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

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.

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 → ℝ be a bifunction, where ℝ is the set of real numbers. The equilibrium problem for F: C × C → ℝ is to find x* ∈ C such that

$$F\left(x^{*},y\right)\ge 0,\forall y\in C.$$

(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 ℝ, where ℝ 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

$$\left\{\begin{array}{c}{F}_1\left(x^{*},y\right)\ge 0,\forall y\in C,\hfill \\ F_2\left(x^{*},y\right)\ge 0,\forall y\in C,\hfill \\ ⋮\hfill \\ F_N\left(x^{*},y\right)\ge 0,\forall y\in C.\hfill \end{array}\right.$$

(1.2)

We denote the set of solutions of (1.2) by ${\cap }_{i=1}^{N}SEP\left(F_i\right)$, where SEP(F _i ) is the set of solutions to the equilibrium problems, that is,

$$F_i\left(x^{*},y\right)\ge 0,\forall y\in C.$$

(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

$$⟨B{x}^{*},y-x^{*}⟩\ge 0,\forall y\in C.$$

(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 [1–22] 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: C → C 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: C → C 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: C → H be nonlinear mappings. Then B is called

• (1) monotone if 〈Bx - By, x - y〉 ≥ 0, ∀x, y ∈ C,

• (2) ξ-inverse-strongly monotone if there exists a constant ξ > 0 such that

  $$⟨Bx-By,x-y⟩\ge \xi {∥Bx-By∥}^2,\forall x,y\in C.$$

• (3) A set-valued mapping Q: H → 2^His 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^His called maximal monotone, if it is monotone and if for any (x, f) ∈ H × H

  $$⟨x-y,f-g⟩\ge 0,\forall \left(y,g\right)\in Graph\left(Q\right)$$

(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:

$$\underset{x\in F}{\text{min}}\left[\frac{1}{2}⟨Ax,x⟩-⟨x,b⟩\right],$$

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 $\bar{\gamma }>0$ with the property

$$⟨Ax,x⟩\ge \bar{\gamma }{∥x∥}^2,\forall x\in H.$$

Marino and Xu [27] introduced a new iterative scheme by the viscosity approximation method:

$$x_{n+1}={\epsilon }_n\gamma f\left(x_n\right)+\left(1-{\epsilon }_{n}A\right)S{x}_n.$$

(1.5)

They proved that the sequences {x _n } generated by (1.5) converges strongly to the unique solution of the variational inequality:

$$⟨\gamma fz-Az,x-z⟩\le 0,\forall x\in F\left(S\right),$$

which is the optimality condition for the minimization problem:

$$\underset{x\in F\left(S\right)}{\text{min}}\left[\frac{1}{2}⟨Ax,x⟩-h\left(x\right)\right],$$

(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:

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen}}\mathsf{\text{arbitrary}},\hfill \\ x_{n+1}={\gamma }_n{x}_n+\left(1-{\gamma }_n\right)S{P}_C\left(x_n-{\alpha }_{n}B{x}_n\right),\forall n\ge 0,\hfill \end{array}\right.$$

(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: H → H be a nonexpansive mapping, Yamada [29] introduced the following iterative scheme called the hybrid steepest descent method:

$$x_{n+1}=S{x}_n-{\alpha }_n\mu BS{x}_n,\forall n\ge 1,$$

(1.8)

where x₁ = x ∈ H, {α _n } ⊂ (0,1), let B: H → H 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 $J_r^F:H\to C$ 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 in [3] that under suitable hypotheses on F (to be stated precisely in Section 2), $J_r^F:H\to C$ is single-valued and firmly nonexpansive and satisfied

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

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

$$x_n={\epsilon }_n\gamma f\left(x_n\right)+\left(1-{\epsilon }_{n}A\right)W_n{J}_{r_{M,n}}^{F_M}{J}_{r_{M-1,n}}^{F_{M-1}}{J}_{r_{M-2,n}}^{F_{M-2}}\cdots J_{r_{2,n}}^{F_2}{J}_{r_{1,n}}^{F_1}{x}_n,$$

(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 $z\in {\cap }_{n=1}^{\infty }F\left(T_n\right)\cap \left({\cap }_{k=1}^{M}SEP\left(F_k\right)\right)$, 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

$$x_{n+1}={\epsilon }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)W_n{J}_{r_n}^F{x}_n.$$

(1.10)

They prove that sequence {x _n } and $\left\{J_{r_n}^F\right\}$ converge strongly to $z\in {\cap }_{n=1}^{\infty }F\left(T_n\right)\cap EP\left(F\right)$, where z is an equilibrium point for F and is the unique solution of the variational inequality:

$$⟨\gamma fz-Az,x-z⟩\le 0,\forall x\in {\cap }_{n=1}^{\infty }F\left(T_n\right)\cap EP\left(F\right)$$

or, equivalently, the unique solution of the minimization problem

$$\underset{x\in {\cap }_{n=1}^{\infty }F\left(T_n\right)\cap EP\left(F\right)}{\text{min}}\left[\frac{1}{2}⟨Ax,x⟩-h\left(x\right)\right],$$

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 : H → C 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

$$x^{*}\in VI\left(C,B\right)\iff x^{*}=P_C\left(x^{*}-\lambda B{x}^{*}\right),\lambda >0.$$

(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: C → H 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:

• (i) for each x ∈ H and x* ∈ C, x* = P _C x ⇔ 〈x - x*, y - x*〉 ≤ 0 for all y ∈ C;

• (ii) P _C : H → C is nonexpansive, that is, ||P _C x - P _C y|| ≤ ||x - y|| for all x, y ∈ H;

• (iii) P _C is firmly nonexpansive, that is, ||P _C x - P _C y||² ≤ 〈P _C x - P _C y, x - y〉 for all x,y ∈ H;

• (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;

• (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

$$\underset{n\to \infty }{\text{lim inf}}∥x_n-x∥<\underset{n\to \infty }{\text{lim inf}}∥x_n-y∥,$$

hold for each y ∈ H with y ≠ x.

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 $\bar{\gamma }>0$. Then, for $0<\gamma <\frac{\bar{\gamma }}{\eta }$,

$$⟨x-y,\left(A-\gamma f\right)x-\left(A-\gamma f\right)y⟩\ge \left(\bar{\gamma }-\eta \gamma \right){∥x-y∥}^2,\forall x,y\in H.$$

That is, A - γf is a strongly monotone with coefficient $\bar{\gamma }-\gamma \eta$.

Lemma 2.5. [27] Assume A be a strongly positive linear bounded operator on H with coefficient $\bar{\gamma }>0$ and 0 < ρ ≤ ||A||^-1. Then $∥I-\rho A∥\le 1-\rho \bar{\gamma }$.

Throughout this article, we assume that a bifunction F : C × C → ℝ 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, lim_t↓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 ℝ satisfying (A1)-(A4). Let r > 0 and x ∈ H. Then, there exists z ∈ C such that

$$F\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0,\forall y\in C.$$

Lemma 2.7. [3] Assume that F : C × C → ℝ satisfies (A1)-(A4). For r > 0 and x ∈ H, define a mapping $J_r^F:H\to C$ as follows:

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

for all z ∈ H. Then, the following hold:

• (1) $J_r^F$ is single-valued;

• (2) $J_r^F$ is firmly nonexpansive, that is, for any x,y ∈ H,

  $${∥J_r^{F}x-J_r^{F}y∥}^2\le ⟨J_r^{F}x-J_r^{F}y,x-y⟩;$$

• (3) $F\left(J_r^F\right)=EP\left(F\right)$;

• (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 inf_n→∞β _n ≤ lim sup_n→∞β _n < 1. Suppose x_n+1= (1 - β _n )l _n + β _n x _n for all integers n ≥ 0 and lim sup_n→∞(||l_n+1- l _n || - ||x_n+1- x _n ||) ≤ 0. Then, lim_n→∞||l _n - x _n || = 0.

Lemma 2.9. [33] Assume {a _n } is a sequence of nonnegative real numbers such that

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

where {b _n } is a sequence in (0,1) and {c _n } is a sequence in ℝ such that

• (1) ${\sum }_{n=1}^{\infty }{b}_n=\infty$,

• (2) $\text{lim}\underset{n\to \infty }{\text{sup}}\frac{c_n}{b_n}\le 0$ or ${\sum }_{n=1}^{\infty }\left|c_n\right|<\infty$,

Then, lim_n→∞a _n = 0.

3. Main results

Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\left\{T_n\right\}}_{n=1}^{\infty }$ be a family of infinitely of nonexpansive mappings of C into itself and let ${\left\{{\mu }_n\right\}}_{n=1}^{\infty }$ be a sequence of nonnegative numbers in [0,1]. For any n ≥ 1, define a mapping W _n : C → C as follows:

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

(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 ${\cap }_{n=1}^{\infty }F\left(T_n\right)\ne \varnothing$, let μ₁, μ₂, ... be real numbers such that 0 ≤ μ _n ≤ b < 1 for every n ≥ 1. Then,

• (1) for every x ∈ C and k ∈ ℕ, the limit lim_n→∞U _n,k x exists;

• (2) the mapping W of C into itself as follows:

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

  (3.2)

is a nonexpansive mapping satisfying $F\left(W\right)={\cap }_{n=1}^{\infty }F\left(T_n\right)$, which it is called the W-mapping generated by T₁, T₂, ... and μ₁, μ₂, ...;

• (3) $F\left(W_n\right)={\cap }_{n=1}^{\infty }F\left(T_n\right)$, for each n ≥ 1;

• (4) If E is any bounded subset of C, then $\underset{n\to \infty }{\text{lim}}\underset{x\in E}{\text{sup}}∥Wx-W_{n}x∥=0.$.

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 ℝ 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

$$\Theta :={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap \left({\cap }_{k=1}^{M}SEP\left(F_k\right)\right)\cap VI\left(C,B\right)\ne \varnothing .$$

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 $\bar{\gamma }>0$ and $0<\gamma <\frac{\bar{\gamma }}{\eta }$. Let {x _n }, {y _n } and {u _n } be sequences generated by

$$\left\{\begin{array}{c}{x}_1=x\in Cchosenarbitrary,\hfill \\ y_n=\left(1-{\delta }_n\right)x_n+{\delta }_n{P}_C\left(x_n-{\alpha }_{n}B{x}_n\right),\hfill \\ u_n=J_{r_{M,n}}^{F_M}{J}_{r_{M-1,n}}^{F_{M-1}}{J}_{r_{M-2,n}}^{F_{M-2}}\dots J_{r_{2,n}}^{F_2}{J}_{r_{1,n}}^{F_1}{y}_n,\hfill \\ x_{n+1}={\epsilon }_n\gamma f\left(u_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)P_C\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right),\forall n\ge 1,\hfill \end{array}\right.$$

(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) lim_n→∞ϵ _n = 0 and ${\sum }_{n=1}^{\infty }{\epsilon }_n=\infty$,

(C2) 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1,

(C3) {α _n }, {λ _n } ⊂ [e, g] ⊂ (0, 2ξ), lim_n→∞α _n = 0 and lim_n→∞λ _n = 0,

(C4) {δ _n } ⊂ [0, b], for some b ∈ (0,1) and lim_n→∞|δ_n+1- δ _n | = 0,

(C5) lim inf_n→∞r _k,n > 0 and lim_n→∞|r_k,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

$$⟨\left(A-\gamma f\right)z,x-z⟩\ge 0,\forall x\in \Theta .$$

(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 $∥I-\rho A∥\le 1-\rho \bar{\gamma }$. We will assume that $∥I-A∥\le 1-\bar{\gamma }$. Since A is a strongly positive bounded linear operator on H, we have

$$∥A∥=\text{sup}\left\{\left|⟨Ax,x⟩\right|:x\in H,∥x∥=1\right\}.$$

Observe that

$$⟨\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)x,x⟩=1-{\beta }_n-{\epsilon }_n⟨Ax,x⟩\ge 1-{\beta }_n-{\epsilon }_n∥A∥\ge 0,$$

this show that (1 - β _n )I - ϵ _n A is positive. It follows that

$$\begin{array}{ll}\hfill ∥\left(1-{\beta }_n\right)I-{\epsilon }_{n}A∥& =\text{sup}\left\{\left|⟨\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)x,x⟩\right|:x\in H,∥x∥=1\right\}\\ =\text{sup}\left\{1-{\beta }_n-{\epsilon }_n⟨Ax,x⟩:x\in H,∥x∥=1\right\}\\ \le 1-{\beta }_n-{\epsilon }_n\bar{\gamma }.\end{array}$$

We divide the proof of Theorem 3.2 into seven steps.

Step 1. We show that the mapping P_Θ(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

$$\begin{array}{ll}\hfill ∥P_{\Theta }\left(\gamma f+\left(I-A\right)\right)\left(x\right)-P_{\Theta }\left(\gamma f+\left(I-A\right)\right)\left(y\right)∥& \le ∥\left(\gamma f+\left(I-A\right)\right)\left(x\right)-\left(\gamma f+\left(I-A\right)\right)\left(y\right)∥\\ \le \gamma ∥f\left(x\right)-f\left(y\right)∥+∥I-A∥∥x-y∥\\ \le \gamma \eta ∥x-y∥+\left(1-\bar{\gamma }\right)∥x-y∥\\ =\left(1-\left(\bar{\gamma }-\eta \gamma \right)\right)∥x-y∥,\forall x,y\in C.\end{array}$$

Since $0<1-\left(\bar{\gamma }-\eta \gamma \right)<1$, it follows that P_Θ (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,

$$z=P_{\Theta }\left(\gamma f+\left(I-A\right)\right)\left(z\right).$$

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,

$$\begin{array}{l}{∥\left(W_n-{\lambda }_{n}B{W}_n\right)x-\left(W_n-{\lambda }_{n}B{W}_n\right)y∥}^2\\ ={∥\left(W_{n}x-W_{n}y\right)-{\lambda }_n\left(B{W}_{n}x-B{W}_{n}y\right)∥}^2\\ ={∥W_{n}x-W_{n}y∥}^2-2{\lambda }_n⟨W_{n}x-W_{n}y,B{W}_{n}x-B{W}_{n}y⟩+{\lambda }_n^2{∥B{W}_{n}x-B{W}_{n}y∥}^2\\ \le {∥x-y∥}^2-2{\lambda }_n\xi ∥B{W}_{n}x-B{W}_{n}y∥+{\lambda }_n^2{∥B{W}_{n}x-B{W}_{n}y∥}^2\\ ={∥x-y∥}^2-{\lambda }_n\left({\lambda }_n-2\xi \right){∥B{W}_{n}x-B{W}_{n}y∥}^2\\ \le {∥x-y∥}^2,\end{array}$$

(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 $\tilde{x}\in \Theta$, then

$$\tilde{x}=P_C\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right).$$

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

$$\begin{array}{ll}\hfill ∥v_n-\tilde{x}∥& =∥P_C\left(x_n-{\alpha }_{n}B{x}_n\right)-P_C\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)∥\\ \le ∥\left(x_n-{\alpha }_{n}B{x}_n\right)-\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)∥\\ =∥\left(I-{\alpha }_{n}B\right)x_n-\left(I-{\alpha }_{n}B\right)\tilde{x}∥\\ \le ∥x_n-\tilde{x}∥\end{array}$$

(3.6)

and

$$\begin{array}{ll}\hfill ∥y_n-\tilde{x}∥& \le \left(1-{\delta }_n\right)∥x_n-\tilde{x}∥+{\delta }_n∥v_n-\tilde{x}∥\\ \le \left(1-{\delta }_n\right)∥x_n-\tilde{x}∥+{\delta }_n∥x_n-\tilde{x}∥\\ =∥x_n-\tilde{x}∥.\end{array}$$

(3.7)

Let ${\Im }_n^k=J_{r_{k,n}}^{F_k}{J}_{r_{k-1,n}}^{F_{k-1}}{J}_{r_{k-2,n}}^{F_{k-2}}\dots J_{r_{2,n}}^{F_2}{J}_{r_{1,n}}^{F_1}$ for k ∈ {1, 2, 3,..., M} and ${\Im }_n^0=I$ for all n. Because $J_{r_{k,n}}^{F_k}$ is nonexpansive for each k = 1, 2, 3,..., M, $\tilde{x}={\Im }_n^k\tilde{x}$ and (3.7), we note that $u_n={\Im }_n^M{y}_n$. It follows that

$$∥u_n-\tilde{x}∥=∥{\Im }_n^M{y}_n-{\Im }_n^M\tilde{x}∥\le ∥y_n-\tilde{x}∥\le ∥x_n-\tilde{x}∥.$$

(3.8)

Let e _n = P _C (W _n u _n - λ _n BW _n u _n ), we can prove that

$$\begin{array}{ll}\hfill ∥e_n-\tilde{x}∥& =∥P_C\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-P_C\left(W_n\tilde{x}-{\lambda }_{n}B{W}_n\tilde{x}\right)∥\\ \le ∥\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-\left(W_n\tilde{x}-{\lambda }_{n}B{W}_n\tilde{x}\right)∥\\ =∥\left(W_n-{\lambda }_{n}B{W}_n\right)u_n-\left(W_n-{\lambda }_{n}B{W}_n\right)\tilde{x}∥\\ \le ∥u_n-\tilde{x}∥\le ∥x_n-\tilde{x}∥,\end{array}$$

(3.9)

which yields that

$$\begin{array}{ll}\hfill ∥x_{n+1}-\tilde{x}∥& =∥{\epsilon }_n\left(\gamma f\left(u_n\right)-A\tilde{x}\right)+{\beta }_n\left(x_n-\tilde{x}\right)+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)\left(e_n-\tilde{x}\right)∥\\ \le {\epsilon }_n∥\gamma f\left(u_n\right)-A\tilde{x}∥+{\beta }_n∥x_n-\tilde{x}∥+∥\left(1-{\beta }_n\right)I-{\epsilon }_{n}A∥∥e_n-\tilde{x}∥\\ \le {\epsilon }_n\gamma ∥f\left(u_n\right)-f\left(\tilde{x}\right)∥+{\epsilon }_n∥\gamma f\left(\tilde{x}\right)-A\tilde{x}∥+{\beta }_n∥x_n-\tilde{x}∥+\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)∥e_n-\tilde{x}∥\\ \le {\epsilon }_n\gamma \eta ∥u_n-\tilde{x}∥+{\epsilon }_n∥\gamma f\left(\tilde{x}\right)-A\tilde{x}∥+{\beta }_n∥x_n-\tilde{x}∥+\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)∥x_n-\tilde{x}∥\\ \le {\epsilon }_n\gamma \eta ∥x_n-\tilde{x}∥+{\epsilon }_n∥\gamma f\left(\tilde{x}\right)-A\tilde{x}∥+{\beta }_n∥x_n-\tilde{x}∥+\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)∥x_n-\tilde{x}∥\\ =\left(1-\left(\bar{\gamma }-\gamma \eta \right){\epsilon }_n\right)∥x_n-\tilde{x}∥+\frac{\left(\bar{\gamma }-\gamma \eta \right){\epsilon }_n}{\left(\bar{\gamma }-\gamma \eta \right)}∥\gamma f\left(\tilde{x}\right)-A\tilde{x}∥.\end{array}$$

By induction, we have

$$∥x_n-\tilde{x}∥\le \text{max}\left\{∥x_1-\tilde{x}∥,\frac{∥\gamma f\left(\tilde{x}\right)-A\tilde{x}∥}{\bar{\gamma }-\gamma \eta }\right\},\forall n\in \mathbb{N}.$$

(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 $\underset{n\to \infty }{\text{lim}}∥x_{n+1}-x_n∥=0$.

We claim that if ω _n be a bounded sequence in C, then

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

(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},

$$\underset{n\to \infty }{\text{lim}}∥J_{r_{k,n+1}}^{F_k}{\omega }_n-J_{r_{k,n}}^{F_k}{\omega }_n∥=0.$$

(3.12)

Note that for every k ∈ {1,2,3,...,M}, we obtain

$${\Im }_n^k=J_{r_{k,n}}^{F_k}{J}_{r_{k-1,n}}^{F_{k-1}}{J}_{r_{k-2,n}}^{F_{k-2}}\dots J_{r_{2,n}}^{F_2}{J}_{r_{1,n}}^{F_1}=J_{r_{k,n}}^{F_k}{\Im }_n^{k-1}.$$

Thus,

$$\begin{array}{l}∥{\Im }_n^k{\omega }_n-{\Im }_{n+1}^k{\omega }_n∥\\ =∥J_{r_{k,n}}^{F_k}{\Im }_n^{k-1}{\omega }_n-J_{r_{k,n+1}}^{F_k}{\Im }_{n+1}^{k-1}{\omega }_n∥\\ \le ∥J_{r_{k,n}}^{F_k}{\Im }_n^{k-1}{\omega }_n-J_{r_{k,n+1}}^{F_k}{\Im }_n^{k-1}{\omega }_n∥+∥J_{r_{k,n+1}}^{F_k}{\Im }_n^{k-1}{\omega }_n-J_{r_{k,n+1}}^{F_k}{\Im }_{n+1}^{k-1}{\omega }_n∥\\ \le ∥J_{r_{k,n}}^{F_k}{\Im }_n^{k-1}{\omega }_n-J_{r_{k,n+1}}^{F_k}{\Im }_n^{k-1}{\omega }_n∥+∥{\Im }_n^{k-1}{\omega }_n-{\Im }_{n+1}^{k-1}{\omega }_n∥\\ \le ∥J_{r_{k,n}}^{F_k}{\Im }_n^{k-1}{\omega }_n-J_{r_{k,n+1}}^{F_k}{\Im }_n^{k-1}{\omega }_n∥+∥J_{r_{k-1,n}}^{F_{k-1}}{\Im }_n^{k-2}{\omega }_n-J_{r_{k-1,n+1}}^{F_{k-1}}{\Im }_n^{k-2}{\omega }_n∥\\ +∥{\Im }_n^{k-2}{\omega }_n-{\Im }_{n+1}^{k-2}{\omega }_n∥\\ \le ∥J_{r_{k,n}}^{F_k}{\Im }_n^{k-1}{\omega }_n-J_{r_{k,n+1}}^{F_k}{\Im }_n^{k-1}{\omega }_n∥+∥J_{r_{k-1,n}}^{F_{k-1}}{\Im }_n^{k-2}{\omega }_n-J_{r_{k-1,n+1}}^{F_{k-1}}{\Im }_n^{k-2}{\omega }_n∥\\ +\dots +∥J_{r_{2,n}}^{F_2}{\Im }_n^1{\omega }_n-J_{r_{2,n+1}}^{F_2}{\Im }_n^1{\omega }_n∥+∥J_{r_{1,n}}^{F_1}{\omega }_n-J_{r_{1,n+1}}^{F_1}{\omega }_n∥.\end{array}$$

(3.13)

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

Since T _n and U _n,n are nonexpansive, we have

$$\begin{array}{ll}\hfill ∥W_{n+1}{x}_n-W_n{x}_n∥& =∥{\mu }_1{T}_1{U}_{n+1,2}{x}_n-{\mu }_1{T}_1{U}_{n,2}{x}_n∥\\ \le {\mu }_1∥U_{n+1,2}{x}_n-U_{n,2}{x}_n∥\\ ={\mu }_1∥{\mu }_2{T}_2{U}_{n+1,3}{x}_n-{\mu }_2{T}_2{U}_{n,3}{x}_n∥\\ \le {\mu }_1{\mu }_2∥U_{n+1,3}{x}_n-U_{n,3}{x}_n∥\\ \le \dots \\ \le {\mu }_1{\mu }_2\dots {\mu }_n∥U_{n+1,n+1}{x}_n-U_{n,n+1}{x}_n∥\\ \le M_1\prod _{i=1}^n{\mu }_i,\end{array}$$

(3.14)

where M₁ ≥ 0 is an appropriate constant such that ||U_n+1,n+1x _n - U_n,n+1x _n || ≤ M₁ for all n ≥ 0. From I - α _n B is nonexpansive, we have

$$\begin{array}{ll}\hfill ∥v_{n+1}-v_n∥& =∥P_C\left(x_{n+1}-{\alpha }_{n+1}B{x}_{n+1}\right)-P_C\left(x_n-{\alpha }_{n}B{x}_n\right)∥\\ \le ∥\left(x_{n+1}-{\alpha }_{n+1}B{x}_{n+1}\right)-\left(x_n-{\alpha }_{n}B{x}_n\right)∥\\ \le ∥\left(x_{n+1}-{\alpha }_{n+1}B{x}_{n+1}\right)-\left(x_n-{\alpha }_{n+1}B{x}_n\right)∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥\\ \le ∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥.\end{array}$$

(3.15)

From (3.3) and (3.15), we have

$$\begin{array}{ll}\hfill ∥y_{n+1}-y_n∥& =∥\left(1-{\delta }_{n+1}\right)\left(x_{n+1}-x_n\right)+{\delta }_{n+1}\left(v_{n+1}-v_n\right)+\left({\delta }_{n+1}-{\delta }_n\right)\left(v_n-x_n\right)∥\\ \le \left(1-{\delta }_{n+1}\right)∥x_{n+1}-x_n∥+{\delta }_{n+1}∥v_{n+1}-v_n∥+\left|{\delta }_{n+1}-{\delta }_n\right|∥v_n-x_n∥\\ \le \left(1-{\delta }_{n+1}\right)∥x_{n+1}-x_n∥+{\delta }_{n+1}\left\{∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥\right\}\\ +\left|{\delta }_n-{\delta }_{n+1}\right|∥x_n-v_n∥\\ =∥x_{n+1}-x_n∥+{\delta }_{n+1}\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥+\left|{\delta }_n-{\delta }_{n+1}\right|∥x_n-v_n∥.\end{array}$$

(3.16)

Now, we compute ||u_n+1- u _n || and ||e_n+1- e _n ||. Consider the following computation:

$$\begin{array}{ll}\hfill ∥u_{n+1}-u_n∥& =∥{\Im }_{n+1}^M{y}_{n+1}-{\Im }_n^M{y}_n∥\\ \le ∥{\Im }_{n+1}^M{y}_{n+1}-{\Im }_{n+1}^M{y}_n∥+∥{\Im }_{n+1}^M{y}_n-{\Im }_n^M{y}_n∥\\ \le ∥y_{n+1}-y_n∥+∥{\Im }_{n+1}^M{y}_n-{\Im }_n^M{y}_n∥\\ \le ∥x_{n+1}-x_n∥+{\delta }_{n+1}\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥+\left|{\delta }_n-{\delta }_{n+1}\right|∥x_n-v_n∥\\ +∥{\Im }_{n+1}^M{y}_n-{\Im }_n^M{y}_n∥\end{array}$$

(3.17)

and

$$\begin{array}{ll}\hfill ∥e_{n+1}-e_n∥& =∥p_C\left(W_{n+1}{u}_{n+1}-{\lambda }_{n+1}B{W}_{n+1}{u}_{n+1}\right)-P_C\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)∥\\ \le ∥\left(W_{n+1}{u}_{n+1}-{\lambda }_{n+1}B{W}_{n+1}{u}_{n+1}\right)-\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)∥\\ =∥\left(W_{n+1}{u}_{n+1}-{\lambda }_{n+1}B{W}_{n+1}{u}_{n+1}\right)-\left(W_{n+1}{u}_n-{\lambda }_{n+1}B{W}_{n+1}{u}_n\right)\\ +\left(W_{n+1}{u}_n-{\lambda }_{n+1}B{W}_{n+1}{u}_n\right)-\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)∥\\ \le ∥\left(W_{n+1}{u}_{n+1}-{\lambda }_{n+1}B{W}_{n+1}{u}_{n+1}\right)-\left(W_{n+1}{u}_n-{\lambda }_{n+1}B{W}_{n+1}{u}_n\right)∥\\ +∥W_{n+1}{u}_n-W_n{u}_n∥+∥{\lambda }_{n}B{W}_n{u}_n-{\lambda }_{n+1}B{W}_{n+1}{u}_n∥\\ \le ∥u_{n+1}-u_n∥+M_1\prod _{i=1}^n{\mu }_i+{\lambda }_n∥B{W}_n{u}_n∥+{\lambda }_{n+1}∥B{W}_{n+1}{u}_n∥\\ \le ∥x_{n+1}-x_n∥+{\delta }_{n+1}\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥+\left|{\delta }_n-{\delta }_{n+1}\right|∥x_n-v_n∥\\ +∥{\Im }_{n+1}^M{y}_n-{\Im }_n^M{y}_n∥+M_1\prod _{i=1}^n{\mu }_i+{\lambda }_n∥B{W}_n{u}_n∥+{\lambda }_{n+1}∥B{W}_{n+1}{u}_n∥.\end{array}$$

(3.18)

Setting

$$l_n=\frac{x_{n+1}-{\beta }_n{x}_n}{1-{\beta }_n}=\frac{{\epsilon }_n\gamma f\left(u_n\right)+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)e_n}{1-{\beta }_n},$$

we have x_n+1= (1 - β _n )l _n + β _n x _n , n ≥ 1. It follows that

$$\begin{array}{ll}\hfill l_{n+1}-l_n& =\frac{{\epsilon }_{n+1}\gamma f\left(u_{n+1}\right)+\left(\left(1-{\beta }_{n+1}\right)I-{\epsilon }_{n+1}A\right)e_{n+1}}{1-{\beta }_{n+1}}\\ -\frac{{\epsilon }_n\gamma f\left(u_n\right)+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)e_n}{1-{\beta }_n}\\ =\frac{{\epsilon }_{n+1}}{1-{\beta }_{n+1}}\left(\gamma f\left(u_{n+1}\right)-A{e}_{n+1}\right)+\frac{{\epsilon }_n}{1-{\beta }_n}\left(A{e}_n-\gamma f\left(u_n\right)\right)+\left(e_{n+1}-e_n\right).\end{array}$$

(3.19)

It follows from (3.18) and (3.19) that

$$\begin{array}{ll}\hfill ∥l_{n+1}-l_n∥-∥x_{n+1}-x_n∥& \le \frac{{\epsilon }_{n+1}}{1-{\beta }_{n+1}}∥\gamma f\left(u_{n+1}\right)-A{e}_{n+1}∥+\frac{{\epsilon }_n}{1-{\beta }_n}∥A{e}_n-\gamma f\left(u_n\right)∥\\ +{\delta }_{n+1}\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥+\left|{\delta }_n-{\delta }_{n+1}\right|∥x_n-v_n∥\\ +∥{\Im }_{n+1}^M{y}_n-{\Im }_n^M{y}_n∥+M_1\prod _{i=1}^n{\mu }_i+{\lambda }_n∥B{W}_n{u}_n∥\\ +{\lambda }_{n+1}∥B{W}_{n+1}{u}_n∥\\ \le \frac{{\epsilon }_{n+1}}{1-{\beta }_{n+1}}\left(∥\gamma f\left(u_{n+1}\right)∥+∥A{e}_{n+1}∥\right)+\frac{{\epsilon }_n}{1-{\beta }_n}\left(∥A{e}_n∥+∥\gamma f\left(u_n\right)∥\right)\\ +{\delta }_{n+1}\left|{\alpha }_{n+1}-{\alpha }_n\right|∥B{x}_n∥+\left|{\delta }_n-{\delta }_{n+1}\right|∥x_n-v_n∥\\ +∥{\Im }_{n+1}^M{y}_n-{\Im }_n^M{y}_n∥+M_1\prod _{i=1}^n{\mu }_i+{\lambda }_n∥B{W}_n{u}_n∥\\ +{\lambda }_{n+1}∥B{W}_{n+1}{u}_n∥.\end{array}$$

(3.20)

This together with conditions (C1)-(C4) and (3.11) imply that

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

By Lemma 2.8, we obtain

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

Consequently,

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

(3.21)

Applying (3.11), (3.21) and conditions (C3), (C4) to (3.15) and (3.17), we obtain that

$$\underset{n\to \infty }{\text{lim}}∥u_{n+1}-u_n∥=\underset{n\to \infty }{\text{lim}}∥v_{n+1}-v_n∥=0.$$

(3.22)

Step 5. We show that $\underset{n\to \infty }{\text{lim}}∥W_n{e}_n-e_n∥=0$.

For any $\tilde{x}\in \Theta$ and (3.5), we obtain

$$\begin{array}{ll}\hfill {∥v_n-\tilde{x}∥}^2& ={∥P_C\left(x_n-{\alpha }_{n}B{x}_n\right)-P_C\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)∥}^2\\ \le {∥\left(x_n-{\alpha }_{n}B{x}_n\right)-\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)∥}^2\\ \le {∥x_n-\tilde{x}∥}^2+\left({\alpha }_n^2-2{\alpha }_n\xi \right){∥B{x}_n-B\tilde{x}∥}^2.\end{array}$$

(3.23)

By Lemma 2.2(iv) and (3.23), we have

$$\begin{array}{ll}\hfill {∥y_n-\tilde{x}∥}^2& \le \left(1-{\delta }_n\right){∥x_n-\tilde{x}∥}^2+{\delta }_n{∥v_n-\tilde{x}∥}^2\\ \le \left(1-{\delta }_n\right){∥x_n-\tilde{x}∥}^2+{\delta }_n\left\{{∥x_n-\tilde{x}∥}^2+\left({\alpha }_n^2-2{\alpha }_n\xi \right){∥B{x}_n-B\tilde{x}∥}^2\right\}\\ ={∥x_n-\tilde{x}∥}^2+\left({\alpha }_n^2-2{\alpha }_n\xi \right){\delta }_n∥B{x}_n-B\tilde{x}∥{.}^2\end{array}$$

(3.24)

So, from (3.8) and (3.24), we derive

$${∥e_n-\tilde{x}∥}^2\le {∥u_n-\tilde{x}∥}^2\le {∥y_n-\tilde{x}∥}^2\le {∥x_n-\tilde{x}∥}^2+\left({\alpha }_n^2-2{\alpha }_n\xi \right){\delta }_n∥B{x}_n-B\tilde{x}∥{.}^2$$

(3.25)

From (3.3), we have

$$\begin{array}{c}{\Vert x_{n+1}-\tilde{x}\Vert }^2={\Vert ((1-{\beta }_n)I-{\epsilon }_{n}A)(e_n-\tilde{x})+{\beta }_n(x_n-\tilde{x})+{\epsilon }_n(\gamma f(u_n)-A\tilde{x})\Vert }^2\\ ={\Vert ((1-{\beta }_n)I-{\epsilon }_{n}A)(e_n-\tilde{x})+{\beta }_n(x_n-\tilde{x})\Vert }^2\\ +{\epsilon }_n^2{\Vert \gamma f(u_n)-A\tilde{x}\Vert }^2+2{\beta }_n{\epsilon }_n〈x_n-\tilde{x},\gamma f(u_n)-A\tilde{x}〉\\ +2{\epsilon }_n〈((1-{\beta }_n)I-{\epsilon }_{n}A)(e_n-\tilde{x}),\gamma f(u_n)-A\tilde{x}〉\\ \le {\left((1-{\beta }_n-{\epsilon }_n\overline{\gamma })\Vert e_n-\tilde{x}\Vert +{\beta }_n\Vert x_n-\tilde{x}\Vert \right)}^2+{\epsilon }_n{L}_n\\ \le {(1-{\beta }_n-{\epsilon }_n\overline{\gamma })}^2{\Vert e_n-\tilde{x}\Vert }^2+{\beta }_n^2{\Vert x_n-\tilde{x}\Vert }^2\\ +2(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\beta }_n\Vert e_n-\tilde{x}\Vert \Vert x_n-\tilde{x}\Vert +{\epsilon }_n{L}_n\\ \le \left[{(1-{\epsilon }_n\overline{\gamma })}^2-2(1-{\epsilon }_n\overline{\gamma }){\beta }_n+{\beta }_n^2\right]{\Vert e_n-\tilde{x}\Vert }^2\\ +(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\beta }_n\left\{{\Vert e_n-\tilde{x}\Vert }^2+{\Vert x_n-\tilde{x}\Vert }^2\right\}+{\beta }_n^2{\Vert x_n-\tilde{x}\Vert }^2+{\epsilon }_n{L}_n\\ =(1-{\epsilon }_n\overline{\gamma })(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\Vert e_n-\tilde{x}\Vert }^2+(1-{\epsilon }_n\overline{\gamma }){\beta }_n{\Vert x_n-\tilde{x}\Vert }^2+{\epsilon }_n{L}_n\\ \le (1-{\epsilon }_n\overline{\gamma })(1-{\beta }_n-{\epsilon }_n\overline{\gamma })\left\{{\Vert x_n-\tilde{x}\Vert }^2+({\alpha }_n^2-2{\alpha }_n\xi ){\delta }_n{\Vert B{x}_n-B\tilde{x}\Vert }^2\right\}\\ +(1-{\epsilon }_n\overline{\gamma }){\beta }_n{\Vert x_n-\tilde{x}\Vert }^2+{\epsilon }_n{L}_n\\ ={(1-{\epsilon }_n\overline{\gamma })}^2{\Vert x_n-\tilde{x}\Vert }^2\\ +(1-{\epsilon }_n\overline{\gamma })(1-{\beta }_n-{\epsilon }_n\overline{\gamma })({\alpha }_n^2-2{\alpha }_n\xi ){\delta }_n{\Vert B{x}_n-B\tilde{x}\Vert }^2+{\epsilon }_n{L}_n\\ \le {\Vert x_n-\tilde{x}\Vert }^2+(1-{\epsilon }_n\overline{\gamma })(1-{\beta }_n-{\epsilon }_n\overline{\gamma })({\alpha }_n^2-2{\alpha }_n\xi ){\delta }_n{\Vert B{x}_n-B\tilde{x}\Vert }^2+{\epsilon }_n{L}_n.\end{array}$$

(3.26)

It follows that

$$\begin{array}{l}\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left(2g\xi -e^2\right)b{∥B{x}_n-B\tilde{x}∥}^2\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left(2{\alpha }_n\xi -{\alpha }_n^2\right){\delta }_n{∥B{x}_n-B\tilde{x}∥}^2\\ \le {∥x_n-\tilde{x}∥}^2-{∥x_{n+1}-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le ∥x_n-x_{n+1}∥\left(∥x_n-\tilde{x}∥+∥x_{n+1}-\tilde{x}∥\right)+{\epsilon }_n{L}_n,\end{array}$$

where

$$\begin{array}{c}{L}_n={\epsilon }_n{∥\gamma f\left(u_n\right)-A\tilde{x}∥}^2+2{\beta }_n⟨x_n-\tilde{x},\gamma f\left(u_n\right)-A\tilde{x}⟩\\ +2⟨\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)\left(e_n-\tilde{x}\right),\gamma f\left(u_n\right)-A\tilde{x}⟩.\end{array}$$

By conditions (C1), (C2) and (3.21), we obtain

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

(3.27)

Since P _C is firmly nonexpansive mapping, we have

$$\begin{array}{ll}\hfill {∥v_n-\tilde{x}∥}^2& ={∥P_C\left(x_n-{\alpha }_{n}B{x}_n\right)-P_C\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)∥}^2\\ \le ⟨\left(x_n-{\alpha }_{n}B{x}_n\right)-\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right),v_n-\tilde{x}⟩\\ =\frac{1}{2}\left\{{∥\left(x_n-{\alpha }_{n}B{x}_n\right)-\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)∥}^2+{∥v_n-\tilde{x}∥}^2\right.\\ \left.-{∥\left(x_n-{\alpha }_{n}B{x}_n\right)-\left(\tilde{x}-{\alpha }_{n}B\tilde{x}\right)-\left(v_n-\tilde{x}\right)∥}^2\right\}\\ \le \frac{1}{2}\left\{{∥x_n-\tilde{x}∥}^2+{∥v_n-\tilde{x}∥}^2-{∥\left(x_n-v_n\right)-{\alpha }_n\left(B{x}_n-B\tilde{x}\right)∥}^2\right\}\\ \le \frac{1}{2}\left\{{∥x_n-\tilde{x}∥}^2+{∥v_n-\tilde{x}∥}^2-{∥x_n-v_n∥}^2\right.\\ \left.-{\alpha }_n^2{∥B{x}_n-B\tilde{x}∥}^2+2{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥\right\}.\end{array}$$

Hence, we have

$${∥v_n-\tilde{x}∥}^2\le {∥x_n-\tilde{x}∥}^2-{∥x_n-v_n∥}^2+2{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥$$

and so

$$\begin{array}{ll}\hfill {∥y_n-\tilde{x}∥}^2& \le \left(1-{\delta }_n\right){∥x_n-\tilde{x}∥}^2+{\delta }_n{∥v_n-\tilde{x}∥}^2\\ \le \left(1-{\delta }_n\right){∥x_n-\tilde{x}∥}^2+{\delta }_n\left\{{∥x_n-\tilde{x}∥}^2-{∥x_n-v_n∥}^2+2{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥\right\}\\ ={∥x_n-\tilde{x}∥}^2-{\delta }_n{∥x_n-v_n∥}^2+2{\delta }_n{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥.\end{array}$$

(3.28)

Using (3.26) and (3.28), we also have

$$\begin{array}{ll}\hfill {∥x_{n+1}-\tilde{x}∥}^2& \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥e_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥u_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥y_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left\{{∥x_n-\tilde{x}∥}^2-{\delta }_n{∥x_n-v_n∥}^2+2{\delta }_n{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥\right\}\\ +\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le {∥x_n-\tilde{x}∥}^2-\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){\delta }_n{∥x_n-v_n∥}^2\\ +2\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){\delta }_n{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥+{\epsilon }_n{L}_n.\end{array}$$

It follow that

$$\begin{array}{l}\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){\delta }_n{∥x_n-v_n∥}^2\\ \le ∥x_n-x_{n+1}∥\left(∥x_n-\tilde{x}∥+∥x_{n+1}-\tilde{x}∥\right)\\ +2\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){\delta }_n{\alpha }_n∥x_n-v_n∥∥B{x}_n-B\tilde{x}∥+{\epsilon }_n{L}_n.\end{array}$$

From conditions (C1), C(4), (3.21) and (3.27), we obtain

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

(3.29)

Observe also that if e _n = P _C (W _n u _n - λ _n BW _n u _n ), then

$$\begin{array}{ll}\hfill {∥e_n-\tilde{x}∥}^2& ={∥P_C\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-P_C\left(\tilde{x}-{\lambda }_{n}B\tilde{x}\right)∥}^2\\ \le {∥\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-\left(\tilde{x}-{\lambda }_{n}B\tilde{x}\right)∥}^2\\ ={∥\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-\left(W_n\tilde{x}-{\lambda }_{n}B{W}_n\tilde{x}\right)∥}^2\\ \le {∥u_n-\tilde{x}∥}^2+\left({\lambda }_n^2-2{\lambda }_n\xi \right){∥B{W}_n{u}_n-B\tilde{x}∥}^2\\ \le {∥x_n-\tilde{x}∥}^2+\left({\lambda }_n^2-2{\lambda }_n\xi \right){∥B{W}_n{u}_n-B\tilde{x}∥}^2.\end{array}$$

(3.30)

Substituting (3.30) in (3.26), we have

$$\begin{array}{ll}\hfill {∥x_{n+1}-\tilde{x}∥}^2& \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥e_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left\{{∥x_n-\tilde{x}∥}^2+\left({\lambda }_n^2-2{\lambda }_n\xi \right){∥B{W}_n{u}_n-B\tilde{x}∥}^2\right\}\\ +\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le {∥x_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left({\lambda }_n^2-2{\lambda }_n\xi \right){∥B{W}_n{u}_n-B\tilde{x}∥}^2+{\epsilon }_n{L}_n.\end{array}$$

It follows that

$$\begin{array}{l}\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left(2g\xi -e^2\right){∥B{W}_n{u}_n-B\tilde{x}∥}^2\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left(2{\lambda }_n\xi -{\lambda }_n^2\right){∥B{W}_n{u}_n-B\tilde{x}∥}^2\\ \le ∥x_n-x_{n+1}∥\left(∥x_n-\tilde{x}∥+∥x_{n+1}-\tilde{x}∥\right)+{\epsilon }_n{L}_n\end{array}$$

Since ||x_n+1- x _n || → 0 (n → ∞) and conditions (C1) and (C2), we obtain

$$\underset{n\to \infty }{\text{lim}}∥B{W}_n{u}_n-B\tilde{x}∥=0.$$

(3.31)

Since P _C is firmly nonexpansive (Lemma 2.2 (iii)), we have

$$\begin{array}{ll}\hfill {∥e_n-\tilde{x}∥}^2& ={∥P_C\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-P_C\left(\tilde{x}-{\lambda }_{n}B\tilde{x}\right)∥}^2\\ \le ⟨\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-\left(\tilde{x}-{\lambda }_{n}B\tilde{x}\right),e_n-\tilde{x}⟩\\ =\frac{1}{2}\left\{{∥\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-\left(\tilde{x}-{\lambda }_{n}B\tilde{x}\right)∥}^2+{∥e_n-\tilde{x}∥}^2\right.\\ \left.-{∥\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right)-\left(\tilde{x}-{\lambda }_{n}B\tilde{x}\right)-\left(e_n-\tilde{x}\right)∥}^2\right\}\\ \le \frac{1}{2}\left\{{∥u_n-\tilde{x}∥}^2+{∥e_n-\tilde{x}∥}^2-{∥\left(W_n{u}_n-e_n\right)-{\lambda }_n\left(B{W}_n{u}_n-B\tilde{x}\right)∥}^2\right\}\\ \le \frac{1}{2}\left\{{∥x_n-\tilde{x}∥}^2+{∥e_n-\tilde{x}∥}^2-{∥W_n{u}_n-e_n∥}^2\right.\\ \left.-{\lambda }_n^2{∥B{W}_n{u}_n-B\tilde{x}∥}^2+2{\lambda }_n∥W_n{u}_n-e_n∥∥B{W}_n{u}_n-B\tilde{x}∥\right\}.\end{array}$$

Hence, we have

$${∥e_n-\tilde{x}∥}^2\le {∥x_n-\tilde{x}∥}^2-{∥W_n{u}_n-e_n∥}^2+2{\lambda }_n∥W_n{u}_n-e_n∥∥B{W}_n{u}_n-B\tilde{x}∥.$$

(3.32)

Using (3.26) and (3.32), we also have

$$\begin{array}{ll}\hfill {∥x_{n+1}-\tilde{x}∥}^2& \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥e_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left\{{∥x_n-\tilde{x}∥}^2-{∥W_n{u}_n-e_n∥}^2\right.\\ \left.+2{\lambda }_n∥W_n{u}_n-e_n∥∥B{W}_n{u}_n-B\tilde{x}∥\right\}+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le {∥x_n-\tilde{x}∥}^2-\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥W_n{u}_n-e_n∥}^2\\ +2\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){\lambda }_n∥W_n{u}_n-e_n∥∥B{W}_n{u}_n-B\tilde{x}∥+{\epsilon }_n{L}_n.\end{array}$$

It follow that

$$\begin{array}{l}\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥W_n{u}_n-e_n∥}^2\\ \le ∥x_n-x_{n+1}∥\left(∥x_n-\tilde{x}∥+∥x_{n+1}-\tilde{x}∥\right)\\ +2\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){\lambda }_n∥W_n{u}_n-e_n∥∥B{W}_n{u}_n-B\tilde{x}∥+{\epsilon }_n{L}_n.\end{array}$$

From conditions (C1), (3.21), and (3.31), we obtain

$$\underset{n\to \infty }{\text{lim}}∥W_n{u}_n-e_n∥=0.$$

(3.33)

For any $\tilde{x}\in \Theta$, note that $J_{r_{k,n}}^{F_k}$ is firmly nonexpansive (Lemma 2.7(2)) for k ∈ {1, 2, 3,..., M}, then we have

$$\begin{array}{ll}\hfill {∥{\Im }_n^k{y}_n-\tilde{x}∥}^2& ={∥J_{r_{k,n}}^{F_n}{\Im }_n^{k-1}{y}_n-J_{r_{k,n}}^{F_k}\tilde{x}∥}^2\\ \le ⟨J_{r_{k,n}}^{F_n}{\Im }_n^{k-1}{y}_n-J_{r_{k,n}}^{F_k}\tilde{x},{\Im }_n^{k-1}{y}_n-\tilde{x}⟩\\ =⟨{\Im }_n^k{y}_n-\tilde{x},{\Im }_n^{k-1}{y}_n-\tilde{x}⟩\\ =\frac{1}{2}\left({∥{\Im }_n^k{y}_n-\tilde{x}∥}^2+{∥{\Im }_n^{k-1}{y}_n-\tilde{x}∥}^2-{∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2\right).\end{array}$$

So, we obtain

$${∥{\Im }_n^k{y}_n-\tilde{x}∥}^2\le {∥{\Im }_n^{k-1}{y}_n-\tilde{x}∥}^2-{∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2,k=1,2,3,\dots ,M$$

which implies that for each k ∈ {1, 2, 3,..., M - 1},

$$\begin{array}{ll}\hfill {∥{\Im }_n^k{y}_n-\tilde{x}∥}^2& \le {∥{\Im }_n^0{y}_n-\tilde{x}∥}^2-{∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2\\ -{∥{\Im }_n^{k-1}{y}_n-{\Im }_n^{k-2}{y}_n∥}^2-\cdots -{∥{\Im }_n^2{y}_n-{\Im }_n^1{y}_n∥}^2-{∥{\Im }_n^1{y}_n-{\Im }_n^0{y}_n∥}^2\\ \le {∥y_n-\tilde{x}∥}^2-{∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2\\ \le {∥x_n-\tilde{x}∥}^2-{∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2.\end{array}$$

Consequently, from (3.26) we derive that

$$\begin{array}{ll}\hfill {∥x_{n+1}-\tilde{x}∥}^2& \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥e_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥u_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ =\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥{\Im }_n^k{y}_n-\tilde{x}∥}^2+\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le \left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right)\left\{{∥x_n-\tilde{x}∥}^2-{∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2\right\}\\ +\left(1-{\epsilon }_n\bar{\gamma }\right){\beta }_n{∥x_n-\tilde{x}∥}^2+{\epsilon }_n{L}_n\\ \le {∥x_n-\tilde{x}∥}^2-\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2+{\epsilon }_n{L}_n.\end{array}$$

Thus, we have

$$\left(1-{\epsilon }_n\bar{\gamma }\right)\left(1-{\beta }_n-{\epsilon }_n\bar{\gamma }\right){∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥}^2\le ∥x_n-x_{n+1}∥\left(∥x_n-\tilde{x}∥+∥x_{n+1}-\tilde{x}∥\right)+{\epsilon }_n{L}_n.$$

By $\underset{n\to \infty }{\text{lim inf}}{\beta }_n>0,{\epsilon }_n\to 0$ as n → ∞ and (3.21), so we deduce that

$$\underset{n\to \infty }{\text{lim}}∥{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n∥=0,k=1,2,\dots ,M-1,$$

(3.34)

that is,

$$∥u_n^{\left(k\right)}-u_n^{\left(k-1\right)}∥\to 0\mathsf{\text{as}}n\to \infty .$$

Therefore, we have

$$\begin{array}{ll}\hfill ∥y_n-u_n∥& =∥{\Im }_n^0{y}_n-{\Im }_n^k{y}_n∥\\ \le ∥{\Im }_n^0{y}_n-{\Im }_n^1{y}_n∥+∥{\Im }_n^1{y}_n-{\Im }_n^2{y}_n∥+\cdots +∥{\Im }_n^{M-1}{y}_n-{\Im }_n^M{y}_n∥.\end{array}$$

From (3.34), we have

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

(3.35)

Since x_n+1= ϵ _n γf(u _n ) + β _n x _n + ((1 - β _n )I - ϵ _n A)e _n , we have

$$\begin{array}{ll}\hfill ∥x_n-e_n∥& \le ∥x_n-x_{n+1}∥+∥x_{n+1}-e_n∥\\ =∥x_n-x_{n+1}∥+∥{\epsilon }_n\gamma f\left(u_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)e_n-e_n∥\\ =∥x_n-x_{n+1}∥+∥{\epsilon }_n\left(\gamma f\left(u_n\right)-A{e}_n\right)+{\beta }_n\left(x_n-e_n\right)∥\\ \le ∥x_n-x_{n+1}∥+{\epsilon }_n\left(∥\gamma f\left(u_n\right)∥+∥A{e}_n∥\right)+{\beta }_n∥x_n-e_n∥,\end{array}$$

that is,

$$∥x_n-e_n∥\le \frac{1}{1-{\beta }_n}∥x_n-x_{n+1}∥+\frac{{\epsilon }_n}{1-{\beta }_n}\left(∥\gamma f\left(u_n\right)∥+∥A{e}_n∥\right).$$

By conditions (C1), (C2), and (3.21) it follows that

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

(3.36)

On the other hand, from (3.3), we have

$$∥y_n-x_n∥={\delta }_n∥v_n-x_n∥.$$

Since $\underset{n\to \infty }{\text{lim}}∥x_n-v_n∥=0$, we get

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

(3.37)

We observe that

$$\begin{array}{ll}\hfill ∥W_n{e}_n-e_n∥& \le ∥W_n{e}_n-W_n{u}_n∥+∥W_n{u}_n-e_n∥\\ \le ∥e_n-x_n+x_n-y_n+y_n-u_n∥+∥W_n{u}_n-e_n∥\\ \le ∥e_n-x_n∥+∥x_n-y_n∥+∥y_n-u_n∥+∥W_n{u}_n-e_n∥.\end{array}$$

Consequently, we obtain

$$\underset{n\to \infty }{\text{lim}}∥W_n{e}_n-e_n∥=0.$$

(3.38)

Let W be the mapping defined by (3.2). Since {e _n } is bounded, applying Lemma 3.1(4) and (3.38), we have

$$∥W{e}_n-e_n∥\le ∥W{e}_n-W_n{e}_n∥+∥W_n{e}_n-e_n∥\to 0\mathsf{\text{as}}n\to \infty .$$

(3.39)

Step 6. We show that q ∈ Θ, where $\Theta :={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap \left({\cap }_{k=1}^{M}SEP\left(F_k\right)\right)\cap VI\left(C,B\right)$.

Since {x _n } is bounded, we see that there exits a subsequence $\left\{x_{n_i}\right\}$ of {x _n } which converges weakly to q. It follows from (3.37) and (3.36) that $y_{n_i}\rightharpoonup q$ and $e_{n_i}\rightharpoonup q$. From (3.35), we obtain that ${\Im }_{n_i}^k{y}_{n_i}\rightharpoonup q$ for k = 1, 2,...,M.

First, we show that $q\in {\cap }_{k=1}^{M}SEP\left(F_k\right)$. Since $u_n={\Im }_n^k{y}_n$ for k = 1, 2, 3,..., M, we also have

$$F_k\left({\Im }_n^k{y}_n,y\right)+\frac{1}{r_n}⟨y-{\Im }_n^k{y}_n,{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n⟩\ge 0,\forall y\in C.$$

If follows from (A2) that,

$$\frac{1}{r_n}⟨y-{\Im }_n^k{y}_n,{\Im }_n^k{y}_n-{\Im }_n^{k-1}{y}_n⟩\ge -F_k\left({\Im }_n^k{y}_n,y\right)\ge F_k\left(y,{\Im }_n^k{y}_n\right).$$

Replacing n by n _i , we have

$$⟨y-{\Im }_{n_i}^k{y}_{n_i},\frac{{\Im }_{n_i}^k{y}_{n_i}-{\Im }_{n_i}^{k-1}{y}_{n_i}}{r_{n_i}}⟩\ge F_k\left(y,{\Im }_{n_i}^k{y}_{n_i}\right).$$

Since $\frac{{\Im }_{n_i}^k{y}_{n_i}-{\Im }_{n_i}^{k-1}{y}_{n_i}}{r_{n_i}}\to 0$ and ${\Im }_{n_i}^k{y}_{n_i}\rightharpoonup q$, it follows by (A4) that

$$F_k\left(y,q\right)\le 0\forall y\in C,$$

for each k = 1,2,3, ...,M.

For t with 0 < t ≤ 1 and y ∈ H, let y _t = ty + (1 - t)q. Since y ∈ C and q ∈ C, we have y _t ∈ C and hence F _k (y _t , q) ≤ 0. So, from (A1) and (A4) we have

$$0=F_k\left(y_t,y_t\right)\le t{F}_k\left(y_t,y\right)+\left(1-t\right)F_k\left(y_t,q\right)\le t{F}_k\left(y_t,y\right)$$

and hence F _k (y _t ,y) ≥ 0. From (A3), we have F _k (q,y) ≥ 0 for all y ∈ C and hence q ∈ SEP(F _k ) for k = 1, 2, 3,..., M, that is, $q\in {\cap }_{k=1}^{M}SEP\left(F_k\right)$.

Next, we show that $q\in {\cap }_{n=1}^{\infty }F\left(T_n\right)$. By Lemma 3.1(2), we have $F\left(W\right)={\cap }_{n=1}^{\infty }F\left(T_n\right)$. Assume q ∉ F(W). Since $e_{n_i}\rightharpoonup q$ and q ≠ Wq, it follows by the Opial's condition (Lemma 2.3) that

$$\begin{array}{ll}\hfill \underset{i\to \infty }{\text{lim inf}}∥e_{n_i}-q∥& <\underset{i\to \infty }{\text{lim inf}}∥e_{n_i}-Wq∥\\ \le \underset{i\to \infty }{\text{lim inf}}\left\{∥e_{n_i}-W{e}_{n_i}∥+∥W{e}_{n_i}-W_q∥\right\}\\ \le \underset{i\to \infty }{\text{lim inf}}∥e_{n_i}-q∥\end{array}$$

which derives a contradiction. Thus, we have $q\in F\left(W\right)={\cap }_{n=1}^{\infty }F\left(T_n\right)$.

Finally, now we prove that q ∈ VI(C, B).

We define the maximal monotone operator

$$Q{q}_1=\left\{\begin{array}{cc}B{q}_1+N_C{q}_1,\hfill & q_1\in C,\hfill \\ \varnothing ,\hfill & q_1\notin C.\hfill \end{array}\right.$$

For any given (q₁, q₂) ∈ G(Q), hence q₂ - Bq₁ ∈ N _C q₁. Since e _n ∈ C we see from the definition of N _C that

$$⟨q_1-e_n,q_2-B{q}_1⟩\ge 0.$$

On the other hand, from e _n = P _C (W _n u _n - α _n BW _n u _n ), we have

$$⟨q_1-e_n,e_n-\left(W_n{u}_n-{\alpha }_{n}B{W}_n{u}_n\right)⟩\ge 0,$$

that is

$$⟨q_1-e_n,\frac{e_n-W_n{u}_n}{{\alpha }_n}+B{W}_n{u}_n⟩\ge 0.$$

Therefore, we obtain

$$\begin{array}{ll}\hfill ⟨q_1-e_{n_i},q_2⟩& \ge ⟨q_1-e_{n_i},B{q}_1⟩\\ \ge ⟨q_1-e_{n_i},B{q}_1⟩-⟨q_1-e_{n_i},\frac{e_{n_i}-W_n{u}_{n_i}}{{\alpha }_{n_i}}+B{W}_n{u}_{n_i}⟩\\ =⟨q_1-e_{n_i},B{q}_1-B{W}_n{u}_{n_i}-\frac{e_{n_i}-W_n{u}_{n_i}}{{\alpha }_{n_i}}⟩\\ =⟨q_1-e_{n_i},B{q}_1-B{e}_{n_i}⟩+⟨q_1-e_{n_i},B{e}_{n_i}-B{W}_n{u}_{n_i}⟩\\ -⟨q_1-e_{n_i},\frac{e_{n_i}-W_n{u}_{n_i}}{{\alpha }_{n_i}}⟩\\ \ge ⟨q_1-e_{n_i},B{e}_{n_i}-B{W}_n{u}_{n_i}⟩-⟨q_1-e_{n_i},\frac{e_{n_i}-W_n{u}_{n_i}}{{\alpha }_{n_i}}⟩,\end{array}$$

(3.40)

Since $∥e_{n_i}-W_n{u}_{n_i}∥\to 0$ as i → ∞ and B is Lipschitz continuous we obtain that

$$⟨q_1-q,q_2⟩\ge 0.$$

Notice that Q is maximal monotone, we obtain that q ∈ Q^-10 and hence q ∈ VI(C,B). This implies q ∈ Θ. Since z = P_Θ(γf + (I - A))(z), we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨x_n-z,\gamma f\left(z\right)-Az⟩=\underset{i\to \infty }{\text{lim}}⟨x_{n_i}-z,\gamma f\left(z\right)-Az⟩=⟨q-z,\gamma f\left(z\right)-Az⟩\le 0.$$

(3.41)

On the other hand, we have

$$\begin{array}{ll}\hfill ⟨e_n-z,\gamma f\left(z\right)-Az⟩& =⟨e_n-x_n,\gamma f\left(z\right)-Az⟩+⟨x_n-z,\gamma f\left(z\right)-Az⟩\\ \le ∥e_n-x_n∥∥\gamma f\left(z\right)-Az∥+⟨x_n-z,\gamma f\left(z\right)-Az⟩.\end{array}$$

From (3.36) and (3.41), we obtain that

$$\underset{n\to \infty }{\text{lim}\text{sup}}⟨e_n-z,\gamma f\left(z\right)-Az⟩\le 0.$$

(3.42)

Step 7. Finally, we show that {x _n } converges strongly to z = P_Θ(I - A + γf)(z). Indeed, from (3.3), we have

$$\begin{array}{c}{\Vert x_{n+1}-z\Vert }^2={\Vert ((1-{\beta }_n)I-{\epsilon }_{n}A)(e_n-z)+{\beta }_n(x_n-z)+{\epsilon }_n(\gamma f(u_n)-Az)\Vert }^2\\ ={\Vert ((1-{\beta }_n)I-{\epsilon }_{n}A)(e_n-z)+{\beta }_n(x_n-z)\Vert }^2\\ +{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2+2{\beta }_n{\epsilon }_n〈x_n-z,\gamma f(u_n)-Az〉\\ +2{\epsilon }_n〈((1-{\beta }_n)I-{\epsilon }_{n}A)(e_n-z),\gamma f(u_n)-Az〉\\ \le {\left((1-{\beta }_n-{\epsilon }_n\overline{\gamma })\Vert e_n-z\Vert +{\beta }_n\Vert x_n-z\Vert \right)}^2+{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2\\ +2{\beta }_n{\epsilon }_n\gamma 〈x_n-z,f(u_n)-f(z)〉+2{\beta }_n{\epsilon }_n〈x_n-z,\gamma f(z)-Az〉\\ +2(1-{\beta }_n){\epsilon }_n\gamma 〈e_n-z,f(u_n)-f(z)〉+2{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉\\ -2{\beta }_n{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉-2{\epsilon }_n^2〈(A(e_n-z),\gamma f(u_n)-Az〉\\ \le {(1-{\beta }_n-{\epsilon }_n\overline{\gamma })}^2{\Vert e_n-z\Vert }^2+{\beta }_n^2{\Vert x_n-z\Vert }^2+2(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\beta }_n\Vert e_n-z\Vert \Vert x_n-z\Vert \\ +{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2+2{\beta }_n{\epsilon }_n\gamma \Vert x_n-z\Vert \Vert f(u_n)-f(z)\Vert \\ +2(1-{\beta }_n){\epsilon }_n\gamma \Vert e_n-z\Vert \Vert f(u_n)-f(z)\Vert +2{\beta }_n{\epsilon }_n\Vert x_n-z\Vert \Vert \gamma f(z)-Az\Vert \\ -2{\beta }_n{\epsilon }_n\Vert e_n-z\Vert \Vert \gamma f(z)-Az\Vert -2{\epsilon }_n^2\Vert A(e_n-z)\Vert \Vert \gamma f(u_n)-Az\Vert \\ +2{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉\\ \le \left[{(1-{\epsilon }_n\overline{\gamma })}^2-2(1-{\epsilon }_n\overline{\gamma }){\beta }_n+{\beta }_n^2\right]{\Vert e_n-z\Vert }^2+{\beta }_n^2{\Vert x_n-z\Vert }^2\\ +(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\beta }_n\left\{{\Vert e_n-z\Vert }^2+{\Vert x_n-z\Vert }^2\right\}+{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2\\ +2{\beta }_n{\epsilon }_n\gamma \eta \Vert x_n-z\Vert \Vert u_n-z\Vert +2(1-{\beta }_n){\epsilon }_n\gamma \eta \Vert e_n-z\Vert \Vert u_n-z\Vert \\ +2{\beta }_n{\epsilon }_n\Vert x_n-z\Vert \Vert \gamma f(z)-Az\Vert -2{\beta }_n{\epsilon }_n\Vert e_n-z\Vert \Vert \gamma f(z)-Az\Vert \\ -2{\epsilon }_n^2\Vert A(e_n-z)\Vert \Vert \gamma f(u_n)-Az\Vert +2{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉\\ \le (1-{\epsilon }_n\overline{\gamma })(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\Vert e_n-z\Vert }^2+(1-{\epsilon }_n\overline{\gamma }){\beta }_n{\Vert x_n-z\Vert }^2\\ +{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2+2{\beta }_n{\epsilon }_n\gamma \eta {\Vert x_n-z\Vert }^2+2(1-{\beta }_n){\epsilon }_n\gamma \eta {\Vert x_n-z\Vert }^2\\ +2{\beta }_n{\epsilon }_n\Vert x_n-z\Vert \Vert \gamma f(z)-Az\Vert -2{\beta }_n{\epsilon }_n\Vert e_n-z\Vert \Vert \gamma f(z)-Az\Vert \\ +2{\epsilon }_n^2\Vert A(e_n-z)\Vert \Vert \gamma f(u_n)-Az\Vert +2{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉\\ \le (1-{\epsilon }_n\overline{\gamma })(1-{\beta }_n-{\epsilon }_n\overline{\gamma }){\Vert x_n-z\Vert }^2+(1-{\epsilon }_n\overline{\gamma }){\beta }_n{\Vert x_n-z\Vert }^2\\ +{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2+2{\epsilon }_n\gamma \eta {\Vert x_n-z\Vert }^2\\ +2{\beta }_n{\epsilon }_n\Vert x_n-z\Vert \Vert \gamma f(z)-Az\Vert -2{\beta }_n{\epsilon }_n\Vert x_n-z\Vert \Vert \gamma f(z)-Az\Vert \\ +2{\epsilon }_n^2\Vert A(e_n-z)\Vert \Vert \gamma f(u_n)-Az\Vert +2{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉\\ =(1-2{\epsilon }_n\overline{\gamma }+{\epsilon }_n^2{\overline{\gamma }}^2+2{\epsilon }_n\gamma \eta ){\Vert x_n-z\Vert }^2+{\epsilon }_n^2{\Vert \gamma f(u_n)-Az\Vert }^2\\ +2{\epsilon }_n^2\Vert A(e_n-z)\Vert \Vert \gamma f(u_n)-Az\Vert +2{\epsilon }_n〈e_n-z,\gamma f(z)-Az〉\\ =[1-2(\overline{\gamma }-\gamma \eta ){\epsilon }_n]{\Vert x_n-z\Vert }^2+{\epsilon }_n\left\{2〈e_n-z,\gamma f(z)-Az〉+{\epsilon }_{n}K\right\}.\end{array}$$

where K is an appropriate constant such that

$$K\ge \text{max}\left\{\underset{n\ge 1}{\text{sup}}\left\{{\bar{\gamma }}^2{∥x_n-z∥}^2+{∥\gamma f\left(u_n\right)-Az∥}^2+2∥A\left(e_n-z\right)∥∥\gamma f\left(u_n\right)-Az∥\right\}\right\},$$

Set $b_n=2\left(\bar{\gamma }-\gamma \eta \right){\epsilon }_n$ and c _n = ϵ _n {2〈e _n - z,γf(z) - Az〉 + ϵ _n K}. Then we have

$${∥x_{n+1}-z∥}^2\le \left(1-b_n\right){∥x_n-z∥}^2+c_n,\forall n\ge 0.$$

(3.43)

From the conditions (C1) and (3.42), we see that

$$\underset{n\to \infty }{\text{lim}}{b}_n=0,\sum _{n=0}^{\infty }{b}_n=\infty \mathsf{\text{and}}\underset{n\to \infty }{\text{lim}\text{sup}}{c}_n\le 0.$$

Therefore, applying Lemma 2.8 to (3.43), we get that {x _n } converges strongly to z ∈ Θ. This completes the proof.

Corollary 3.3. 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 ℝ satisfying (A1)-(A4) and let B be ξ-inverse strongly monotone such that

$$\Theta :=\left({\cap }_{k=1}^{M}SEP\left(F_k\right)\right)\cap VI\left(C,B\right)\ne \varnothing .$$

Let f be a contraction of H into itself with η ∈ (0,1). Let {x _n }, {y _n } and {u _n } be sequences generated by

$$\left\{\begin{array}{c}{x}_1=x\in Cchosenarbitrary,\hfill \\ y_n=\left(1-{\delta }_n\right)x_n+{\delta }_n{P}_C\left(x_n-{\alpha }_{n}B{x}_n\right),\hfill \\ u_n=J_{r_{M,n}}^{F_M}{J}_{r_{M-1,n}}^{F_{M-1}}{J}_{r_{M-2,n}}^{F_{M-2}}\dots J_{r_{2,n}}^{F_2}{J}_{r_{1,n}}^{F_1}{y}_n,\hfill \\ x_{n+1}={\epsilon }_{n}f\left(u_n\right)+{\beta }_n{x}_n+\left(1-{\beta }_n-{\epsilon }_n\right)P_C\left(u_n-{\lambda }_{n}B{u}_n\right),\forall n\ge 1,\hfill \end{array}\right.$$

where {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) lim_n→∞ϵ _n = 0 and ${\sum }_{n=1}^{\infty }{\epsilon }_n=\infty$,

(C2) 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1,

(C3) {α _n }, {λ _n } ⊂ [e, g] ⊂ (0, 2ξ), lim_n→∞α _n = 0 and lim_n→∞λ _n = 0,

(C4) {δ _n } ⊂ [0, b], for some b ∈ (0,1) and lim_n→∞|δ_n+1- δ _n | = 0,

(C5) lim inf_n→∞r _k,n > 0 and lim_n→∞|r_k,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

$$〈(f(z)-z,x-z〉\ge 0,\forall x\in \Theta .$$

Equivalently, we have z = P_Θf(z).

Proof. Put T _n ≡ I for all n ≥ 1 and for all x ∈ C. Then W _n = I, A = I and γ = 1. The conclusion follows from Theorem 3.2. This completes the proof.

If δ _n = 0 and M = 1, in Theorem 3.2, then we can obtain the following result immediately.

Corollary 3.4. 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 ℝ 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

$$\Theta :={\cap }_{n=1}^{\infty }F\left(T_n\right)\cap EP\left(F\right)\cap VI\left(C,B\right)\ne \varnothing .$$

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 $\bar{\gamma }>0$ and $0<\gamma <\frac{\bar{\gamma }}{\eta }$. Let {x _n }, {y _n } and {u _n } be sequences generated by

$$\left\{\begin{array}{c}{x}_1=x\in Cchosenarbitrary,\hfill \\ F\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,u_n-x_n⟩\ge 0,\forall y\in C,\hfill \\ x_{n+1}={\epsilon }_n\gamma f\left(u_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\epsilon }_{n}A\right)P_C\left(W_n{u}_n-{\lambda }_{n}B{W}_n{u}_n\right),\forall n\ge 1,\hfill \end{array}\right.$$

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

(C1) lim_n→∞ϵ _n = 0 and ${\sum }_{n=1}^{\infty }{\epsilon }_n=\infty$,

(C2) 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1,

(C3) {λ _n } ⊂ [e, g] ⊂ (0, 2ξ) and lim_n→∞λ _n = 0,

(C4) lim inf_n→∞r _n > 0 and lim_n→∞|r_n+1- r _n | = 0.

Then, {x _n } and {u _n } converge strongly to a point z ∈ Θ which is the unique solution of the variational inequality

$$⟨\left(A-\gamma f\right)z,x-z⟩\ge 0,\forall x\in \Theta .$$

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