Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Abstract

Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the set ${\bigcap }_{i=1}^N\text{Fix (}{S}_i)$ of fixed points of a finite family of nonexpansive mappings ${\left\{S_i\right\}}_{i=1}^N$ in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of ${\bigcap }_{i=1}^N\text{Fix (}{S}_i)\cap GMEP$, which is the unique solution of a variational inequality.

AMS subject classifications: 49J40; 47J20; 47H09.

1 Introduction

The theory of equilibrium problems has played an important role in the study of a wide class of problems arising in economics, finance, transportation, network and structural analysis, elasticity and optimization, and has numerous applications, including but not limited to problems in economics, game theory, finance, traffic analysis, circuit network analysis and mechanics. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be productive and innovative. It is remarkable that the variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems [1, 2].

Let H be a real Hilbert space. Throughout this paper, we write x _n ⇀ x to indicate that the sequence {x _n } converges weakly to x. The x _n → x indicates that {x _n } converges strongly to x. Let C be a nonempty closed convex subset of H and Θ be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for Θ: C × C → R is to find $\overline{x}\in C$ such that

$$\Theta \left(\overline{x},y\right)\ge 0,\forall y\in C.$$

(1.1)

The set of solutions of problem (1.1) is denoted by EP(Θ). Given a mapping T: C → H, let Θ (x, y) = 〈Tx, y - x〉 for all x, y ∈ C. Then, z ∈ EP(Θ) if and only if 〈Tz, y - z〉 ≥ 0 for all y ∈ C. Numerous problems in physics, optimization, and economics reduce to finding a solution of problem (1.1). Equilibrium problems have been studied extensively [2–18]. Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP(Θ) is nonempty and derived a strong convergence theorem. Very recently, Peng and Yao [4] introduced the following generalized mixed equilibrium problem of finding $\overline{x}\in C$ such that

$$\Theta \left(\overline{x},y\right)+\phi \left(y\right)-\phi \left(\overline{x}\right)+〈A\overline{x},y-\overline{x}〉\ge 0,\forall y\in C,$$

(1.2)

where A: H → H is a nonlinear mapping, φ: C → R is a function and Θ: C × C → R is a bifunction. The set of solutions of problem (1.2) is denoted by GMEP.

In particular, whenever A = 0, problem (1.2) reduces to the following mixed equilibrium problem of finding $\overline{x}\in C$ such that

$$\Theta \left(\overline{x},y\right)+\phi \left(y\right)-\phi \left(\overline{x}\right)\ge 0,\forall y\in C,$$

which was considered by Ceng and Yao [5]. The set of solutions of this problem is denoted by MEP.

Whenever φ = 0, problem (1.2) reduces to the following generalized equilibrium problem of finding $\overline{x}\in C$ such that

$$\Theta \left(\overline{x},y\right)+〈A\overline{x},y-\overline{x}〉\ge 0,\forall y\in C,$$

(1.3)

which was introduced and studied by Takahashi and Takahashi [13]. The set of solutions of problem (1.3) is denoted by GEP. Obviously, the generalized equilibrium problem covers the equilibrium problem as a special case. It is assumed in [4] that Θ: C ×C → R is a bifunction satisfying conditions (H1)-(H4) and φ: C → R is a lower semicontinuous and convex function with restriction (A1) or (A2), where

(H1) Θ (x, x) = 0, ∀x ∈ C;

(H2) Θ is monotone, i.e., Θ (x, y) + Θ (y, x) ≤ 0, ∀x, y ∈ C;

(H3) for each y ∈ C, x ↦ Θ (x, y) is weakly upper semicontinuous;

(H4) for each x ∈ C, y ↦ Θ (x, y) is convex and lower semicontinuous;

(A1) for each x ∈ H and r > 0, there exist a bounded subset D _x ⊂ C and y _x ∈ C such that for any z ∈ C \ D _x ,

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

(A2) C is a bounded set.

It is worth pointing out that, related iterative methods for solving fixed point problems, variational inequalities and optimization problems can be found in [19–35].

Recall that a ρ-Lipschitzian mapping T: C → H is a mapping on C such that

$$ǁTx-Tyǁ\le \rho ǁx-yǁ,\forall x,y\in C,$$

where ρ ≥ 0 is a constant. In particular, if ρ ∈ [0, 1) then T is called a contraction on C; if ρ = 1 then T is called a nonexpansive mapping on C. Denote the set of fixed points of T by Fix(T). It is well known that if C is a nonempty bounded closed convex subset of H and S: C → C is nonexpansive, then Fix(S) ≠ Ø. Let P _C be the metric projection of H onto C, that is, for every point x ∈ H, there exists a unique nearest point of C, denoted by P _C x, such that ǀǀ x - P _C x ǀǀ ≤ ǀǀ x - y ǀǀ for all y ∈ C. Recall also that a mapping A of C into H is called

• (i) monotone if

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

• (ii) η-strongly monotone if there exists a constant η > 0 such that

  $$〈Ax-Ay,x-y〉\ge \eta {ǁx-yǁ}^2,\forall x,y\in C;$$

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

  $$〈Ax-Ay,x-y〉\ge \delta {ǁAx-Ayǁ}^2,\forall x,y\in C.$$

Furthermore, let A be a strongly positive bounded linear operator on H, that is, there exists a constant $\overline{\gamma }>0$ such that

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

(1.4)

1.1 The W-mappings

The concept of W-mappings was introduced in Atsushiba and Takahashi [22]. It is very useful in establishing the convergence of iterative methods for computing a common fixed point of nonlinear mappings (see, for instance, [23, 25, 27]).

Let λ_n,1, λ_n,2..., λ _{n, N} ∈ (0, 1], n ≥ 1. Given the nonexpansive mappings S₁, S₂,..., S _N on H, Atsushiba and Takahashi defines, for each n ≥ 1, mappings U_n,1, U_n,2,..., U _{n, N} by

$$\begin{array}{c}{U}_{n,1}={\lambda }_{n,1}{S}_1+\left(1-{\lambda }_{n,1}\right)I,\hfill \\ U_{n,2}={\lambda }_{n,2}{S}_2{U}_{n,1}+\left(1-{\lambda }_{n,2}\right)I,\hfill \\ ⋮\hfill \\ U_{n,N-1}={\lambda }_{n,N-1}{S}_{N-1}{U}_{n,N-2}+\left(1-{\lambda }_{n,N-1}\right)I,\hfill \\ W_n:=U_{n,N}={\lambda }_{n,N}{S}_N{U}_{n,N-1}+\left(1-{\lambda }_{n,N}\right)I.\hfill \end{array}$$

(1.5)

The W _n is called the W-mapping generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} . Note that Nonexpansivity of S _i implies the nonexpansivity of W _n .

Colao et al. [14] introduced an iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space H. Moreover, they proved the strong convergence of the proposed iterative algorithm.

1.2 Theorem CMX

(See [[14], Theorem 3.1]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let ${\left\{S_i\right\}}_{i=1}^N$ be a finite family of nonexpansive mappings on H, A a strongly positive bounded linear operator on H with coefficient $\overline{\gamma }$ and f an α-contraction on H for some α ∈ (0, 1). Moreover, let {α _n } be a sequence in (0, 1), ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ a sequence in [a, b] with 0 < a ≤ b < 1, {r _n } a sequence in (0, ∞) and γ and β two real numbers such that 0 < β < 1 and $0<\gamma <\overline{\gamma }/\alpha$. Let Θ: C × C → R be a bifunction satisfying assumptions (H1)-(H4) and ${\cap }_{i=1}^N\text{Fix(}{S}_i)\cap EP\left(\Theta \right)\ne \varnothing$. For every n ≥ 1, let W _n be the W-mapping generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} . Given x₁ ∈ H arbitrarily, suppose the sequences {x _n } and {u _n } are generated iteratively by

$$\left\{\begin{array}{c}\Theta \left(u_n,y\right)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C,\hfill \\ x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+\beta x_n+\left(\left(1-\beta \right)I-{\alpha }_{n}A\right)W_n{u}_n,\forall n\ge 1,\hfill \end{array}\right.$$

(1.6)

where the sequences {α _n }, {r _n } and the finite family of sequences ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ satisfy the conditions:

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

• (ii) lim inf_n→∞r _n > 0 and lim_n→∞r _n /r_n+1= 1 (or lim_n→∞ǀr_n+1- r _n ǀ = 0);

• (iii) lim_n→∞ǀλ _{n, i} - λ_{n- 1, i}ǀ = 0 for every i ∈ {1,..., N}.

Then both {x _n } and {u _n } converge strongly to $x^{*}\in {\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap EP\left(\Theta \right)$, which is the unique fixed point of the composite mapping $P_{{\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap EP\left(\Theta \right)}\left(I-A+\gamma f\right)$, i.e.,

$$x^{*}=P_{{\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap EP\left(\Theta \right)}\left(I-A+\gamma f\right)x^{*}.$$

Very recently, Yao et al. [10] relaxed the β in Colao, Marino and Xu's iterative scheme (1.6) by a sequence of {β _n }. They showed that if with additional condition 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1 holds, then the sequences {x _n } and {u _n } generated by (1.6) (but now with β _n in the place of β) still converge strongly to $x^{*}\in {\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap EP\left(\Theta \right)$, which is the unique fixed point of the composite mapping $P_{{\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap EP\left(\Theta \right)}\left(I-A+\gamma f\right)$, i.e.,

$$x^{*}=P_{{\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap EP\left(\Theta \right)}\left(I-A+\gamma f\right)x^{*}.$$

1.3 Hybrid steepest-descent method

Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, and let T: H → H be nonexpansive such that Fix(T) ≠ Ø. Yamada [20] introduced the so-called hybrid steepest-descent method for solving the variational inequality problem: finding $\tilde{x}\in \text{Fix}\left(T\right)$ such that

$$〈F\tilde{x},x-\tilde{x}〉\ge 0,\forall x\in \text{Fix}\left(T\right).$$

This method generates a sequence {x _n } via the following iterative scheme:

$$x_{n+1}=T{x}_n-{\lambda }_{n+1}\mu F\left(T{x}_n\right),\forall n\ge 0,$$

(1.7)

where 0 < μ < 2η/κ², the initial guess x₀ ∈ H is arbitrary and the sequence {λ _n } in (0, 1) satisfies the conditions:

$${\lambda }_n\to 0,\sum _{n=0}^{\infty }{\lambda }_n=\infty \text{and}\sum _{n=0}^{\infty }ǀ{\lambda }_{n+1}-{\lambda }_nǀ<\infty .$$

A key fact in Yamada's argument is that, for small enough λ > 0, the mapping

$$T^{\lambda }x:=Tx-\lambda \mu F\left(Tx\right),\forall x\in H$$

is a contraction, due to the κ-Lipschitz continuity and η-strong monotonicity of F.

1.4 Our hybrid model

In this paper, assume Θ: C × C → R is a bifunction satisfying assumptions (H1)-(H4) and φ: C → R is a lower semicontinuous and convex function with restriction (A1) or (A2). Let the mapping A: H → H be δ-inverse strongly monotone, and ${\left\{S_i\right\}}_{i=1}^N$ be a finite family of nonexpansive mappings on H such that ${\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP\ne \varnothing$. Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ² and 0 ≤ γρ < τ, where $\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}$. By combining Yamada's hybrid steepest-descent method [20] and Colao, Marino and Xu's hybrid viscosity approximation method [14] (see also [10]), we propose the following hybrid iterative method for finding a common element of the set of solutions of generalized mixed equilibrium problem (1.2) and the set of fixed points of finitely many nonexpansive mappings ${\left\{S_i\right\}}_{i=1}^N$, that is, for given x₁ ∈ H arbitrarily, let {x _n } and {u _n } be generated iteratively by

$$\left\{\begin{array}{c}\Theta \left(u_n,y\right)+\phi \left(y\right)-\phi \left(u_n\right)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C,\hfill \\ x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n,\forall n\ge 1,\hfill \end{array}\right.$$

(1.8)

where {a _n }, {β _n } ⊂ (0, 1), {r _n } ⊂ (0, 2δ], ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N\subset \left[a,b\right]$ with 0 < a ≤ b < 1, and W _n is the W-mapping generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} . We shall prove that under quite mild hypotheses, both sequences {x _n } and {u _n } converge strongly to $x^{*}\in {\cap }_{i=1}^N\text{Fix(}{S}_i)\cap GMEP$, where $x^{*}=P_{{\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP}\left(I-\mu F+\gamma f\right)x^{*}$ is a unique solution of the variational inequality:

$$〈\left(\mu F-\gamma f\right)x^{*},x^{*}-x〉\le 0,\forall x\in \bigcap _{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP.$$

(1.9)

Compared with Theorem 3.2 of Yao et al. [10], our Theorem 3.1 improves and extends their Theorem 3.2 [10] in the following aspects:

• (i) The contraction f: H → H with coefficient ρ ∈ (0, 1) in [[10], Theorem 3.2] is extended to the case of general Lipschitzian mapping f on H with constant ρ ≥ 0.

• (ii) The strongly positive bounded linear operator A: H → H with coefficient $\tilde{\gamma }>\text{0}$ in [[10], Theorem 3.2] is extended to the case of general κ-Lipschitzian and η-strongly monotone operator F: H → H with constants κ, η > 0.

• (iii) The equilibrium problem in [[10], Theorem 3.2] is extended to the case of generalized mixed equilibrium problem (1.2). Obviously, the problem (1.2) is more complicated than their problem (1.1).

• (iv) The hybrid viscosity approximation method in [[10], Theorem 3.2] (see also [[14], Theorem 3.1]) is extended to develop our iterative method by virtue of Yamada's hybrid steepest-descent method [20].

2 Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉, and norm ǀǀ · ǀǀ. Let C be a nonempty closed convex subset of H. Recall that the metric (or nearest point) projection from H onto C is the mapping P _C : H → C which assigns to each point x ∈ H the unique point P _C x ∈ C satisfying the property

$$ǁx-P_{C}xǁ=\underset{y\in C}{\text{inf}}ǁx-yǁ=:d\left(x,C\right).$$

In order to prove our main results in the next section, we need the following lemmas and propositions.

Lemma 2.1 (See [36]). Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C, we then have

• (i) z = P _C x if and only if 〈 x - z, y - z 〉 ≤ 0, ∀y ∈ C.

• (ii) z = P _C x if and only if ǀǀ x - z ǀǀ² ≤ ǀǀ x - y ǀǀ² - ǀǀ y - z ǀǀ², ∀ _y ∈ C.

• (iii) 〈P _C x - P _C y, x - y 〉 ≥ ǀǀ P _C x - P _C y ǀǀ², ∀x, y ∈ H.

Consequently, P _C is nonexpansive and monotone.

Lemma 2.2 (See [5]). Let C be a nonempty closed convex subset of H. Let Θ: C×C → R be a bifunction satisfying conditions (H1)-(H4) and let φ: C → R be a lower semicontinuous and convex function. For r > 0 and x ∈ H, define a mapping $T_r^{\left(\Theta ,\phi \right)}:H\to C$ as follows:

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

for all x ∈ H. Assume that either (A1) or (A2) holds. Then the following assertions hold:

(i)$T_r^{\left(\Theta ,\phi \right)}\left(x\right)\ne \varnothing$ for each x ∈ H and $T_r^{\left(\Theta ,\phi \right)}$ is single-valued;

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

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

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

• (iv) MEP(Θ, φ)is closed and convex.

Remark 2.1. If φ = 0, then $T_r^{\left(\Theta ,\phi \right)}$ is rewritten as $T_r^{\Theta };\text{if}\Theta =0$ additionally, then $T_r^{\Theta }=P_C$.

Lemma 2.3 (See [21]). Let {x _n } and {y _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 )y _n +β _n x _n for all integers n ≥ 0 and lim sup_n→∞(ǀǀy_n+1- y _n ǀǀ - ǀǀx_n+1- x _n ǀǀ) ≤ 0. Then, lim_n→∞ǀǀy _n - x _n ǀǀ = 0.

Proposition 2.1 (See [[6], Proposition 2.1]). Let C, H, Θ, φ and $T_r^{\left(\Theta ,\phi \right)}$ be as in Lemma 2.2. Then the following inequality holds:

$${ǁT_s^{\left(\Theta ,\phi \right)}x-T_t^{\left(\Theta ,\phi \right)}xǁ}^2\le \frac{s-t}{s}〈T_s^{\left(\Theta ,\phi \right)}x-T_t^{\left(\Theta ,\phi \right)}x,T_s^{\left(\Theta ,\phi \right)}x-x〉$$

for all s, t > 0 and x ∈ H.

Lemma 2.4 (See [19]). Let {a _n } be a sequence of nonnegative numbers satisfying the condition

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

where {δ _n }, {σ _n } are sequences of real numbers such that

• (i) {δ _n } ⊂ [0, 1] and ${\sum }_{n=1}^{\infty }{\delta }_n=\infty$, or equivalently,

  $$\prod _{n=1}^{\infty }\left(1-{\delta }_n\right):=\underset{n\to \infty }{\text{lim}}\prod _{k=1}^n\left(1-{\delta }_k\right)=0;$$

• (ii) lim sup_n→∞σ _n ≤ 0, or

(ii)' ${\sum }_{n=1}^{\infty }{\delta }_n{\sigma }_n$ is convergent.

Then lim_n→∞a _n = 0.

We will need the following result concerning the W-mapping W _n generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} in (1.5).

Proposition 2.2 (See [23]). Let C be a nonempty closed convex subset of a Banach space X. Let S₁, S₂,..., S _N be a finite family of nonexpansive mappings of C into itself such that ${\cap }_{i=1}^N\text{Fix(}{S}_i)\ne \varnothing$, and let λ_n,1, λ_n,2,..., λ _{n, N} be real numbers such that 0 < λ _{n, i} ≤ b < 1 for i = 1, 2,..., N. For any n ≥ 1, let W _n be the W-mapping of C into itself generated by S₁,..., S _N and λ_n,1,..., λ _{n, N} . If X is strictly convex, then $\text{Fix(}{W}_n)={\cap }_{i=1}^N\text{Fix(}{S}_i)$.

Proposition 2.3 (See [[14], Lemma 2.8]). Let C be a nonempty convex subset of a Banach space. Let ${\left\{S_i\right\}}_{i=1}^N$ be a finite family of nonexpansive mappings of C into itself and ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ be sequences in [0, 1] such that λ _{n, i} → λ _i (i = 1,..., N). Moreover for every integer n ≥ 1, let W and W _n be the W-mappings generated by S₁,..., S _N and λ₁,..., λ _N and S₁,..., S _N and λ_n,1,..., λ _{n, N} respectively. Then for every x ∈ C, it follows that

$$\underset{n\to \infty }{\text{lim}}ǁW_{n}x-Wxǁ=0.$$

The following two lemmas are the immediate consequences of the inner product on H.

Lemma 2.5. For all x, y ∈ H, there holds the inequality

$${ǁx+yǁ}^2\le {ǁxǁ}^2+2〈y,x+y〉.$$

Lemma 2.6 (See [36]). For all x, y, z ∈ H and α, β, γ ∈ [0, 1] with α + β + γ = 1, there holds the equality

$${ǁ\alpha x+\beta y+\gamma zǁ}^2=\alpha {ǁxǁ}^2+\beta {ǁyǁ}^2+\gamma {ǁzǁ}^2-\alpha \beta {ǁx-yǁ}^2-\beta \gamma {ǁy-zǁ}^2-\gamma \alpha {ǁz-xǁ}^2.$$

The following lemma plays a crucial role in proving strong convergence of our iterative schemes.

Lemma 2.7 (See [[19], Lemma 3.1]). Let λ be a number in (0, 1] and let μ > 0. Let F: H → H be an operator on H such that, for some constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone. Associating with a nonexpansive mapping T: H → H, define the mapping T^λ: H → H by

$$T^{\lambda }x:=Tx-\lambda \mu F\left(Tx\right),\forall x\in H.$$

Then T^λ is a contraction provided μ < 2η/κ², that is,

$$ǁT^{\lambda }x-T^{\lambda }yǁ\le \left(1-\lambda \tau \right)ǁx-yǁ,\forall x,y\in H,$$

where $\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}\in \left(0,1\right]$.

Remark 2.2. Put $F=\frac{1}{2}I$, where I is the identity operator of H. Then we have μ < 2η/κ² = 4. Also, put μ = 2. Then it is easy to see that $\kappa =\eta =\frac{1}{2}$ and

$$\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}=1-\sqrt{1-2\left(2\cdot \frac{1}{2}-2{\left(\frac{1}{2}\right)}^2\right)}=1.$$

In particular, whenever λ > 0, we have T^λx: = Tx - λμF(Tx) = (1 - λ) Tx.

3 Iterative scheme and strong convergence

In this section, based on Yamada's hybrid steepest-descent method [20] and Colao, Marino and Xu's hybrid viscosity approximation method [14] (see also [10]), we introduce a hybrid iterative method for finding a common element of the set of solutions of generalized mixed equilibrium problem (1.2) and the set of fixed points of finitely many nonexpansive mappings in a real Hilbert space. Moreover, we derive the strong convergence of the proposed iterative algorithm to a common solution of problem (1.2) and the fixed point problem of finitely many nonexpansive mappings.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ: C × C → R be a bifunction satisfying assumptions (H1)-(H4) and φ: C → R be a lower semicontinuous and convex function with restriction (A1) or (A2). Let the mapping A: H → H be δ-inverse strongly monotone, and ${\left\{S_i\right\}}_{i=1}^N$ be a finite family of nonexpansive mappings on H such that ${\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP\ne \varnothing$. Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ- Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ² and 0 ≤ γρ < τ, where $\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}$. Suppose {a _n } and {β _n } are two sequences in (0, 1), {r _n } is a sequence in (0, 2δ] and ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ is a sequence in [a, b] with 0 < a ≤ b < 1. For every n ≥ 1, let W _n be the W-mapping generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} . Given x₁ ∈ H arbitrarily, suppose the sequences {x _n } and {u _n } are generated iteratively by

$$\left\{\begin{array}{c}\Theta \left(u_n,y\right)+\phi \left(y\right)-\phi \left(u_n\right)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C,\hfill \\ x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n,\forall n\ge 1,\hfill \end{array}\right.$$

(3.1)

where the sequences {a _n }, {β _n }, {r _n } and the finite family of sequences ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ satisfy the conditions:

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

• (ii) 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1;

• (iii) 0 < lim inf_n→∞r _n ≤ lim sup_n→∞r _n < 2δ and lim_n→∞(r_n+1- r _n ) = 0;

• (iv) lim_n→∞(λ_{n+1, i}- λ _{n, i} ) = 0 for all i = 1, 2,..., N.

Then both {x _n } and {u _n } converge strongly to $x^{*}\in {\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP$, where $x^{*}=P_{{\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP}\left(I-\mu F+\gamma f\right)x^{*}$ is a unique solution of the variational inequality:

$$〈\left(\mu F-\gamma f\right)x^{*},x^{*}-x〉\le 0,\forall x\in \bigcap _{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP.$$

(3.2)

Proof. Let $Q=P_{{\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP}$. Note that F: H → H is a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H is a ρ-Lipschitzian mapping with constant ρ ≥ 0. Then, we have

$$\begin{array}{cc}\hfill {ǁ\left(I-\mu F\right)x-\left(I-\mu F\right)yǁ}^2& ={ǁx-yǁ}^2-2\mu 〈x-y,Fx-Fy〉+{\mu }^2{ǁFx-Fyǁ}^2\hfill \\ \le \left(1-2\mu \eta +{\mu }^2{\kappa }^2\right){ǁx-yǁ}^2\hfill \\ ={\left(1-\tau \right)}^2{ǁx-yǁ}^2,\hfill \end{array}$$

where $\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}$, and hence

$$\begin{array}{cc}\hfill ǁQ\left(I-\mu F+\gamma f\right)\left(x\right)-Q\left(I-\mu F+\gamma f\right)\left(y\right)ǁ& \le ǁ\left(I-\mu F+\gamma f\right)\left(x\right)-\left(I-\mu F+\gamma f\right)\left(y\right)ǁ\hfill \\ \le ǁ\left(I-\mu F\right)x-\left(I-\mu F\right)yǁ+\gamma ǁf\left(x\right)-f\left(y\right)ǁ\hfill \\ \le \left(1-\tau \right)ǁx-yǁ+\gamma \rho ǁx-yǁ\hfill \\ =\left(1-\left(\tau -\gamma \rho \right)\right)ǁx-yǁ,\hfill \end{array}$$

for all x, y ∈ H. Since 0 ≤ γρ < τ ≤ 1, it is known that 1 - (τ - γρ) ∈ [0, 1). Therefore, Q(I - μF + γf) is a contraction of H into itself, which implies that there exists a unique element x* ∈ H such that $x^{*}=Q\left(I-\mu F+\gamma f\right)x^{*}=P_{{\cap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP}\left(I-\mu F+\gamma f\right)x^{*}$.

From the definition of $T_r^{\left(\Theta ,\phi \right)}$, we know that $u_n=T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)$. Take $p\in {\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP$ arbitrarily. Since $p=T_{r_n}^{\left(\Theta ,\phi \right)}\left(p-r_{n}Ap\right)=S_{i}p$, A is δ-inverse strongly monotone and 0 < r _n ≤ 2δ, we deduce that, for any n ≥ 1,

$$\begin{array}{cc}\hfill {ǁu_n-pǁ}^2& ={ǁT_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(p-r_{n}Ap\right)ǁ}^2\hfill \\ \le {ǁ\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right)ǁ}^2\hfill \\ ={ǁx_n-p-r_n\left(A{x}_n-Ap\right)ǁ}^2\hfill \\ ={ǁx_n-pǁ}^2-2r_n〈x_n-p,A{x}_n-Ap〉+r_n^2{ǁA{x}_n-Apǁ}^2\hfill \\ \le {ǁx_n-pǁ}^2+r_n\left(r_n-2\delta \right){ǁA{x}_n-Apǁ}^2\hfill \\ \le {ǁx_n-pǁ}^2.\hfill \end{array}$$

(3.3)

First we will prove that both {x _n } and {u _n } are bounded.

Indeed, taking into account the control conditions (i) and (ii), we may assume, without loss of generality, that α _n ≤ 1 - β _n for all n ≥ 1. Now, by Proposition 2.2 we have p ∈ Fix(W _n ).

Then utilizing Lemma 2.7, from (3.1) and (3.3) we obtain

$$\begin{array}{c}ǁx_{n+1}-pǁ\hfill \\ =ǁ{\alpha }_n\left(\gamma f\left(x_n\right)-\mu Fp\right)+{\beta }_n\left(x_n-p\right)+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n-\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_{n}pǁ\hfill \\ \le {\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁ+{\beta }_nǁx_n-pǁ+ǁ\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n-\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_{n}pǁ\hfill \\ ={\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁ+{\beta }_nǁx_n-pǁ+\left(1-{\beta }_n\right)ǁ\left(I-\frac{{\alpha }_n}{1-{\beta }_n}\mu F\right)W_n{u}_n-\left(I-\frac{{\alpha }_n}{1-{\beta }_n}\mu F\right)W_{n}pǁ\hfill \\ \le \left(1-{\beta }_n\right)\left(1-\frac{{\alpha }_n\tau }{1-{\beta }_n}\right)ǁu_n-pǁ+{\beta }_nǁx_n-pǁ+{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁ\hfill \\ \le \left(1-{\beta }_n-{\alpha }_n\tau \right)ǁu_n-pǁ+{\beta }_nǁx_n-pǁ+{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁ\hfill \\ \le \left(1-{\alpha }_n\tau \right)ǁx_n-pǁ+{\alpha }_n\gamma ǁf\left(x_n\right)-f\left(p\right)ǁ+{\alpha }_nǁ\gamma f\left(p\right)-\mu Fpǁ\hfill \\ \le \left(1-{\alpha }_n\tau \right)ǁx_n-pǁ+{\alpha }_n\gamma \rho ǁx_n-pǁ+{\alpha }_nǁ\gamma f\left(p\right)-\mu Fpǁ\hfill \\ =\left(1-\left(\tau -\gamma \rho \right){\alpha }_n\right)ǁx_n-pǁ+{\alpha }_nǁ\gamma f\left(p\right)-\mu Fpǁ\hfill \\ =\left(1-\left(\tau -\gamma \rho \right){\alpha }_n\right)ǁx_n-pǁ+\left(\tau -\gamma \rho \right){\alpha }_n\frac{ǁ\gamma f\left(p\right)-\mu Fpǁ}{\tau -\gamma \rho }\hfill \\ \le \text{max}\left\{ǁx_n-pǁ,\frac{ǁ\gamma f\left(p\right)-\mu Fpǁ}{\tau -\gamma \rho }\right\}.\hfill \end{array}$$

(3.4)

It follows from (3.4) and induction that

$$ǁx_n-pǁ\le \text{max}\left\{ǁx_0-pǁ,\frac{ǁ\gamma f\left(p\right)-\mu Fpǁ}{\tau -\gamma \rho }\right\},\forall n\ge 1.$$

Therefore {x _n } is bounded. We also obtain that {u _n }, {Ax _n }, {W _n u _n } and {f (x _n )} are all bounded. We shall use M to denote the possible different constants appearing in the following reasoning.

Next, we show that ǀǀ x_n+1- x _n ǀǀ → 0.

Indeed, set x_n+1= β _n x _n + (1 - β _n )z _n for all n ≥ 1. Then from the definition of z _n we obtain

$$\begin{array}{cc}\hfill z_{n+1}-z_n& =\frac{x_{n+2}-{\beta }_{n+1}{x}_{n+1}}{1-{\beta }_{n+1}}-\frac{x_{n+1}-{\beta }_n{x}_n}{1-{\beta }_n}\hfill \\ =\frac{{\alpha }_{n+1}\gamma f\left(x_{n+1}\right)+\left(\left(1-{\beta }_{n+1}\right)I-{\alpha }_{n+1}\mu F\right)W_{n+1}{u}_{n+1}}{1-{\beta }_{n+1}}-\frac{{\alpha }_n\gamma f\left(x_n\right)+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n}{1-{\beta }_n}\hfill \\ =\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\gamma f\left(x_{n+1}\right)-\frac{{\alpha }_n}{1-{\beta }_n}\gamma f\left(x_n\right)+W_{n+1}{u}_{n+1}\hfill \\ -W_n{u}_n+\frac{{\alpha }_n}{1-{\beta }_n}\mu F{W}_n{u}_n-\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\mu F{W}_{n+1}{u}_{n+1}\hfill \\ =\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left[\gamma f\left(x_{n+1}\right)-\mu F{W}_{n+1}{u}_{n+1}\right]+\frac{{\alpha }_n}{1-{\beta }_n}\left[\mu F{W}_n{u}_n-\gamma f\left(x_n\right)\right]\hfill \\ +W_{n+1}{u}_{n+1}-W_{n+1}{u}_n+W_{n+1}{u}_n-W_n{u}_n.\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill ǁz_{n+1}-z_nǁ-ǁx_{n+1}-x_nǁ& \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left(\gamma ǁf\left(x_{n+1}\right)ǁ+\mu ǁF{W}_{n+1}{u}_{n+1}ǁ\right)\hfill \\ +\frac{{\alpha }_n}{1-{\beta }_n}\left(\mu ǁF{W}_n{u}_nǁ+\gamma ǁf\left(x_n\right)ǁ\right)+ǁW_{n+1}{u}_{n+1}\hfill \\ -W_{n+1}{u}_nǁ+ǁW_{n+1}{u}_n-W_n{u}_nǁ-ǁx_{n+1}-x_nǁ\hfill \\ \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left(\gamma ǁf\left(x_{n+1}\right)ǁ+\mu ǁF{W}_{n+1}{u}_{n+1}ǁ\right)\hfill \\ +\frac{{\alpha }_n}{1-{\beta }_n}\left(\mu ǁF{W}_n{u}_nǁ+\gamma ǁf\left(x_n\right)ǁ\right)+ǁW_{n+1}{u}_n-W_n{u}_nǁ\hfill \\ +ǁu_{n+1}-u_nǁ-ǁx_{n+1}-x_nǁ.\hfill \end{array}$$

(3.5)

From (1.5), since S _i and U _{n, i} for all i = 1, 2,..., N are nonexpansive,

$$\begin{array}{c}ǁW_{n+1}{u}_n-W_n{u}_nǁ\hfill \\ =ǁ{\lambda }_{n+1,N}{S}_N{U}_{n+1,N-1}{u}_n+\left(1-{\lambda }_{n+1,N}\right)u_n-{\lambda }_{n,N}{S}_N{U}_{n,N-1}{u}_n-\left(1-{\lambda }_{n,N}\right)u_nǁ\hfill \\ \le ǀ{\lambda }_{n+1,N}-{\lambda }_{n,N}ǀǁu_nǁ+ǁ{\lambda }_{n+1,N}{S}_N{U}_{n+1,N-1}{u}_n-{\lambda }_{n,N}{S}_N{U}_{n,N-1}{u}_nǁ\hfill \\ \le ǀ{\lambda }_{n+1,N}-{\lambda }_{n,N}ǀǁu_nǁ+ǁ{\lambda }_{n+1,N}\left(S_N{U}_{n+1,N-1}{u}_n-S_N{U}_{n,N-1}{u}_n\right)ǁ\hfill \\ +ǀ{\lambda }_{n+1,N}-{\lambda }_{n,N}ǀǁS_N{U}_{n,N-1}{u}_nǁ\hfill \\ \le 2Mǀ{\lambda }_{n+1,N}-{\lambda }_{n,N}ǀ+{\lambda }_{n+1,N}ǁU_{n+1,N-1}{u}_n-U_{n,N-1}{u}_nǁ.\hfill \end{array}$$

(3.6)

Again, from (1.5),

$$\begin{array}{c}ǁU_{n+1,N-1}{u}_n-U_{n,N-1}{u}_nǁ\hfill \\ =ǁ{\lambda }_{n+1,N-1}{S}_{N-1}{U}_{n+1,N-2}{u}_n+\left(1-{\lambda }_{n+1,N-1}\right)u_n-{\lambda }_{n,N-1}{S}_{N-1}{U}_{n,N-2}{u}_n-\left(1-{\lambda }_{n,N-1}\right)u_nǁ\hfill \\ \le ǀ{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}ǀǁu_nǁ+ǁ{\lambda }_{n+1,N-1}{S}_{N-1}{U}_{n+1,N-2}{u}_n-{\lambda }_{n,N-1}{S}_{N-1}{U}_{n,N-2}{u}_nǁ\hfill \\ \le ǀ{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}ǀǁu_nǁ+{\lambda }_{n+1,N-1}ǁS_{N-1}{U}_{n+1,N-2}{u}_n-S_{N-1}{U}_{n,N-2}{u}_nǁ\hfill \\ +ǀ{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}ǀM\hfill \\ \le 2Mǀ{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}ǀ+{\lambda }_{n+1,N-1}ǁU_{n+1,N-2}{u}_n-U_{n,N-2}{u}_nǁ\hfill \\ \le 2Mǀ{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}ǀ+ǁU_{n+1,N-2}{u}_n-U_{n,N-2}{u}_nǁ.\hfill \end{array}$$

(3.7)

Therefore, we have

$$\begin{array}{c}ǁU_{n+1,N-1}{u}_n-U_{n,N-1}{u}_nǁ\hfill \\ \le 2Mǀ{\lambda }_{n+1,N-1}-{\lambda }_{n,N-1}ǀ+2Mǀ{\lambda }_{n+1,N-2}-{\lambda }_{n,N-2}ǀ+ǁU_{n+1,N-3}{u}_n-U_{n,N-3}{u}_nǁ\hfill \\ \le 2M\sum _{i=2}^{N-1}ǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ+ǁU_{n+1,1}{u}_n-U_{n,1}{u}_nǁ\hfill \\ =ǁ{\lambda }_{n+1,1}{S}_1{u}_n+\left(1-{\lambda }_{n+1,1}\right)u_n-{\lambda }_{n,1}{S}_1{u}_n-\left(1-{\lambda }_{n,1}\right)u_nǁ+2M\sum _{i=2}^{N-1}ǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ,\hfill \end{array}$$

and then

$$\begin{array}{c}ǁU_{n+1,N-1}{u}_n-U_{n,N-1}{u}_nǁ\hfill \\ \le ǀ{\lambda }_{n+1,1}-{\lambda }_{n,1}ǀǁu_nǁ+ǁ{\lambda }_{n+1,1}{S}_1{u}_n-{\lambda }_{n,1}{S}_1{u}_nǁ+2M\sum _{i=2}^{N-1}ǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ\hfill \\ \le 2M\sum _{i=1}^{N-1}ǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ.\hfill \end{array}$$

(3.8)

Substituting (3.8) into (3.6), we have

$$\begin{array}{cc}\hfill ǁW_{n+1}{u}_n-W_n{u}_nǁ& \le 2Mǀ{\lambda }_{n+1,N}-{\lambda }_{n,N}ǀ+2{\lambda }_{n+1,N}M\sum _{i=1}^{N-1}ǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ\hfill \\ \le 2M\sum _{i=1}^Nǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ.\hfill \end{array}$$

(3.9)

On the other hand, utilizing the δ-inverse strongly monotonicity of A we have

$$\begin{array}{cc}\hfill ǁ\left(x_{n+1}-r_{n+1}A{x}_{n+1}\right)-\left(x_n-r_{n}A{x}_n\right)ǁ& =ǁx_{n+1}-x_n-r_{n+1}\left(A{x}_{n+1}-A{x}_n\right)+\left(r_n-r_{n+1}\right)A{x}_nǁ\hfill \\ \le ǁx_{n+1}-x_n-r_{n+1}\left(A{x}_{n+1}-A{x}_n\right)ǁ+ǀr_{n+1}-r_nǀǁA{x}_nǁ\hfill \\ \le ǁx_{n+1}-x_nǁ+ǀr_{n+1}-r_nǀǁA{x}_nǁ,\hfill \end{array}$$

(3.10)

Since $u_n=T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)$ and $u_{n+1}=T_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_{n+1}-r_{n+1}A{x}_{n+1}\right)$, we get

$$\begin{array}{c}ǁu_{n+1}-u_nǁ\hfill \\ =ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_{n+1}-r_{n+1}A{x}_{n+1}\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ\hfill \\ =ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_{n+1}-r_{n+1}A{x}_{n+1}\right)-T_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)+T_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ\hfill \\ \le ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_{n+1}-r_{n+1}A{x}_{n+1}\right)-T_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ+ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ\hfill \\ \le ǁ\left(x_{n+1}-r_{n+1}A{x}_{n+1}\right)-\left(x_n-r_{n}A{x}_n\right)ǁ+ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ\hfill \\ \le ǁx_{n+1}-x_nǁ+ǀr_{n+1}-r_nǀǁA{x}_nǁ+ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ,\hfill \end{array}$$

(3.11)

Using (3.9) and (3.11) in (3.5), we get

$$\begin{array}{c}ǁz_{n+1}-z_nǁ-ǁx_{n+1}-x_nǁ\hfill \\ \le \frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left(\gamma ǁf\left(x_{n+1}\right)ǁ+\mu ǁF{W}_{n+1}{u}_{n+1}ǁ\right)+\frac{{\alpha }_n}{1-{\beta }_n}\left(\mu ǁF{W}_n{u}_nǁ+\gamma ǁf\left(x_n\right)ǁ\right)\hfill \\ +2M\sum _{i=1}^Nǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ+ǁx_{n+1}-x_nǁ+ǀr_{n+1}-r_nǀǁA{x}_nǁ\hfill \\ +ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ-ǁx_{n+1}-x_nǁ\hfill \\ =\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left(\gamma ǁf\left(x_{n+1}\right)ǁ+\mu ǁF{W}_{n+1}{u}_{n+1}ǁ\right)+\frac{{\alpha }_n}{1-{\beta }_n}\left(\mu ǁF{W}_n{u}_nǁ+\gamma ǁf\left(x_n\right)ǁ\right)\hfill \\ +2M\sum _{i=1}^Nǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ+ǀr_{n+1}-r_nǀǁA{x}_nǁ\hfill \\ +ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ.\hfill \end{array}$$

(3.12)

Note that 0 < lim inf_n→∞r _n ≤ lim sup_n→∞r _n < 2δ and lim_n→∞(r_n+1- r _n ) = 0. Then utilizing Proposition 2.1 we have

$$\underset{n\to \infty }{\text{lim}}ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ=0.$$

(3.13)

Consequently, it follows from (3.13) and conditions (i), (iii), (iv) that

$$\begin{array}{c}\underset{n\to \infty }{\text{lim }\text{sup}}\left(ǁz_{n+1}-z_nǁ-ǁx_{n+1}-x_nǁ\right)\hfill \\ \le \underset{n\to \infty }{\text{lim }\text{sup}}\{\frac{{\alpha }_{n+1}}{1-{\beta }_{n+1}}\left(\gamma ǁf\left(x_{n+1}\right)ǁ+\mu ǁF{W}_{n+1}{u}_{n+1}ǁ\right)+\frac{{\alpha }_n}{1-{\beta }_n}\left(\mu ǁF{W}_n{u}_nǁ+\gamma ǁf\left(x_n\right)ǁ\right)\hfill \\ +2M\sum _{i=1}^Nǀ{\lambda }_{n+1,i}-{\lambda }_{n,i}ǀ+ǀr_{n+1}-r_nǀǁA{x}_nǁ+ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ\}\hfill \\ \text{ = }\text{0}\text{.}\hfill \end{array}$$

Hence by Lemma 2.3 we have

$$\underset{n\to \infty }{\text{lim}}ǁz_n-x_nǁ=0.$$

Consequently

$$\underset{n\to \infty }{\text{lim}}ǁx_{n+1}-x_nǁ=\underset{n\to \infty }{\text{lim}}\left(1-{\beta }_n\right)ǁz_n-x_nǁ=0.$$

(3.14)

From (3.11), (3.13), (3.14) and condition (iii) we have

$$\underset{n\to \infty }{\text{lim}}ǁu_{n+1}-u_nǁ=0.$$

Since x_n+1= a _n γ f(x _n ) + β _n x _n + ((1 - β _n )I - a _n μF)W _n u _n , we have

$$\begin{array}{cc}\hfill ǁx_n-W_n{u}_nǁ& \le ǁx_n-x_{n+1}ǁ+ǁx_{n+1}-W_n{u}_nǁ\hfill \\ \le ǁx_n-x_{n+1}ǁ+{\alpha }_nǁ\gamma f\left(x_n\right)-\mu F{W}_n{u}_nǁ+{\beta }_nǁx_n-W_n{u}_nǁ,\hfill \end{array}$$

that is

$$ǁx_n-W_n{u}_nǁ\le \frac{1}{1-{\beta }_n}ǁx_n-x_{n+1}ǁ+\frac{{\alpha }_n}{1-{\beta }_n}ǁ\gamma f\left(x_n\right)-\mu F{W}_n{u}_nǁ.$$

It follows that

$$\underset{n\to \infty }{\text{lim}}ǁx_n-W_n{u}_nǁ=0.$$

(3.15)

On the other hand, from (3.3) and (3.4) we get

$$\begin{array}{cc}\hfill {ǁx_{n+1}-pǁ}^2& \le {\left[\left(1-{\beta }_n-{\alpha }_n\tau \right)ǁu_n-pǁ+{\beta }_nǁx_n-pǁ+{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁ\right]}^2\hfill \\ \le \left(1-{\beta }_n-{\alpha }_n\tau \right){ǁu_n-pǁ}^2+{\beta }_n{ǁx_n-pǁ}^2+\frac{{\alpha }_n}{\tau }{ǁ\gamma f\left(x_n\right)-\mu Fpǁ}^2\hfill \\ \le \left(1-{\beta }_n-{\alpha }_n\tau \right)\left[{ǁx_n-pǁ}^2+r_n\left(r_n-2\delta \right){ǁA{x}_n-Apǁ}^2\right]+{\beta }_n{ǁx_n-pǁ}^2\hfill \\ +\frac{{\alpha }_n}{\tau }{ǁ\gamma f\left(x_n\right)-\mu Fpǁ}^2\hfill \\ =\left(1-{\alpha }_n\tau \right){ǁx_n-pǁ}^2+r_n\left(r_n-2\delta \right)\left(1-{\beta }_n-{\alpha }_n\tau \right){ǁA{x}_n-Apǁ}^2\hfill \\ +\frac{{\alpha }_n}{\tau }ǁ\gamma f\left(x_n\right)-\mu Fp{ǁ}^2\hfill \\ \le {ǁx_n-pǁ}^2+r_n\left(r_n-2\delta \right)\left(1-{\beta }_n-{\alpha }_n\tau \right){ǁA{x}_n-Apǁ}^2+\frac{{\alpha }_n}{\tau }{ǁ\gamma f\left(x_n\right)-\mu Fpǁ}^2,\hfill \end{array}$$

and hence

$$\begin{array}{c}{r}_n\left(2\delta -r_n\right)\left(1-{\beta }_n-{\alpha }_n\tau \right){ǁA{x}_n-Apǁ}^2\hfill \\ \le {ǁx_n-pǁ}^2-{ǁx_{n+1}-pǁ}^2+\frac{{\alpha }_n}{\tau }{ǁ\gamma f\left(x_n\right)-\mu Fpǁ}^2\hfill \\ =\left(ǁx_n-pǁ+ǁx_{n+1}-pǁ\right)ǁx_n-x_{n+1}ǁ+\frac{{\alpha }_n}{\tau }{ǁ\gamma f\left(x_n\right)-\mu Fpǁ}^2.\hfill \end{array}$$

Obviously, conditions (i), (ii), (iii) guarantee that α _n → 0, 0 < lim inf_n→∞β _n ≤ lim sup_n→∞β _n < 1 and 0 < lim inf_n→∞r _n ≤ lim sup_n→∞r _n < 2δ. Thus from ǀǀ x _n - x_n+1ǀǀ → 0 we conclude that

$$\underset{n\to \infty }{\text{lim}}ǁA{x}_n-Apǁ=0.$$

(3.16)

Note that $T_r^{\left(\Theta ,\phi \right)}$ is firmly nonexpansive. Hence we have

$$\begin{array}{c}{ǁu_n-pǁ}^2\hfill \\ ={ǁT_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(p-r_{n}Ap\right)ǁ}^2\hfill \\ \le 〈\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right),u_n-p〉\hfill \\ =\frac{1}{2}\left[{ǁ\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right)ǁ}^2+{ǁu_n-pǁ}^2-{ǁ\left(x_n-r_{n}A{x}_n\right)-\left(p-r_{n}Ap\right)-\left(u_n-p\right)ǁ}^2\right]\hfill \\ \le \frac{1}{2}\left[{ǁx_n-pǁ}^2+{ǁu_n-pǁ}^2-{ǁx_n-u_n-r_n\left(A{x}_n-Ap\right)ǁ}^2\right]\hfill \\ =\frac{1}{2}\left[{ǁx_n-pǁ}^2+{ǁu_n-pǁ}^2-{ǁx_n-u_nǁ}^2+2r_n〈A{x}_n-Ap,x_n-u_n〉-r_n^2{ǁA{x}_n-Apǁ}^2\right],\hfill \end{array}$$

which implies that

$${ǁu_n-pǁ}^2\le {ǁx_n-pǁ}^2-{ǁx_n-u_nǁ}^2+2r_nǁA{x}_n-Apǁǁx_n-u_nǁ.$$

(3.17)

Therefore, utilizing Lammas 2.5 and 2.7 we deduce from (3.17) that

$$\begin{array}{c}{ǁx_{n+1}-pǁ}^2\hfill \\ ={ǁ{\alpha }_n\left(\gamma f\left(x_n\right)-\mu Fp\right)+{\beta }_n\left(x_n-W_n{u}_n\right)+\left(I-{\alpha }_n\mu F\right)W_n{u}_n-\left(I-{\alpha }_n\mu F\right)W_{n}pǁ}^2\hfill \\ \le {ǁ\left(I-{\alpha }_n\mu F\right)W_n{u}_n-\left(I-{\alpha }_n\mu F\right)W_{n}p+{\beta }_n\left(x_n-W_n{u}_n\right)ǁ}^2+2{\alpha }_n〈\gamma f\left(x_n\right)-\mu Fp,x_{n+1}-p〉\hfill \\ \le {\left[ǁ\left(I-{\alpha }_n\mu F\right)W_n{u}_n-\left(I-{\alpha }_n\mu F\right)W_{n}p+{\beta }_nǁx_n-W_n{u}_nǁ\right]}^2+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ\hfill \\ \le {\left[\left(1-{\alpha }_n\tau \right)ǁu_n-pǁ+{\beta }_nǁx_n-W_n{u}_nǁ\right]}^2+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ\hfill \\ \le {\left(ǁu_n-pǁ+ǁx_n-W_n{u}_nǁ\right)}^2+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ\hfill \\ ={ǁu_n-pǁ}^2+{ǁx_n-W_n{u}_nǁ}^2+2ǁu_n-pǁǁx_n-W_n{u}_nǁ+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ\hfill \\ \le {ǁx_n-pǁ}^2-{ǁx_n-u_nǁ}^2+2r_nǁA{x}_n-Apǁǁx_n-u_nǁ+{ǁx_n-W_n{u}_nǁ}^2\hfill \\ +2ǁu_n-pǁǁx_n-W_n{u}_nǁ+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ.\hfill \end{array}$$

Then we have

$$\begin{array}{cc}\hfill {ǁx_n-u_nǁ}^2& \le {ǁx_n-pǁ}^2-{ǁx_{n+1}-pǁ}^2+2r_nǁA{x}_n-Apǁǁx_n-u_nǁ+{ǁx_n-W_n{u}_nǁ}^2\hfill \\ +2ǁu_n-pǁǁx_n-W_n{u}_nǁ+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ\hfill \\ \le \left(ǁx_n-pǁ+ǁx_{n+1}-pǁ\right)ǁx_n-x_{n+1}ǁ+2r_nǁA{x}_n-Apǁǁx_n-u_nǁ\hfill \\ +{ǁx_n-W_n{u}_nǁ}^2+2ǁu_n-pǁǁx_n-W_n{u}_nǁ+2{\alpha }_nǁ\gamma f\left(x_n\right)-\mu Fpǁǁx_{n+1}-pǁ.\hfill \end{array}$$

So, from (3.14)-(3.16) and α _n → 0, we have

$$\underset{n\to \infty }{\text{lim}}ǁx_n-u_nǁ=0.$$

Since

$$ǁW_n{u}_n-u_nǁ\le ǁW_n{u}_n-x_nǁ+ǁx_n-u_nǁ,$$

we also have

$$\underset{n\to \infty }{\text{lim}}ǁW_n{u}_n-u_nǁ=0.$$

Next, let us show that

$$\underset{n\to \infty }{\text{limsup}}〈\left(\mu F-\gamma f\right)x^{*},x^{*}-x_n〉\le 0,$$

where $x^{*}=P_{{\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP}\left(I-\mu F+\gamma f\right)x^{*}$is a unique solution of the variational inequality (3.2). To show this, we can choose a subsequence $\left\{u_{n_i}\right\}$ of {u _n } such that

$$\underset{n\to \infty }{\text{lim sup}}〈\left(\mu F-\gamma f\right)x^{*},x^{*}-u_n〉=\underset{i\to \infty }{\text{lim}}〈\left(\mu F-\gamma f\right)x^{*},x^{*}-u_{n_i}〉.$$

Since $\left\{u_{n_i}\right\}$ is bounded, there exists a subsequence {u _ij } of $\left\{u_{n_i}\right\}$ which converges weakly to w. Without loss of generality, we may assume that $u_{n_i}\rightharpoonup w$. From ǀǀW _n u _n - u _n ǀǀ → 0, we obtain $W_n{u}_{n_i}\rightharpoonup w$. Now we show that w ∈ GMEP. From $u_n=T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)$, we know that

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

From (H2) it follows that

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

Replacing n by n _i , we have

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

(3.18)

Put u _t = ty + (1 - t)w for all t ∈ (0, 1] and y ∈ C. Then, we have u _t ∈ C. So, from (3.18) we have

$$\begin{array}{cc}\hfill 〈u_t-u_{n_i},A{u}_t〉& \ge 〈u_t-u_{n_i},A{u}_t〉-\phi \left(u_t\right)+\phi \left(u_{n_i}\right)-〈u_t-u_{n_i},A{x}_{n_i}〉\hfill \\ -〈u_t-u_{n_i},\frac{u_{n_i}-x_{n_i}}{r_{n_i}}〉+\Theta \left(u_t,u_{n_i}\right)\hfill \\ =〈u_t-u_{n_i},A{u}_t-A{u}_{n_i}〉+〈u_t-u_{n_i},A{u}_{n_i}-A{x}_{n_i}〉-\phi \left(u_t\right)+\phi \left(u_{n_i}\right)\hfill \\ -〈u_t-u_{n_i},\frac{u_{n_i}-x_{n_i}}{r_{n_i}}〉+\Theta \left(u_t,u_{n_i}\right).\hfill \end{array}$$

Since $ǁu_{n_i}-x_{n_i}ǁ\to 0$, we have $ǁA{u}_{n_i}-A{x}_{n_i}ǁ\to 0$. Further, from the monotonicity of A, we have $〈u_t-u_{n_i},A{u}_t-A{u}_{n_i}〉\ge 0$. So, from (H4), the weakly lower semicontinuity of $\phi ,\frac{u_{n_i}-x_{n_i}}{r_{n_i}}\to 0$ and $u_{n_i}\rightharpoonup w$, we have

$$〈u_t-w,A{u}_t〉\ge -\phi \left(u_t\right)+\phi \left(w\right)+\Theta \left(u_t,w\right),$$

(3.19)

as i → ∞. From (H1), (H4) and (3.19), we also have

$$\begin{array}{cc}\hfill 0& =\Theta \left(u_t,u_t\right)+\phi \left(u_t\right)-\phi \left(u_t\right)\hfill \\ \le t\Theta \left(u_t,y\right)+\left(1-t\right)\Theta \left(u_t,w\right)+t\phi \left(y\right)+\left(1-t\right)\phi \left(w\right)-\phi \left(u_t\right)\hfill \\ =t\left[\Theta \left(u_t,y\right)+\phi \left(y\right)-\phi \left(u_t\right)\right]+\left(1-t\right)\left[\Theta \left(u_t,w\right)+\phi \left(w\right)-\phi \left(u_t\right)\right]\hfill \\ \le t\left[\Theta \left(u_t,y\right)+\phi \left(y\right)-\phi \left(u_t\right)\right]+\left(1-t\right)〈u_t-w,A{u}_t〉\hfill \\ =t\left[\Theta \left(u_t,y\right)+\phi \left(y\right)-\phi \left(u_t\right)\right]+\left(1-t\right)t〈y-w,A{u}_t〉,\hfill \end{array}$$

and hence

$$0\le \Theta \left(u_t,y\right)+\phi \left(y\right)-\phi \left(u_t\right)+\left(1-t\right)〈y-w,A{u}_t〉.$$

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

$$0\le \Theta \left(w,y\right)+\phi \left(y\right)-\phi \left(w\right)+〈y-w,Aw〉.$$

This implies that w ∈ GMEP.

We shall show $w\in {\cap }_{i=1}^N\text{Fix}\left(S_i\right)$. To see this, we observe that we may assume (by passing to a further subsequence if necessary)

$${\lambda }_{n_m,k}\to {\lambda }_k\in \left(0,1\right)\left(k=1,2,\dots ,N\right).$$

Let W be the W-mapping generated by S₁,..., S _N and λ₁,..., λ _N . Then by Proposition 2.3, we have, for every x ∈ H,

$$W_{n_m}x\to Wx.$$

(3.20)

Moreover, from Proposition 2.2 it follows that $\text{Fix}\left(W\right)={\cap }_{i=1}^N\text{Fix}\left(S_i\right)$. Assume that $w\notin {\cap }_{i=1}^N\text{Fix}\left(S_i\right)$; then w ≠ Ww. Since w ∈ GMEP, in terms of ǀǀ x _n - W _n u _n ǀǀ → 0 and Opial's property of a Hilbert space, we conclude from (3.20) that

$$\begin{array}{cc}\hfill \underset{m\to \infty }{\text{lim }\text{inf}}ǁx_{n_m}-wǁ& <\underset{m\to \infty }{\text{lim }\text{inf}}ǁx_{n_m}-Wwǁ\hfill \\ \le \underset{m\to \infty }{\text{lim }\text{inf}}\left(ǁx_{n_m}-W_{n_m}{u}_{n_m}ǁ+ǁW_{n_m}{u}_{n_m}-W_{n_m}wǁ+ǁW_{n_m}w-Wwǁ\right)\hfill \\ =\underset{m\to \infty }{\text{lim }\text{inf}}ǁW_{n_m}{u}_{n_m}-W_{n_m}wǁ\hfill \\ \le \underset{m\to \infty }{\text{lim }\text{inf}}ǁu_{n_m}-wǁ\hfill \\ =\underset{m\to \infty }{\text{lim }\text{inf}}ǁT_{r_{n_m}}^{\left(\Theta ,\phi \right)}\left(x_{n_m}-r_{n_m}A{x}_{n_m}\right)-T_{r_{n_m}}^{\left(\Theta ,\phi \right)}\left(w-r_{n_m}Aw\right)ǁ\hfill \\ \le \underset{m\to \infty }{\text{lim }\text{inf}}ǁ\left(x_{n_m}-r_{n_m}A{x}_{n_m}\right)-\left(w-r_{n_m}Aw\right)ǁ\hfill \\ =\underset{m\to \infty }{\text{lim}\text{inf}}ǁx_{n_m}-w-r_{n_m}\left(A{x}_{n_m}-Aw\right)ǁ\hfill \\ \le \underset{m\to \infty }{\text{lim }\text{inf}}ǁx_{n_m}-wǁ,\hfill \end{array}$$

due to the δ-inverse strong monotonicity of A. This is a contradiction. So, we get $w\in {\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)$. Therefore $w\in {\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP$. Since $x^{*}=P_{{\bigcap }_{i=1}^N\text{Fix}\left(S_i\right)\cap GMEP}\left(I-\mu F+\gamma f\right)x^{*}$, we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\text{lim }\text{sup}}〈\left(\mu F-\gamma f\right)x^{*},x^{*}-x_n〉& =\underset{n\to \infty }{\text{lim }\text{sup}}〈\left(\mu F-\gamma f\right)x^{*},x^{*}-u_n〉\hfill \\ =\underset{i\to \infty }{\text{lim}}〈\left(\mu F-\gamma f\right)x^{*},x^{*}-u_{n_i}〉\hfill \\ =〈\left(\mu F-\gamma f\right)x^{*},x^{*}-w〉\le 0.\hfill \end{array}$$

(3.21)

Finally, we prove that {x _n } and {u _n } converge strongly to x*. From (3.1), utilizing Lemmas 2.5 and 2.7 we have

$$\begin{array}{c}{ǁx_{n+1}-x^{*}ǁ}^2\hfill \\ ={ǁ{\alpha }_n\left(\gamma f\left(x_n\right)-\mu F{x}^{*}\right)+{\beta }_n\left(x_n-x^{*}\right)+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n-\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{x}^{*}ǁ}^2\hfill \\ \le {ǁ{\beta }_n\left(x_n-x^{*}\right)+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n-\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{x}^{*}ǁ}^2\hfill \\ +2{\alpha }_n〈\gamma f\left(x_n\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left[{\beta }_nǁx_n-x^{*}ǁ+ǁ\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{u}_n-\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{x}^{*}ǁ\right]}^2\hfill \\ +2{\alpha }_n〈\gamma f\left(x_n\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left[{\beta }_nǁx_n-x^{*}ǁ+ǁ\left(\left(1-{\beta }_n\right)I-\frac{{\alpha }_n}{1-{\beta }_n}\mu F\right)W_n{u}_n-\left(I-\frac{{\alpha }_n}{1-{\beta }_n}\mu F\right)W_n{x}^{*}ǁ\right]}^2\hfill \\ +2{\alpha }_n\gamma 〈f\left(x_n\right)-f\left(x^{*}\right),x_{n+1}-x^{*}〉+2{\alpha }_n〈\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left[{\beta }_nǁx_n-x^{*}ǁ+\left(1-{\beta }_n\right)\left(1-\frac{{\alpha }_n}{1-{\beta }_n}\tau \right)ǁx_n-x^{*}ǁ\right]}^2\hfill \\ +2{\alpha }_n\gamma \rho ǁx_n-x^{*}ǁǁx_{n+1}-x^{*}ǁ+2{\alpha }_n〈\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ \le {\left(1-{\alpha }_n\tau \right)}^2{ǁx_n-x^{*}ǁ}^2+{\alpha }_n\gamma \rho \left({ǁx_n-x^{*}ǁ}^2+{ǁx_{n+1}-x^{*}ǁ}^2\right)\hfill \\ +2{\alpha }_n〈\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ ={\left(1-{\alpha }_n\tau \right)}^2{ǁx_n-x^{*}ǁ}^2+{\alpha }_n\gamma \rho {ǁx_n-x^{*}ǁ}^2+{\alpha }_n\gamma \rho {ǁx_{n+1}-x^{*}ǁ}^2\hfill \\ +2{\alpha }_n〈\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉.\hfill \end{array}$$

This implies that

$$\begin{array}{cc}\hfill {ǁx_{n+1}-x^{*}ǁ}^2& \le \frac{{\left(1-{\alpha }_n\tau \right)}^2+{\alpha }_n\gamma \rho }{1-{\alpha }_n\gamma \rho }{ǁx_n-x^{*}ǁ}^2+\frac{2{\alpha }_n}{1-{\alpha }_n\gamma \rho }〈\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ =\left[1-\frac{2\left(\tau -\gamma \rho \right){\alpha }_n}{1-{\alpha }_n\gamma \rho }\right]{ǁx_n-x^{*}ǁ}^2+\frac{{\left({\alpha }_n\tau \right)}^2}{1-{\alpha }_n\gamma \rho }{ǁx_n-x^{*}ǁ}^2\hfill \\ +\frac{2{\alpha }_n}{1-{\alpha }_n\gamma \rho }〈\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\hfill \\ \le \left[1-\frac{2\left(\tau -\gamma \rho \right){\alpha }_n}{1-{\alpha }_n\gamma \rho }\right]{ǁx_n-x^{*}ǁ}^2+\frac{2\left(\tau -\gamma \rho \right){\alpha }_n}{1-{\alpha }_n\gamma \rho }\hfill \\ \times \{\frac{\left({\alpha }_n{\tau }^2\right)M_1}{2\left(\tau -\gamma \rho \right)}+\frac{1}{\tau -\gamma \rho }\gamma f\left(x^{*}\right)-\mu F{x}^{*},x_{n+1}-x^{*}〉\}\hfill \\ =\left(1-{\delta }_n\right){ǁx_n-x^{*}ǁ}^2+{\delta }_n{\sigma }_n,\hfill \end{array}$$

where M₁ = sup{ǀǀx _n - p ǀǀ²: n ≥ 1}, ${\delta }_n=\frac{2\left(\tau -\gamma \rho \right){\alpha }_n}{1-{\alpha }_n\gamma \rho }$ and ${\sigma }_n=\frac{\left({\alpha }_n{\tau }^2\right)M_1}{2\left(\tau -\gamma p\right)}+\frac{1}{\tau -\gamma \rho }\gamma f\left(x^{*}\right)-\mu F{x}^{*},{x_n}_{+1}-x^{*}〉$. It is easy to see that ${\delta }_n\to 0,{\sum }_{n=1}^{\infty }{\delta }_n=\infty$ and $\text{lim }\underset{n\to \infty }{\text{sup}}{\sigma }_n\le 0$. Hence, by Lemma 2.4, the sequence {x _n } converges strongly to x*. Consequently, we can obtain from ǀǀx _n - u _n ǀǀ → 0 that {u _n } also converges strongly to x*. This completes the proof. □

Remark 3.1.

• (i) The new technique of argument is applied to derive our Theorem 3.1. For instance, Lemma 2.7 for deriving the convergence of hybrid steepest-descent method plays an important role in proving the strong convergence of the sequences {x _n }, {u _n } in our Theorem 3.1. In addition, utilizing Proposition 2.1 and r_n+1- r _n → 0 we can obtain $\underset{n\to \infty }{\text{lim}}ǁT_{r_{n+1}}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)-T_{r_n}^{\left(\Theta ,\phi \right)}\left(x_n-r_{n}A{x}_n\right)ǁ=0$.

• (ii) In order to show $w\in {\cap }_{i=1}^N\text{Fix}\left(S_i\right)$, the proof of Theorem 3.2 [10] directly asserts that ǀǀu _n - W _n u _n ǀǀ→ 0 (n → ∞) implies $ǁu_{n_j}-W_n{u}_{n_j}ǁ\to 0\left(j\to \infty \right)$ for all n. Actually, this assertion seems impossible under their assumptions imposed on ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$. However, following Colao, Marino and Xu's Step 7 of the proof in [[14], Theorem 3.1] and utilizing Proposition 2.3 (i.e., Lemma 2.8 in [14]), we successively derive $w\in {\cap }_{i=1}^N\text{Fix}\left(S_i\right)$ by the condition ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N\subset \left[a,b\right]$ with 0 < a ≤ b < 1.

Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A: H → H be δ-inverse strongly monotone, Θ: C × C → R be a bifunction satisfying assumptions (H1)-(H4) and φ: C → R be a lower semicontinuous and convex function with restriction (A1) or (A2) such that GMEP ≠ Ø. Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ² and 0 ≤ γρ < τ, where $\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}$. Suppose {α _n } and {β _n } are two sequences in (0, 1) and {r _n } is a sequence in (0, 2δ]. Given x₁ ∈ H arbitrarily, suppose the sequences {x _n } and {u _n } are generated iteratively by

$$\left\{\begin{array}{c}\Theta \left(u_n,y\right)+\phi \left(y\right)-\phi \left(u_n\right)+〈A{x}_n,y-u_n〉+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\forall y\in C,\hfill \\ x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)u_n,\forall n\ge 1,\hfill \end{array}\right.$$

(3.22)

where the sequences {α _n }, {β _n }, {r _n } satisfy the conditions:

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

• (ii) 0 < lim inf_{n →∞}β _n ≤ lim sup_{n →∞}β _n < 1;

• (iii) 0 < lim inf_{n →∞}r _n ≤ lim sup_n→∞r _n < 2δ and lim_n→∞(r_n+1- r _n ) = 0.

Then both {x _n } and {u _n } converge strongly to x* ∈ GMEP, where x* = P _GMEP (I - μF + γ f)x*.

Proof. Put S _i x = x for all i = 1, 2,..., N and x ∈ H and take the finite family of sequences ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ in [a, b] with 0 < a ≤ b < 1 such that lim_n→∞(λ_{n+1, i}- λ _{n, i} ) = 0 for all i = 1, 2,..., N. In this case, the W-mapping W _n generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} , is the identity mapping I of H. It is easy to see that all conditions of Theorem 3.1 are satisfied. Thus, the desired result follows from Theorem 3.1. □

Theorem 3.3. Let H be a real Hilbert space. Let ${\left\{S_i\right\}}_{i=1}^N$ be a finite family of nonexpansive mappings on H such that ${\cap }_{i=1}^N\text{Fix}\left(S_i\right)\ne \varnothing$. Let F: H → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0 and f: H → H a ρ-Lipschitzian mapping with constant ρ ≥ 0. Let 0 < μ < 2η/κ² and 0 ≤ γρ < τ, where $\tau =1-\sqrt{1-\mu \left(2\eta -\mu {\kappa }^2\right)}$. Suppose {α _n } and {β _n } are two sequences in (0, 1) and ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ is a sequence in [a, b] with 0 < a ≤ b < 1. For every n ≥ 1, let W _n be the W-mapping generated by S₁,..., S _N and λ_n,1, λ_n,2,..., λ _{n, N} . Given x₁ ∈ H arbitrarily, let {x _n } be a sequence generated by

$$x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+\left(\left(1-{\beta }_n\right)I-{\alpha }_n\mu F\right)W_n{x}_n,\forall n\ge 1,$$

where the sequences {α _n }, {β _n } and the finite family of sequences ${\left\{{\lambda }_{n,i}\right\}}_{i=1}^N$ satisfy the conditions:

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

• (ii) 0 < lim inf_{n →∞}β _n ≤ lim sup_{n →∞}β _n < 1;

• (iii) lim_n→∞(λ_{n+1, i}- λ _{n, i} ) = 0 for all i = 1, 2,..., N.

Then {x _n } converges strongly to $x^{*}\in {\cap }_{i=1}^N\text{Fix}\left(S_i\right)$, where $x^{*}=P_{{\cap }_{i=1}^N\mathsf{\text{Fix}}\left(S_i\right)}\left(I-\mu F+\gamma f\right)x^{*}$.

Proof. Put C = H and r _n = 1, and take Θ(x, y) = 0, Ax = 0 and φ(x) = 0 for all x, y ∈ H. Then Θ: H × H → R is a bifunction satisfying assumptions (H1)-(H4) and φ: H → R is a lower semicontinuous and convex function with restriction (A1). Moreover the mapping A: H → H is δ-inverse strongly monotone for any $\delta >\frac{1}{2}$. In this case, from Theorem 3.1 we deduce that u _n = x _n , 0 < lim inf_n→∞r _n ≤ lim sup_n→∞r _n < 2δ and lim_n→∞(r_n+1-r _n ) = 0. Beyond question, all conditions of Theorem 3.1 are satisfied. Therefore the conclusion follows. □