A new hybrid general iterative algorithm for common solutions of generalized mixed equilibrium problems and variational inclusions

Abstract

In this article, we introduce a new general iterative method for finding a common element of the set of solutions generalized for mixed equilibrium problems, the set of solution for fixed point for nonexpansive mappings and the set of solutions for the variational inclusions for β₁, β₂-inverse-strongly monotone mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some suitable conditions. Our results improve and extend the corresponding results of Marino and Xu, Su et al., Tan and Chang and some authors.

Mathematics subject Classification: 46C05; 47H09; 47H10.

1. Introduction

Let C be a closed convex subset of a real Hilbert space H with the inner product 〈·, ·〉 and the norm || · ||. Let F be a bifunction of C × C into , where is the set of real numbers, Ψ : C → H be a mapping and $\phi :C\to \mathcal{R}$ be a real-valued function. The generalized mixed equilibrium problem for finding x ∈ C such that

$$F\left(x,y\right)+〈\mathrm{\Psi }x,y-x〉+\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C.$$

(1.1)

The set of solutions of (1.1) is denoted by GMEP(F, φ, Ψ), that is

$$\mathsf{\text{GMEP}}(F,\phi ,\mathrm{\Psi })=\{x\in C:F(x,y)+〈\mathrm{\Psi }x,y-x〉+\phi (y)-\phi (x)\ge 0,\forall y\in C\}.$$

If F ≡ 0, the problem (1.1) is reduced into the mixed variational inequality of Browder type[1] for finding x ∈ C such that

$$〈\mathrm{\Psi }x,y-x〉+\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C.$$

(1.2)

The set of solutions of (1.2) is denoted by MVI(C, φ, Ψ).

If Ψ ≡ 0, the problem (1.1) is reduced into the mixed equilibrium problem for finding x ∈ C such that

$$F\left(x,y\right)+\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C.$$

(1.3)

The set of solutions of (1.3) is denoted by MEP(F, φ).

If φ ≡ 0, the problem (1.3) is reduced into the equilibrium problem[2] for finding x ∈ C such that

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

(1.4)

The set of solutions of (1.4) is denoted by EP(F). See, e.g. [3–6] and the references therein.

If F ≡ 0 and φ ≡ 0, the problem (1.1) is reduced into the Hartmann-Stampacchia variational inequality[7] for finding x ∈ C such that

$$〈\mathrm{\Psi }x,y-x〉\ge 0,\forall y\in C.$$

(1.5)

The set of solutions of (1.5) is denoted by VI(C, Ψ).

If F ≡ 0 and Ψ ≡ 0, the problem (1.1) is reduced into the minimize problem for finding x ∈ C such that

$$\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C.$$

(1.6)

The set of solutions of (1.6) is denoted by Argmin(φ).

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:

$$\theta \left(x\right)=\frac{1}{2}〈Ax,x〉-〈x,y〉,\forall x\in F\left(S\right),$$

(1.7)

where A is a linear bounded operator, F(S) is the fixed point set of a nonexpansive mapping S and y is a given point in H[8].

Recall, a mapping S : C → C is said to be nonexpansive if

$$∥Sx-Sy∥\le ∥x-y∥,\forall x,y\in C.$$

If C is bounded closed convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty [9]. A mapping S : C → C is said to be a k-strictly pseudo-contraction[10] if there exists 0 ≤ k < 1 such that

$${∥Sx-Sy∥}^2\le {∥x-y∥}^2+k{∥\left(I-S\right)x-\left(I-S\right)y∥}^2,\forall x,y\in C,$$

where I denotes the identity operator on C. A mapping A of C into H is called monotone if

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

A mapping A of C into H is called an α-inverse-strongly monotone if there exists a positive real number α such that

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

A mapping A of C into H is called α-strongly monotone if there exists a positive real number α such that

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

A linear bounded operator A is called strongly positive if there exists a constant $\bar{\gamma }>0$ with the property

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

A self mapping f : C → C is called contraction on C if there exists a constant α ∈ (0, 1) such that

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

Let B : H → H be a single-valued nonlinear mapping and M : H → 2^Hbe a set-valued mapping. The variational inclusion problem is to find x ∈ H such that

$$\theta \in B\left(x\right)+M\left(x\right),$$

(1.8)

where θ is the zero vector in H. The set of solutions of problem (1.8) is denoted by I(B, M). The variational inclusion has been extensively studied in the literature. See, e.g. [11–15] and the reference therein.

A set-valued mapping M : H → 2^His called monotone if for all x, y ∈ H, f ∈ M(x) and g ∈ M(y) imply 〈x - y, f - g〉 ≥ 0. A monotone mapping M is maximal if its graph G(M) := {(f, x) ∈ H × H : f ∈ M(x)} of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for (x, f) ∈ H × H, 〈x - y, f - g〉 > 0 for all (y, g) ∈ G(M) imply f ∈ M(x).

Let B be an inverse-strongly monotone mapping of C into H and let N _C v be normal cone to C at v ∈ C, i.e., N _C v = {w ∈ H : 〈v - u, w〉 ≥ 0, ∀u ∈ C}, and define

$$Mv=\left\{\begin{array}{cc}Bv+N_{C}v,\hfill & \mathsf{\text{if}}v\in C,\hfill \\ \varnothing ,\hfill & \mathsf{\text{if}}v\notin C.\hfill \end{array}\right.$$

Then M is a maximal monotone and θ ∈ Mν if and only if v ∈ VI(C, B) [16].

Let M : H → 2^Hbe a set-valued maximal monotone mapping, then the single-valued mapping J_M,λ: H → H defined by

$$J_{M,\lambda }\left(x\right)={\left(I+\lambda M\right)}^{-1}\left(x\right),x\in H$$

(1.9)

is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping. In the worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone and that a solution of problem (1.8) is a fixed point of the operator J_M,λ(I - λB) for all λ > 0 (see also [17]).

In 2000, Moudafi [18] introduced the viscosity approximation method for nonexpansive mapping and proved that if H is a real Hilbert space, the sequence {x _n } defined by the iterative method below, with the initial guess x₀ ∈ C is chosen arbitrarily,

$$x_{n+1}={\alpha }_{n}f\left(x_n\right)+\left(1-{\alpha }_n\right)S{x}_n,n\ge 0,$$

(1.10)

where {α _n } ⊂ (0, 1) satisfies certain conditions, converge strongly to a fixed point of S (say $\bar{x}\in C$) which is the unique solution of the following variational inequality.

In 2005, Iiduka and Takahashi [19] introduced following iterative process x₀ ∈ C,

$$x_{n+1}={\alpha }_{n}u+\left(1-{\alpha }_n\right)S{P}_C\left(x_n-{\lambda }_{n}A{x}_n\right),\forall n\ge 0,$$

(1.11)

where u ∈ C, {α _n } ⊂ (0, 1) and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < 2β. They proved that under certain appropriate conditions imposed on {α _n } and {λ _n }, the sequence {x _n } converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say $\bar{x}\in C$) which solve some variational inequality.

In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansive mapping. They defined the sequence {x _n } generated by the algorithm x₀ ∈ C,

$$x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{x}_n,n\ge 0$$

(1.12)

where {α _n } ⊂ (0, 1) and A is a strongly positive linear bounded operator. They proved that if C = H then the sequence {x _n } converges strongly to a fixed point of S (say $\bar{x}\in H$) which is the unique solution of the following variational inequality.

In 2008, Su et al. [20] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: x₁, u _n ∈ H

$$\left\{\begin{array}{c}F\left(u_n,y\right)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C,\hfill \\ x_{n+1}={\alpha }_{n}f\left(x_n\right)+\left(1-{\alpha }_n\right)S{P}_C\left(u_n-{\lambda }_{n}A{u}_n\right),\hfill \end{array}\right.$$

(1.13)

for all n ∈ ℕ, where {α _n } ⊂ [0, 1) and {r _n } ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they proved {x _n } and {u _n } converge strongly to the same point z, where z = P_{F(S)∩VI(C,A)∩ EP(F)}f(z).

In 2011, Tan and Chang [14] introduced following iterative process for {T _n : C → C} be a sequence of nonexpansive mappings. Let {x _n } be the sequence defined by

$$x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)\left(S{P}_C\left(\left(1-t_n\right)J_{M,\lambda }\left(I-\lambda A\right)T_n\left(I-\mu B\right)\right)x_n\right),\forall n\ge 0,$$

(1.14)

where {α _n } ⊂ (0, 1), λ ∈ (0, 2α] and μ ∈ (0, 2β]. The sequence {x _n } converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solutions of the variational inequality and the generalized equilibrium problem.

In this article, we modify by Marino and Xu [8], Su et al. [20] and Tan and Chang [14], the purpose of this article, we show that under some control conditions the sequence {x _n } converges strongly to a common element of the set of fixed points of nonexpansive mappings, the solution of the generalized mixed equilibrium problems and the solution of the variational inclusions in a real Hilbert space.

2. Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. Recall that the metric (nearest point) projection P _C from H onto C assigns to each x ∈ H, the unique point in P _C x ∈ C satisfying the property

$$∥x-P_{C}x∥=\underset{y\in C}{\mathsf{\text{inf}}}∥x-y∥.$$

The following characterizes the projection P _C . We recall some lemmas which will be needed in the rest of this article.

Lemma 2.1. The function u ∈ C is a solution of the variational inequality (1.5) if and only if u ∈ C satisfies the relation u = P _C (u - λ Ψu) for all λ > 0.

Lemma 2.2. For a given z ∈ H, u ∈ C, u = P _C z ⇔ 〈u - z, v - u〉 ≥ 0, ∀v ∈ C.

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

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

(2.1)

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

$$〈x-P_{C}x,y-P_{C}x〉\le 0.$$

(2.2)

Lemma 2.3. [21]Let M : H → 2^Hbe a maximal monotone mapping and let B : H → H be a monotone and Lipshitz continuous mapping. Then the mapping L = M + B : H → 2^His a maximal monotone mapping.

Lemma 2.4. [22]Each Hilbert space H satisfies Opial's condition, that is, for any sequence {x _n } ⊂ H with x _n ⇀ x, the inequality lim inf_n→∞||x _n - x|| < lim inf_n→∞||x _n - y||, hold for each y ∈ H with y ≠ x.

Lemma 2.5. [23]Assume {a _n } is a sequence of nonnegative real numbers such that

$$a_{n+1}\le \left(1-{\gamma }_n\right)a_n+{\delta }_n,\forall n\ge 0,$$

where {γ _n } ⊂ (0, 1) and {δ _n } is a sequence insuch that

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$.

• (ii) $\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{sup}}}\frac{{\delta }_n}{{\gamma }_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\delta }_n\right|<\mathrm{\infty }$.

Then lim_n→∞a _n = 0.

Lemma 2.6. [24]Let C be a closed convex subset of a real Hilbert space H and let S : C → C be a nonexpansive mapping. Then I - S is demiclosed at zero, that is,

$$x_n\rightharpoonup x,x_n-S{x}_n\to 0$$

implies x = Sx.

For solving the mixed equilibrium problem, let us assume that the bifunction $F:C\times C\to \mathcal{R}$ and $\phi :C\to \mathcal{R}$ 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 any x, y ∈ C;

(A3) for each fixed y ∈ C, x ↦ F(x, y) is weakly upper semicontinuous;

(A4) for each fixed x ∈ C, y ↦ F(x, y) is convex and lower semicontinuous;

(B1) for each x ∈ C and r > 0, there exist a bounded subset D _x ⊆ C and y _x ∈ C such that for any z ∈ C \ D _x ,

$$F\left(z,y_x\right)+\phi \left(y_x\right)-\phi \left(z\right)+\frac{1}{r}〈y_x-z,z-x〉<0,$$

(B2) C is a bounded set.

Lemma 2.7. [25]Let C be a nonempty closed convex subset of a real Hilbert space H. Let$F:C\times C\to \mathcal{R}$be a bifunction mapping satisfies (A1)-(A4) and let$\phi :C\to \mathcal{R}$is convex and lower semicontinuous such that$C\cap dom\phi \ne \varnothing$. Assume that either (B1) or (B2) holds. For r > 0 and x ∈ H, then there exists z ∈ C such that

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

Define a mapping $T_r^{\left(F,\phi \right)}:H\to C$ as follows:

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

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

• (i) $T_r^{\left(F,\phi \right)}$ is single-valued;

• (ii) $T_r^{\left(F,\phi \right)}$ is firmly nonexpansive, i.e., for any x, y ∈ H,

  $${∥T_r^{\left(F,\phi \right)}x-T_r^{\left(F,\phi \right)}y∥}^2\le 〈T_r^{\left(F,\phi \right)}x-T_r^{\left(F,\phi \right)}y,x-y〉;$$

• (iii) $F\left(T_r^{\left(F,\phi \right)}\right)=\mathsf{\text{MEP}}\left(F,\phi \right)$;

• (iv) MEP(F, φ) is closed and convex.

Lemma 2.8. [8]Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient$\bar{\gamma }>0$and 0 < ρ ≤ ||A||^-1, then$∥I-\rho A∥\le 1-\rho \bar{\gamma }$.

Lemma 2.9. [26]Let H be a real Hilbert space and A : H → H a mapping.

• (i) If A is a δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1, then I - A is a contraction with constant $\sqrt{\left(1-\delta \right)/\mu }$.

• (ii) If A is a δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1, then for any fixed number τ ∈ (0, 1), I - τA is a contraction with constant $1-\tau \left(1-\sqrt{\left(1-\delta \right)/\mu }\right)$.

3. Strong convergence theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of F(S), GMEP(F₁, φ₁, B₁), GMEP(F₂, φ₂, B₂), I(A₁, M₁) and I(A₂, M₂).

Theorem 3.1. Let H be a real Hilbert space, C be a closed convex subset of H. Let F₁, F₂be bifunctions of C × C intosatisfying (A1)-(A4) and A₁, A₂, B₁, B₂ : C → H be β₁, β₂, η, ρ-inverse-strongly monotone mappings, ${\phi }_1,{\phi }_2:C\to \mathcal{R}$be convex and lower semicontinuous functions, f : C → C be a contraction with coefficient α (0 < α < 1), M₁, M₂ : H → 2^Hbe maximal monotone mappings and A is a δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1, γ is a positive real number such that$\gamma <\frac{1}{\alpha }\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)$. Assume that either (B1) or (B2) holds. Let S be a nonexpansive mapping of C into itself such that

$$\mathrm{\Theta }:=F\left(S\right)\cap \mathsf{\text{GMEP}}\left(F_1,{\phi }_1,B_1\right)\cap \mathsf{\text{GMEP}}\left(F_2,{\phi }_2,B_2\right)\cap I\left(A_1,M_1\right)\cap I\left(A_2,M_2\right)\ne \varnothing .$$

Suppose {x _n } is a sequences generated by the following algorithm x₀ ∈ C arbitrarily:

$$\left\{\begin{array}{c}{u}_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)\hfill \\ v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)\hfill \\ x_{n+1}={\xi }_n{P}_C\left\{{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{P}_C\left[J_{M_1,{\lambda }_1}\left(1-{\lambda }_1{A}_1\right)u_n\right]\right\}\hfill \\ +\left(1-{\xi }_n\right)P_C\left[J_{M_2,{\lambda }_2}\left(I-{\lambda }_2{A}_2\right)v_n\right],\forall n\ge 0,\hfill \end{array}\right.$$

(3.1)

where {α _n }, {ξ _n } ⊂ (0, 1), λ₁ ∈ (0, 2β₁) such that 0 < a₁ ≤ λ₁ < b₁ < 2β₁, λ₂ ∈ (0, 2β₂) such that 0 < a₁ ≤ λ₂ ≤ b₂ < 2β₂, r _n ∈ (0, 2η) with 0 < c ≤ d ≤ 1 - η and s _n ∈ (0, 2ρ) with 0 < e ≤ f ≤ 1 - ρ satisfy the following conditions:

(C1): lim_n→∞α _n = 0, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },{\sum }_{n=1}^{\mathrm{\infty }}\left|{\alpha }_{n+1}-{\alpha }_n\right|<\mathrm{\infty }$,

(C2): 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1, ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$,

(C3): lim inf_n→∞r _n > 0 and lim_n→∞|r_n+1- r _n | = 0,

(C4): lim inf_n→∞s _n > 0 and lim_n→∞|s_n+1- s _n | = 0.

Then {x _n } converges strongly to q ∈ Θ, where q = P_Θ(γf + I - A)(q) which solves the following variational inequality:

$$〈\left(\gamma f-A\right)q,p-q〉\le 0,\forall p\in \mathrm{\Theta }$$

which is the optimality condition for the minimization problem

$$\underset{q\in \mathrm{\Theta }}{\mathsf{\text{min}}}\frac{1}{2}〈Aq,q〉-h\left(q\right),$$

(3.2)

where h is a potential function for γf (i.e., h'(q) = γf(q) for q ∈ H).

Proof. Since A₁, A₂ are β₁, β₂-inverse-strongly monotone mappings, we have

$$\begin{array}{ll}\hfill {∥\left(I-{\lambda }_1{A}_1\right)x-\left(I-{\lambda }_1{A}_1\right)y∥}^2& ={∥\left(x-y\right)-{\lambda }_1\left(A_{1}x-A_{1}y\right)∥}^2\\ ={∥x-y∥}^2-2{\lambda }_1〈x-y,A_{1}x-A_{1}y〉+{\lambda }_1^2{∥A_{1}x-A_{1}y∥}^2\\ \le {∥x-y∥}^2+{\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_{1}x-A_{1}y∥}^2\\ \le {∥x-y∥}^2.\end{array}$$

In similar way, we can obtain

$${∥\left(I-{\lambda }_2{A}_2\right)x-\left(I-{\lambda }_2{A}_2\right)y∥}^2\le {∥x-y∥}^2.$$

And B₁, B₂ are η, ρ-inverse-strongly monotone mappings, we have

$$\begin{array}{ll}\hfill {∥\left(I-r_n{B}_1\right)x-\left(I-r_n{B}_1\right)y∥}^2& ={∥\left(x-y\right)-r_n\left(B_{1}x-B_{1}y\right)∥}^2\\ ={∥x-y∥}^2-2r_n〈x-y,B_{1}x-B_{1}y〉+r_n^2{∥B_{1}x-B_{1}y∥}^2\\ \le {∥x-y∥}^2+r_n\left(r_n-2\eta \right){∥B_{1}x-B_{1}y∥}^2\\ \le {∥x-y∥}^2.\end{array}$$

In similar way, we can obtain

$${∥\left(I-s_n{B}_2\right)x-\left(I-s_n{B}_2\right)y∥}^2\le {∥x-y∥}^2.$$

It is clear that if 0 < λ₁ < 2β₁, 0 < λ₂ < 2β₂, 0 < r _n < 2η, 0 < s _n ≤ 2ρ then I - λ₁A₁, I - λ₂A₂, I - r _n B₁, I - s _n B₂ are all nonexpansive. We will divide the proof into six steps.

Step 1. We will show {x _n } is bounded. Put $y_n=J_{M_1,{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)$ for all n ≥ 0 and $w_n=J_{M_2,{\lambda }_2}\left(v_n-{\lambda }_2{A}_2{v}_n\right)$ for all n ≥ 0. It follows that

$$\begin{array}{ll}\hfill ∥y_n-q∥& =∥J_{M_1,{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)-J_{M_1,{\lambda }_1}\left(q-{\lambda }_1{A}_{1}q\right)∥\\ \le ∥u_n-q∥.\end{array}$$

(3.3)

In similar way, we can obtain

$$\begin{array}{ll}\hfill ∥w_n-q∥& =∥J_{M_{2,}{\lambda }_2}\left(v_n-{\lambda }_2{A}_2{v}_n\right)-J_{M_2,{\lambda }_2}\left(q-{\lambda }_2{A}_{2}q\right)∥\\ \le ∥v_n-q∥.\end{array}$$

(3.4)

Put $y_n^{\prime }=P_C{y}_n,n\ge 0$. It follows that

$$\begin{array}{ll}\hfill ∥{y^{\prime }}_n-q∥& =∥P_C{y}_n-P_{C}q∥\\ \le ∥y_n-q∥.\end{array}$$

(3.5)

By Lemma 2.7, we have $u_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right),v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)$ for all n ≥ 0. Then, we have

$$\begin{array}{ll}\hfill {∥u_n-q∥}^2& ={∥T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)-T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(q-r_n{B}_{1}q\right)∥}^2\\ \le {∥\left(x_n-r_n{B}_1{x}_n\right)-\left(q-r_n{B}_{1}q\right)∥}^2\\ \le {∥x_n-q∥}^2+r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2\\ \le {∥x_n-q∥}^2.\end{array}$$

(3.6)

In similar way, we can obtain

$$\begin{array}{ll}\hfill {∥v_n-q∥}^2& ={∥T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)-T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(q-s_n{B}_{2}q\right)∥}^2\\ \le {∥\left(x_n-s_n{B}_2{x}_n\right)-\left(q-s_n{B}_{2}q\right)∥}^2\\ \le {∥x_n-q∥}^2+s_n\left(s_n-2\rho \right){∥B_2{x}_n-B_{2}q∥}^2\\ \le {∥x_n-q∥}^2.\end{array}$$

(3.7)

Put z _n = P _C [α _n γf(x _n ) + (I - α _n A)SP _C y _n ] for all n ≥ 0. From (3.1) and by Lemma 2.9

(ii), we deduce that

$$\begin{array}{ll}\hfill ∥x_{n+1}-q∥& =∥{\xi }_n\left(z_n-q\right)+\left(1-{\xi }_n\right)\left(P_C{w}_n-q\right)∥\\ \le {\xi }_n∥P_C\left[{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{P}_C{y}_n\right]-P_{C}q∥+\left(1-{\xi }_n\right)∥P_C{w}_n-P_{C}q∥\\ \le {\xi }_n∥{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{P}_C{y}_n-q∥+\left(1-{\xi }_n\right)∥w_n-q∥\\ ={\xi }_n∥{\alpha }_n\left(\gamma f\left(x_n\right)-Aq\right)+\left(I-{\alpha }_{n}A\right)\left(S{P}_C{y}_n-q\right)∥+\left(1-{\xi }_n\right)∥w_n-q∥\\ \le {\xi }_n{\alpha }_n∥\gamma f\left(x_n\right)-Aq∥\\ +{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥P_C{y}_n-q∥+\left(1-{\xi }_n\right)∥w_n-q∥\\ \le {\xi }_n{\alpha }_n∥\gamma f\left(x_n\right)-\gamma f\left(q\right)∥+{\xi }_n{\alpha }_n∥\gamma f\left(q\right)-Aq∥\\ +{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥P_C{y}_n-q∥+\left(1-{\xi }_n\right)∥x_n-q∥\\ \le {\xi }_n{\alpha }_n\gamma \alpha ∥x_n-q∥+{\xi }_n{\alpha }_n∥\gamma f\left(q\right)-Aq∥\\ +{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥y_n-q∥+\left(1-{\xi }_n\right)∥x_n-q∥\\ =\left(1-\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right){\xi }_n{\alpha }_n\right)∥x_n-q∥+{\xi }_n{\alpha }_n∥\gamma f\left(q\right)-Aq∥\\ \le \left(1-\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right){\xi }_n{\alpha }_n\right)∥x_n-q∥\\ +\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right){\xi }_n{\alpha }_n\frac{∥\gamma f\left(q\right)-Aq∥}{\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right)}\\ \le \mathsf{\text{max}}\left\{∥x_n-q∥,\frac{∥\gamma f\left(q\right)-Aq∥}{\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha }\right\}.\end{array}$$

(3.8)

It follows from induction that

$$∥x_n-q∥\le \mathsf{\text{max}}\left\{∥x_0-q∥,\frac{∥\gamma f\left(q\right)-Aq∥}{\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha }\right\},n\ge 0.$$

Therefore {x _n } is bounded, so are {y _n },{z _n },{P _C w _n },{SP _C y _n },{f(x _n )} and {ASP _C y _n }.

Step 2. We claim that lim_n→∞||x_n+2- x_n+1|| = 0. From (3.1), we have

$$\begin{array}{ll}\hfill ∥x_{n+2}-x_{n+1}∥& =∥{\xi }_{n+1}{z}_{n+1}+\left(1-{\xi }_{n+1}\right)P_C{w}_{n+1}-{\xi }_n{z}_n-\left(1-{\xi }_n\right)P_C{w}_n∥\\ =∥{\xi }_{n+1}\left(z_{n+1}-z_n\right)+\left({\xi }_{n+1}-{\xi }_n\right)z_n\\ +\left(1-{\xi }_{n+1}\right)\left(P_C{w}_{n+1}-P_C{w}_n\right)+\left({\xi }_n-{\xi }_{n+1}\right)P_C{w}_n∥\\ \le {\xi }_{n+1}∥z_{n+1}-z_n∥+\left(1-{\xi }_{n+1}\right)∥w_{n+1}-w_n∥\\ +\left|{\xi }_{n+1}-{\xi }_n\right|\left(∥z_n∥+∥P_C{w}_n∥\right).\end{array}$$

(3.9)

Since I - λ₂A₂ be nonexpansive, we have

$$\begin{array}{ll}\hfill ∥w_{n+1}-w_n∥& =∥J_{M_2,{\lambda }_2}\left(v_{n+1}-{\lambda }_2{A}_2{v}_{n+1}\right)-J_{M_2,{\lambda }_2}\left(v_n-{\lambda }_2{A}_2{v}_n\right)∥\\ \le ∥\left(v_{n+1}-{\lambda }_2{A}_2{v}_{n+1}\right)-\left(v_n-{\lambda }_2{A}_2{v}_n\right)∥\\ \le ∥v_{n+1}-v_n∥.\end{array}$$

(3.10)

On the other hand, from $v_{n-1}=T_{s_{n-1}}^{\left(F_2,{\phi }_2\right)}\left(x_{n-1}-s_{n-1}{B}_2{x}_{n-1}\right)$ and $v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)$, it follows that

$$\begin{array}{c}{F}_2\left(v_{n-1},y\right)+〈B_2{x}_{n-1},y-v_{n-1}〉+{\phi }_2\left(y\right)-{\phi }_2\left(v_{n-1}\right)\\ +\frac{1}{s_{n-1}}〈y-v_{n-1},v_{n-1}-x_{n-1}〉\ge 0,\forall y\in C\end{array}$$

(3.11)

and

$$F_2\left(v_n,y\right)+〈B_2{x}_n,y-v_n〉+{\phi }_2\left(y\right)-{\phi }_2\left(v_n\right)+\frac{1}{s_n}〈y-v_n,v_n-x_n〉\ge 0,\forall y\in C.$$

(3.12)

Substituting y = v _n in (3.11) and y = v_n-1in (3.12), we get

$$F_2\left(v_{n-1},v_n\right)+〈B_2{x}_{n-1},v_n-v_{n-1}〉+{\phi }_2\left(v_n\right)-{\phi }_2\left(v_{n-1}\right)+\frac{1}{s_{n-1}}〈v_n-v_{n-1},v_{n-1}-x_{n-1}〉\ge 0$$

and

$$F_2\left(v_n,v_{n-1}\right)+〈B_2{x}_n,v_{n-1}-v_n〉+{\phi }_2\left(v_{n-1}\right)-{\phi }_2\left(v_n\right)+\frac{1}{s_n}〈v_{n-1}-v_n,v_n-x_n〉\ge 0.$$

From (A2), we obtain

$$〈v_n-v_{n-1},B_2{x}_{n-1}-B_2{x}_n+\frac{v_{n-1}-x_{n-1}}{s_{n-1}}-\frac{v_n-x_n}{s_n}〉\ge 0,$$

and then

$$〈v_n-v_{n-1},s_{n-1}\left(B_2{x}_{n-1}-B_2{x}_n\right)+v_{n-1}-x_{n-1}-\frac{s_{n-1}}{s_n}\left(v_n-x_n\right)〉\ge 0,$$

so

$$〈v_n-v_{n-1},s_{n-1}{B}_2{x}_{n-1}-s_{n-1}{B}_2{x}_n+v_{n-1}-v_n+v_n-x_{n-1}-\frac{s_{n-1}}{s_n}\left(v_n-x_n\right)〉\ge 0.$$

It follows that

$$\begin{array}{c}〈v_n-v_{n-1},\left(I-s_{n-1}{B}_2\right)x_n-\left(I-s_{n-1}{B}_2\right)x_{n-1}\\ +v_{n-1}-v_n+v_n-x_n-\frac{s_{n-1}}{s_n}\left(v_n-x_n\right)〉\ge 0,\\ 〈v_n-v_{n-1},v_{n-1}-v_n〉+〈v_n-v_{n-1},x_n-x_{n-1}+\left(1-\frac{s_{n-1}}{s_n}\right)\left(v_n-x_n\right)〉\ge 0.\end{array}$$

Without loss of generality, let us assume that there exists a real number e such that s_{n- 1}> e > 0, for all n ∈ ℕ. Then, we have

$$\begin{array}{ll}\hfill {∥v_n-v_{n-1}∥}^2& \le 〈v_n-v_{n-1},x_n-x_{n-1}+\left(1-\frac{s_{n-1}}{s_n}\right)\left(v_n-x_n\right)〉\\ \le ∥v_n-v_{n-1}∥\left\{∥x_n-x_{n-1}∥+\left|1-\frac{s_{n-1}}{s_n}\right|∥v_n-x_n∥\right\}\end{array}$$

and hence

$$\begin{array}{ll}\hfill ∥v_n-v_{n-1}∥& \le ∥x_n-x_{n-1}∥+\frac{1}{s_n}\left|s_n-s_{n-1}\right|∥v_n-x_n∥\\ \le ∥x_n-x_{n-1}∥+\frac{M_1}{e}\left|s_n-s_{n-1}\right|,\end{array}$$

(3.13)

where M₁ = sup{||v _n - x _n || : n ∈ ℕ}. Substituting (3.13) into (3.9) and (3.10) that

$$\begin{array}{ll}\hfill ∥x_{n+2}-x_{n+1}∥& \le {\xi }_{n+1}∥z_{n+1}-z_n∥+\left(1-{\xi }_{n+1}\right)\left\{∥x_{n+1}-x_n∥+\frac{M_1}{e}\left|s_n-s_{n-1}\right|\right\}\\ +\left|{\xi }_{n+1}-{\xi }_n\right|\left(∥z_n∥+∥P_C{w}_n∥\right).\end{array}$$

(3.14)

By Lemma 2.9 (ii), it follow that

$$\begin{array}{l}∥z_{n+1}-z_n∥\\ =∥P_C\left[{\alpha }_{n+1}\gamma f\left(x_{n+1}\right)+\left(I-{\alpha }_{n+1}A\right)S{P}_C{y}_{n+1}\right]\\ -P_C\left[{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{P}_C{y}_n\right]∥\\ \le ∥{\alpha }_{n+1}\gamma f\left(x_{n+1}\right)+\left(I-{\alpha }_{n+1}A\right)S{P}_C{y}_{n+1}-{\alpha }_n\gamma f\left(x_n\right)-\left(I-{\alpha }_{n}A\right)S{P}_C{y}_n∥\\ =∥{\alpha }_{n+1}\gamma \left(f\left(x_{n+1}\right)-f\left(x_n\right)\right)+\left({\alpha }_{n+1}-{\alpha }_n\right)\gamma f\left(x_n\right)\\ +\left(I-{\alpha }_{n+1}A\right)\left(S{P}_C{y}_{n+1}-S{P}_C{y}_n\right)+\left({\alpha }_n-{\alpha }_{n+1}\right)AS{P}_C{y}_n∥\\ \le {\alpha }_{n+1}\gamma \alpha ∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|∥\gamma f\left(x_n\right)∥\\ +\left(1-{\alpha }_{n+1}\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥y_{n+1}-y_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|∥AS{P}_C{y}_n∥\\ ={\alpha }_{n+1}\gamma \alpha ∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|\left(∥\gamma f\left(x_n\right)∥+∥AS{P}_C{y}_n∥\right)\\ +\left(1-{\alpha }_{n+1}\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥y_{n+1}-y_n∥.\end{array}$$

(3.15)

Since I - λ₁A₁ be nonexpansive, we have

$$\begin{array}{ll}\hfill ∥y_{n+1}-y_n∥& =∥J_{M_1,{\lambda }_1}\left(u_{n+1}-{\lambda }_1{A}_1{u}_{n+1}\right)-J_{M_1,{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)∥\\ \le ∥\left(u_{n+1}-{\lambda }_1{A}_1{u}_{n+1}\right)-\left(u_n-{\lambda }_1{A}_1{u}_n\right)∥\\ \le ∥\left(I-{\lambda }_1{A}_1\right)u_{n+1}-\left(I-{\lambda }_1{A}_1\right)u_n∥\\ \le ∥u_{n+1}-u_n∥.\end{array}$$

(3.16)

On the other hand, from $u_{n-1}=T_{r_{n-1}}^{\left(F_1,{\phi }_1\right)}\left(x_{n-1}-r_{n-1}{B}_1{x}_{n-1}\right)$ and $u_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)$, it follows that

$$\begin{array}{l}{F}_1\left(u_{n-1},y\right)+〈B_1{x}_{n-1},y-u_{n-1}〉+{\phi }_1\left(y\right)-{\phi }_1\left(u_{n-1}\right)\\ +\frac{1}{r_{n-1}}〈y-u_{n-1},u_{n-1}-x_{n-1}〉\ge 0,\forall y\in C\end{array}$$

(3.17)

and

$$F_1\left(u_n,y\right)+〈B_1{x}_n,y-u_n〉+{\phi }_1\left(y\right)-{\phi }_1\left(u_n\right)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C.$$

(3.18)

Substituting y = u _n in (3.17) and y = u_n-1in (3.18), we get

$$F_1\left(u_{n-1},u_n\right)+〈B_1{x}_{n-1},u_n-u_{n-1}〉+{\phi }_1\left(u_n\right)-{\phi }_1\left(u_{n-1}\right)+\frac{1}{r_{n-1}}〈u_n-u_{n-1},u_{n-1}-x_{n-1}〉\ge 0$$

and

$$F_1\left(u_n,u_{n-1}\right)+〈B_1{x}_n,u_{n-1}-u_n〉+{\phi }_1\left(u_{n-1}\right)-{\phi }_1\left(u_n\right)+\frac{1}{r_n}〈u_{n-1}-u_n,u_n-x_n〉\ge 0.$$

From (A2), we obtain

$$〈u_n-u_{n-1},B_1{x}_{n-1}-B_1{x}_n+\frac{u_{n-1}-x_{n-1}}{r_{n-1}}-\frac{u_n-x_n}{r_n}〉\ge 0,$$

and then

$$〈u_n-u_{n-1},r_{n-1}\left(B_1{x}_{n-1}-B_1{x}_n\right)+u_{n-1}-x_{n-1}-\frac{r_{n-1}}{r_n}\left(u_n-x_n\right)〉\ge 0,$$

so

$$〈u_n-u_{n-1},r_{n-1}{B}_1{x}_{n-1}-r_{n-1}{B}_1{x}_n+u_{n-1}-u_n+u_n-x_{n-1}-\frac{r_{n-1}}{r_n}\left(u_n-x_n\right)〉\ge 0.$$

It follows that

$$\begin{array}{c}〈u_n-u_{n-1},\left(I-r_{n-1}{B}_1\right)x_n-\left(I-r_{n-1}{B}_1\right)x_{n-1}\\ +u_{n-1}-u_n+u_n-x_n-\frac{r_{n-1}}{r_n}\left(u_n-x_n\right)〉\ge 0,\\ 〈u_n-u_{n-1},u_{n-1}-u_n〉+〈u_n-u_{n-1},x_n-x_{n-1}+\left(1-\frac{r_{n-1}}{r_n}\right)\left(u_n-x_n\right)〉\ge 0.\end{array}$$

Without loss of generality, let us assume that there exists a real number c such that r_{n- 1}> c > 0, for all n∈ℕ. Then, we have

$$\begin{array}{ll}\hfill {∥u_n-u_{n-1}∥}^2& \le 〈u_n-u_{n-1},x_n-x_{n-1}+\left(1-\frac{r_{n-1}}{r_n}\right)\left(u_n-x_n\right)〉\\ \le ∥u_n-u_{n-1}∥\left\{∥x_n-x_{n-1}∥+\left|1-\frac{r_{n-1}}{r_n}\right|∥u_n-x_n∥\right\}\end{array}$$

and hence

$$\begin{array}{ll}\hfill ∥u_n-u_{n-1}∥& \le ∥x_n-x_{n-1}∥+\frac{1}{r_n}\left|r_n-r_{n-1}\right|∥u_n-x_n∥\\ \le ∥x_n-x_{n-1}∥+\frac{M_2}{c}\left|r_n-r_{n-1}\right|,\end{array}$$

(3.19)

where M₂ = sup{||u _n - x _n || : n ∈ ℕ}. Substituting (3.19) into (3.16), we have

$$∥y_n-y_{n-1}∥\le ∥x_n-x_{n-1}∥+\frac{M_2}{c}\left|r_n-r_{n-1}\right|.$$

(3.20)

Substituting (3.20) into (3.15), we obtain that

$$\begin{array}{ll}\hfill ∥z_{n+1}-z_n∥& \le {\alpha }_{n+1}\gamma \alpha ∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|\left(∥\gamma f\left(x_n\right)∥+∥AS{P}_C{y}_n∥\right)\\ +\left(1-{\alpha }_{n+1}\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\left\{∥x_n-x_{n-1}∥+\frac{M_2}{c}\left|r_n-r_{n-1}\right|\right\}\end{array}$$

(3.21)

And substituting (3.10), (3.13), (3.21) into (3.9), we get

$$\begin{array}{ll}\hfill ∥x_{n+2}-x_{n+1}∥& \le {\xi }_{n+1}\left\{{\alpha }_{n+1}\gamma \alpha \right.∥x_{n+1}-x_n∥+\left|{\alpha }_{n+1}-{\alpha }_n\right|\left(∥\gamma f\left(x_n\right)∥+∥AS{P}_C{y}_n∥\right)\\ \left.+\left(1-{\alpha }_{n+1}\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥x_n-x_{n-1}∥+\frac{M_2}{c}\left|r_n-r_{n-1}\right|\right\}\\ +\left(1-{\xi }_{n+1}\right)\left\{∥x_n-x_{n-1}∥+\frac{M_1}{e}\left|s_n-s_{n-1}\right|\right\}\\ +\left|{\xi }_{n+1}-{\xi }_n\right|\left(∥z_n∥+∥P_C{w}_n∥\right)\\ \le \left(1-\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right){\xi }_{n+1}{\alpha }_{n+1}\right)∥x_{n+1}-x_n∥\\ +\left(\left|{\alpha }_{n+1}-{\alpha }_n\right|+\left|{\xi }_{n+1}-{\xi }_n\right|\right)M_3\\ +\frac{M_2}{c}\left|r_n-r_{n-1}\right|+\frac{M_1}{e}\left|s_n-s_{n-1}\right|,\end{array}$$

(3.22)

where M₃ > 0 is a constant satisfying

$$\underset{n}{\mathsf{\text{sup}}}\left\{∥\gamma f\left(x_n\right)∥+∥AS{P}_C{y}_n∥,∥z_n∥+∥P_C{w}_n∥\right\}\le M_3.$$

This together with (C1)-(C4) and Lemma 2.5, imply that

$$\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥x_{n+2}-x_{n+1}∥=0.$$

(3.23)

From (3.20) and (C3), we also have ||y_n+1- y _n || → 0 as n → ∞.

Step 3. We show the followings:

• (i) lim_n→∞||A ₁ u _n - A ₁ q|| = 0;

• (ii) lim_n→∞||A ₂ v _n - A ₂ q|| = 0;

• (iii) lim_n→∞||B ₁ x _n - B ₁ q|| = 0;

• (iv) lim_n→∞||B ₂ x _n - B ₂ q|| = 0.

For q ∈ Θ and $q=J_{M_1,{\lambda }_1}\left(q-{\lambda }_1{A}_{1}q\right)$, then we get

$$\begin{array}{ll}\hfill {∥y_n-q∥}^2& ={∥J_{M_1,{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)-J_{M_1,{\lambda }_1}\left(q-{\lambda }_1{A}_{1}q\right)∥}^2\\ \le {∥\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right)∥}^2\\ \le {∥u_n-q∥}^2+{\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2\\ \le {∥x_n-q∥}^2+{\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2.\end{array}$$

Using (3.5), it follows that

$$\begin{array}{l}{∥z_n-q∥}^2\\ ={∥P_C\left[{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{y^{\prime }}_n\right]-P_C\left(q\right)∥}^2\\ \le {∥{\alpha }_n\left(\gamma f\left(x_n\right)-Aq\right)+\left(I-{\alpha }_{n}A\right)\left(S{y^{\prime }}_n-q\right)∥}^2\\ \le {∥{\alpha }_n\left(\gamma f\left(x_n\right)-Aq\right)+\left(I-{\alpha }_{n}A\right)\left({y^{\prime }}_n-q\right)∥}^2\\ \le {∥{\alpha }_n\left(\gamma f\left(x_n\right)-Aq\right)+\left(I-{\alpha }_{n}A\right)\left(y_n-q\right)∥}^2\\ \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){∥y_n-q∥}^2\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\left\{{∥x_n-q∥}^2+{\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2\right\}\\ \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +{∥x_n-q∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2.\end{array}$$

(3.24)

By the convexity of the norm ||·||, we have

$$\begin{array}{ll}\hfill {∥x_{n+1}-q∥}^2& ={∥{\xi }_n{z}_n+\left(1-{\xi }_n\right)P_C{w}_n-q∥}^2\\ ={∥{\xi }_n\left(z_n-q\right)+\left(1-{\xi }_n\right)\left(P_C{w}_n-q\right)∥}^2\\ \le {\xi }_n{∥z_n-q∥}^2+\left(1-{\xi }_n\right){∥w_n-q∥}^2.\end{array}$$

(3.25)

Substituting (3.4), (3.7), (3.24) into (3.25), we obtain

$$\begin{array}{l}{∥x_{n+1}-q∥}^2\\ \le {\xi }_n\left\{{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\right.\\ \times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥+{∥x_n-q∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \left.\times {\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2\right\}+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ ={\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥+{\xi }_n{∥x_n-q∥}^2+{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times {\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2+\left(1-{\xi }_n\right){∥x_n-q∥}^2.\end{array}$$

So, we obtain

$$\begin{array}{l}{\xi }_n\left(1-{\alpha }_n\left(1-\frac{1-\delta }{\mu }\right)\right){\lambda }_1\left(2{\beta }_1-{\lambda }_1\right){∥A_1{u}_n-A_{1}q∥}^2\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\epsilon }_n+∥x_n-x_{n+1}∥\left(∥x_n-q∥+∥x_{n+1}-q∥\right),\end{array}$$

where ${\epsilon }_n=2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥$. Since conditions (C1), (C2) and lim_n→∞||x_n+1-x _n || = 0, then we obtain that ||A₁u _n - A₁q|| → 0 as n → ∞. For q ∈ Θ and $q=J_{M_2,{\lambda }_2}\left(q-{\lambda }_2{A}_{2}q\right)$, then we get

$$\begin{array}{ll}\hfill {∥w_n-q∥}^2& ={∥J_{M_2,{\lambda }_2}\left(v_n-{\lambda }_2{A}_2{v}_n\right)-J_{M_2,{\lambda }_2}\left(q-{\lambda }_2{A}_{2}q\right)∥}^2\\ \le {∥\left(v_n-{\lambda }_2{A}_2{v}_n\right)-\left(q-{\lambda }_2{A}_{2}q\right)∥}^2\\ \le {∥v_n-q∥}^2+{\lambda }_2\left({\lambda }_2-2{\beta }_2\right){∥A_2{v}_n-A_{2}q∥}^2\\ \le {∥x_n-q∥}^2+{\lambda }_2\left({\lambda }_2-2{\beta }_2\right){∥A_2{v}_n-A_{2}q∥}^2.\end{array}$$

(3.26)

Substituting (3.24), (3.26) into (3.25), we obtain

$$\begin{array}{l}{∥x_{n+1}-q∥}^2\\ \le {\xi }_n\left\{{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\right.∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ \left.+{∥x_n-q∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2\right\}\\ +\left(1-{\xi }_n\right)\left\{{∥x_n-q∥}^2+{\lambda }_2\left({\lambda }_2-2{\beta }_2\right){∥A_2{v}_n-A_{2}q∥}^2\right\}\\ ={\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +{\xi }_n{∥x_n-q∥}^2+{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1\left({\lambda }_1-2{\beta }_1\right){∥A_1{u}_n-A_{1}q∥}^2\\ +\left(1-{\xi }_1\right){∥x_n-q∥}^2+\left(1-{\xi }_n\right){\lambda }_2\left({\lambda }_2-2{\beta }_2\right){∥A_2{v}_n-A_{2}q∥}^2.\end{array}$$

So, we obtain

$$\begin{array}{l}\left(1-{\xi }_n\right){\lambda }_2\left(2{\beta }_2-{\lambda }_2\right){∥A_2{v}_n-A_{2}q∥}^2\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\epsilon }_n+{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1\left({\lambda }_1-2{\beta }_1\right)\\ \times {∥A_1{u}_n-A_{1}q∥}^2+∥x_n-x_{n+1}∥\left(∥x_n-q∥+∥x_{n+1}-q∥\right),\end{array}$$

where ${\epsilon }_n=2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥$. Since conditions (C1), (C2), lim_n→∞||x_n+1- x _n || = 0 and lim_n→∞||A₁u _n - A₁q|| → 0 then we obtain that ||A₂v _n - A₂q|| → 0 as n → ∞. We consider this inequality in (3.24) that

$$\begin{array}{ll}\hfill {∥z_n-q∥}^2& \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){∥y_n-q∥}^2\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥.\end{array}$$

(3.27)

Substituting (3.3) and (3.6) into (3.27), we have

$$\begin{array}{ll}\hfill {∥z_n-q∥}^2& \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times \left\{{∥x_n-q∥}^2+r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2\right\}\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ ={\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){∥x_n-q∥}^2\\ +\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2\\ +2{\alpha }_n\left(1+{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2\\ +\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥.\end{array}$$

(3.28)

Substituting (3.4), (3.7) and (3.28) into (3.25), we obtain

$$\begin{array}{l}{∥x_{n+1}-q∥}^2\\ \le {\xi }_n\left\{{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\right.\\ \times r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\}+(1-{\xi }_n){∥x_n-q∥}^2\\ ={\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\xi }_n{∥x_n-q∥}^2+{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2+2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥+\left(1-{\xi }_n\right){∥x_n-q∥}^2.\end{array}$$

(3.29)

So, we also have

$$\begin{array}{l}{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n\left(2\eta -r_n\right){∥B_1{x}_n-B_{1}q∥}^2\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\epsilon }_n+∥x_n-x_{n+1}∥\left(∥x_n-q∥+∥x_{n+1}-q∥\right),\end{array}$$

where ${\epsilon }_n=2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥$. Since conditions (C1), (C2), (C3), lim_n→∞||x_n+1- x _n || = 0, then we obtain that ||B₁x _n - B₁q|| → 0 as n → ∞. Substituting (3.4), (3.7) and (3.28) into (3.25), we obtain

$$\begin{array}{l}{∥x_{n+1}-q∥}^2\\ \le {\xi }_n\left\{{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\right.\\ \times r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \left.\times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\right\}\\ +\left(1-{\xi }_n\right)\{{∥x_n-q∥}^2+s_n\left(s_n-2\rho \right){∥B_2{x}_n-B_{2}q∥}^2\}\\ ={\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\xi }_n{∥x_n-q∥}^2+{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2+2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ +\left(1-{\xi }_n\right)s_n\left(s_n-2\rho \right){∥B_2{x}_n-B_{2}q∥}^2\\ ={\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2+{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2+2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\\ \times ∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +\left(1-{\xi }_n\right)s_n\left(s_n-2\rho \right){∥B_2{x}_n-B_{2}q∥}^2.\end{array}$$

So, we also have

$$\begin{array}{l}\left(1-{\xi }_n\right)s_n\left(2\rho -s_n\right){∥B_2{x}_n-B_{2}q∥}^2\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\epsilon }_n+∥x_n-x_{n+1}∥\left(∥x_n-q∥+∥x_{n+1}-q∥\right)\\ +{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n\left(r_n-2\eta \right){∥B_1{x}_n-B_{1}q∥}^2,\end{array}$$

where ${\epsilon }_n=2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥$. Since conditions (C1), (C2), (C4), lim_n→∞||x_n+1- x _n || = 0 and lim_n→∞||B₁x _n - B₁q|| = 0, then we obtain that ||B₂x _n - B₂q|| → 0 as n → ∞.

Step 4. We show the followings:

• (i) lim_n→∞||x _n - u _n || = 0;

• (ii) lim_n→∞||u _n - y _n || = 0;

• (iii) lim_n→∞||x _n - v _n || = 0;

• (iv) lim_n→∞||v _n - w _n || = 0;

• (v) $\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥u_n-{y^{\prime }}_n∥=0$;

• (vi) $\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥{y^{\prime }}_n-S{y^{\prime }}_n∥=0$.

Since $T_{r_n}^{\left(F_1,{\phi }_1\right)}$ is firmly nonexpansive, we observe that

$$\begin{array}{l}{∥u_n-q∥}^2\\ ={∥T_{r_n}^{(F_1,{\phi }_1)}\left(x_n-r_n{B}_1{x}_n\right)-T_{r_n}^{(F_1,{\phi }_1)}\left(q-r_n{B}_{1}q\right)∥}^2\\ \le 〈\left(x_n-r_n{B}_1{x}_n\right)-\left(q-r_n{B}_{1}q\right),u_n-q〉\\ =\frac{1}{2}({∥\left(x_n-r_n{B}_1{x}_n\right)-\left(q-r_n{B}_{1}q\right)∥}^2+{∥u_n-q∥}^2-∥\left(x_n-r_n{B}_1{x}_n\right)\\ {-\left(q-r_n{B}_{1}q\right)-\left(u_n-q\right)∥}^2)\\ \le \frac{1}{2}({∥x_n-q∥}^2+{∥u_n-q∥}^2-{∥\left(x_n-u_n\right)-r_n\left(B_1{x}_n-B_{1}q\right)∥}^2)\\ =\frac{1}{2}({∥x_n-q∥}^2+{∥u_n-q∥}^2-{∥x_n-u_n∥}^2\\ +2r_n〈B_1{x}_n-B_{1}q,x_n-u_n〉-r_n^2{∥B_1{x}_n-B_{1}q∥}^2).\end{array}$$

Hence, we have

$${∥u_n-q∥}^2\le {∥x_n-q∥}^2-{∥x_n-u_n∥}^2+2r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥.$$

(3.30)

Since $J_{M_1,{\lambda }_1}$ is 1-inverse-strongly monotone, we compute

$$\begin{array}{ll}\hfill {∥y_n-q∥}^2& ={∥J_{M_1,{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)-J_{M_1,{\lambda }_1}\left(q-{\lambda }_1{A}_{1}q\right)∥}^2\\ \le 〈\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right),y_n-q〉\\ =\frac{1}{2}({∥\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right)∥}^2+{∥y_n-q∥}^2\\ -{∥\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right)-\left(y_n-q\right)∥}^2\\ =\frac{1}{2}({∥u_n-q∥}^2+{∥y_n-q∥}^2-{∥\left(u_n-y_n\right)-{\lambda }_1\left(A_1{u}_n-A_{1}q\right)∥}^2)\\ \le \frac{1}{2}({∥u_n-q∥}^2+{∥y_n-q∥}^2-{∥u_n-y_n∥}^2\\ +2{\lambda }_1〈u_n-y_n,A_1{u}_n-A_{1}q〉-{\lambda }_1^2{∥A_1{u}_n-A_{1}q∥}^2),\end{array}$$

which implies that

$${∥y_n-q∥}^2\le {∥u_n-q∥}^2-{∥u_n-y_n∥}^2+2{\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥.$$

(3.31)

Substitute (3.30) into (3.31), we have

$$\begin{array}{ll}\hfill {∥y_n-q∥}^2& \le \{{∥x_n-q∥}^2-{∥x_n-u_n∥}^2+2r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\}\\ -{∥u_n-y_n∥}^2+2{\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥.\end{array}$$

(3.32)

Substitute (3.32) into (3.27), we have

$$\begin{array}{ll}\hfill {∥z_n-q∥}^2& \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\{{∥x_n-q∥}^2-{∥x_n-u_n∥}^2\\ +2r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥-{∥u_n-y_n∥}^2+2{\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥\}\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2-{∥x_n-u_n∥}^2\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥-{∥u_n-y_n∥}^2\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥.\end{array}$$

(3.33)

Since $T_{s_n}^{\left(F_2,{\phi }_2\right)}$ is firmly nonexpansive, we observe that

$$\begin{array}{l}{∥v_n-q∥}^2\\ ={∥T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)-T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(q-s_n{B}_{2}q\right)∥}^2\\ \le 〈\left(x_n-s_n{B}_2{x}_n\right)-\left(q-s_n{B}_{2}q\right),v_n-q〉\\ =\frac{1}{2}({∥\left(x_n-s_n{B}_2{x}_n\right)-\left(q-s_n{B}_{2}q\right)∥}^2+{∥v_n-q∥}^2\\ -{∥\left(x_n-s_n{B}_2{x}_n\right)-\left(q-s_n{B}_{2}q\right)-\left(v_n-q\right)∥}^2)\\ \le \frac{1}{2}({∥x_n-q∥}^2+{∥v_n-q∥}^2-{∥\left(x_n-v_n\right)-s_n\left(B_2{x}_n-B_{2}q\right)∥}^2)\\ =\frac{1}{2}({∥x_n-q∥}^2+{∥v_n-q∥}^2-{∥x_n-v_n∥}^2\\ +2s_n〈B_2{x}_n-B_{2}q,x_n-v_n〉-s_n^2{∥B_2{x}_n-B_{2}q∥}^2).\end{array}$$

Hence, we have

$${∥v_n-q∥}^2\le {∥x_n-q∥}^2-{∥x_n-v_n∥}^2+2s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥.$$

(3.34)

Since $J_{M_2,{\lambda }_2}$ is 1-inverse-strongly monotone, we compute

$$\begin{array}{ll}\hfill {∥w_n-q∥}^2& ={∥J_{M_2,{\lambda }_2}\left(v_n-{\lambda }_2{A}_2{v}_n\right)-J_{M_2,{\lambda }_2}\left(q-{\lambda }_2{A}_{2}q\right)∥}^2\\ \le 〈\left(v_n-{\lambda }_2{A}_2{v}_n\right)-\left(q-{\lambda }_2{A}_{2}q\right),w_n-q〉\\ =\frac{1}{2}({∥\left(v_n-{\lambda }_2{A}_2{v}_n\right)-\left(q-{\lambda }_2{A}_{2}q\right)∥}^2+{∥w_n-q∥}^2\\ -{∥\left(v_n-{\lambda }_2{A}_2{v}_n\right)-\left(q-{\lambda }_2{A}_{2}q\right)-\left(w_n-q\right)∥}^2)\\ =\frac{1}{2}({∥v_n-q∥}^2+{∥w_n-q∥}^2-{∥\left(v_n-w_n\right)-{\lambda }_2\left(A_2{v}_n-A_{2}q\right)∥}^2)\\ \le \frac{1}{2}({∥v_n-q∥}^2+{∥w_n-q∥}^2-{∥v_n-w_n∥}^2\\ +2{\lambda }_2〈v_n-w_n,A_2{v}_n-A_{2}q〉-{\lambda }_2^2{∥A_2{v}_n-A_{2}q∥}^2),\end{array}$$

which implies that

$${∥w_n-q∥}^2\le {∥v_n-q∥}^2-{∥v_n-w_n∥}^2+2{\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥.$$

(3.35)

Substitute (3.34) into (3.35), we have

$$\begin{array}{ll}\hfill {∥w_n-q∥}^2& \le \{{∥x_n-q∥}^2-{∥x_n-v_n∥}^2+2s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥\}\\ -{∥v_n-w_n∥}^2+2{\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥.\end{array}$$

(3.36)

Substitute (3.33) and (3.36) into (3.25), we obtain

$$\begin{array}{ll}\hfill {∥x_{n+1}-q∥}^2& \le {\xi }_n{∥z_n-q∥}^2+\left(1-{\xi }_n\right){∥w_n-q∥}^2\\ \le {\xi }_n\left\{{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2-{∥x_n-u_n∥}^2-{∥u_n-y_n∥}^2\right.\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥\\ \left.+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\right\}\\ +\left(1-{\xi }_n\right)\{{∥x_n-q∥}^2-{∥x_n-v_n∥}^2\\ +2s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥-{∥v_n-w_n∥}^2\\ +2{\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥\}\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\xi }_n{∥x_n-q∥}^2-{∥x_n-u_n∥}^2-{∥u_n-y_n∥}^2\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥\\ +2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +\left(1-{\xi }_n\right){∥x_n-q∥}^2-{∥x_n-v_n∥}^2+2\left(1-{\xi }_n\right)s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥\\ -{∥v_n-w_n∥}^2+2\left(1-{\xi }_n\right){\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥.\end{array}$$

Then, we derive

$$\begin{array}{l}{∥x_n-u_n∥}^2+{∥u_n-y_n∥}^2+{∥x_n-v_n∥}^2+{∥v_n-w_n∥}^2\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2-{∥x_{n+1}-q∥}^2\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥\\ +2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +2\left(1-{\xi }_n\right)s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥+2\left(1-{\xi }_n\right){\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥\\ ={\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+∥x_{n+1}-x_n∥\left(∥x_n-q∥+∥x_{n+1}-q∥\right)\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n∥∥A_1{u}_n-A_{1}q∥\\ +2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥y_n-q∥\\ +2\left(1-{\xi }_n\right)s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥+2\left(1-{\xi }_n\right){\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥.\end{array}$$

By conditions (C1)-( C4), lim_n→∞||x _n -x_n+1|| = 0, lim_n→∞||B₁x _n -B₁q|| = 0, lim_n→∞||B₂x _n -B₂q|| = 0, lim_n→∞||A₁u _n - A₁q|| = 0 and lim_n→∞||A₂v _n - A₂q|| = 0. So, we have ||x _n - u _n || → 0, ||u _n - y _n || → 0, ||x _n - v _n || → 0, ||v _n - w _n || → 0 as n → ∞.

From (2.1), we have

$$\begin{array}{ll}\hfill {∥{y^{\prime }}_n-q∥}^2& ={∥P_C{J}_{M_1{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)-P_C{J}_{M_1{\lambda }_1}\left(q-{\lambda }_1{A}_{1}q\right)∥}^2\\ \le {∥J_{M_1,{\lambda }_1}\left(u_n-{\lambda }_1{A}_1{u}_n\right)-J_{M_1,{\lambda }_1}\left(q-{\lambda }_1{A}_{1}q\right)∥}^2\\ \le 〈\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right),y_n^{\text{'}}-q〉\\ =\frac{1}{2}({∥\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right)∥}^2+{∥{y^{\prime }}_n-q∥}^2\\ -{∥\left(u_n-{\lambda }_1{A}_1{u}_n\right)-\left(q-{\lambda }_1{A}_{1}q\right)-\left({y^{\prime }}_n-q\right)∥}^2)\\ =\frac{1}{2}({∥u_n-q∥}^2+{∥{y^{\prime }}_n-q∥}^2-{∥\left(u_n-y_n^{\text{'}}\right)-{\lambda }_1\left(A_1{u}_n-A_{1}q\right)∥}^2)\\ \le \frac{1}{2}({∥u_n-q∥}^2+{∥{y^{\prime }}_n-q∥}^2-{∥u_n-y_n^{\text{'}}∥}^2\\ +2{\lambda }_1〈u_n-y_n^{\text{'}},A_1{u}_n-A_{1}q〉-{\lambda }_1^2{∥A_1{u}_n-A_{1}q∥}^2),\end{array}$$

which implies that

$${∥{y^{\prime }}_n-q∥}^2\le {∥u_n-q∥}^2-{∥u_n-y_n^{\text{'}}∥}^2+2{\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥.$$

(3.37)

Substitute (3.30) into (3.37), we have

$$\begin{array}{ll}\hfill {∥{y^{\prime }}_n-q∥}^2& \le \{{∥x_n-q∥}^2-{∥x_n-u_n∥}^2+2r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\}\\ -{∥u_n-y_n^{\text{'}}∥}^2+2{\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥.\end{array}$$

(3.38)

We consider this inequality in (3.24) that

$$\begin{array}{ll}\hfill {∥z_n-q∥}^2& \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){∥{y^{\prime }}_n-q∥}^2\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥.\end{array}$$

(3.39)

Substitute (3.38) into (3.39), we have

$$\begin{array}{ll}\hfill {∥z_n-q∥}^2& \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)\{{∥x_n-q∥}^2-{∥x_n-u_n∥}^2\\ +2r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥-{∥u_n-y_n^{\text{'}}∥}^2+2{\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥\}\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥\\ \le {\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2-{∥x_n-u_n∥}^2\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥-{∥u_n-y_n^{\text{'}}∥}^2\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥\\ +2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥.\end{array}$$

(3.40)

Substitute (3.36) and (3.40) into (3.25), we obtain

$$\begin{array}{ll}\hfill {∥x_{n+1}-q∥}^2& \le {\xi }_n{∥z_n-q∥}^2+\left(1-{\xi }_n\right){∥w_n-q∥}^2\\ \le {\xi }_n\left\{{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2-{∥x_n-u_n∥}^2-{∥u_n-y_n^{\text{'}}∥}^2\right.\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥\\ \left.+2{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥\right\}\\ +\left(1-{\xi }_n\right)\{{∥x_n-q∥}^2-{∥x_n-v_n∥}^2+2s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥-{∥v_n-w_n∥}^2\\ +2{\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥\}\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{\xi }_n{∥x_n-q∥}^2-{∥x_n-u_n∥}^2-{∥u_n-y_n^{\text{'}}∥}^2\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥\\ +2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥\\ +\left(1-{\xi }_n\right){∥x_n-q∥}^2-{∥x_n-v_n∥}^2+2\left(1-{\xi }_n\right)s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥\\ -{∥v_n-w_n∥}^2+2\left(1-{\xi }_n\right){\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥.\end{array}$$

Then, we derive

$$\begin{array}{l}{∥x_n-u_n∥}^2+{∥u_n-y_n^{\text{'}}∥}^2+{∥x_n-v_n∥}^2+{∥v_n-w_n∥}^2\\ \le {\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2+{∥x_n-q∥}^2-{∥x_{n+1}-q∥}^2\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥\\ +2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥\\ +2\left(1-{\xi }_n\right)s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥+2\left(1-{\xi }_n\right){\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥\\ ={\xi }_n{\alpha }_n{∥\lambda f\left(x_n\right)-Aq∥}^2+∥x_{n+1}-x_n∥\left(∥x_n-q∥+∥x_{n+1}-q∥\right)\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)r_n∥B_1{x}_n-B_{1}q∥∥x_n-u_n∥\\ +2{\xi }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right){\lambda }_1∥u_n-y_n^{\text{'}}∥∥A_1{u}_n-A_{1}q∥\\ +2{\xi }_n{\alpha }_n\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥\gamma f\left(x_n\right)-Aq∥∥{y^{\prime }}_n-q∥\\ +2\left(1-{\xi }_n\right)s_n∥B_2{x}_n-B_{2}q∥∥x_n-v_n∥+2\left(1-{\xi }_n\right){\lambda }_2∥v_n-w_n∥∥A_2{v}_n-A_{2}q∥.\end{array}$$

By conditions (C1)-(C4), lim_n→∞||x _n -x_n+1|| = 0, lim_n→∞||B₁x _n -B₁q|| = 0, lim_n→∞||B₂x _n -B₂q|| = 0, lim_n→∞||A₁u _n - A₁q|| = 0, lim_n→∞||A₂v _n - A₂q|| = 0, ||x _n - u _n || → 0, ||x _n - v _n || → 0, ||v _n - w _n || → 0 as n → ∞. So, we have $∥u_n-{y^{\prime }}_n∥\to 0$. It follows that

$$∥x_n-w_n∥\le ∥x_n-v_n∥+∥v_n-w_n∥\to 0,\mathsf{\text{as}}n\to \mathrm{\infty }.$$

We compute that

$$\begin{array}{ll}\hfill ∥x_{n+1}-x_n∥& =∥{\xi }_n\left(z_n-x_n\right)+\left(1-{\xi }_n\right)\left(P_C{w}_n-x_n\right)∥\\ \le {\xi }_n∥z_n-x_n∥+\left(1-{\xi }_n\right)∥w_n-x_n∥.\end{array}$$

From lim_n→∞||x _n - w _n || = 0, lim_n→∞||x_n+1- x _n || = 0, and hence

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

(3.41)

It follows by step 4 (i) and (ii),

$$∥x_n-y_n∥\le ∥x_n-u_n∥+∥u_n-y_n∥\to 0,\mathsf{\text{as}}n\to \mathrm{\infty }.$$

Since

$$∥z_n-y_n∥\le ∥z_n-x_n∥+∥x_n-y_n∥.$$

So, by (3.41) and lim_n→∞||x _n - y _n || = 0, we obtain

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

(3.42)

We show $∥S{y^{\prime }}_n-z_n∥\to 0$ as n → ∞. By nonexpansiveness of P _C notice that

$$\begin{array}{ll}\hfill ∥S{y^{\prime }}_n-z_n∥& =∥S{y^{\prime }}_n-P_C\left[{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{y^{\prime }}_n\right]∥\\ =∥P_{C}S{y^{\prime }}_n-P_C\left[{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{y^{\prime }}_n\right]∥\\ \le ∥S{y^{\prime }}_n-{\alpha }_n\gamma f\left(x_n\right)-\left(I-{\alpha }_{n}A\right)S{y^{\prime }}_n∥\\ \le {\alpha }_n∥\gamma f\left(x_n\right)-AS{y^{\prime }}_n∥.\end{array}$$

By condition (C1), we get $\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥S{y^{\prime }}_n-z_n∥=0$. Since $∥S{y^{\prime }}_n-{y^{\prime }}_n∥\le ∥S{y^{\prime }}_n-z_n∥+∥z_n-y_n∥+∥y_n-u_n∥+∥u_n-{y^{\prime }}_n∥$, so by (3.42), $\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥S{y^{\prime }}_n-z_n∥=0,\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥y_n-u_n∥=0$ and $\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥u_n-{y^{\prime }}_n∥=0$, we obtain $\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}∥S{y^{\prime }}_n-{y^{\prime }}_n∥=0$.

Step 5. We show that q ∈ Θ := F(S) ∩ GMEP(F₁, φ₁, B₁) ∩ GMEP(F₂, φ₂, B₂) ∩ I(A_l, M₁) ∩ I(A₂, M₂) and $\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{sup}}}〈\left(\gamma f-A\right)q,S{y^{\prime }}_n-q〉\le 0$. It is easy to see that P_Θ(γf + (I - A)) is a contraction of H into itself. In fact, from Lemma 2.9, we have

$$\begin{array}{ll}\hfill ∥P_{\mathrm{\Theta }}\left(\gamma f+\left(I-A\right)\right)x-P_{\mathrm{\Theta }}\left(\gamma f+\left(I-A\right)\right)y∥& \le ∥\left(\gamma f+\left(I-A\right)\right)x-\left(\gamma f+\left(I-A\right)\right)y∥\\ \le \gamma ∥f\left(x\right)-f\left(y\right)∥+\left(I-A\right)∥x-y∥\\ \le \gamma \alpha ∥x-y∥+\left(1-\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)∥x-y∥\\ =\left(\sqrt{\frac{1-\delta }{\mu }}+\gamma \alpha \right)∥x-y∥.\end{array}$$

Hence H is complete, there exists a unique fixed point q ∈ H such that q = P_Θ(γf + (I - A))(q). By Lemma 2.2 we obtain that 〈(γf - A)q, w - q〉 ≤ 0 for all w ∈ Θ.

Next, we show that $\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{sup}}}〈\left(\gamma f-A\right)q,S{y}_n^{\text{'}}-q〉\le 0$, where q = P_Θ(γf + I - A)(q) is the unique solution of the variational inequality 〈(γf - A)q, p - q〉 ≥ 0, ∀p ∈ Θ. We can choose a subsequence $\left\{y_{n_i}^{\text{'}}\right\}$ of $\left\{y_n^{\text{'}}\right\}$ such that

$$\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈\left(\gamma f-A\right)q,S{y^{\prime }}_n-q〉=\underset{i\to \mathrm{\infty }}{\mathsf{\text{lim}}}〈\left(\gamma f-A\right)q,S{y^{\prime }}_{n_i}-q〉.$$

Since $\left\{y_{n_i}^{\text{'}}\right\}$ is bounded, there exists a subsequence $\left\{y_{n_{i_j}}^{\text{'}}\right\}$ of $\left\{y_{n_i}^{\text{'}}\right\}$ which converges weakly to w. We may assume without loss of generality that $y_{n_i}^{\prime }\rightharpoonup w$.

We claim that w ∈ Θ. Since $∥{y^{\prime }}_n-S{y^{\prime }}_n∥\to 0$ and by Lemma 2.6, we have w ∈ F(S).

Next, we show that w ∈ GMEP(F₁, φ₁, B₁). Since $u_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)$, we know that

$$F_1\left(u_n,y\right)+{\phi }_1\left(y\right)-{\phi }_1\left(u_n\right)+〈B_1{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C.$$

It follows by (A2) that

$${\phi }_1\left(y\right)-{\phi }_1\left(u_n\right)+〈B_1{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge F_1\left(y,u_n\right),\forall y\in C.$$

Hence,

$${\phi }_1\left(y\right)-{\phi }_1\left(u_{n_i}\right)+〈B_1{x}_{n_i},y-u_{n_i}〉+\frac{1}{r_{n_i}}〈y-u_{n_i},u_{n_i}-x_{n_i}〉\ge F_1\left(y,u_{n_i}\right),\forall y\in C.$$

(3.43)

For t ∈ (0, 1] and y ∈ H, let y _t = ty + (1 - t)w. From (3.43), we have

$$\begin{array}{ll}\hfill 〈y_t-u_{n_i},B_1{y}_t〉& \ge 〈y_t-u_{n_i},B_1{y}_t〉-{\phi }_1\left(y_t\right)+{\phi }_1\left(u_{n_i}\right)-〈B_1{x}_{n_i},y_t-u_{n_i}〉\\ -\frac{1}{r_{n_i}}〈y_t-u_{n_i},u_{n_i}-x_{n_i}〉+F_1\left(y_t,u_{n_i}\right)\\ =〈y_t-u_{n_i},B_1{y}_t-B_1{u}_{n_i}〉+〈y_t-u_{n_i},B_1{u}_{n_i}-B_1{x}_{n_i}〉-{\phi }_1\left(y_t\right)+{\phi }_1\left(u_{n_i}\right)\\ -\frac{1}{r_{n_i}}〈y_t-u_{n_i},u_{n_i}-x_{n_i}〉+F_1\left(y_t,u_{n_i}\right).\end{array}$$

From $∥u_{n_i}-x_{n_i}∥\to 0$, we have $∥B_1{u}_{n_i}-B_1{x}_{n_i}∥\to 0$. Further, from (A4) and the weakly lower semicontinuity of ${\phi }_1,\frac{u_{n_i}-x_{n_i}}{r_{n_i}}\to 0$ and $u_{n_i}\rightharpoonup w$, we have

$$〈y_t-w,B_1{y}_t〉\ge -{\phi }_1\left(y_t\right)+{\phi }_1\left(w\right)+F_1\left(y_t,w\right).$$

(3.44)

From (A1), (A4) and (3.44), we have

$$\begin{array}{ll}\hfill 0& =F_1\left(y_t,y_t\right)-{\phi }_1\left(y_t\right)+{\phi }_1\left(y_t\right)\\ \le t{F}_1\left(y_t,y\right)+\left(1-t\right)F_1\left(y_t,w\right)+t{\phi }_1\left(y\right)+\left(1-t\right){\phi }_1\left(w\right)-{\phi }_1\left(y_t\right)\\ =t\left[F_1\left(y_t,y\right)+{\phi }_1\left(y\right)-{\phi }_1\left(y_t\right)\right]+\left(1-t\right)\left[F_1\left(y_t,w\right)+{\phi }_1\left(w\right)-{\phi }_1\left(y_t\right)\right]\\ \le t\left[F_1\left(y_t,y\right)+{\phi }_1\left(y\right)-{\phi }_1\left(y_t\right)\right]+\left(1-t\right)〈y_t-w,B_1{y}_t〉\\ =t\left[F_1\left(y_t,y\right)+{\phi }_1\left(y\right)-{\phi }_1\left(y_t\right)\right]+\left(1-t\right)t〈y-w,B_1{y}_t〉,\end{array}$$

and hence

$$0\le F_1\left(y_t,y\right)+{\phi }_1\left(y\right)-{\phi }_1\left(y_t\right)+\left(1-t\right)〈y-w,B_1{y}_t〉.$$

Letting t → 0, we have, for each y ∈ C,

$$F_1\left(w,y\right)+{\phi }_1\left(y\right)-{\phi }_1\left(w\right)+〈y-w,B_{1}w〉\ge 0.$$

This implies that w ∈ GMEP(F₁, φ₁, B₁). By following the same arguments, we can show that w ∈ GMEP(F₂, φ₂, B₂).

Lastly, we show that w ∈ I(A₁, M₂). In fact, since A₁ is a β₁-inverse-strongly monotone, A₁ is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that M₁ + A₁ is a maximal monotone. Let (v, g) ∈ G(M₁ + A₁), since g - A₁v ∈ M₁(v). Again since $y_{n_i}=J_{M_1{\lambda }_1}\left(u_{n_i}-{\lambda }_1{A}_1{u}_{n_i}\right)$, we have $u_{n_i}-{\lambda }_1{A}_1{u}_{n_i}\in \left(I+{\lambda }_1{M}_1\right)\left(y_{n_i}\right)$, that is, $\frac{1}{{\lambda }_1}\left(u_{n_i}-y_{n_i}-{\lambda }_1{A}_1{u}_{n_i}\right)\in M_1\left(y_{n_i}\right)$. By virtue of the maximal monotonicity of M₁ + A₁, we have

$$〈v-y_{n_i},g-A_{1}v-\frac{1}{{\lambda }_1}\left(u_{n_i}-y_{n_i}-{\lambda }_1{A}_1{u}_{n_i}\right)〉\ge 0,$$

and hence

$$\begin{array}{ll}\hfill 〈v-y_{n_i},g〉& \ge 〈v-y_{n_i},A_{1}v+\frac{1}{\lambda }(u_{n_i}-y_{n_i}-{\lambda }_1{A}_1{u}_{n_i})〉\\ =〈v-y_{n_i},A_{1}v-A_1{y}_{n_i}〉+〈v-y_{n_i},A_1{y}_{n_i}-A_1{u}_{n_i}〉\\ +〈v-y_{n_i},\frac{1}{{\lambda }_1}\left(u_{n_i}-y_{n_i}\right)〉.\end{array}$$

It follows from lim_n→∞||u _n - y _n || = 0, we have lim_n→∞||A₁u _n - A₁y _n || = 0 and $y_{n_i}\rightharpoonup w$ that

$$\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈v-y_n,g〉=〈v-w,g〉\ge 0.$$

It follows from the maximal monotonicity of A₁ + M₁ that θ ∈ (M₁ + A₁)(w), that is, w ∈ I(A₁, M₁). By following the same arguments, we can show that w ∈ I(A₂, M₂). Therefore, w ∈ Θ. It follows that

$$\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}\mathsf{\text{sup}}}〈\left(\gamma f-A\right)q,S{y^{\prime }}_n-q〉=\underset{i\to \mathrm{\infty }}{\mathsf{\text{lim}}}〈\left(\gamma f-A\right)q,S{y^{\prime }}_{n_i}-q〉=〈\left(\gamma f-A\right)q,w-q〉\le 0.$$

Step 6. We prove x _n → q. By using (3.1), Lemma 2.9 (ii) and together with Schwarz inequality, we have

$$\begin{array}{ll}\hfill {∥x_{n+1}-q∥}^2& ={∥{\xi }_n{P}_C\left[\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{y^{\prime }}_n\right)-q\right]+\left(1-{\xi }_n\right)\left(P_C{w}_n-q\right)∥}^2\\ \le {\xi }_n{∥P_C\left[\left({\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{y^{\prime }}_n\right)-P_C\left(q\right)\right]∥}^2+\left(1-{\xi }_n\right){∥w_n-q∥}^2\\ \le {\xi }_n{∥{\alpha }_n\left(\gamma f\left(x_n\right)-Aq\right)+\left(I-{\alpha }_{n}A\right)\left(S{y^{\prime }}_n-q\right)∥}^2+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le {\xi }_n{\left(I-{\alpha }_{n}A\right)}^2{∥S{y^{\prime }}_n-q∥}^2+{\xi }_n{\alpha }_n^2{∥\gamma f\left(x_n\right)-Aq∥}^2\\ +2{\xi }_n{\alpha }_n〈\left(I-{\alpha }_{n}A\right)\left(S{y^{\prime }}_n-q\right),\gamma f\left(x_n\right)-Aq〉+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le {\xi }_n{\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)}^2{∥y_n-q∥}^2+{\xi }_n{\alpha }_n^2{∥\gamma f\left(x_n\right)-Aq∥}^2\\ +2{\xi }_n{\alpha }_n〈S{y^{\prime }}_n-q,\gamma f\left(x_n\right)-Aq〉\\ -2{\xi }_n{\alpha }_n^2〈A\left(S{y^{\prime }}_n-q\right),\gamma f\left(x_n\right)-Aq〉+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le {\xi }_n{\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)}^2{∥x_n-q∥}^2+{\xi }_n{\alpha }_n^2{∥\gamma f\left(x_n\right)-Aq∥}^2\\ +2{\xi }_n{\alpha }_n〈S{y^{\prime }}_n-q,\gamma f\left(x_n\right)-\gamma f\left(q\right)〉+2{\xi }_n{\alpha }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉\\ -2{\xi }_n{\alpha }_n^2〈A\left(S{y^{\prime }}_n-q\right),\gamma f\left(x_n\right)-Aq〉+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le {\xi }_n{\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)}^2{∥x_n-q∥}^2+{\xi }_n{\alpha }_n^2{∥\gamma f\left(x_n\right)-Aq∥}^2\\ +2{\xi }_n{\alpha }_n∥S{y^{\prime }}_n-q∥∥\gamma f\left(x_n\right)-\gamma f\left(q\right)∥+2{\xi }_n{\alpha }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉\\ -2{\xi }_n{\alpha }_n^2〈A\left(S{y^{\prime }}_n-q\right),\gamma f\left(x_n\right)-Aq〉+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le {\xi }_n{\left(1-{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)\right)}^2{∥x_n-q∥}^2+{\xi }_n{\alpha }_n^2{∥\gamma f\left(x_n\right)-Aq∥}^2\\ +2{\xi }_n\gamma \alpha {\alpha }_n∥y_n-q∥∥x_n-q∥+2{\xi }_n{\alpha }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉\\ -2{\xi }_n{\alpha }_n^2〈A\left({y^{\prime }}_n-q\right),\gamma f\left(x_n\right)-Aq〉+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le \left({\xi }_n-2{\xi }_n{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)+{\xi }_n{\alpha }_n^2{\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)}^2\right){∥x_n-q∥}^2\\ +{\xi }_n{\alpha }_n^2{∥\gamma f\left(x_n\right)-Aq∥}^2+2{\xi }_n\gamma \alpha {\alpha }_n{∥x_n-q∥}^2\\ +2{\xi }_n{\alpha }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉\\ -2{\xi }_n{\alpha }_n^2〈A\left(S{y^{\prime }}_n-q\right),\gamma f\left(x_n\right)-Aq〉+\left(1-{\xi }_n\right){∥x_n-q∥}^2\\ \le \left(1-2{\xi }_n{\alpha }_n\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)+2{\xi }_n\gamma \alpha {\alpha }_n\right){∥x_n-q∥}^2+{\alpha }_n\left\{{\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2\right.\\ +2{\xi }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉-2{\xi }_n{\alpha }_n∥A\left(S{y^{\prime }}_n-q\right)∥∥\gamma f\left(x_n\right)-Aq∥\\ \left.+{\xi }_n{\alpha }_n{\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)}^2{∥x_n-q∥}^2\right\}\\ =\left(1-2\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right){\xi }_n{\alpha }_n\right){∥x_n-q∥}^2+{\alpha }_n\left\{{\xi }_n{\alpha }_n{∥\gamma f\left(x_n\right)-Aq∥}^2\right.\\ +2{\xi }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉-2{\xi }_n{\alpha }_n∥A\left(S{y^{\prime }}_n-q\right)∥∥\gamma f\left(x_n\right)-Aq∥\\ \left.+{\xi }_n{\alpha }_n{\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)}^2{∥x_n-q∥}^2\right\}.\end{array}$$

Since {x _n } is bounded, where $\eta \ge {\xi }_n{∥\gamma f\left(x_n\right)-Aq∥}^2-2{\xi }_n∥A\left(S{y^{\prime }}_n-q\right)∥∥\gamma f\left(x_n\right)-Aq∥+{\xi }_n{\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)}^2{∥x_n-q∥}^2$ for all n ≥ 0. It follows that

$${∥x_{n+1}-q∥}^2\le \left(1-2\left(\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)-\gamma \alpha \right){\xi }_n{\alpha }_n\right){∥x_n-q∥}^2{\alpha }_n{\varsigma }_n,$$

where ${\varsigma }_n=2{\xi }_n〈S{y^{\prime }}_n-q,\gamma f\left(q\right)-Aq〉+\eta {\alpha }_n$. By $\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{sup}}}〈\left(\gamma f-A\right)q,S{y^{\prime }}_n-q〉\le 0$, we get lim sup_n→∞ς _n ≤ 0. Applying Lemma 2.5, we can conclude that x _n → q. This completes the proof. □

Next, the following example shows that all conditions of Theorem 3.1 are satisfied.

Example 3.2. For instance, let ${\alpha }_n=\frac{1}{2\left(n+1\right)},{\xi }_n=\frac{2n+2}{2\left(2n\right)},r_n=\frac{n}{n+1}$ and $s_n=\frac{n}{n+1}$, then, we will show that the sequences {α _n } satisfy the condition (C1). Indeed, we take ${\alpha }_n=\frac{1}{2\left(n+1\right)}$ then we have

$$\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}{\alpha }_n=\underset{n\to \mathrm{\infty }}{\mathsf{\text{lim}}}\frac{1}{2\left(n+1\right)}=0$$

$$\sum _{n=1}^{\mathrm{\infty }}{\alpha }_n=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2\left(n+1\right)}=\mathrm{\infty }$$

and

$$\sum _{n=1}^{\mathrm{\infty }}\left|{\alpha }_n-{\alpha }_{n-1}\right|=\sum _{n=1}^{\mathrm{\infty }}\left|\frac{1}{2\left(n+1\right)}-\frac{1}{2n}\right|\le \left|\frac{1}{2.2}-\frac{1}{2.1}\right|+\left|\frac{1}{2.3}-\frac{1}{2.2}\right|+\left|\frac{1}{2.4}-\frac{1}{2.3}\right|+\cdots =\frac{1}{2}.$$

Then, the sequence {α _n } satisfy the condition (C1).

We will show that the sequences {ξ _n } satisfy the condition (C2). Indeed, we set ${\xi }_n=\frac{2n+2}{2\left(2n\right)}=\frac{1}{2}+\frac{1}{2n}$. By similarly (C1), it is easy to see that 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1 and ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$.

Next, we will show that the condition (C3) is achieves. We take $r_n=\frac{n}{n+1}$, then we compute

$$\begin{array}{ll}\hfill \sum _{n=1}^{\mathrm{\infty }}\left|r_n-r_{n-1}\right|& =\sum _{n=1}^{\mathrm{\infty }}\left|\frac{n}{n+1}-\frac{n-1}{\left(n-1\right)+1}\right|=\sum _{n=1}^{\mathrm{\infty }}\left|\frac{n\left(n\right)-\left(n-1\right)\left(n+1\right)}{\left(n+1\right)n}\right|\\ =\sum _{n=1}^{\mathrm{\infty }}\left|\frac{n^2-n^2+1}{\left(n+1\right)n}\right|=\sum _{n=1}^{\mathrm{\infty }}\left|\frac{1}{n\left(n+1\right)}\right|.\end{array}$$

Then, we have $\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{inf}}}{r}_n=\mathsf{\text{lim}}\underset{n\to \mathrm{\infty }}{\mathsf{\text{inf}}}\frac{n}{n+1}=1$ and also lim_n→∞|r_n+1- r _n | = 0.

The sequence {r _n } satisfy the condition (C3). In the same way with (C4). □

Corollary 3.3. Let H be a real Hilbert space, C be a closed convex subset of H. Let F₁, F₂be bifunctions of C × C intosatisfying (A1)-(A4) and B, B₁, B₂ : C → H be β, η, ρ-inverse-strongly monotone mappings, ${\phi }_1,{\phi }_2:C\to \mathcal{R}$be convex and lower semicontinuous functions, f : C → C be a contraction with coefficient α (0 < α < 1), M : H → 2^Hbe a maximal monotone mapping. Assume that either (B1) or (B2) holds. Let S be a nonexpansive mapping of C into itself such that

$${\mathrm{\Theta }}_1:=F\left(S\right)\cap \mathsf{\text{GMEP}}\left(F_1,{\phi }_1,B_1\right)\cap \mathsf{\text{GMEP}}\left(F_2,{\phi }_2,B_2\right)\cap I\left(B,M\right)\ne \varnothing .$$

Suppose {x _n } is a sequences generated by the following algorithm x₀ ∈ C arbitrarily:

$$\left\{\begin{array}{c}{u}_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)\hfill \\ v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)\hfill \\ x_{n+1}={\xi }_n{P}_C\left[{\alpha }_{n}f\left(x_n\right)+\left(I-{\alpha }_n\right)S{P}_C\left(J_{M,\lambda }\left(I-\lambda B\right)u_n\right)\right]+\left(1-{\xi }_n\right)P_C{v}_n,\hfill \end{array}\right.$$

where {α _n }, {ξ _n } ⊂ (0, 1), λ ∈ (0, 2β) such that 0 < a ≤ λ ≤ b < 2β, r _n ∈ (0, 2η) with 0 < c ≤ d ≤ 1 - η and s _n ∈ (0, 2ρ) with 0 < e ≤ f ≤ 1 - ρ satisfy the following conditions:

(C1): lim_n→∞α _n = 0, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },{\sum }_{n=1}^{\mathrm{\infty }}\left|{\alpha }_{n+1}-{\alpha }_n\right|<\mathrm{\infty }$,

(C2): 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1, ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$,

(C3): lim inf_n→∞r _n > 0 and lim_n→∞|r_n+1- r _n | = 0,

(C4): lim inf_n→∞s _n > 0 and lim_n→∞|s_n+1- s _n | = 0.

Then {x _n } converges strongly to q ∈ Θ₁, where$q=P_{{\mathrm{\Theta }}_1}\left(f+I\right)\left(q\right)$which solves the following variational inequality:

$$〈\left(f-I\right)q,p-q〉\le 0,\forall p\in {\mathrm{\Theta }}_1.$$

Proof. Putting A ≡ I, γ ≡ 1, $J_{M_2,{\lambda }_2}\equiv I$ and A₂ ≡ 0 in Theorem 3.1, we can obtain desired conclusion immediately. □

Corollary 3.4. Let H be a real Hilbert space, C be a closed convex subset of H. Let F₁, F₂be bifunctions of C × C intosatisfying (A1)-(A4) and B,B₁,B₂ : C → H be β, η, ρ-inverse-strongly monotone mappings, ${\phi }_1,{\phi }_2:C\to \mathcal{R}$be convex and lower semicontinuous functions. Assume that either (B1)or (B2) holds. Let S be a nonexpansive mapping of C into itself such that

$${\mathrm{\Theta }}_1:=F\left(S\right)\cap \mathsf{\text{GMEP}}\left(F_1,{\phi }_1,B_1\right)\cap \mathsf{\text{GMEP}}\left(F_2,{\phi }_2,B_2\right)\cap I\left(B,M\right)\ne \varnothing .$$

Suppose {x _n } is a sequences generated by the following algorithm x₀ ∈ C arbitrarily:

$$\left\{\begin{array}{c}{u}_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)\hfill \\ v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)\hfill \\ x_{n+1}={\xi }_n{P}_C\left[{\alpha }_{n}u+\left(I-{\alpha }_n\right)S{P}_C\left(J_{M,\lambda }\left(I-\lambda B\right)u_n\right)\right]+\left(1-{\xi }_n\right)P_C{v}_n,\hfill \end{array}\right.$$

where {α _n }, {ξ _n } ⊂ (0, 1), λ ∈ (0, 2β) such that 0 < a ≤ λ ≤ b < 2β, r _n ∈ (0, 2η) with 0 < c ≤ d ≤ 1 - η and s _n ∈ (0, 2ρ) with 0 < e ≤ f ≤ 1 - ρ satisfy the following conditions:

(C1): lim_n→∞α _n = 0, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },{\sum }_{n=1}^{\mathrm{\infty }}\left|{\alpha }_{n+1}-{\alpha }_n\right|<\mathrm{\infty }$,

(C2): 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1, ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$,

(C3): lim inf_n→∞r _n > 0 and lim_n→∞|r_n+1- r _n | = 0,

(C4): lim inf_n→∞s _n > 0 and lim_n→∞|s_n+1- s _n | = 0.

Then {x _n } converges strongly to q ∈ Θ₁, where$q=P_{{\mathrm{\Theta }}_1}\left(q\right)$which solves the following variational inequality:

$$〈u-q,p-q〉\le 0,\forall p\in {\mathrm{\Theta }}_1.$$

Proof. Putting f ≡ u ∈ C in Corollary 3.3, we can obtain desired conclusion immediately. □

Corollary 3.5. Let H be a real Hilbert space, C be a closed convex subset of H. Let F₁, F₂be bifunctions of C × C intosatisfying (A1)-(A4) and B,B₁,B ₂ : C → H be β, η, ρ-inverse-strongly monotone mappings, ${\phi }_1,{\phi }_2:C\to \mathcal{R}$be convex and lower semicontinuous functions, f : C → C be a contraction with coefficient α (0 < α < 1) and A is a δ-strongly monotone and μ-strictly pseudo-contraction with δ + μ > 1, γ is a positive real number such that$\gamma <\frac{1}{\alpha }\left(1-\sqrt{\frac{1-\delta }{\mu }}\right)$. Assume that either (B1) or (B2) holds. Let S be a nonexpansive mapping of C into itself such that

$${\mathrm{\Theta }}_2:=F\left(S\right)\cap \mathsf{\text{GMEP}}\left(F_1,{\phi }_1,B_1\right)\cap \mathsf{\text{GMEP}}\left(F_2,{\phi }_2,B_2\right)\cap VI\left(C,B\right)\ne \varnothing .$$

Suppose {x _n } is a sequences generated by the following algorithm x₀ ∈ C arbitrarily:

$$\left\{\begin{array}{c}{u}_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)\hfill \\ v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)\hfill \\ x_{n+1}={\xi }_n{P}_C\left[{\alpha }_n\gamma f\left(x_n\right)+\left(I-{\alpha }_{n}A\right)S{P}_C\left(I-\lambda B\right)u_n\right]+\left(1-{\xi }_n\right)P_C{v}_n,\hfill \end{array}\right.$$

where {α _n }, {ξ _n } ⊂ (0, 1), λ ⊂ (0, 2β) such that 0 < a ≤ λ ≤ b < 2β, r _n ∈ (0, 2η) with 0 < c ≤ d ≤ 1 - η and s _n ∈ (0, 2ρ) with 0 < e ≤ f ≤ 1 - ρ satisfy the following conditions:

(C1): lim_n→∞α _n = 0, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },{\sum }_{n=1}^{\mathrm{\infty }}\left|{\alpha }_{n+1}-{\alpha }_n\right|<\mathrm{\infty }$,

(C2): 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1, ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$,

(C3): lim inf_n→∞r _n > 0 and lim_n→∞|r_{n+ 1}- r _n | = 0,

(C4): lim inf_n→∞s _n > 0 and lim_n→∞|s_n+1- s _n | = 0.

Then {x _n } converges strongly to q ∈ Θ₂, where$q=P_{{\mathrm{\Theta }}_2}\left(\gamma f+I-A\right)\left(q\right)$which solves the following variational inequality:

$$〈\left(\gamma f-A\right)q,p-q〉\le 0,\forall p\in {\mathrm{\Theta }}_2.$$

Proof. Taking $J_{M_1,{\lambda }_1}\equiv I,J_{M_2,{\lambda }_2}\equiv I,A_1\equiv B$ and A₂ ≡ 0 in Theorem 3.1, we can obtain desired conclusion immediately. □

Corollary 3.6. Let H be a real Hilbert space, C be a closed convex subset of H. Let F₁, F₂be bifunctions of C × C intosatisfying (A1)-(A4) and B₁, B₂ : C → H be η, ρ-inverse-strongly monotone mappings, ${\phi }_1,{\phi }_2:C\to \mathcal{R}$be convex and lower semicontinuous functions, f : C → C be a contraction with coefficient α (0 < α < 1). Assume that either (B1) or (B2) holds. Let S be a nonexpansive mapping of C into itself such that

$${\mathrm{\Theta }}_3:=F\left(S\right)\cap \mathsf{\text{GMEP}}\left(F_1,{\phi }_1,B_1\right)\cap \mathsf{\text{GMEP}}\left(F_2,{\phi }_2,B_2\right)\ne \varnothing .$$

Suppose {x _n } is a sequences generated by the following algorithm x₀ ∈ C arbitrarily:

$$\left\{\begin{array}{c}{u}_n=T_{r_n}^{\left(F_1,{\phi }_1\right)}\left(x_n-r_n{B}_1{x}_n\right)\hfill \\ v_n=T_{s_n}^{\left(F_2,{\phi }_2\right)}\left(x_n-s_n{B}_2{x}_n\right)\hfill \\ x_{n+1}={\xi }_n{P}_C\left[{\alpha }_{n}f\left(x_n\right)+\left(I-{\alpha }_n\right)S{P}_C{u}_n\right]+\left(1-{\xi }_n\right)P_C{v}_n,\hfill \end{array}\right.$$

where {α _n }, {ξ _n } ⊂ (0, 1), r _n ∈ (0, 2η) with 0 < c ≤ d ≤ 1 - η and s _n ∈ (0, 2ρ) with 0 < e ≤ f ≤ 1 - ρ satisfy the following conditions:

(C1): lim_n→∞α _n = 0, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty },{\sum }_{n=1}^{\mathrm{\infty }}\left|{\alpha }_{n+1}-{\alpha }_n\right|<\mathrm{\infty }$,

(C2): 0 < lim inf_n→∞ξ _n < lim sup_n→∞ξ _n < 1, ${\sum }_{n=1}^{\mathrm{\infty }}\left|{\xi }_{n+1}-{\xi }_n\right|<\mathrm{\infty }$,

(C3): lim inf_n→∞r _n > 0 and lim_n→∞|r_n+1- r _n | = 0,

(C4): lim inf_n→∞s _n > 0 and lim_n→∞|s_n+1- s _n | = 0.

Then {x _n } converges strongly to q ∈ Θ₃, where$q=P_{{\mathrm{\Theta }}_3}\left(f+I\right)\left(q\right)$which solves the following variational inequality:

$$〈\left(f-I\right)q,p-q〉\le 0,\forall p\in {\mathrm{\Theta }}_3$$

Proof. Taking γ ≡ 1, A ≡ I and B ≡ 0 in Corollary 3.5, we can obtain desired conclusion immediately. □