right]+\left(\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}\right)f\left(T_{{\mu }_n}{W}_n{y}_n\right)\hfill \\ +\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}\left[T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_n}{W}_n{y}_n\right]\hfill \\ +\left[\frac{{\gamma }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\gamma }_n}{1-{\beta }_n}\right]T_{{\mu }_n}{W}_n{y}_n.\hfill \end{array}$$

Therefore, we have

$$\begin{array}{c}∥t_{n+1}-t_n∥\hfill \\ \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left[\alpha ∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_n}{W}_n{y}_n∥+∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_n}{W}_n{y}_n∥\right]\hfill \\ +\left|\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}\right|\left[∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)∥+∥T_{{\mu }_n}{W}_n{y}_n∥\right]\hfill \\ +∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_n}{W}_n{y}_n∥.\hfill \end{array}$$

On the other hand

$$\begin{array}{c}∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_n}{W}_n{y}_n∥\hfill \\ \le ∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_{n+1}}{W}_{n+1}{y}_n∥\hfill \\ +∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_n-T_{{\mu }_{n+1}}{W}_n{y}_n∥\hfill \\ +∥T_{{\mu }_{n+1}}{W}_n{y}_n-T_{{\mu }_n}{W}_n{y}_n∥\hfill \\ \le ∥y_{n+1}-y_n∥+∥W_{n+1}{y}_n-W_n{y}_n∥+∥T_{{\mu }_{n+1}}{W}_n{y}_n-T_{{\mu }_n}{W}_n{y}_n∥\hfill \end{array}$$

Observing that $z_n={\mathcal{J}}_n^M{x}_n$, $z_{n+1}={\mathcal{J}}_{n+1}^M{x}_{n+1}$ and ${\mathcal{J}}_n^M{x}_n=J_{r_M,n}^{F_M}{\mathcal{J}}_n^{M-1}{x}_n$ we get

$$\frac{1}{r_{M,n}}〈y-z_n,z_n-{\mathcal{J}}_n^{M-1}{x}_n〉+F_M\left(z_n,y\right)\ge 0,\forall y\in C,$$

(17)

and

$$\frac{1}{r_{M,n+1}}〈y-z_{n+1},z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}〉+F_M\left(z_{n+1},y\right)\ge 0,\forall y\in C,$$

(18)

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

$$〈z_{n+1}-z_n,z_n-{\mathcal{J}}_n^{M-1}{x}_n-\frac{r_{M,n}}{r_{M,n+1}}\left(z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}\right)〉\ge 0,$$

and hence

$$\begin{array}{c}〈z_{n+1}-z_n,z_n-{\mathcal{J}}_n^{M-1}{x}_n-z_{n+1}+{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}\hfill \\ +\left(1-\frac{r_{M,n}}{r_{M,n+1}}\right)\left(z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}\right)〉\ge 0,\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}∥z_{n+1}-z_n∥\hfill \\ \le ∥{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}-{\mathcal{J}}_n^{M-1}{x}_n∥+\left|1-\frac{r_{M,n}}{r_{M,n+1}}\right|∥z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}∥\hfill \\ \le ∥{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_n∥+∥{\mathcal{J}}_{n+1}^{M-1}{x}_n-{\mathcal{J}}_n^{M-1}{x}_n∥\hfill \\ +\left|1-\frac{r_{M,n}}{r_{M,n+1}}\right|∥z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}∥\hfill \\ \le ∥x_{n+1}-x_n∥+∥{\mathcal{J}}_{n+1}^{M-1}{x}_n-{\mathcal{J}}_n^{M-1}{x}_n∥\hfill \\ +\left|1-\frac{r_{M,n}}{r_{M,n+1}}\right|∥z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}∥.\hfill \end{array}$$

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

$$\begin{array}{c}∥y_{n+1}-y_n∥\hfill \\ =∥{\eta }_{n+1}{v}_{n+1}+\left(1-{\eta }_{n+1}\right)w_{n+1}-{\eta }_n{v}_n-\left(1-{\eta }_n\right)w_n∥\hfill \\ =∥{\eta }_{n+1}\left(v_{n+1}-v_n\right)+\left({\eta }_{n+1}-{\eta }_n\right)v_n+\left(1-{\eta }_{n+1}\right)w_{n+1}\hfill \\ -\left(1-{\eta }_{n+1}\right)w_n+\left({\eta }_n-{\eta }_{n+1}\right)w_n∥\hfill \\ \le {\eta }_{n+1}∥v_{n+1}-v_n∥+\left|{\eta }_{n+1}-{\eta }_n\right|\left(∥v_n∥+∥w_n∥\right)\hfill \\ +\left(1-{\eta }_{n+1}\right)∥w_{n+1}-w_n∥\hfill \\ ={\eta }_{n+1}∥P_C\left(z_{n+1}-{\zeta }_{n+1}A{z}_{n+1}\right)-P_C\left(z_n-{\zeta }_{n}A{z}_n\right)∥\hfill \\ +\left|{\eta }_{n+1}-{\eta }_n\right|\left(∥v_n∥+∥w_n∥\right)\hfill \\ +\left(1-{\eta }_{n+1}\right)∥P_C\left(z_{n+1}-{\delta }_{n+1}B{z}_{n+1}\right)-P_C\left(z_n-{\delta }_{n}B{z}_n\right)∥\hfill \\ ={\eta }_{n+1}∥P_C\left(z_{n+1}-{\zeta }_{n+1}A{z}_{n+1}\right)-P_C\left(z_n-{\zeta }_{n+1}A{z}_n\right)\hfill \\ +P_C\left(z_n-{\zeta }_{n+1}A{z}_n\right)-P_C\left(z_n-{\zeta }_{n}A{z}_n\right)∥\hfill \\ +\left|{\eta }_{n+1}-{\eta }_n\right|\left(∥v_n∥+∥w_n∥\right)\hfill \\ +\left(1-{\eta }_{n+1}\right)∥P_C\left(z_{n+1}-{\delta }_{n+1}B{z}_{n+1}\right)-P_C\left(z_n-{\delta }_{n+1}B{z}_n\right)\hfill \\ +P_C\left(z_n-{\delta }_{n+1}B{z}_n\right)-P_C\left(z_n-{\delta }_{n}B{z}_n\right)∥\hfill \\ \le {\eta }_{n+1}∥z_{n+1}-z_n∥+{\eta }_{n+1}\left|{\zeta }_{n+1}-{\zeta }_n\right|∥A{z}_n∥\hfill \\ +\left|{\eta }_{n+1}-{\eta }_n\right|\left(∥v_n∥+∥w_n∥\right)+\left(1-{\eta }_{n+1}\right)∥z_{n+1}-z_n∥\hfill \\ +\left(1-{\eta }_{n+1}\right)\left|{\delta }_{n+1}-{\delta }_n\right|∥B{z}_n∥\hfill \\ \le ∥z_{n+1}-z_n∥+{\eta }_{n+1}\left|{\zeta }_{n+1}-{\zeta }_n\right|∥A{z}_n∥\hfill \\ +\left|{\eta }_{n+1}-{\eta }_n\right|\left(∥v_n∥+∥w_n∥\right)+\left|{\delta }_{n+1}-{\delta }_n\right|∥B{z}_n∥.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}∥t_{n+1}-t_n∥-∥x_{n+1}-x_n∥\hfill \\ \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left[∥y_{n+1}-y_n∥+∥T_{{\mu }_{n+1}}{W}_{n+1}{y}_{n+1}-T_{{\mu }_n}{W}_n{y}_n∥\right]\hfill \\ +\left|\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n}{1-{\beta }_n}\right|\left[∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)∥+∥T_{{\mu }_n}{W}_n{y}_n∥\right]\hfill \\ +∥{\mathcal{J}}_{n+1}^{M-1}{x}_n-{\mathcal{J}}_n^{M-1}{x}_n∥+\left|1-\frac{r_{M,n}}{r_{M,n+1}}\right|∥z_{n+1}-{\mathcal{J}}_{n+1}^{M-1}{x}_{n+1}∥\hfill \\ +{\eta }_{n+1}\left|{\zeta }_{n+1}-{\zeta }_n\right|∥A{z}_n∥+\left|{\eta }_{n+1}-{\eta }_n\right|\left(∥v_n∥+∥w_n∥\right)\hfill \\ +\left|{\delta }_{n+1}-{\delta }_n\right|∥B{z}_n∥+∥W_{n+1}{y}_n-W_n{y}_n∥+∥T_{{\mu }_{n+1}}{W}_n{y}_n-T_{{\mu }_n}{W}_n{y}_n∥.\hfill \end{array}$$

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

$$\underset{n\to \infty }{\text{lim}\text{sup}}\left(∥t_{n+1}-t_n∥-∥x_{n+1}-x_n∥\right)\le 0.$$

Hence by Lemma 2.4, we obtain lim _{n →∞} ∥ t _n - x _n ∥ = 0. Consequently,

$$\underset{n\to \infty }{\text{lim}}∥x_{n+1}-x_n∥=\left(1-{\beta }_n\right)∥t_n-x_n∥=0.$$

Step 5. ${\text{lim}}_{n\to \infty }∥{\mathcal{J}}_n^{k+1}{x}_n-{\mathcal{J}}_n^k{x}_n∥=0$, ∀k ∈ {0, 1, 2, ..., M - 1}.

Proof of Step 5. Let $p\in \mathcal{F}$ and k ∈ {1, 2, ..., M - 1}. Since $J_{r_{k+1,n}}^{F_{k+1}}$ is firmly nonexpansive, we obtain

$$\begin{array}{c}{∥{\mathcal{J}}_n^{k+1}{x}_n-p∥}^2\hfill \\ ={∥J_{r_{k+1,n}}^{F_{k+1}}{\mathcal{J}}_n^k{x}_n-J_{r_{k+1,n}}^{F_{k+1}}p∥}^2=〈J_{r_{k+1,n}}^{F_{k+1}}{\mathcal{J}}_n^k{x}_n-p,{\mathcal{J}}_n^k{x}_n-p〉\hfill \\ =\frac{1}{2}\left[{∥J_{r_{k+1,n}}^{F_{k+1}}{\mathcal{J}}_n^k{x}_n-p∥}^2+{∥{\mathcal{J}}_n^k{x}_n-p∥}^2-{∥J_{r_{k+1,n}}^{F_{k+1}}{\mathcal{J}}_n^k{x}_n-{\mathcal{J}}_n^k{x}_n∥}^2\right].\hfill \end{array}$$

It follows that

$${∥{\mathcal{J}}_n^{k+1}{x}_n-p∥}^2\le {∥x_n-p∥}^2-{∥{\mathcal{J}}_n^{k+1}{x}_n-{\mathcal{J}}_n^k{x}_n∥}^2.$$

(19)

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

$$\begin{array}{c}{∥x_{n+1}-p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥T_{{\mu }_n}{W}_n{y}_n-p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥y_n-p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥z_n-p∥}^2\hfill \\ ={\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥{\mathcal{J}}_n^M{x}_n-p∥}^2\hfill \\ ={\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2\hfill \\ +{\gamma }_n{∥J_{r_M,n}^{F_M}\dots J_{r_{k+2},n}^{F_{k+2}}{\mathcal{J}}_n^{k+1}{y}_n-J_{r_M,n}^{F_M}\dots J_{r_{k+2},n}^{F_{k+2}}p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥{\mathcal{J}}_n^{k+1}{y}_n-p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥{\mathcal{J}}_n^{k+1}{x}_n-p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2\hfill \\ +{\gamma }_n\left[{∥x_n-p∥}^2-∥{\mathcal{J}}_n^{k+1}{x}_n-{\mathcal{J}}_n^k{x}_n∥\right].\hfill \end{array}$$

Then, we have

$$\begin{array}{c}{\gamma }_n{∥{\mathcal{J}}_n^{k+1}{x}_n-{\mathcal{J}}_n^k{x}_n∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2\hfill \\ +\left(1-{\alpha }_n-{\beta }_n\right){∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2\hfill \\ ={\alpha }_n\left[{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2-{∥x_n-p∥}^2\right]+{∥x_n-p∥}^2-{∥x_{n+1}-p∥}^2\hfill \\ \le {\alpha }_n\left[{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2-{∥x_n-p∥}^2\right]\hfill \\ +∥x_n-x_{n+1}∥\left[∥x_n-p∥+∥x_{n+1}-p∥\right].\hfill \end{array}$$

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

$$\underset{n\to \infty }{\text{lim}}∥{\mathcal{J}}_n^{k+1}{x}_n-{\mathcal{J}}_n^k{x}_n∥=0.$$

Step 6. ${\text{lim}}_{n\to \infty }∥x_n-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥=0$.

Proof of Step 6. Observe that

$$\begin{array}{c}∥x_n-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥\hfill \\ \le ∥x_n-x_{n+1}∥+∥x_{n+1}-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥\hfill \\ =∥x_n-x_{n+1}∥+∥{\alpha }_n\left[f\left(T_{{\mu }_n}{W}_n{y}_n\right)-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n\right]\hfill \\ +{\beta }_n\left[x_n-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n\right]∥\hfill \\ \le ∥x_n-x_{n+1}∥+{\alpha }_n\left[∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)∥+∥T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥\right]\hfill \\ +{\beta }_n∥x_n-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥,\hfill \end{array}$$

hence

$$\begin{array}{c}∥x_n-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥\hfill \\ \le \frac{1}{1-{\beta }_n}∥x_{n+1}-x_n∥+\frac{{\alpha }_n}{1-{\beta }_n}\left[∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)∥+∥T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥\right].\hfill \end{array}$$

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

$$\underset{n\to \infty }{\text{lim}}∥x_n-T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n∥=0.$$

Step 7. lim_n→∞∥ x _n - T _t x _n ∥ = 0, for all t ∈ S.

Proof of Step 7. Let $p\in \mathcal{F}$ and set $M_0=\text{max}\left\{∥x_1-p∥,\frac{1}{1-\alpha }∥f\left(p\right)-p∥\right\}$ and D = {y ∈ H : ∥ y - p ∥ ≤ M₀}, we remark that D is bounded closed convex set, {y _n } ⊂ D and it is invariant under $\left\{J_{r_{k,n}}^{F_k}:k=1,2,...,M,\forall n\in \mathbb{N}\right\}$, φ and W _n for all n ∈ ℕ. We will show that

$$\underset{n\to \infty }{\text{lim}\text{sup}}\underset{y\in D}{\text{sup}}∥T_{{\mu }_n}y-T_t{T}_{{\mu }_n}y∥=0,\forall t\in S$$

(20)

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

$$\overline{co}{F}_{\delta }\left(T_t;D\right)+B_{\delta }\subset F_{\epsilon }\left(T_t;D\right),\forall t\in S.$$

(21)

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

$$∥\frac{1}{N+1}\sum _{i=0}^N{T}_{t^{i}s}y-T_t\left(\frac{1}{N+1}\sum _{i=0}^N{T}_{t^{i}s}y\right)∥\le \delta ,$$

(22)

for all t, s ∈ S and y ∈ D. Let t ∈ S. Since {μ _n } is strongly left regular, there exists N₀ ∈ ℕ such that $∥{\mu }_n-l_{t^i}^{*}{\mu }_n∥\le \frac{\delta }{\left(M_0+\left|\right|p\left|\right|\right)}$ for n ≥ N₀ and i = 1, 2, ..., N. Then, we have

$$\begin{array}{l}\underset{y\in D}{\sup }\Vert T_{{\mu }_n}y-{\displaystyle \int }\frac{1}{N+1}{\displaystyle \sum }_{i=0}^N{T}_{t^{i}s}yd{\mu }_n(s)\Vert \hfill \\ =\underset{y\in D}{\sup }\underset{\left|\right|z\left|\right|=1}{\sup }|〈T_{{\mu }_n}y,z〉-〈{\displaystyle \int }\frac{1}{N+1}{\displaystyle \sum }_{i=0}^N{T}_{t^{i}s}yd{\mu }_n(s),z〉|\hfill \\ =\underset{y\in D}{\sup }\underset{\left|\right|z\left|\right|=1}{\sup }|\frac{1}{N+1}{\displaystyle \sum }_{i=0}^N{({\mu }_n)}_s〈T_{s}y,z〉-\frac{1}{N+1}{\displaystyle \sum }_{i=0}^N{({\mu }_n)}_s〈T_{t^{i}s}y,z〉|\hfill \\ \le \frac{1}{N+1}{\displaystyle \sum }_{i=0}^N\underset{y\in D}{\sup }\underset{\left|\right|z\left|\right|=1}{\sup }\left|{({\mu }_n)}_s〈T_{s}y,z〉-{(l_{t^i}^{\ast }{\mu }_n)}_s〈T_{s}y,z〉\right|\hfill \\ \le \underset{i=1,2,\dots ,N}{\max }\left|\right|{\mu }_n-l_{t^i}^{\ast }{\mu }_n\left|\right|(M_0+|\left|p\right||)\le \delta ,\forall n\ge N_0.\hfill \end{array}$$

(23)

By Lemma 2.1 we have

$$\int \frac{1}{N+1}\sum _{i=0}^N{T}_{t^{i_S}}yd{\mu }_n\left(s\right)\in \overline{co}\left\{\frac{1}{N+1}\sum _{i=0}^N{T}_{t^i}\left(T_{s}y\right):s\in S\right\}.$$

(24)

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

$$\begin{array}{c}{T}_{{\mu }_n}y\in \overline{co}\left\{\frac{1}{N+1}\sum _{i=0}^N{T}_{t^{i_S}}y:s\in S\right\}+B_{\delta }\hfill \\ \subset \overline{co}{F}_{\delta }\left(T_t;D\right)+B_{\delta }\subset F_{\epsilon }\left(T_t;D\right),\hfill \end{array}$$

for all y ∈ D and n ≥ N₀. Therefore,

$$\underset{n\to \infty }{\text{lim}\text{sup}}\underset{y\in D}{\text{sup}}∥T_t\left(T_{{\mu }_n}y\right)-T_{{\mu }_n}y∥\le \epsilon .$$

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 ∈ ℕ such that ${\alpha }_n<\frac{\delta }{4M_0}$, $T_{{\mu }_n}y\in F_{\delta }\left(T_t,D\right)$ for all y ∈ D and $∥x_n-T_{{\mu }_n}{W}_n{y}_n∥<\frac{\delta }{2}$ for all n ≥ N₁. We note that

$$\begin{array}{c}{\alpha }_n∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-T_{{\mu }_n}{W}_n{y}_n∥\hfill \\ \le {\alpha }_n\left[∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-f\left(p\right)∥+∥f\left(p\right)-p∥+∥p-T_{{\mu }_n}{W}_n{y}_n∥\right]\hfill \\ \le {\alpha }_n\left[\alpha ∥y_n-p∥+∥f\left(p\right)-p∥+∥p-y_n∥\right]\le 2M_0{\alpha }_n\le \frac{\delta }{2},\hfill \end{array}$$

for all n ≥ N₁. Therefore, we have

$$\begin{array}{ll}\hfill x_{n+1}& =T_{{\mu }_n}{W}_n{y}_n+{\alpha }_n\left(f\left(T_{{\mu }_n}{W}_n{y}_n\right)-T_{{\mu }_n}{W}_n{y}_n\right)\\ +{\beta }_n\left(x_n-T_{{\mu }_n}{W}_n{y}_n\right)\\ \in F_{\delta }\left(T_t;D\right)+B_{\frac{\delta }{2}}+B_{\frac{\delta }{2}}\subset F_{\delta }\left(T_t;D\right)+B_{\delta }\subset F_{\epsilon }\left(T_t;D\right),\end{array}$$

for all n ≥ N₁. This shows that

$$∥x_n-T_t{x}_n∥\le \epsilon ,\forall n\ge N_1.$$

Since ϵ > 0 is arbitrary, we get lim _{n →∞} ∥ x _n - T _t (x _n ) ∥ = 0.

Step 8. The weak ω-limit set of {x _n }, ω _ω {x _n }, is a subset of .

Proof of Step 8. Let z ∈ ω _ω {x _n } and let $\left\{x_{n_m}\right\}$ be a subsequence of {x _n } weakly converging to z, we need to show that $z\in \mathcal{F}$. Noting Step 5, with no loss of generality, we may assume that ${\mathcal{J}}_{n_m}^k{x}_{n_m}\rightharpoonup z,\forall k\in \left\{1,2,\dots ,M\right\}$. At first, note that by (A₂) and given y ∈ C and k ∈ {1, 2, ..., M}, we have

$$〈y-{\mathcal{J}}_{n_m}^{k+1}{x}_{n_m},\frac{1}{r_{k+1,n_m}}\left({\mathcal{J}}_{n_m}^{k+1}{x}_{n_m}-{\mathcal{J}}_{n_m}^k{x}_{n_m}\right)〉\ge F_{k+1}\left(y,{\mathcal{J}}_{n_m}^{k+1}{x}_{n_m}\right).$$

Step 5 and condition(B₅) imply that

$$\frac{{\mathcal{J}}_{n_m}^{k+1}{x}_{n_m}-{\mathcal{J}}_{n_m}^k{x}_{n_m}}{r_{k+1,n_m}}\to 0.$$

Since ${\mathcal{J}}_{n_m}^k{x}_{n_m}\rightharpoonup z$, from the lower semi-continuity of F _{k +1} on the second variable, we have F _{k +1}(y, z) ≤ 0 for all y ∈ C and for all k ∈ {0, 1, 2, ..., M - 1}. For t with 0 < t ≤ 1 and y ∈ C, let y _t = ty + (1 - t)z. Since y ∈ C and z ∈ C, we have y _t ∈ C and hence F _{k +1}(y _t , z) ≤ 0. So from the convexity of F _{k +1} on second variable, we have

$$0=F_{k+1}\left(y_t,y_t\right)\le t{F}_{k+1}\left(y_t,y\right)+\left(1-t\right)F_{k+1}\left(y_t,z\right)\le t{F}_{k+1}\left(y_t,y\right)\le F_{k+1}\left(y_t,y\right).$$

hence F _{k +1}(y _t , y) ≥ 0. therefore, we have F _{k +1}(z, y) ≥ 0 for all y ∈ C and k ∈ {0, 1, 2, ..., M- 1}. Therefore $z\in {\cap }_{k=1}^M\text{EP}\left(F_k\right)=\text{EP}\left(\mathcal{J}\right)$.

Since $x_{n_m}\rightharpoonup z$, it follows by Step 7 and Lemma 2.2 that z ∈ Fix(T _t ) for all t ∈ S. Therefore, z ∈ Fix(φ). We will show z ∈ Fix(W). Assume z ∉ Fix(W) Since $z\in \text{Fix}\left(\phi \right)\cap \text{EP}\left(\mathcal{J}\right)$, by our assumption, we have T _i z ∈ Fix(φ),∀i ∈ ℕ and then W _n z ∈ Fix(φ). Hence by Lemma 2.1, $T_{{\mu }_n}{W}_{n}z=W_{n}z$, therefore by Lemma 2.5, we get

$$T_{{\mu }_n}{W}_n{\mathcal{J}}_n^{M}z=W_{n}z,\forall n\in \mathbb{N}.$$

(25)

Now, by (25), Step 6, Lemma 2.6 and Opial's condition, we have

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}\text{inf}}∥x_{n_m}-z∥\hfill \\ <\underset{n\to \infty }{\text{lim}\text{inf}}∥x_{n_m}-Wz∥\hfill \\ \le \underset{n\to \infty }{\text{lim}\text{inf}}\left(∥x_{n_m}-T_{{\mu }_{n_m}}{W}_{n_m}{\mathcal{J}}_{n_m}^M{x}_{n_m}∥\right.\hfill \\ \left.+∥T_{{\mu }_{n_m}}{W}_{n_m}{\mathcal{J}}_{n_m}^M{x}_{n_m}-T_{{\mu }_{n_m}}{W}_{n_m}{\mathcal{J}}_{n_m}^{M}z∥+∥T_{{\mu }_{n_m}}{W}_{n_m}{\mathcal{J}}_{n_m}^{M}z-Wz∥\right)\hfill \\ \le \underset{n\to \infty }{\text{lim}\text{inf}}\left(∥x_{n_m}-T_{{\mu }_{n_m}}{W}_{n_m}{\mathcal{J}}_{n_m}^M{x}_{n_m}∥+∥x_{n_m}-z∥+∥W_{n_m}z-Wz∥\right)\hfill \\ \le \underset{n\to \infty }{\text{lim}\text{inf}}∥x_{n_m}-z∥.\hfill \end{array}$$

This is a contradiction. So we get $z\in \text{Fix}\left(W\right)=\bigcap _{i=1}^{\infty }\text{Fix}\left(T_i\right)$.

Now, let us show that z ∈ VI(C, A) ∩ VI(C, B). Observe that,

$$\begin{array}{l}{\Vert x_{n+1}-p\Vert }^2\hfill \\ \le {\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2+{\gamma }_n{\Vert T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n-p\Vert }^2\hfill \\ \le {\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2+{\gamma }_n{\Vert y_n-p\Vert }^2\hfill \\ ={\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2+{\gamma }_n\Vert {\eta }_n{P}_C(z_n-{\zeta }_{n}A{z}_n)\hfill \\ {+(1-{\eta }_n)P_C(z_n-{\delta }_{n}B{z}_n)-p\Vert }^2\hfill \\ ={\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2+{\gamma }_n\Vert {\eta }_n[P_C(z_n-{\zeta }_{n}A{z}_n)\hfill \\ {-P_C(p-{\zeta }_{n}Ap)]+(1-{\eta }_n)[P_C(z_n-{\delta }_{n}B{z}_n)-P_C(p-{\delta }_{n}Bp)]\Vert }^2.\hfill \end{array}$$

(26)

From (26), we have

$$\begin{array}{l}{\Vert x_{n+1}-p\Vert }^2\hfill \\ \le {\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2\hfill \\ +{\gamma }_n[{\eta }_n{\Vert (z_n-p)-{\zeta }_n(A{z}_n-Ap)\Vert }^2+(1-{\eta }_n){\Vert z_n-p\Vert }^2]\hfill \\ ={\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2\hfill \\ +{\gamma }_n(1-{\eta }_n){\Vert z_n-p\Vert }^2+{\gamma }_n{\eta }_n[{\Vert z_n-p\Vert }^2+{\zeta }_n^2{\Vert A{z}_n-Ap\Vert }^2\hfill \\ -2{\zeta }_n〈A{z}_n-Ap,z_n-p〉;]\hfill \\ \le {\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2\hfill \\ +{\gamma }_n(1-{\eta }_n){\Vert z_n-p\Vert }^2+{\gamma }_n{\eta }_n[{\Vert z_n-p\Vert }^2+{\zeta }_n^2{\Vert A{z}_n-Ap\Vert }^2\hfill \\ -2{\zeta }_n\beta {\Vert A{z}_n-Ap,z_n-p\Vert }^2]\hfill \\ ={\alpha }_n{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2+{\beta }_n{\Vert x_n-p\Vert }^2\hfill \\ +{\gamma }_n{\Vert z_n-p\Vert }^2+{\zeta }_n({\zeta }_n-2\beta ){\Vert A{z}_n-Ap\Vert }^2\hfill \\ \le {\Vert x_n-p\Vert }^2+{\alpha }_n[{\Vert f(T_{{\mu }_n}{W}_n{y}_n)-p\Vert }^2-{\Vert x_n-p\Vert }^2]\hfill \\ +{\zeta }_n({\zeta }_n-2\beta ){\Vert A{z}_n-Ap\Vert }^2,\hfill \end{array}$$

which implies that

$$\begin{array}{c}-{\zeta }_n\left({\zeta }_n-2\beta \right){∥A{z}_n-Ap∥}^2\hfill \\ \le \left[∥x_n-p∥+∥x_{n+1}-p∥\right]∥x_n-x_{n+1}∥\hfill \\ +{\alpha }_n\left[{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2-{∥x_n-p∥}^2\right].\hfill \end{array}$$

Therefore, from step 4 and condition B₁, we obtain

$$\underset{n\to \infty }{\text{lim}}∥A{z}_n-Ap∥=0.$$

(27)

On the other hand from (26), we have

$$\begin{array}{c}{∥x_{n+1}-p∥}^2\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n\left[{\eta }_n{∥z_n-p∥}^2\right.\hfill \\ \left.+\left(1-{\eta }_n\right){∥\left(z_n-p\right)-{\delta }_n\left(B{z}_n-Bp\right)∥}^2\right]\hfill \\ ={\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n\left[{\eta }_n{∥z_n-p∥}^2\right.\hfill \\ \left.+\left(1-{\eta }_n\right)\left({∥z_n-p∥}^2-2{\delta }_n〈B{z}_n-Bp,z_n-p〉+{\delta }_n^2{∥B{z}_n-Bp∥}^2\right)\right]\hfill \\ \le {\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n\left[{\eta }_n{∥z_n-p∥}^2\right.\hfill \\ \left.+\left(1-{\eta }_n\right)\left({∥z_n-p∥}^2-2{\delta }_n\gamma {∥B{z}_n-Bp∥}^2+{\delta }_n^2{∥B{z}_n-Bp∥}^2\right)\right]\hfill \\ ={\alpha }_n{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2+{\beta }_n{∥x_n-p∥}^2+{\gamma }_n{∥z_n-p∥}^2\hfill \\ +{\delta }_n\left({\delta }_n-2\gamma \right){\gamma }_n\left(1-{\eta }_n\right){∥B{z}_n-Bp∥}^2\hfill \\ \le {∥x_n-p∥}^2+{\alpha }_n\left[{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2-{∥x_n-p∥}^2\right]\hfill \\ +{\delta }_n\left({\delta }_n-2\gamma \right){∥B{z}_n-Bp∥}^2\hfill \end{array}$$

which implies that

$$\begin{array}{c}-{\delta }_n\left({\delta }_n-2\gamma \right){∥B{z}_n-Bp∥}^2\hfill \\ \le \left[∥x_n-p∥+∥x_{n+1}-p∥\right]∥x_n-x_{n+1}∥\hfill \\ +{\alpha }_n\left[{∥f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p∥}^2-{∥x_n-p∥}^2\right].\hfill \end{array}$$

Therefore, from step 4 and condition B₁, we obtain

$$\underset{n\to \infty }{\text{lim}}∥B{z}_n-Bp∥=0.$$

(28)

From (6) and (12), we have

$$\begin{array}{c}{∥v_n-p∥}^2\hfill \\ ={∥P_C\left(z_n-{\zeta }_{n}A{z}_n\right)-P_C\left(p-{\zeta }_{n}Ap\right)∥}^2\hfill \\ \le 〈\left(z_n-{\zeta }_{n}A{z}_n\right)-\left(p-{\zeta }_{n}Ap\right),v_n-p〉\hfill \\ =\frac{1}{2}\left[{∥\left(z_n-{\zeta }_{n}A{z}_n\right)-\left(p-{\zeta }_{n}Ap\right)∥}^2+{∥v_n-p∥}^2\right.\hfill \\ \left.-{∥\left(z_n-{\zeta }_{n}A{z}_n\right)-\left(p-{\zeta }_{n}Ap\right)-\left(v_n-p\right)∥}^2\right]\hfill \\ =\frac{1}{2}\left[{∥z_n-p∥}^2+{∥v_n-p∥}^2-{∥z_n-v_n∥}^2\right.\hfill \\ \left.+2{\zeta }_n〈z_n-v_n,A{z}_n-Ap〉-{\zeta }_n^2{∥A{z}_n-Ap∥}^2\right].\hfill \end{array}$$

So we obtain

$$\begin{array}{ll}\hfill {∥v_n-p∥}^2& \le {∥z_n-p∥}^2-{∥z_n-v_n∥}^2\\ +2{\zeta }_n〈z_n-v_n,A{z}_n-Ap〉-{\zeta }_n^2{∥A{z}_n-Ap∥}^2.\end{array}$$

(29)

By using the same method as (29), we have

$$\begin{array}{ll}\hfill {∥w_n-p∥}^2& \le {∥z_n-p∥}^2-{∥z_n-w_n∥}^2\\ +2{\delta }_n〈z_n-w_n,B{z}_n-Bp〉-{\delta }_n^2{∥B{z}_n-Bp∥}^2.\end{array}$$

(30)

From (29), (30) and definition of y _n , we have,

$$\begin{array}{l}{\Vert y_n-p\Vert }^2\hfill \\ =\Vert {\eta }_n[P_C(z_n-{\zeta }_{n}A{z}_n)-p]\hfill \\ {+(1-{\eta }_n)[P_C(z_n-{\delta }_{n}B{z}_n)-p]\Vert }^2\hfill \\ ={\Vert {\eta }_n(v_n-p)+(1-{\eta }_n)(w_n-p)\Vert }^2\hfill \\ \le {\eta }_n{\Vert v_n-p\Vert }^2+(1-{\eta }_n){\Vert w_n-p\Vert }^2\hfill \\ \le {\eta }_n[{\Vert z_n-p\Vert }^2-{\Vert z_n-v_n\Vert }^2+2{\zeta }_n〈z_n-v_n,A{z}_n-Ap〉;\hfill \\ -{\zeta }_n^2{\Vert A{z}_n-Ap\Vert }^2]+(1-{\eta }_n)[{\Vert z_n-p\Vert }^2-{\Vert z_n-w_n\Vert }^2\hfill \\ +2{\delta }_n〈z_n-w_n,B{z}_n-Bp〉;-{\delta }_n^2{\Vert B{z}_n-Bp\Vert }^2]\hfill \\ \le {\Vert z_n-p\Vert }^2+{\eta }_n[-{\Vert z_n-v_n\Vert }^2+2{\zeta }_n\Vert z_n-v_n\Vert \Vert A{z}_n-Ap\Vert \hfill \\ -{\zeta }_n^2{\Vert A{z}_n-Ap\Vert }^2]+(1-{\eta }_n)[-{\Vert z_n-w_n\Vert }^2\hfill \\ +2{\delta }_n\Vert z_n-w_n\Vert \Vert B{z}_n-Bp\Vert -{\delta }_n^2{\Vert B{z}_n-Bp\Vert }^2]\hfill \end{array}$$

(31)

By (31), we have

$$\begin{array}{l}\left|\right|x_{n+1}-p|{|}^2\hfill \\ \le {\alpha }_n\left|\right|f(T_{{\mu }_n}{W}_n{y}_n)-p|{|}^2+{\beta }_n||x_n-p|{|}^2+{\gamma }_n\left|\right|T_{{\mu }_n}{W}_n{\mathcal{J}}_n^M{y}_n-p|{|}^2\hfill \\ \le {\alpha }_n\left|\right|f(T_{{\mu }_n}{W}_n{y}_n)-p|{|}^2+{\beta }_n||x_n-p|{|}^2+{\gamma }_n\left|\right|z_n-p|{|}^2\hfill \\ +{\gamma }_n{\eta }_n[-||z_n-v_n|{|}^2+2{\zeta }_n\left|\right|z_n-v_n\left|\right|\left|\right|A{z}_n-Ap\left|\right|\hfill \\ -{\zeta }_n^2\left|\right|A{z}_n-Ap|{|}^2]+{\gamma }_n(1-{\eta }_n)[-||z_n-w_n|{|}^2\hfill \\ +2{\delta }_n\left|\right|\left|\right|z_n-w_n\left|\right|\left|\right|B{z}_n-Bp\left|\right|-{\delta }_n^2\left|\right|B{z}_n-Bp|{|}^2]\hfill \\ \le {\alpha }_n\left|\right|f(T_{{\mu }_n}{W}_n{y}_n)-p|{|}^2+{\beta }_n||x_n-p|{|}^2+{\gamma }_n\left|\right|x_n-p|{|}^2\hfill \\ -{\gamma }_n{\eta }_n\left|\right|z_n-v_n|{|}^2+{\gamma }_n{\eta }_n[2{\zeta }_n\left|\right|z_n-v_n\left|\right|\left|\right|A{z}_n-Ap\left|\right|\hfill \\ -{\zeta }_n^2\left|\right|A{z}_n-Ap|{|}^2]-{\gamma }_n(1-{\eta }_n)\left|\right|z_n-w_n|{|}^2\hfill \\ +{\gamma }_n(1-{\eta }_n)[2{\delta }_n|\left|\right||z_n-w_n|\left|\right||B{z}_n-Bp||-{\delta }_n^2||B{z}_n-Bp|{|}^2]\hfill \\ =\left|\right|x_n-p|{|}^2+{\alpha }_n[\left|\right|f(T_{{\mu }_n}{W}_n{y}_n)-p|{|}^2-||x_n-p|{|}^2]\hfill \\ -{\gamma }_n{\eta }_n\left|\right|z_n-v_n|{|}^2+{\gamma }_n{\eta }_n[2{\zeta }_n\left|\right|z_n-v_n\left|\right|\left|\right|A{z}_n-Ap\left|\right|\hfill \\ -{\zeta }_n^2\left|\right|A{z}_n-Ap|{|}^2]-{\gamma }_n(1-{\eta }_n)\left|\right|z_n-w_n|{|}^2\hfill \\ +{\gamma }_n(1-{\eta }_n)[2{\delta }_n|\left|\right||z_n-w_n|\left|\right||B{z}_n-Bp||-{\delta }_n^2||B{z}_n-Bp|{|}^2],\hfill \end{array}$$

which implies that

$$\begin{array}{c}{\gamma }_n{\eta }_n\left|\right|z_n-v_n|{|}^2\hfill \\ \le \left[\left|\right|x_n-p\left|\right|+\left|\right|x_{n+1}-p\left|\right|\right]\left|\right|x_{n+1}-x_n\left|\right|\hfill \\ +{\alpha }_n\left[\left|\right|f\left(T_{{\mu }_n}{W}_n{y}_n\right)-p|{|}^2-||x_n-p|{|}^2\right]\hfill \\ +{\gamma }_n{\eta }_n\left[2{\zeta }_n\left|\right|z_n-v_n\left|\right|\left|\right|A{z}_n-Ap\left|\right|-{\zeta }_n^2\left|\right|A{z}_n-Ap|{|}^2\right]\hfill \\ +{\gamma }_n\left(1-{\eta }_n\right)\left[2{\delta }_n\left|\right|\left|\right|z_n-w_n\left|\right|\left|\right|B{z}_n-Bp\left|\right|-{\delta }_n^2\left|\right|B{z}_n-Bp|{|}^2\right],\hfill \end{array}$$

and

$$\begin{array}{c}{\gamma }_n\left(1-{\eta }_n\right)\left|\right|z_n-w_n|{|}^2\hfill \\ \le \left[\left|\right|x_n-p\left|\right|+\left|\right|x_{n+1}-p\left|\right|\right]\left|\right|x_{n+1}-x_n\left|\right|\hfill \\ +{\gamma }_n{\eta }_n\left[2{\zeta }_n\left|\right|z_n-v_n\left|\right|\left|\right|A{z}_n-Ap\left|\right|-{\zeta }_n^2\left|\right|A{z}_n-Ap|{|}^2\right]\hfill \\ +{\gamma }_n\left(1-{\eta }_n\right)\left[2{\delta }_n\left|\right|\left|\right|z_n-w_n\left|\right|\left|\right|B{z}_n-Bp\left|\right|-{\delta }_n^2\left|\right|B{z}_n-Bp|{|}^2\right].\hfill \end{array}$$

Therefore, from 0 < lim inf _{n →∞} γ _n ≤ lim sup _{n →∞} γ _n < 1, condition B₁, step 4, (27) and (28) we get

$$\underset{n\to \infty }{\text{lim}}\left|\right|z_n-v_n\left|\right|=0and\underset{n\to \infty }{\text{lim}}\left|\right|z_n-w_n\left|\right|=0.$$

(32)

Let U : H → 2^Hbe a set-valued mapping is defined by

$$Ux=\left\{\begin{array}{cc}Ax+N_{C}x,\hfill & x\in C,\hfill \\ \varnothing ,\hfill & x\notin C,\hfill \end{array}\right.$$

where N _C x is the normal cone to C at x ∈ C. Since A is monotone. Thus U is maximal monotone see [1]. Let (x, y) ∈ G(U), hence y - Ax ∈ N _C x and since v _n = P _C (z _n - ζ _n Az _n ) therefore, 〈x - v _n , y - Ax〉 ≥ 0. On the other hand from (7), we have

$$〈x-v_n,v_n-\left(z_n-{\zeta }_{n}A{z}_n\right)〉\ge 0,$$

i.e.,

$$〈x-v_n,\frac{v_n-z_n}{{\zeta }_n}+A{z}_n〉\ge 0$$

Therefore, we have

$$\begin{array}{c}〈x-v_{n_i},y〉\hfill \\ \ge 〈x-v_{n_i},Ax〉\hfill \\ \ge 〈x-v_{n_i},Ax〉-〈x-v_{n_i},\frac{v_{n_i}-z_{n_i}}{{\zeta }_{n_i}}+A{z}_{n_i}〉\hfill \\ =〈x-v_{n_i},Ax-\frac{v_{n_i}-z_{n_i}}{{\zeta }_{n_i}}-A{z}_{n_i}〉\hfill \\ =〈x-v_{n_i},Ax-A{v}_{n_i}〉+〈x-v_{n_i},A{v}_{n_i}-A{z}_{n_i}〉-〈x-v_{n_i},\frac{v_{n_i}-z_{n_i}}{{\zeta }_{n_i}}〉\hfill \\ \ge 〈x-v_{n_i},A{v}_{n_i}-A{z}_{n_i}〉-〈x-v_{n_i},\frac{v_{n_i}-z_{n_i}}{{\zeta }_{n_i}}〉\hfill \\ \ge 〈x-v_{n_i},A{v}_{n_i}-A{z}_{n_i}〉-\left|\right|x-v_{n_i}\left|\right|∥\frac{v_{n_i}-z_{n_i}}{{\zeta }_{n_i}}∥.\hfill \end{array}$$

From (32), we get $\underset{i\to \infty }{\text{lim}}\left|\right|v_{n_i}-z_{n_i}\left|\right|=0$. Noting that $x_{n_i}\rightharpoonup z$ and A is $\frac{1}{\beta }$-lipschitzian, we obtain

$$〈x-z,y〉\ge 0.$$

(33)

Since U is maximal monotone, we have z ∈ U^-10, and hence z ∈ VI(C, A). Let V : H → 2^Hbe a set-valued mapping is defined by

$$Vx=\left\{\begin{array}{cc}Bx+N_{C}x,\hfill & x\in C,\hfill \\ \varnothing ,\hfill & x\notin C,\hfill \end{array}\right.$$

where N _C x is the normal cone to C at x ∈ C. Since B is monotone. Thus U is maximal monotone see [1]. Repeating the same argument as above, we can derive z ∈ VI(C, B). Therefore, $z\in \mathcal{F}$.

Step 9. There exists a unique x* ∈ C such that

$$\underset{n\to \infty }{\text{lim}\text{sup}}〈f\left(x^{*}\right)-x^{*},x_n-x^{*}〉\le 0.$$

Proof of Step 9. Note that f is a contraction mapping with coefficient α ∈ (0, 1). Then $\left|\right|P_{\mathcal{F}}f\left(x\right)-P_{\mathcal{F}}f\left(y\right)\left|\right|\le \left|\right|f\left(x\right)-f\left(y\right)\left|\right|\le \alpha \left|\right|x-y\left|\right|$ for all x, y ∈ H. Therefore $P_{\mathcal{F}}$ is a contraction of H into itself, which implies that there exists a unique element x* ∈ H such that $x^{*}=P_{\mathcal{F}}f\left(x^{*}\right)$. at the same time, we note that x* ∈ C. Using Lemma 2.3, we have

$$〈f\left(x^{*}\right)-x^{*},x^{*}-z〉\ge 0,\forall z\in \mathcal{F}.$$

(34)

We can choose a subsequence $\left\{x_{n_k}\right\}$ of {x _n } such that

$$\underset{n\to \infty }{\text{lim}\text{sup}}〈f\left(x^{*}\right)-x^{*},x_n-x^{*}〉=\underset{k\to \infty }{\text{lim}}〈f\left(x^{*}\right)-x^{*},x_{n_k}-x^{*}〉.$$

Since $\left\{x_{n_k}\right\}$ is bounded, therefore, $\left\{x_{n_k}\right\}$ has subsequence $\left\{x_{n_{k_j}}\right\}$ such that $x_{n_{k_j}}\rightharpoonup z^{*}$. With no loss of generality, we may assume that $x_{n_k}\rightharpoonup z^{*}$. Applying Step 8 and (34), we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}〈f\left(x^{*}\right)-x^{*},x_n-x^{*}〉=〈f\left(x^{*}\right)-x^{*},z^{*}-x^{*}〉\le 0.$$

Step 10, The sequences {x _n } converges strongly to x*, which is obtained in Steep 9.

Proof of Step 10. We have

$$\begin{array}{c}\left|\right|x_{n+1}-x^{*}|{|}^2\hfill \\ =\left|\right|{\alpha }_n\left(f\left(T_{{\mu }_n}{W}_n{y}_n\right)-x^{*}\right)+{\beta }_n\left(x_n-x^{*}\right)+{\gamma }_n\left(T_{{\mu }_n}{W}_n{y}_n-x^{*}\right)|{|}^2\hfill \\ \le \left|\right|{\beta }_n\left(x_n-x^{*}\right)+{\gamma }_n\left(T_{{\mu }_n}{W}_n{y}_n-x^{*}\right)|{|}^2+2{\alpha }_n〈f\left(T_{{\mu }_n}{W}_n{y}_n\right)-x^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left[{\beta }_n\left|\right|x_n-x^{*}\left|\right|+{\gamma }_n\left|\right|T_{{\mu }_n}{W}_n{y}_n-x^{*}\left|\right|\right]}^2\hfill \\ +2{\alpha }_n〈f\left(T_{{\mu }_n}{W}_n{y}_n\right)-f\left(x^{*}\right),x_{n+1}-x^{*}〉+2{\alpha }_n〈f\left(x^{*}\right)-x^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left[{\beta }_n\left|\right|x_n-x^{*}\left|\right|+{\gamma }_n\left|\right|y_n-x^{*}\left|\right|\right]}^2\hfill \\ +2{\alpha }_n\alpha \left|\right|y_n-x^{*}\left|\right|\left|\right|x_{n+1}-x^{*}\left|\right|+2{\alpha }_n〈f\left(x^{*}\right)-x^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left[{\beta }_n\left|\right|x_n-x^{*}\left|\right|+{\gamma }_n\left|\right|x_n-x^{*}\left|\right|\right]}^2\hfill \\ +2{\alpha }_n\alpha \left|\right|x_n-x^{*}\left|\right|\left|\right|x_{n+1}-x^{*}\left|\right|+2{\alpha }_n〈f\left(x^{*}\right)-x^{*},x_{n+1}-x^{*}〉\hfill \\ ={\left(1-{\alpha }_n\right)}^2\left|\right|x_n-x^{*}|{|}^2+{\alpha }_n\alpha \left[\left|\right|x_n-x^{*}|{|}^2+||x_{n+1}-x^{*}|{|}^2\right]\hfill \\ +2{\alpha }_n〈f\left(x^{*}\right)-x^{*},x_{n+1}-x^{*}〉\hfill \end{array}$$

Which implies that

$$\begin{array}{c}\left|\right|x_{n+1}-x^{*}|{|}^2\hfill \\ \le \frac{{\left(1-{\alpha }_n\right)}^2+{\alpha }_n\alpha }{1-{\alpha }_n\alpha }\left|\right|x_n-x^{*}|{|}^2+\frac{2{\alpha }_n}{1-{\alpha }_n\alpha }〈f\left(x^{*}\right)-x^{*},x_{n+1}-x^{*}〉\hfill \\ =\left(1-{\alpha }_n\frac{2-2\alpha }{1-{\alpha }_n\alpha }\right)\left|\right|x_n-x^{*}|{|}^2+{\alpha }_n{\tau }_n,\hfill \end{array}$$

(35)

where

$${\tau }_n=\frac{{\alpha }_n}{1-{\alpha }_n\alpha }\left|\right|x_n-x^{*}|{|}^2+\frac{2}{1-{\alpha }_n\alpha }〈f\left(x^{*}\right)-x^{*},x_{n+1}-x^{*}〉,$$

By Step 9, and condition (B₁), we get lim sup _{n →∞} τ _n ≤ 0. Now applying Lemma 2.8 to (35), we conclude that x _n → x*. Consequently, from $\left|\right|J_{r_k,n}^{F_k}{x}_n-x^{*}\left|\right|\le \left|\right|x_n-x^{*}\left|\right|$, we have $J_{r_k,n}^{F_k}{x}_n\to x^{*}$, for all k ∈ {1, 2, ..., M}.

Corollary 3.2. (see Yao et al. [8]) Let C be a nonempty closed convex subset of a real Hilbert space H, F a bi-functions from C×C into ℝ which satisfy (A₁) - (A₄) and ${\left\{T_i\right\}}_{i=1}^{\infty }$ an infinite family of nonexpansive mapping of C into C such that ${\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{EP}\left(F\right)\ne \varnothing$. Let {α _n }, {β _n } and {γ _n } are three sequences in (0, 1) such that α _n + β _n + γ _n = 1 and {r _n } ⊂ (0, ∞). Suppose the following conditions are satisfied:

(B₁) lim _{n →∞} α _n = 0 and ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$,

(B₂) 0 < lim inf _{n →∞} β _n ≤ lim sup _{n →∞} β _n < 1,

(B₃) lim inf _{n →∞} r _n > 0 and lim _{n →∞} (r _{n +1} - r _n ) = 0.

Let f be a contraction of C into itself with coefficient α ∈ (0, 1) and given x₁ ∈ C arbitrarily. Then the sequence {x _n } generated by

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+{\gamma }_n{W}_n{J}_{r_n}^F{x}_n,n\ge 1.$$

converge strongly to $x^{*}\in {\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{EP}\left(F\right)$, where $x^{*}\in P_{{\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{EP}\left(F\right)}f\left(x^{*}\right)$.

Proof. Take A = B = 0, φ = {I}, F₁ = F and F _k = 0 for k ∈ {2, ..., M} in Theorem 3.1, then we have $T_{{\mu }_n}=I$ and $y_n=z_n=J_{r_m}^k{x}_n$. So from Theorem 3.1 the sequence {x _n } converges strongly to $x^{*}\in {\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{EP}\left(F\right)$, where $x^{*}\in P_{{\cap }_{i=1}^{\infty }\text{Fix}\left(T_i\right)\cap \text{EP}\left(F\right)}f\left(x^{*}\right)$.

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H, $\mathcal{J}=\left\{F_k:k=1,2,\dots ,M\right\}$ be a finite family of bi-functions from C × C into ℝ which satisfy (A₁)-(A₄), T a nonexpansive mappings on C such that $\text{Fix}\left(T\right)\cap \text{EP}\left(\mathcal{J}\right)\ne \varnothing$. Let {α _n }, {β _n } and {γ _n } are three sequences in (0, 1) such that α _n + β _n + γ _n = 1 and ${\left\{r_{k,n}\right\}}_{k=1}^M$ be sequences in (0, ∞). Suppose the following conditions are satisfied:

(B₁) lim _{n →∞} α _n = 0 and ${\sum }_{n=1}^{\infty }{\alpha }_n=\infty$,

(B₂) 0 < lim inf _{n →∞} β _n ≤ lim sup _{n →∞} β _n < 1,

(B₃) 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 H into itself and given x₁ ∈ H arbitrarily. If the sequences {x _n } generated iteratively by

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+{\beta }_n{x}_n+{\gamma }_n\frac{1}{n}\sum _{k=0}^{\infty }{\left(\frac{n-1}{n}\right)}^k{T}^k{J}_{r_{M,n}}^{F_M}\dots J_{r_{2n}}^{F_2},J_{r_{1n}}^{F_1},x_n,n\ge 1.$$

Then, sequences {x _n } and ${\left\{J_{r_k,n}^{F_k}{x}_n\right\}}_{k=1}^M$ converge strongly to $x^{*}\in \text{Fix}\left(T\right)\cap \text{EP}\left(\mathcal{J}\right)$, where $x^{*}=P_{\text{Fix}\left(T\right)\cap \text{EP}\left(\mathcal{J}\right)}f\left(x^{*}\right)$.

Proof. Let S = {0, 1, ...}, φ = {Tⁱ: i ∈ S} and T⁰ = I. For f = (x₀, x₁, ...) ∈ B(S), define

$${\mu }_n\left(f\right)=\frac{1}{n}\sum _{k=0}^{\infty }{\left(\frac{n-1}{n}\right)}^k{x}_k,\forall n\in \mathbb{N}.$$

Then {μ _n } is a regular sequence of means on B(S) such that lim _{n →∞} ∥ μ _{n +} - μ _n ∥ = 0; for more details, see [34]. Next for each x ∈ H and n ∈ ℕ, we have

$$T_{{\mu }_n}x=\frac{1}{n}\sum _{k=0}^{\infty }{\left(\frac{n-1}{n}\right)}^k{T}^{k}x.$$

Take A = B = 0, T _i = I for all i ∈ ℕ in Theorem 3.1 then we have y _n = z _n and W _n = I for all n ∈ ℕ. Therefore, it follows from Theorem 3.1 that the sequences {x _n } and ${\left\{J_{r_k,n}^{F_k}{x}_n\right\}}_{k=1}^M$ converge strongly, as n → ∞ to a point $x^{*}\in \text{Fix}\left(T\right)\cap \text{EP}\left(\mathcal{J}\right)$, where $x^{*}=P_{\text{Fix}\left(T\right)\cap \text{EP}\left(\mathcal{J}\right)}f\left(x^{*}\right)$.

Remark 3.4. Theorem 3.1 improve [8, Theorem 1.2] in the following aspects.

• (a) Our iterative process (11) is more general than Yao et al. process (14) because it can be applied to solving the problem of finding a common element of the set of solutions of systems of equilibrium problems and systems of variational inequalities.

• (b) Our iterative process (11) is very diffident from Yao et al. process (14) because there are left amenable semigroup of nonexpansive mappings.

• (c) Our method of proof is very different from the on in Yao et al. [8] for example we use Corollary 1.1 and Theorem 1.2 of Bruck [33] fore the proof of Theorem 3.1.