A new modified block iterative algorithm for uniformly quasi-ϕ-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems

Abstract

In this paper, we introduce a new modified block iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, and the set of solutions of a system of generalized mixed equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Our results extend and improve ones from several earlier works.

2000 MSC: 47H05; 47H09; 47H10.

1 Introduction

Let C be a nonempty closed convex subset of a real Banach space E with ||·|| and let E* be the dual space of E. Let {f _i }_i∈Γ: C × C → ℝ be a bifunction, {φ _i }_i∈Γ: C → ℝ be a real-valued function, and {B _i }_i∈Γ: C → E* be a monotone mapping, where Γ is an arbitrary index set. The system of generalized mixed equilibrium problems is to find x ∈ C such that

$$f_i\left(x,y\right)+⟨B_{i}x,y-x⟩+{\phi }_i\left(y\right)-{\phi }_i\left(x\right)\ge 0,i\in \Gamma ,\forall y\in C.$$

(1.1)

If Γ is a singleton, then problem (1.1) reduces to the generalized mixed equilibrium problem, which is to find x ∈ C such that

$$f\left(x,y\right)+⟨Bx,y-x⟩+\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C.$$

(1.2)

The set of solutions to (1.2) is denoted by GMEP(f, B, φ ), i.e.,

$$\mathsf{\text{GMEP}}\left(f,B,\phi \right)=\left\{x\in C:f\left(x,y\right)+⟨Bx,y-x⟩+\phi \left(y\right)-\phi \left(x\right)\ge 0,\forall y\in C\right\}.$$

(1.3)

If B ≡ 0, the problem (1.2) reduces into the mixed equilibrium problem for f, denoted by MEP (f, φ), which is to find x ∈ C such that

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

(1.4)

If f ≡ 0, the problem (1.2) reduces into the mixed variational inequality of Browder type, denoted by VI(C, B, φ), which is to find x ∈ C such that

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

(1.5)

If B ≡ 0 and φ ≡ 0 the problem (1.2) reduces into the equilibrium problem for f, denoted by EP(f ), which is to find x ∈ C such that

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

(1.6)

If f ≡ 0, the problem (1.4) reduces into the minimize problem, denoted by Argmin(φ), which is to find x ∈ C such that

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

(1.7)

The above formulation (1.5) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an EP(f). In other words, the EP(f) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(f); see, for example [1, 2] and references therein. Some solution methods have been proposed to solve the EP(f); see, for example, [1–15] and references therein.

The normalized duality mapping J : E → 2 ^E* is defined by

$$J\left(x\right)=\left\{x^{*}\in E^{*}:⟨x,x^{*}⟩=\parallel x{\parallel }^2,\parallel x^{*}\parallel =\parallel x\parallel \right\}$$

for all x ∈ E. If E is a Hilbert space, then J = I, where I is the identity mapping. Consider the functional defined by

$$\varphi \left(x,y\right)=\parallel x{\parallel }^2-2⟨x,Jy⟩+\parallel y{\parallel }^2,\forall x,y\in E.$$

(1.8)

As well known that if C is a nonempty closed convex subset of a Hilbert space H and P _C : H → C is the metric projection of H onto C, then P _C is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function ϕ that

$${\left(\parallel x\parallel -\parallel y\parallel \right)}^2\le \varphi \left(x,y\right)\le {\left(\parallel x\parallel +\parallel y\parallel \right)}^2,\forall x,y\in E.$$

(1.9)

If E is a Hilbert space, then ϕ(x, y) = ||x - y||², for all x, y ∈ E. On the other hand, the generalized projection[16] Π _C : E → C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional ϕ(x, y), that is, ${\Pi }_{C}x=\bar{x}$, where $\bar{x}$ is the solution to the minimization problem

$$\varphi \left(\bar{x},x\right)=\underset{y\in C}{inf}\varphi \left(y,x\right),$$

(1.10)

existence and uniqueness of the operator Π _C follows from the properties of the functional ϕ(x, y) and strict monotonicity of the mapping J (see, for example, [16–20]).

Remark 1.1. If E is a reflexive, strictly convex, and smooth Banach space, then for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y. It is sufficient to show that if ϕ(x, y) = 0 then x = y. From (1.8), we have ||x|| = ||y||. This implies that 〈x, Jy〉 = ||x||² = ||Jy||². From the definition of J, one has Jx = Jy. Therefore, we have x = y; see [18, 20] for more details.

Let C be a closed convex subset of E, a mapping T : C → C is said to be L-Lipschitz continuous if ||Tx - Ty|| ≤ L||x - y||, ∀x, y ∈ C and a mapping T is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y||, ∀x, y ∈ C. A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T; that is, F(T) = {x ∈ C : Tx = x}. Recall that a point p in C is said to be an asymptotic fixed point of T[21] if C contains a sequence {x _n } which converges weakly to p such that lim_n→∞||x _n - Tx _n || = 0. The set of asymptotic fixed points of T will be denoted by $\widetilde{F\left(T\right)}$.

A mapping T from C into itself is said to be relatively nonexpansive[22–24] if $\widetilde{F\left(T\right)}=F\left(T\right)$ and ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T). T is said to be relatively quasi-nonexpansive if F(T) ≠ ∅ and ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T). T is said to be ϕ-nonexpansive, if ϕ(Tx, Ty) ≤ ϕ(x, y) for x, y ∈ C. T is said to be quasi-ϕ-asymptotically nonexpansive if F(T) ≠ ∅ and there exists a real sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that ϕ(p, Tⁿx) ≤ k _n ϕ(p, x) for all n ≥ 1 x ∈ C and p ∈ F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in [25–27].

We note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [25–29] which requires the strong restriction: $F\left(T\right)=\widetilde{F\left(T\right)}$. A mapping T is said to be closed if for any sequence {x _n } ⊂ C with x _n → x and Tx _n → y, then Tx = y. It is easy to know that each relatively nonexpansive mapping is closed.

Definition 1.2. (Chang et al. [30]) (1) Let ${\left\{T_i\right\}}_{i=1}^{\infty }:C\to C$ be a sequence of mapping. ${\left\{T_i\right\}}_{i=1}^{\infty }$ is said to be a family of uniformly quasi-ϕ-asymptotically nonexpansive mappings, if $ℱ:={\cap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$, and there exists a sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that for each i ≥ 1

$$\begin{array}{ccc}\hfill \varphi \left(p,T_i^{n}x\right)\hfill & \hfill \le \hfill & \hfill k_n\varphi \left(p,x\right),\forall p\in ℱ,x\in C,\forall n\ge 1.\hfill \end{array}$$

(1.11)

• (2) A mapping T : C → C is said to be uniformly L-Lipschitz continuous, if there exists a constant L > 0 such that

  $$\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y\parallel ,\forall x,y\in C.$$

  (1.12)

Recall that let A : C → E* be a mapping. Then A is called

• (i) monotone if

  $$⟨Ax-Ay,x-y⟩\ge 0,\forall x,y\in C,$$

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

  $$⟨Ax-Ay,x-y⟩\ge \alpha \parallel Ax-Ay{\parallel }^2,\forall x,y\in C.$$

Remark 1.3. It is easy to see that an α-inverse-strongly monotone is monotone and $\frac{1}{\alpha }$-Lipschitz continuous.

In 2004, Matsushita and Takahashi [31] introduced the following iteration: a sequence {x _n } defined by

$$x_{n+1}={\Pi }_C{J}^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)JT{x}_n\right),$$

(1.13)

where the initial guess element x₀ ∈ C is arbitrary, {α _n } is a real sequence in [0, 1], T is a relatively nonexpansive mapping and Π _C denotes the generalized projection from E onto a closed convex subset C of E. They proved that the sequence {x _n } converges weakly to a fixed point of T.

In 2005, Matsushita and Takahashi [28] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping T in a Banach space E:

$$\left\{\begin{array}{c}\hfill x_0\in C\mathsf{\text{chosen arbitrarily}},\hfill \\ \hfill y_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)JT{x}_n\right),\hfill \\ \hfill C_n=\left\{z\in C:\varphi \left(z,y_n\right)\le \varphi \left(z,x_n\right)\right\},\hfill \\ \hfill Q_n=\left\{z\in C:⟨x_n-z,J{x}_0-J{x}_n⟩\ge 0\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_n\cap Q_n}{x}_0.\hfill \end{array}\right.$$

(1.14)

They proved that {x _n } converges strongly to Π_F(T)x₀, where Π_F(T)is the generalized projection from C onto F(T).

In 2008, Iiduka and Takahashi [32] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E : x₁ = x ∈ C and

$$x_{n+1}={\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right),$$

(1.15)

for every n = 1, 2, 3, ..., where Π _C is the generalized metric projection from E onto C, J is the duality mapping from E into E* and {λ _n } is a sequence of positive real numbers. They proved that the sequence {x _n } generated by (1.15) converges weakly to some element of VI(A, C). Takahashi and Zembayashi [33, 34] studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Banach spaces.

In 2009, Wattanawitoon and Kumam [14] using the idea of Takahashi and Zembayashi [33] extended the notion from relatively nonexpansive mappings or ϕ-nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Cholamjiak [35] studied the following iterative algorithm:

$$\{\begin{array}{l}{z}_n={\Pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}JT{x}_n+{\gamma }_{n}JS{z}_n),\hfill \\ u_n\in Csuchthatf(u_n\mathrm{,}y)+\frac{1}{r_n}〈y-u_n\mathrm{,}J{u}_n-J{y}_n〉\ge \mathrm{0,}\forall y\in C\mathrm{,}\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z\mathrm{,}{u}_n)\le \varphi (z\mathrm{,}{x}_n)\},\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0\mathrm{,}\hfill \end{array}$$

(1.16)

where J is the duality mapping on E. Assume that {α _n }, {β _n } and {γ _n } are sequences in [0, 1]. Then, he proved that {x _n } converges strongly to q = Π _F x₀, where F := F (T ) ∩ F (S) ∩ EP(f ) ∩ VI(A, C).

In 2010, Saewan et al. [29] introduced a new hybrid projection iterative scheme which is difference from the algorithm (1.16) of Cholamjiak in [[35], Theorem 3.1] for two relatively quasi-nonexpansive mappings in a Banach space. Motivated by the results of Takahashi and Zembayashi [34], Cholumjiak and Suantai [36] proved the following strong convergence theorem by the hybrid iterative scheme for approximation of common fixed point of countable families of relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space: x₀ ∈ E, $x_1={\Pi }_{C_1}{x}_0$, C₁ = C

$$\left\{\begin{array}{c}\hfill y_{n,i}=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)JT{x}_n\right),\hfill \\ \hfill u_{n,i}=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\cdots T_{r_{1,n}}^{f_1}{y}_{n,i},\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:{sup}_{i>1}\varphi \left(z,J{u}_{n,i}\right)\le \varphi \left(w,J{x}_n\right)\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,n\ge 1.\hfill \end{array}\right.$$

(1.17)

Then, they proved that under certain appropriate conditions imposed on {α _n }, and {r_{n, i}}, the sequence {x _n } converges strongly to Π_F(T)∩EP(f)x₀.

We note that the block iterative method is a method which often used by many authors to solve the convex feasibility problem (see, [37, 38], etc.). In 2008, Plubtieng and Ungchittrakool [39] established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Chang et al. [30] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mapping, and they obtained the strong convergence theorems in a Banach space.

In 2010, Saewan and Kumam [40] obtained the following result for the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property.

Theorem SK Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let f be a bifunction from C × C to ℝ satisfying (A1)-(A4). Let B be a continuous monotone mapping of C into E*. Let ${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$ be an infinite family of closed uniformly L _i -Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 such that $F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \mathsf{\text{GEP}}\left(f,B\right)$is a nonempty and bounded subset in C. For an initial point x₀ ∈ E with $x_1={\Pi }_{C_1}{x}_0$ and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill y_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{z}_n\right),\hfill \\ \hfill z_n=J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{x}_n\right),\hfill \\ \hfill u_n\in C\mathsf{\text{such that }}f\left(u_n,y\right)+⟨B{y}_n,y-u_n⟩+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\forall y\in C,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(1.18)

where J is the duality mapping on E, θ _n = sup_q∈F(k _n - 1)ϕ(q, x _n ), {α_{n, i}}, {β _n } are sequences in [0, 1] and {r _n } ⊂ [a, ∞) for some a > 0. If ${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0 and lim inf_{n → ∞}α_{n, 0}α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F , where p = Π _F x₀.

Quite recently, Qin et al. [9] purposed the problem of approximating a common fixed point of two asymptotically quasi-ϕ-nonexpansive mappings based on hybrid projection methods. Strong convergence theorems are established in a real Banach space. Zegeye et al. [15] introduced an iterative process which converges strongly to a common element of set of common fixed points of countably infinite family of closed relatively quasi- nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for an α-inverse-strongly monotone mapping in Banach spaces.

Motivated and inspired by the work of Chang et al. [30], Qin et al. [7], Takahashi and Zembayashi [33], Wattanawitoon and Kumam [14], Zegeye [41] and Saewan and Kumam [40], we introduce a new modified block hybrid projection algorithm for finding a common element of the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the system of generalized mixed equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in the framework Banach spaces. The results presented in this paper improve and generalize some well-known results in the literature.

2 Preliminaries

A Banach space E is said to be strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all x, y ∈ E with ||x|| = ||y|| = 1 and x ≠ y. Let U = {x ∈ E : ||x|| = 1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit

$$\underset{t\to 0}{\lim }\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U. Let E be a Banach space. The modulus of convexity of E is the function δ : [0, 2] → [0, 1] defined by

$$\delta \left(\epsilon \right)=inf\left\{1-\parallel \frac{x+y}{2}\parallel :x,y\in E,\parallel x\parallel =\parallel y\parallel =1,\parallel x-y\parallel \ge \epsilon \right\}.$$

A Banach space E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (0, 2]. Let p be a fixed real number with p ≥ 2. A Banach space E is said to be p-uniformly convex if there exists a constant c > 0 such that δ(ε) ≥ cε^p for all ε ∈ [0, 2]; see [42, 43] for more details. Observe that every p-uniformly convex is uniformly convex. One should note that no a Banach space is p-uniformly convex for 1 < p < 2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

Remark 2.1. The following basic properties can be found in Cioranescu [18].

• (i) If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.

• (ii) If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J : E → 2^E* is single-valued, one-to-one, and onto.

• (iii) A Banach space E is uniformly smooth if and only if E* is uniformly convex.

• (iv) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence {x _n } ⊂ E, if x _n ⇀ x ∈ E and ||x _n || → ||x||, then x _n → x.

We also need the following lemmas for the proof of our main results.

Lemma 2.2. (Beauzamy[44] and Xu[45]). If E be a 2-uniformly convex Banach space, then for all x, y ∈ E we have

$$\parallel x-y\parallel \le \frac{2}{c^2}\parallel Jx-Jy\parallel ,$$

where J is the normalized duality mapping of E and 0 < c ≤ 1.

The best constant $\frac{1}{c}$ in lemma is called the p-uniformly convex constant of E.

Lemma 2.3. (Beauzamy[44] and Zalinescu[46]). If E be a p-uniformly convex Banach space and let p be a given real number with p ≥ 2, then for all x, y ∈ E, j _x ∈ J _p (x) and j _y ∈ J _p (y)

$$⟨x-y,j_x-j_y⟩\ge \frac{c^p}{2^{p-2}p}\parallel x-y{\parallel }^p,$$

where J _p is the generalized duality mapping of E and$\frac{1}{c}$is the p-uniformly convexity constant of E.

Lemma 2.4. (Kamimura and Takahashi[19]). Let E be a uniformly convex and smooth Banach space and let {x _n } and {y _n } be two sequences of E. If ϕ(x _n , y _n ) → 0 and either {x _n } or {y _n } is bounded, then ||x _n -y _n || → 0.

Lemma 2.5. (Alber[16]). Let C be a nonempty closed convex subset of a smooth Banach space and x ∈ E. Then x₀ = Π _C x if and only if

$$⟨x_0-y,Jx-J{x}_0⟩\ge 0,\forall y\in C.$$

Lemma 2.6. (Alber [[16], Lemma 2.4]). Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty closed convex subset of E and let x ∈ E. Then

$$\varphi \left(y,{\Pi }_{C}x\right)+\varphi \left({\Pi }_{C}x,x\right)\le \varphi \left(y,x\right),\forall y\in C.$$

Let E be a reflexive, strictly convex, smooth Banach space and J is the duality mapping from E into E*. Then J^-1 is also single value, one-to-one, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [16]:

$$V\left(x,x^{*}\right)=\parallel x{\parallel }^2-2⟨x,x^{*}⟩+\parallel x^{*}{\parallel }^2,$$

(2.1)

for all x ∈ E and x* ∈ E*, that is, V (x, x*) = ϕ(x, J^-1(x*)).

Lemma 2.7. (Alber[16]). Let E be a reflexive, strictly convex smooth Banach space and let V be as in (2.1). Then

$$V\left(x,x^{*}\right)+2⟨J^{-1}\left(x^{*}\right)-x,y^{*}⟩\le V\left(x,x^{*}+y^{*}\right),$$

for all x ∈ E and x*, y* ∈ E*.

A set valued mapping U : E ⇉ E* with graph G(U) = (x, x*) : x* ∈ Ux}, domain D(U) = {x ∈ E : Ux ≠ ∅}, and range R(U) = ∪{Ux : x ∈ D(U)}. U is said to be monotone if 〈x - y, x* -y*〉 ≥ 0 whenever (x, x*) ∈ G(U), (y, y*) ∈ G(U). We denote a set valued operator U from E to E* by U ⊂ E × E*. A monotone U is said to be maximal if its graph is not property contained in the graph of any other monotone operator. If U is maximal monotone, then the solution set U^-10 is closed and convex. Let E be a reflexive, strictly convex and smooth Banach space, and it is known that U is a maximal monotone if and only if R(J + rU) = E* for all r > 0. Define the resolvent of U by J _r x = x _r . In other words, Jr = (J + r U)^-1 for all r > 0. J _r is a single-valued mapping from E to D(U). Also, U^-1(0) = F(J _r ) for all r > 0, where F(J _r ) is the set of all fixed points of Jr. Define, for r > 0, the Yosida approximation of U by T _r x = (Jx - JJ _r x)/r for all x ∈ C: We know that T _r x ∈ U (J _r x) for all r > 0 and x ∈ E.

Let A be an inverse-strongly monotone mapping of C into E* which is said to be hemicontinuous if for all x, y ∈ C, and the mapping F of [0, 1] into E*, defined by F(t) = A(tx + (1 - t)y), is continuous with respect to the weak* topology of E*. We define by N _C (v) the normal cone for C at a point v ∈ C, that is,

$$N_C\left(v\right)=\left\{x^{*}\in E^{*}:⟨v-y,x^{*}⟩\ge 0,\forall y\in C\right\}.$$

(2.2)

Lemma 2.8. (Rockafellar[47]). Let C be a nonempty, closed convex subset of a Banach space E, and A is a monotone, hemicontinuous operator of C into E*. Let U ⊂ E × E* be an operator defined as follows:

$$Uv=\left\{\begin{array}{cc}\hfill Av+N_C\left(v\right),\hfill & \hfill v\in C;\hfill \\ \hfill \varnothing \hfill & \hfill otherwise.\hfill \end{array}\right.$$

(2.3)

Then U is maximal monotone and U^-10 = VI(A, C).

Lemma 2.9. (Chang et al. [30]). Let E be a uniformly convex Banach space, r > 0 be a positive number and B _r (0) be a closed ball of E. Then, for any given sequence ${\left\{x_i\right\}}_{i=1}^{\infty }\subset B_r\left(0\right)$and for any given sequence ${\left\{{\lambda }_i\right\}}_{i=1}^{\infty }$ of positive number with ${\sum }_{n=1}^{\infty }{\lambda }_n=1$, there exists a continuous, strictly increasing, and convex function g: [0, 2r) → [0, ∞) with g(0) = 0 such that, for any positive integer i, j with i < j,

$${∥\sum _{n=1}^{\infty }{\lambda }_n{x}_n∥}^2\le \sum _{n=1}^{\infty }{\lambda }_n\parallel x_n{\parallel }^2-{\lambda }_i{\lambda }_{j}g\left(\parallel x_i-x_j\parallel \right).$$

(2.4)

Lemma 2.10. (Chang et al. [30]). Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty closed convex subset of E. Let T : C → C be a closed and quasi-ϕ-asymptotically nonexpansive mapping with a sequence {k _n } ⊂ [1, ∞), k _n → 1. Then F (T ) is a closed convex subset of C:

For solving the equilibrium problem for a bifunction f : C × C → ℝ, let us assume that f satisfies the following conditions:

(A1) f(x, x) = 0 for all x ∈ C;

(A2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C;

(A3) for each x, y, z ∈ C,

$$\underset{t\downarrow 0}{\lim }f\left(tz+\left(1-t\right)x,y\right)\le f\left(x,y\right);$$

(A4) for each x ∈ C, y α f(x, y) is convex and lower semicontinuous.

For example, let A be a continuous and monotone operator of C into E* and define

$$f\left(x,y\right)=⟨Ax,y-x⟩,\forall x,y\in C.$$

Then, f satisfies (A1)-(A4). The following result is in Blum and Oettli [1].

Motivated by Combettes and Hirstoaga [2] in a Hilbert space and Takahashi and Zembayashi [33] in a Banach space, Zhang [48] obtained the following lemma.

Lemma 2.11. (Zhang [[48], Lemma 1.5]). Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that f be a bifunction from C × C to ℝ satisfying (A1)-(A4), A : C → E* be a continuous and monotone mapping and φ : C → ℝ be a semicontinuous and convex functional. For r > 0 and let x ∈ E. Then, there exists z ∈ C such that

$$Q\left(z,y\right)+\frac{1}{r}⟨y-z,Jz-Jx⟩\ge 0,\forall y\in C,$$

where Q(z, y) = f(z, y) + 〈Bz, y - z〉 + φ(y) φ(z), x, y ∈ C. Furthermore, define a mapping T _r : E → C as follows:

$$T_{r}x=\left\{z\in C:Q\left(z,y\right)+\frac{1}{r}⟨y-z,Jz-Jx⟩\ge 0,\forall y\in C\right\}.$$

Then the following hold:

1. T _r is single-valued;

2. T _r is firmly nonexpansive, i.e., for all x, y ∈ E, 〈T _r x - T _r y, JT _r x - JT _r y〉 ≤ 〈T _r x - T _r y, Jx -Jy〉;

3. $F\left(T_r\right)=\widetilde{F\left(T_r\right)}=GMEP\left(f,B,\phi \right)$;

4. GMEP(f, B, φ) is closed and convex;

5. ϕ(p, T _r z) + ϕ(T _r z, z) ≤ ϕ(p, z), ∀p ∈ F(T _r ) and z ∈ E.

3 Main results

In this section, we prove the new convergence theorems for finding the set of solutions of system of generalized mixed equilibrium problems, the common fixed point set of a family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings, and the solution set of variational inequalities for an α-inverse strongly monotone mapping in a 2-uniformly convex and uniformly smooth Banach space.

Theorem 3.1. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C i→ ℝ be a lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$be an infinite family of closed uniformly L _i -Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 such that $F:=\left({\cap }_{i=1}^{\infty }F\left(S_i\right)\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\left(\cap \mathsf{\text{VI}}\left(A,C\right)\right)$ is a nonempty and bounded subset in C. For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill v_n={\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right),\hfill \\ \hfill z_n=J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\right),\hfill \\ \hfill y_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{z}_n\right),\hfill \\ \hfill u_n=T_{r_{m,n}}^{Q_m}{T}_{r_{m-1,n}}^{Q_{m-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}{y}_n,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 1,\hfill \end{array}\right.$$

(3.1)

where θ _n = sup_q∈F(k _n -1)ϕ(q, x _n ), for each i ≥ 0, {α _n , _i } and {β _n } are sequences in [0, 1], {r _{j, n} } ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$is the 2-uniformly convexity constant of E. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0, lim inf_{n → ∞}(1 - β _n ) > 0 and lim inf_n→∞α _n ,₀α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Proof. We first show that C_n+1is closed and convex for each n ≥ 0. Clearly, C₁ = C is closed and convex. Suppose that C _n is closed and convex for each n ∈ ℕ. Since for any z ∈ C _n , we know ϕ(z, u _n ) ≤ ϕ(z, x _n ) + θn is equivalent to 2〈z, Jx _n - Ju _n 〉 ≤ ||x _n ||² - ||u _n ||² + θ _n . So, C_n+1is closed and convex.

Next, we show that F ⊂ C _n for all n ≥ 0. Since $u_n={\Omega }_n^m{y}_n$, when ${\Omega }_n^j=T_{r_{j,n}}^{Q_j}{T}_{r_{j-1,n}}^{Q_{j-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}$, j = 1, 2, 3, ..., m, ${\Omega }_n^0=I$, by the convexity of ||·||², property of ϕ, Lemma 2.9 and by uniformly quasi-ϕ-asymptotically nonexpansive of S _n for each q ∈ F ⊂ C _n , we have

$$\begin{array}{c}\varphi (q\mathrm{,}{u}_n)=\varphi (q\mathrm{,}{\Omega }_n^m{y}_n)\\ \le \varphi (q\mathrm{,}{y}_n)\\ =\varphi (q\mathrm{,}{J}^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)\\ =\parallel q{\parallel }^2-2〈q\mathrm{,}{\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n〉+\parallel {\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n{\parallel }^2\\ \le \parallel q{\parallel }^2-2{\beta }_n〈q\mathrm{,}J{x}_n〉-2(1-{\beta }_n)〈q\mathrm{,}J{z}_n〉+{\beta }_n\parallel x_n{\parallel }^2+(1-{\beta }_n)\parallel z_n{\parallel }^2\\ ={\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n)\varphi (q\mathrm{,}{z}_n)\end{array}$$

(3.2)

and

$$\begin{array}{ccc}\hfill \varphi \left(q,z_n\right)\hfill & \hfill =\hfill & \hfill \varphi \left(q,J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\right)\right)\hfill \\ \hfill =\hfill & \hfill \parallel q{\parallel }^2-2⟨q,{\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n⟩+\parallel {\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n{\parallel }^2\hfill \\ \hfill =\hfill & \hfill \parallel q{\parallel }^2-2{\alpha }_{n,0}⟨q,J{x}_n⟩-2{\sum }_{i=1}^{\infty }{\alpha }_{n,i}⟨q,J{S}_i^n{v}_n⟩+\parallel {\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n{\parallel }^2\hfill \\ \hfill \le \hfill & \hfill \parallel q{\parallel }^2-2{\alpha }_{n,0}⟨q,J{x}_n⟩-2{\sum }_{i=1}^{\infty }{\alpha }_{n,i}⟨q,J{S}_i^n{v}_n⟩+{\alpha }_{n,0}\parallel J{x}_n{\parallel }^2+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}\parallel J{S}_i^n{v}_n{\parallel }^2\hfill \\ \hfill -{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill =\hfill & \hfill \parallel q{\parallel }^2-2{\alpha }_{n,0}⟨q,J{x}_n⟩+{\alpha }_{n,0}\parallel J{x}_n{\parallel }^2-2{\sum }_{i=1}^{\infty }{\alpha }_{n,i}⟨q,J{S}_i^n{v}_n⟩\hfill \\ \hfill +{\sum }_{i=1}^{\infty }{\alpha }_{n,i}\parallel J{S}_i^n{v}_n{\parallel }^2-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill =\hfill & \hfill {\alpha }_{n,0}\varphi \left(q,x_n\right)+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}\varphi \left(q,S_i^n{v}_n\right)-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill \le \hfill & \hfill {\alpha }_{n,0}\varphi \left(q,x_n\right)+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi \left(q,v_n\right)-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel .\hfill \end{array}$$

(3.3)

It follows from Lemma 2.7 that

$$\begin{array}{ccc}\hfill \varphi \left(q,v_n\right)\hfill & \hfill =\hfill & \hfill \varphi \left(q,{\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)\right)\hfill \\ \hfill =\hfill & \hfill V\left(q,J{x}_n-{\lambda }_{n}A{x}_n\right)\hfill \\ \hfill \le \hfill & \hfill V\left(q,\left(J{x}_n-{\lambda }_{n}A{x}_n\right)+{\lambda }_{n}A{x}_n\right)-2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-q,{\lambda }_{n}A{x}_n⟩\hfill \\ \hfill =\hfill & \hfill V\left(q,J{x}_n\right)-2{\lambda }_n⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-q,A{x}_n⟩\hfill \\ \hfill =\hfill & \hfill \varphi \left(q,x_n\right)-2{\lambda }_n⟨x_n-q,A{x}_n⟩+2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-x_n,-{\lambda }_{n}A{x}_n⟩.\hfill \end{array}$$

(3.4)

Since q ∈VI(A, C) and A is an α-inverse-strongly monotone mapping, we have

$$\begin{array}{ccc}\hfill -2{\lambda }_n⟨x_n-q,A{x}_n⟩\hfill & \hfill =\hfill & \hfill -2{\lambda }_n⟨x_n-q,A{x}_n-Aq⟩-2{\lambda }_n⟨x_n-q,Aq⟩\hfill \\ \hfill \le \hfill & \hfill -2{\lambda }_n⟨x_n-q,A{x}_n-Aq⟩\hfill \\ \hfill \le \hfill & \hfill -2\alpha {\lambda }_n\parallel A{x}_n-Aq{\parallel }^2.\hfill \end{array}$$

(3.5)

From Lemma 2.2 and ||Ax _n || ≤ ||Ax _n - Aq||, ∀q ∈ VI(A, C), we also have

$$\begin{array}{ccc}\hfill 2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-x_n,-{\lambda }_{n}A{x}_n⟩\hfill & \hfill =\hfill & \hfill 2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-J^{-1}\left(J{x}_n\right),-{\lambda }_{n}A{x}_n⟩\hfill \\ \hfill \le \hfill & \hfill 2\parallel J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-J^{-1}\left(J{x}_n\right)\parallel \parallel {\lambda }_{n}A{x}_n\parallel \hfill \\ \hfill \le \hfill & \hfill \frac{4}{c^2}\parallel J{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-J{J}^{-1}\left(J{x}_n\right)\parallel \parallel {\lambda }_{n}A{x}_n\parallel \hfill \\ \hfill =\hfill & \hfill \frac{4}{c^2}\parallel J{x}_n-{\lambda }_{n}A{x}_n-J{x}_n\parallel \parallel {\lambda }_{n}A{x}_n\parallel \hfill \\ \hfill =\hfill & \hfill \frac{4}{c^2}\parallel {\lambda }_{n}A{x}_n{\parallel }^2\hfill \\ \hfill =\hfill & \hfill \frac{4}{c^2}{\lambda }_n^2\parallel A{x}_n{\parallel }^2\hfill \\ \hfill \le \hfill & \hfill \frac{4}{c^2}{\lambda }_n^2\parallel A{x}_n-Aq{\parallel }^2.\hfill \end{array}$$

(3.6)

Substituting (3.5) and (3.6) into (3.4), we obtain

$$\begin{array}{ccc}\hfill \varphi \left(q,v_n\right)\hfill & \hfill \le \hfill & \hfill \varphi \left(q,x_n\right)-2\alpha {\lambda }_n\parallel A{x}_n-Aq{\parallel }^2+\frac{4}{c^2}{\lambda }_n^2\parallel A{x}_n-Aq{\parallel }^2\hfill \\ \hfill =\hfill & \hfill \varphi \left(q,x_n\right)+2{\lambda }_n\left(\frac{2}{c^2}{\lambda }_n-\alpha \right)\parallel A{x}_n-Aq{\parallel }^2\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_n\right).\hfill \end{array}$$

Substituting (3.7) into (3.3), we also have

$$\begin{array}{ccc}\hfill \varphi \left(q,z_n\right)\hfill & \hfill \le \hfill & \hfill {\alpha }_{n,0}\varphi \left(q,x_n\right)+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi \left(q,x_n\right)-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill \le \hfill & \hfill {\alpha }_{n,0}{k}_n\varphi \left(q,x_n\right)+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi \left(q,x_n\right)-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill =\hfill & \hfill k_n\varphi \left(q,x_n\right)-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_n\right)+{sup}_{q\in F}\left(k_n-1\right)\varphi \left(q,x_n\right)-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill =\hfill & \hfill \varphi \left(q,x_n\right)+{\theta }_n-{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_n\right)+{\theta }_n.\hfill \end{array}$$

(3.8)

and substituting (3.8) into (3.2), we also have

$$\begin{array}{ccc}\hfill \varphi \left(q,u_n\right)\hfill & \hfill \le \hfill & \hfill \varphi \left(q,x_n\right)+{\theta }_n.\hfill \end{array}$$

(3.9)

This shows that q ∈ C_n+1implies that F ⊂ C_n+1and hence, F ⊂ C _n for all n ≥ 0. This implies that the sequence {x _n } is well defined. From definition of C_n+1that $x_n={\Pi }_{C_n}{x}_0$ and $x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\in C_{n+1}\subset C_n$, we have

$$\begin{array}{ccc}\hfill \varphi \left(x_n,x_0\right)\hfill & \hfill \le \hfill & \hfill \varphi \left(x_{n+1},x_0\right),\forall n\ge 0.\hfill \end{array}$$

(3.10)

By Lemma 2.6, we get

$$\begin{array}{ccc}\hfill \varphi \left(x_n,x_0\right)\hfill & \hfill =\hfill & \hfill \varphi \left({\Pi }_{C_n}{x}_0,x_0\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_0\right)-\varphi \left(q,x_n\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_0\right),\forall q\in F.\hfill \end{array}$$

(3.11)

From (3.10) and (3.11), then {ϕ(x _n , x₀)} are nondecreasing and bounded. So, we obtain that $\underset{n\to \infty }{\lim }\varphi \left(x_n,x_0\right)$ exists. In particular, by (1.9), the sequence {(||x _n || - ||x₀||)² is bounded. This implies {x _n } is also bounded. Denote

$$M=\underset{n\ge 0}{sup}\left\{\parallel x_n\parallel \right\}<\infty .$$

(3.12)

Moreover, by the definition of θ _n and (3.12), it follows that

$${\theta }_n\to 0\mathsf{\text{as}}n\to \infty .$$

(3.13)

Next, we show that {x _n } is a Cauchy sequence in C. Since $x_m={\Pi }_{C_m}{x}_0\in C_m\subset C_n$, for m > n, by Lemma 2.6, we have

$$\begin{array}{ccc}\hfill \varphi \left(x_m,x_n\right)\hfill & \hfill =\hfill & \hfill \varphi \left(x_m,{\Pi }_{C_n}{x}_0\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(x_m,x_0\right)-\varphi \left({\Pi }_{C_n}{x}_0,x_0\right)\hfill \\ \hfill =\hfill & \hfill \varphi \left(x_m,x_0\right)-\varphi \left(x_n,x_0\right).\hfill \end{array}$$

Since lim_n→∞ϕ(x _n , x₀) exists and we take m, n → ∞, we get ϕ(x _m , x _n ) → 0. From Lemma 2.4, we have lim_n→∞||x _m - x _n || = 0. Thus, {x _n } is a Cauchy sequence, and by the completeness of E, there exists a point p ∈ C such that x _n → p as n → ∞.

Now, we claim that ||Ju _n - Jx _n || → 0, as n → ∞. By definition of $x_n={\Pi }_{C_n}{x}_0$, we have

$$\begin{array}{ccc}\hfill \varphi \left(x_{n+1},x_n\right)\hfill & \hfill =\hfill & \hfill \varphi \left(x_{n+1},{\Pi }_{C_n}{x}_0\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(x_{n+1},x_0\right)-\varphi \left({\Pi }_{C_n}{x}_0,x_0\right)\hfill \\ \hfill =\hfill & \hfill \varphi \left(x_{n+1},x_0\right)-\varphi \left(x_n,x_0\right).\hfill \end{array}$$

Since lim_n→∞ϕ(x _n , x₀) exists, we also have

$$\underset{n\to \infty }{\lim }\varphi \left(x_{n+1},x_n\right)=0.$$

(3.14)

Again from Lemma 2.4 that

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

(3.15)

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain

$$\underset{n\to \infty }{\lim }\parallel J{x}_{n+1}-J{x}_n\parallel =0.$$

(3.16)

Since $x_{n+1}={\Pi }_{C_{n+1}}{x}_0\in C_{n+1}\subset C_n$ and the definition of C_n+1, we have

$$\varphi \left(x_{n+1},u_n\right)\le \varphi \left(x_{n+1},x_n\right)+{\theta }_n.$$

By (3.13) and (3.14) that

$$\underset{n\to \infty }{\lim }\varphi \left(x_{n+1},u_n\right)=0.$$

(3.17)

Again applying Lemma 2.4, we have

$$\underset{n\to \infty }{\lim }\parallel x_{n+1}-u_n\parallel =0.$$

(3.18)

Since

$$\begin{array}{ccc}\hfill \parallel u_n-x_n\parallel \hfill & \hfill =\hfill & \hfill \parallel u_n-x_{n+1}+x_{n+1}-x_n\parallel \hfill \\ \hfill \le \hfill & \hfill \parallel u_n-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel .\hfill \end{array}$$

It follows from (3.15) and (3.18) that

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

(3.19)

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we also have

$$\underset{n\to \infty }{\lim }\parallel J{u}_n-J{x}_n\parallel =0.$$

(3.20)

Next, we will show that $p\in F:={\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\cap \left({\cap }_{i=1}^{\infty }F\left(S_i\right)\right)\cap \mathsf{\text{VI}}\left(A,C\right).$

• (a) We show that $p\in {\cap }_{i=1}^{\infty }F\left(S_i\right)$. Since $x_{n+1}={\Pi }_{C_{n+1}}{x}_0\in C_{n+1}\subset C_n$, it follow from (3.8), we have

  $$\varphi \left(x_{n+1},z_n\right)\le \varphi \left(x_{n+1},x_n\right)+{\theta }_n,$$

by (3.13) and (3.14), we get

$$\underset{n\to \infty }{\lim }\varphi \left(x_{n+1},z_n\right)=0$$

(3.21)

again from Lemma 2.4 that

$$\underset{n\to \infty }{\lim }\parallel x_{n+1}-z_n\parallel =0.$$

(3.22)

Since J is uniformly norm-to-norm continuous, we obtain

$$\underset{n\to \infty }{\lim }\parallel J{x}_{n+1}-J{z}_n\parallel =0.$$

(3.23)

From (3.50), we note that

$$\begin{array}{lll}\hfill \parallel J{x}_{n+1}-J{z}_n\parallel & =\parallel J{x}_{n+1}-\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\right)\parallel & \hfill \text{(1)}\\ =\parallel {\alpha }_{n,0}J{x}_{n+1}-{\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{x}_{n+1}-{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\parallel & \hfill \text{(2)}\\ =\parallel {\alpha }_{n,0}\left(J{x}_{n+1}-J{x}_n\right)+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}\left(J{x}_{n+1}-J{S}_i^n{v}_n\right)\parallel & \hfill \text{(3)}\\ =\parallel {\sum }_{i=1}^{\infty }{\alpha }_{n,i}\left(J{x}_{n+1}-J{S}_i^n{v}_n\right)-{\alpha }_{n,0}\left(J{x}_n-J{x}_{n+1}\right)\parallel & \hfill \text{(4)}\\ \ge {\sum }_{i=1}^{\infty }{\alpha }_{n,i}\parallel J{x}_{n+1}-J{S}_i^n{v}_n\parallel -{\alpha }_{n,0}\parallel J{x}_n-J{x}_{n+1}\parallel ,& \hfill \text{(5)}\\ \hfill \text{(6) }\end{array}$$

and hence

$$\parallel J{x}_{n+1}-J{S}_i^n{v}_n\parallel \le \frac{1}{{\sum }_{i=1}^{\infty }{\alpha }_{n,i}}\left(\parallel J{x}_{n+1}-J{z}_n\parallel +{\alpha }_{n,0}\parallel J{x}_n-J{x}_{n+1}\parallel \right).$$

(3.24)

From (3.16), (3.23) and ${\lim \inf }_{n\to \infty }{\sum }_{i=1}^{\infty }{\alpha }_{n,i}>0$, we obtain that

$$\underset{n\to \infty }{\lim }\left|\right|J{x}_{n+1}-J{S}_i^n{v}_n\left|\right|=0.$$

(3.25)

Since J^-1 is uniformly norm-to-norm continuous on bounded sets, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel x_{n+1}-S_i^n{v}_n\parallel =0.\hfill \end{array}$$

(3.26)

Using the triangle inequality that

$$\begin{array}{ccc}\hfill \parallel x_n-S_i^n{v}_n\parallel \hfill & \hfill =\hfill & \hfill \parallel x_n-x_{n+1}+x_{n+1}-S_i^n{v}_n\parallel \hfill \\ \hfill \le \hfill & \hfill \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-S_i^n{v}_n\parallel .\hfill \end{array}$$

From (3.15) and (3.26), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel x_n-S_i^n{v}_n\parallel =0.\hfill \end{array}$$

(3.27)

On the other hand, we note that

$$\begin{array}{ccc}\hfill \varphi \left(q,x_n\right)-\varphi \left(q,u_n\right)+{\theta }_n\hfill & \hfill =\hfill & \hfill \parallel x_n{\parallel }^2-\parallel u_n{\parallel }^2-2⟨q,J{x}_n-J{u}_n⟩+{\theta }_n\hfill \\ \hfill \le \hfill & \hfill \parallel x_n-u_n\parallel \left(\parallel x_n\parallel +\parallel u_n\parallel \right)+2\parallel q\parallel \parallel J{x}_n-J{u}_n\parallel +{\theta }_n.\hfill \end{array}$$

It follows from θ _n → 0, ||x _n - u _n || → 0 and ||Jx _n - Ju _n || → 0, that

$$\varphi \left(q,x_n\right)-\varphi \left(q,u_n\right)+{\theta }_n\to 0\mathsf{\text{as }}n\to \infty .$$

(3.28)

From (3.2), (3.3) and (3.7) that

$$\begin{array}{l}\varphi (q\mathrm{,}{u}_n)\le \varphi (q\mathrm{,}{y}_n)\\ \le {\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n)\varphi (q\mathrm{,}{z}_n)\\ \le {\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n)[{\alpha }_{n\mathrm{,0}}\varphi (q\mathrm{,}{x}_n)+{\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}}{k}_n\varphi (q\mathrm{,}{v}_n)\\ -{\alpha }_{n\mathrm{,0}}{\alpha }_{n\mathrm{,}j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel ]\\ ={\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\alpha }_{n\mathrm{,0}}\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}}{k}_n\varphi (q\mathrm{,}{v}_n)\\ -(1-{\beta }_n){\alpha }_{n\mathrm{,0}}{\alpha }_{n\mathrm{,}j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \\ \le {\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\alpha }_{n\mathrm{,0}}\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_n\varphi (q\mathrm{,}{v}_n)}\\ \le {\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\alpha }_{n\mathrm{,0}}\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_n}[\varphi (q\mathrm{,}{x}_n)-2{\lambda }_n(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-Aq{\parallel }^2]\\ \le {\beta }_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\alpha }_{n\mathrm{,0}}{k}_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_n\varphi (q\mathrm{,}{x}_n)}\\ -(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_n}2{\lambda }_n(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-Aq{\parallel }^2\\ \le {\beta }_n{k}_n\varphi (q\mathrm{,}{x}_n)+(1-{\beta }_n)k_n\varphi (q\mathrm{,}{x}_n)-(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_{n}2{\lambda }_n}(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-Aq{\parallel }^2\\ =k_n\varphi (q\mathrm{,}{x}_n)-(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_{n}2{\lambda }_n}(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-Aq{\parallel }^2]\\ \le \varphi (q\mathrm{,}{x}_n)+{\sup }_{q\in F}(k_n-1)\varphi (q\mathrm{,}{x}_n)-(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_{n}2{\lambda }_n}(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-Aq{\parallel }^2\\ =\varphi (q\mathrm{,}{x}_n)+{\theta }_n-(1-{\beta }_n){\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}{k}_{n}2{\lambda }_n}(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-Aq{\parallel }^2\mathrm{,}\end{array}$$

and hence

$$\begin{array}{lll}\hfill 2a\left(\alpha -\frac{2b}{c^2}\right)\parallel A{x}_n-Aq{\parallel }^2& \le 2{\lambda }_n\left(\alpha -\frac{2}{c^2}{\lambda }_n\right)\parallel A{x}_n-Aq{\parallel }^2& \hfill \text{(1) }\\ \le \frac{1}{\left(1-{\beta }_n\right){\sum }_{i=1}^{\infty }{\alpha }_{n,i}{k}_n}\left(\varphi \left(q,x_n\right)-\varphi \left(q,u_n\right)+{\theta }_n\right).& \hfill \text{(2) }\\ \hfill \text{(3) }\end{array}$$

(3.29)

From (3.28), {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, lim inf_{n →∞}(1 - β _n ) > 0 and lim inf_{n → ∞}α _n ,₀α_{n, i}> 0, for i ≥ 0 and k _n → 1 as n → ∞, we obtain that

$$\underset{n\to \infty }{\lim }\parallel A{x}_n-Aq\parallel =0.$$

(3.30)

From Lemmas 2.6, 2.7 and (3.6), we compute

$$\begin{array}{ccc}\hfill \varphi \left(x_n,v_n\right)\hfill & \hfill =\hfill & \hfill \varphi \left(x_n,{\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(x_n,J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)\right)\hfill \\ \hfill =\hfill & \hfill V\left(x_n,J{x}_n-{\lambda }_{n}A{x}_n\right)\hfill \\ \hfill \le \hfill & \hfill V\left(x_n,\left(J{x}_n-{\lambda }_{n}A{x}_n\right)+{\lambda }_{n}A{x}_n\right)-2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-x_n,{\lambda }_{n}A{x}_n⟩\hfill \\ \hfill =\hfill & \hfill \varphi \left(x_n,x_n\right)+2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-x_n,-{\lambda }_{n}A{x}_n⟩\hfill \\ \hfill =\hfill & \hfill 2⟨J^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)-x_n,-{\lambda }_{n}A{x}_n⟩\hfill \\ \hfill \le \hfill & \hfill \frac{4{\lambda }_n^2}{c^2}\parallel A{x}_n-Aq{\parallel }^2\hfill \\ \hfill \le \hfill & \hfill \frac{4b^2}{c^2}\parallel A{x}_n-Aq{\parallel }^2.\hfill \end{array}$$

Applying Lemma 2.4 and (3.30) that

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel x_n-v_n\parallel =0\hfill \end{array}$$

(3.31)

and we also obtain

$$\underset{n\to \infty }{\lim }\left|\right|J{x}_n-J{v}_n\left|\right|=0.$$

(3.32)

Since $S_i^n$ is continuous, for any i ≥ 1

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel S_i^n{x}_n-S_i^n{v}_n\parallel =0.\hfill \end{array}$$

(3.33)

Again by the triangle inequality, we get

$$\begin{array}{ccc}\hfill \parallel x_n-S_i^n{x}_n\parallel \hfill & \hfill \le \hfill & \hfill \parallel x_n-S_i^n{v}_n\parallel +\parallel S_i^n{v}_n-S_i^n{x}_n\parallel .\hfill \end{array}$$

From (3.27) and (3.33), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel x_n-S_i^n{x}_n\parallel =0,\forall i\ge 1.\hfill \end{array}$$

(3.34)

By using triangle inequality, we get

$$\parallel S_i^n{x}_n-p\parallel \le \parallel S_i^n{x}_n-x_n\parallel +\parallel x_n-p\parallel ,\forall i\ge 1.$$

We know that x _n → p as n → ∞ and from (3.34)

$$S_i^n{x}_n\to pforeachi\ge 1.$$

Moreover, by the assumption that ∀i ≥ 1, S _i is uniformly L _i -Lipschitz continuous, and hence we have.

$$\begin{array}{ccc}\hfill \parallel S_i^{n+1}{x}_n-S_i^n{x}_n\parallel \hfill & \hfill \le \hfill & \hfill \parallel S_i^{n+1}{x}_n-S_i^{n+1}{x}_{n+1}\parallel +\parallel S_i^{n+1}{x}_{n+1}-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel +\parallel x_n-S_i^n{x}_n\parallel \hfill \\ \hfill \le \hfill & \hfill \left(L_i+1\right)\parallel x_{n+1}-x_n\parallel +\parallel S_i^{n+1}{x}_{n+1}-x_{n+1}\parallel +\parallel x_n-S_i^n{x}_n\parallel .\hfill \end{array}$$

(3.35)

By (3.15) and (3.34), it yields that $\parallel S_i^{n+1}{x}_n-S_i^n{x}_n\parallel \to 0$. From $S_i^n{x}_n\to p$, we have $S_i^{n+1}{x}_n\to p$, that is $S_i{S}_i^n{x}_n\to p$. In view of closeness of S _i , we have S _i p = p, for all i ≥ 1. This implies that $p\in {\cap }_{i=1}^{\infty }F\left(S_i\right)$.

• (b) We show that $p\in {\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right).$

Let $u_n={\Omega }_n^m{y}_n$, when ${\Omega }_n^j=T_{r_{j,n}}^{Q_j}{T}_{r_{j-1,n}}^{Q_{j-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}$, j = 1,2,3, ..., m and ${\Omega }_n^0=I$, we obtain

$$\begin{array}{ccc}\hfill \varphi \left(q,u_n\right)\hfill & \hfill =\hfill & \hfill \varphi \left(q,{\Omega }_n^m{y}_n\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,{\Omega }_n^{m-1}{y}_n\right)\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,{\Omega }_n^{m-2}{y}_n\right)\hfill \\ \hfill ⋮\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,{\Omega }_n^j{y}_n\right).\hfill \end{array}$$

(3.36)

By Lemma (2.11)(5), we have for j = 1, 2, 3, ..., m

$$\begin{array}{ccc}\hfill \varphi \left({\Omega }_n^j{y}_n,y_n\right)+{\theta }_n\hfill & \hfill \le \hfill & \hfill \varphi \left(q,y_n\right)-\varphi \left(q,{\Omega }_n^j{y}_n\right)+{\theta }_n\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_n\right)-\varphi \left(q,{\Omega }_n^j{y}_n\right)+{\theta }_n\hfill \\ \hfill \le \hfill & \hfill \varphi \left(q,x_n\right)-\varphi \left(q,u_n\right)+{\theta }_n.\hfill \end{array}$$

(3.37)

From (3.13) and (3.28), we get $\varphi \left({\Omega }_n^j{y}_n,y_n\right)\to 0$as n → ∞, for j = 1, 2, 3, ..., m and Lemma 2.4 implies that

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel {\Omega }_n^j{y}_n-y_n\parallel =0,\forall j=1,2,3,\dots ,m.\hfill \end{array}$$

(3.38)

Since $x_{n+1}={\Pi }_{C_{n+1}}{x}_0\in C_{n+1}\subset C_n$, it follows from (3.2) and (3.8) that

$$\varphi \left(x_{n+1},y_n\right)\le \varphi \left(x_{n+1},x_n\right)+{\theta }_n.$$

By (3.13) and (3.14), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\varphi \left(x_{n+1},y_n\right)=0.\hfill \end{array}$$

Applying Lemma 2.4 that

$$\underset{n\to \infty }{\lim }\parallel x_{n+1}-y_n\parallel =0.$$

(3.39)

Using the triangle inequality, we obtain

$$\begin{array}{ccc}\hfill \parallel x_n-y_n\parallel \hfill & \hfill \le \hfill & \hfill \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-y_n\parallel .\hfill \end{array}$$

From (3.15) and (3.39), we get

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel x_n-y_n\parallel =0.\hfill \end{array}$$

Since x _n → p and ||x _n - y _n || → 0, we have y _n → p as n → ∞.

Again by using the triangle inequality, we have for j = 1, 2, 3, ..., m

$$\begin{array}{ccc}\hfill \parallel p-{\Omega }_n^j{y}_n\parallel \hfill & \hfill \le \hfill & \hfill \parallel p-y_n\parallel +\parallel y_n-{\Omega }_n^j{y}_n\parallel .\hfill \end{array}$$

From (3.38) and y _n → p as n → ∞, we get

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel p-{\Omega }_n^j{y}_n\parallel =0,\forall j=1,2,3,\dots ,m.\hfill \end{array}$$

(3.41)

By using the triangle inequality, we obtain

$$\begin{array}{ccc}\hfill \parallel {\Omega }_n^j{y}_n-{\Omega }_n^{j-1}{y}_n\parallel \hfill & \hfill \le \hfill & \hfill \parallel {\Omega }_n^j{y}_n-p\parallel +\parallel p-{\Omega }_n^{j-1}{y}_n\parallel .\hfill \end{array}$$

From (3.41), we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{\lim }\parallel {\Omega }_n^j{y}_n-{\Omega }_n^{j-1}{y}_n\parallel =0,\forall j=1,2,3,\dots ,m.\hfill \end{array}$$

(3.42)

Since {r _{j, n} } ⊂ [d, ∞) and J is uniformly continuous on any bounded subset of E,

$$\begin{array}{cc}\hfill \underset{n\to \infty }{\lim }\frac{\parallel J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n\parallel }{r_{j,n}}=0,\hfill & \hfill \forall j=1,2,3,\dots ,m.\hfill \end{array}$$

(3.43)

From Lemma 2.11, we get for j = 1, 2, 3, ..., m

$$Q_j\left({\Omega }_n^j{y}_n,y\right)+\frac{1}{r_{j,n}}⟨y-{\Omega }_n^j{y}_n,J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n⟩\ge 0,\forall y\in C.$$

From (A2),

$$\frac{1}{r_{j,n}}⟨y-{\Omega }_n^j{y}_n,J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n⟩\ge Q_j\left(y,{\Omega }_n^j{y}_n\right),\forall y\in C,\forall j=1,2,3,\dots ,m.$$

From (3.41) and (3.43), we have

$$\begin{array}{c}\hfill 0\ge Q_j\left(y,p\right),\forall y\in C,\forall j=1,2,3,\dots ,m.\hfill \end{array}$$

(3.44)

For t with 0 < t ≤ 1 and y ∈ C; let y _t = ty + (1 - t)p. Then, we get that y _t ∈ C. From (3.44), and it follows that

$$\begin{array}{c}\hfill Q_j\left(y_t,p\right)\le 0,\forall y\in C,\forall j=1,2,3,\dots ,m.\hfill \end{array}$$

(3.45)

By the conditions (A1) and (A4), we have for j = 1, 2, 3, ..., m

$$\begin{array}{ccc}\hfill 0\hfill & \hfill =\hfill & \hfill Q_j\left(y_t,y_t\right)\hfill \\ \hfill \le \hfill & \hfill t{Q}_j\left(y_t,y\right)+\left(1-t\right)Q_j\left(y_t,p\right)\hfill \\ \hfill \le \hfill & \hfill t{Q}_j\left(y_t,y\right)\hfill \\ \hfill =\hfill & \hfill Q_j\left(y_t,y\right).\hfill \end{array}$$

(3.46)

From (A3) and letting t → 0, This implies that p ∈ GMEP(f _j , B _j , φ _j ), ∀j = 1, 2, 3, ..., m. Therefore $p\in {\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)$

• (c) We show that p ∈ VI(A, C). Indeed, define U ⊂ E × E* by

  $$Uv=\left\{\begin{array}{cc}\hfill Av+N_C\left(v\right),\hfill & \hfill v\in C;\hfill \\ \hfill \varnothing ,\hfill & \hfill v\notin C.\hfill \end{array}\right.$$

  (3.47)

By Lemma 2.8, U is maximal monotone and U^-10 = VI(A, C). Let (v, w) ∈ G(U). Since w ∈ Uv = Av + N _C (v), we get w - Av ∈ N _C (v).

From v _n ∈ C, we have

$$\begin{array}{c}\hfill ⟨v-v_n,w-Av⟩\ge 0.\hfill \end{array}$$

(3.48)

On the other hand, since $v_n={\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right)$. Then, by Lemma 2.5, we have

$$⟨v-v_n,J{v}_n-\left(J{x}_n-{\lambda }_{n}A{x}_n\right)⟩\ge 0,$$

and thus

$$⟨v-v_n,\frac{J{x}_n-J{v}_n}{{\lambda }_n}-A{x}_n⟩\le 0.$$

(3.49)

It follows from (3.48), (3.49) and A is monotone and $\frac{1}{\alpha }$-Lipschitz continuous that

$$\begin{array}{ccc}\hfill ⟨v-v_n,w⟩\hfill & \hfill \ge \hfill & \hfill ⟨v-v_n,Av⟩\hfill \\ \hfill \ge \hfill & \hfill ⟨v-v_n,Av⟩+⟨v-v_n,\frac{J{x}_n-J{v}_n}{{\lambda }_n}-A{x}_n⟩\hfill \\ \hfill =\hfill & \hfill ⟨v-v_n,Av-A{x}_n⟩+⟨v-z{v}_n,\frac{J{x}_n-J{v}_n}{{\lambda }_n}⟩\hfill \\ \hfill =\hfill & \hfill ⟨v-v_n,Av-A{v}_n⟩+⟨v-v_n,A{v}_n-A{x}_n⟩+⟨v-v_n,\frac{J{x}_n-J{v}_n}{{\lambda }_n}⟩\hfill \\ \hfill \ge \hfill & \hfill -\parallel v-v_n\parallel \frac{\parallel v_n-x_n\parallel }{\alpha }-\parallel v-v_n\parallel \frac{\parallel J{x}_n-J{v}_n\parallel }{a}\hfill \\ \hfill \ge \hfill & \hfill -H\left(\frac{\parallel v_n-x_n\parallel }{\alpha }+\frac{\parallel J{x}_n-J{v}_n\parallel }{a}\right),\hfill \end{array}$$

where H = sup_n≥1||v - v _n ||. Take the limit as n i→ ∞, (3.31) and (3.32), we obtain 〈v - p, w〉 ≥ 0. By the maximality of B we have p ∈ B^-10, that is p ∈ VI(A, C). Hence, from (a), (b) and (c), we obtain p ∈ F.

Finally, we show that p = Π _F x₀. From $x_n={\Pi }_{C_n}{x}_0$, we have 〈Jx₀ - Jx _n , x _n - z〉 ≥ 0, ∀z ∈ C _n . Since F ⊂ C _n , we also have

$$⟨J{x}_0-J{x}_n,x_n-y⟩\ge 0,\forall y\in F.$$

Taking limit n → ∞, we obtain

$$⟨J{x}_0-Jp,p-y⟩\ge 0,\forall y\in F.$$

By Lemma 2.5, we can conclude that p = Π _F x₀ and x _n → p as n → ∞. This completes the proof.   □

If S _i = S for each i ∈ ℕ, then Theorem 3.1 is reduced to the following corollary.

Corollary 3.2. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C → ℝ be a lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let S : C → C be a closed L-Lipschitz continuous and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 such that $F:=F\left(S\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\cap \mathsf{\text{VI}}\left(A,C\right)$is a nonempty and bounded subset in C. For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill v_n={\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right),\hfill \\ \hfill z_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{S}^n{v}_n\right),\hfill \\ \hfill y_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{z}_n\right),\hfill \\ \hfill u_n=T_{r_{m,n}}^{Q_m}{T}_{r_{m-1,n}}^{Q_{m-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}{y}_n,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 1,\hfill \end{array}\right.$$

(3.50)

where θ _n = sup_q∈F(k _n - 1)ϕ(q, x _n ), {α _n }, {β _n } are sequences in [0, 1], {r _{j, n} } ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$c is the 2-uniformly convexity constant of E. If lim inf_n→∞(1 - β _n ) > 0 and lim inf_n→∞(1 - α _n ) > 0, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

For a special case that i = 1, 2, we can obtain the following results on a pair of quasi-_-asymptotically nonexpansive mappings immediately from Theorem 3.1.

Corollary 3.3. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C → ℝ be a lower semicontinuous and convex function. Let A be an α-inversestrongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let S, T : C → C be two closed quasi-ϕ-asymptotically nonexpansive mappings and L _S , L _T -Lipschitz continuous, respectively with a sequence {k _n } ⊂ [1, ∞), k _n → 1 such that $F:=F\left(S\right)\cap F\left(T\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\cap \mathsf{\text{VI}}\left(A,C\right)$ is a nonempty and bounded subset in C. For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill v_n={\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right),\hfill \\ \hfill z_n=J^{-1}\left({\alpha }_{n}J{x}_n+{\beta }_{n}J{S}^n{v}_n+{\gamma }_{n}J{T}^n{v}_n\right),\hfill \\ \hfill y_n=J^{-1}\left({\delta }_{n}J{x}_n+\left(1-{\delta }_n\right)J{z}_n\right),\hfill \\ \hfill u_n=T_{r_{m,n}}^{Q_m}{T}_{r_{m-1,n}}^{Q_{m-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}{y}_n,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.51)

where θ _n = sup_q∈F(k _n - 1)ϕ(q, x _n ), {α _n }, {β _n }, {γ _n } and {δ _n } are sequences in [0, 1], {r _{j, n} } ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$is the 2-uniformly convexity constant of E. If α _n + β _n + γ _n = 1 for all n ≥ 0 and lim inf_{n →∞}α _n β _n > 0, lim inf_{n →∞}α _n γ _n > 0, lim inf_{n →∞}β _n γ _n > 0 and lim inf_{n →∞}(1 - δ _n ) > 0, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Corollary 3.4. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C → ℝ be a lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$be an infinite family of closed quasi-ϕ- nonexpansive mappings such that$F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\cap \mathsf{\text{VI}}\left(A,C\right)\ne \varnothing .$For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\{\begin{array}{l}{v}_n={\Pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n\mathrm{,0}}J{x}_n+{\displaystyle {\sum }_{i=1}^{\infty }{\alpha }_{n\mathrm{,}i}}J{S}_i{v}_n),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n),\hfill \\ u_n=T_{r_{m\mathrm{,}n}}^{Q_m}{T}_{r_{m-\mathrm{1,}n}}^{Q_{m-1}}\cdots T_{r_{\mathrm{2,}n}}^{Q_2}{T}_{r_{\mathrm{1,}n}}^{Q_1}{y}_n\mathrm{,}\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z\mathrm{,}{u}_n)\le \varphi (z\mathrm{,}{x}_n),\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0\mathrm{,}\forall n\ge \mathrm{0,}\hfill \end{array}$$

(3.52)

where {α _{n, i} } and {β _n } are sequences in [0, 1], {r _{j, n} } ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$is the 2-uniformly convexity constant of E. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0, lim inf_{n →∞}(1 -β _n ) > 0 and lim inf_{n →∞}α_{n, 0}α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Proof. Since ${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$ is an infinite family of closed quasi-ϕ-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with sequence k _n = 1. Hence, the conditions appearing in Theorem 3.1 F is a bounded subset in C and for each i ≥ 1, S _i is uniformly L _i -Lipschitz continuous are of no use here. By virtue of the closeness of mapping S _i for each i ≥ 1, it yields that p ∈ F (S _i ) for each i ≥ 1, that is, $p\in {\cap }_{i=1}^{\infty }F\left(S_i\right)$. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Corollary 3.4 is obtained from Theorem 3.1 immediately.   □

4 Deduced theorems

Corollary 4.1. [[41], Theorem 3.2] Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)-(A4) and φ : C → ℝ is convex and lower semicontinuous. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let ${\left\{S_i\right\}}_{i=1}^N:C\to C$be a finite family of closed quasi-ϕ- nonexpansive mappings such that $F:={\cap }_{i=1}^{N}F\left(S_i\right)\cap \mathsf{\text{GMEP}}\left(f,B,\phi \right)\cap \mathsf{\text{VI}}\left(A,C\right)\ne \varnothing .$ For an initial point x₀ ∈ E with$\frac{1}{c}$and C₁ = C, we define the sequence {x _n } as follows:

$$\{\begin{array}{l}{z}_n={\Pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ y_n=J^{-1}({\alpha }_{0}J{x}_n+{\displaystyle {\sum }_{i=1}^N{\alpha }_i}J{S}_i{z}_n),\hfill \\ f(u_n\mathrm{,}y)+〈B{u}_n\mathrm{,}y-u_n〉+\phi (y)-\phi (u_n)+\frac{1}{r_n}〈y-u_n\mathrm{,}J{u}_n-J{y}_n〉\ge \mathrm{0,}\forall y\in C\mathrm{,}\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z\mathrm{,}{u}_n)\le \varphi (z\mathrm{,}{x}_n),\hfill \\ x_{n+1}={\Pi }_{C_{n+1}}{x}_0\mathrm{,}\forall n\ge \mathrm{0,}\hfill \end{array}$$

(4.1)

where {α _i } is sequence in [0, 1], {r _n } ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$c is the 2-uniformly convexity constant of E. If α _i ∈ (0, 1) such that${\sum }_{i=0}^N{\alpha }_i=1$then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Remark 4.2. Theorems 3.1, Corollaries 3.4 and 4.1 improve and extend the corresponding results of Wattanawitoon and Kumam [14] and Zegeye [41] in the following senses:

• from a solution of the classical equilibrium problem to the generalized mixed equilibrium problem with an infinite family of quasi-ϕ-asymptotically mappings;

• for the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or quasi-ϕ-nonexpansive mappings and a finite family of closed relatively quasi-nonexpansive mappings to an infinite family of quasi-ϕ-asymptotically nonex-pansive mappings.

Corollary 4.3. Let C be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)-(A4) and φ : C → ℝ is convex and lower semicontinuous. Let B be a continuous monotone mapping of C into E*. Let${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 and uniformly L _i -Lipschitz continuous such that $F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \mathsf{\text{GMEP}}\left(f,B,\phi \right)$is a nonempty and bounded subset in C. For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill y_n=J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{x}_n\right),\hfill \\ \hfill f\left(u_n,y\right)+⟨B{u}_n,y-u_n⟩+\phi \left(y\right)-\phi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\forall y\in C,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(4.2)

where θ _n = sup_q∈F(k _n - 1)ϕ(q, x _n ), {α _n , _i } is sequence in [0, 1], {r _n } ⊂ [a, ∞) for some a > 0. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0 and lim inf_n→∞α_{n, 0}α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Proof. Put A ≡ 0 in Theorem 3.1 Then, we get that z _n = x _n . Thus, the method of proof of Theorem 3.1 gives the required assertion without the requirement that E be 2-uniformly convex.   □

If setting B ≡ 0 and φ ≡ 0 in Corollary 4.3, then we have the following corollary.

Corollary 4.4. Let C be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C × C to ℝ satisfying (A1)-(A4) and φ : C → ℝ is convex and lower semicontinuous. Let${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 and uniformly L _i -Lipschitz continuous such that $F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \mathsf{\text{EP}}\left(f\right)$ is a nonempty and bounded subset in C. For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill y_n=J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{x}_n\right),\hfill \\ \hfill f\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\forall y\in C,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(4.3)

where θ _n = sup_q∈F(k _n - 1)ϕ(q, x _n ), {α_{n, i}} is sequence in [0, 1], {r _n } ⊂ [a, ∞) for some a > 0. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0 and lim inf_n→∞α_{n, 0}α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Remark 4.5. Corollaries 4.3 and 4.4 improve and extend the corresponding results of Zegeye [41] and Wattanawitoon and Kumam [14] in the sense from a finite family of closed relatively quasi-nonexpansive mappings and closed relatively quasi-nonexpansive mappings to more general than an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings.

Remark 4.6. Moreover, Our theorems improve, generalize, unify and extend Qin et al. [9], Zeg-eye et al. [15], Zegeye [41] and Wattanawitoon and Kumam [14, 49] and several results recently announced.

5 Applications

5.1 Application to complementarity problems

Let K be a nonempty, closed convex cone in E. We define the polar K* of K as follows:

$$K^{*}=\left\{y^{*}\in E^{*}:⟨x,y^{*}⟩\ge 0,\forall x\in K\right\}.$$

(5.1)

If A : K → E* is an operator, then an element u ∈ K is called a solution of the complementarity problem[20] if

$$Au\in K^{*}\mathsf{\text{and}}⟨u,Au⟩=0.$$

(5.2)

The set of solutions of the complementarity problem is denoted by CP(A, K).

Theorem 5.1. Let K be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C → ℝ be a lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of K into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ K and u ∈ CP(A, K) ≠ ∅. Let ${\left\{S_i\right\}}_{i=1}^{\infty }:K\to K$ be an infinite family of closed uniformly L _i -Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 such that $F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\cap CP\left(A,K\right)$ is a nonempty and bounded subset in K. For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and K₁ = K, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill v_n={\Pi }_K{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right),\hfill \\ \hfill z_n=J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\right),\hfill \\ \hfill y_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{z}_n\right),\hfill \\ \hfill u_n=T_{r_{m,n}}^{Q_m}{T}_{r_{m-1,n}}^{Q_{m-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}{y}_n,\hfill \\ \hfill K_{n+1}=\left\{z\in K_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{K_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(5.3)

where J is the duality mapping on E, θ _n = sup_q∈F(k _n - 1)ϕ(q, x _n ), for each i ≥ 0, {α_{n, i}} and {β _n } are sequences in [0, 1], {r_{j, n}} ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$is the 2-uniformly convexity constant of E. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0, lim inf_n→∞(1 - β _n ) > 0 and lim inf_{n → ∞}α_{n, 0}α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Proof. As in the proof of Takahashi in [[20], Lemma 7.11], we get that VI(A, K) = CP(A, K). So, we obtain the result. □

5.2 Application to zero points

Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of E into E*. Assume that A satisfies the conditions:

(C1) A is α-inverse-strongly monotone,

(C2) A^-10 = {u ∈ E : Au = 0} ≠ ∅.

Theorem 5.2. Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to R which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C → ℝ be a lower semicontinuous and convex function. Let A be an operator of E into E* satisfying (C1) and (C2). Let${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$be an infinite family of closed uniformly L _i - Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 such that

$$F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\cap A^{-1}0$$

is a nonempty and bounded subset in C: For an initial point x₀ ∈ E with$x_1={\Pi }_{C_1}{x}_0$and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill v_n={\Pi }_C{J}^{-1}\left(J{x}_n-{\lambda }_{n}A{x}_n\right),\hfill \\ \hfill z_n=J^{-1}\left({\alpha }_{n,0}J{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\right),\hfill \\ \hfill y_n=J^{-1}\left({\beta }_{n}J{x}_n+\left(1-{\beta }_n\right)J{z}_n\right),\hfill \\ \hfill u_n=T_{r_{m,n}}^{Q_m}{T}_{r_{m-1,n}}^{Q_{m-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}{y}_n,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\varphi \left(z,u_n\right)\le \varphi \left(z,x_n\right)+{\theta }_n\right\},\hfill \\ \hfill x_{n+1}={\Pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(5.4)

where J is the duality mapping on E, θ _n = sup_q∈F(kn - 1)ϕ(q, x _n ), for each i ≥ 0, {α _{n, i} } and {β _n } are sequences in [0, 1], {r_{j, n}] ⊂ [d, ∞) for some d > 0 and {λ _n } ⊂ [a, b] for some a, b with 0 < a < b < c²α/ 2, where$\frac{1}{c}$is the 2-uniformly convexity constant of E. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0, lim inf_n→∞(1 - β _n ) > 0 and lim inf_{n →∞}α_{n, 0}α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Proof. Setting C = E in Corollary 3.4, we also get Π _E = I. We also have VI(A, C) = VI(A, E) {x ∈ E : Ax = 0} ≠ ∅ and then the condition ||Ay|| ≤ ||Ay - Au|| holds for all y ∈ E and u ∈ A^{- 1}0. So, we obtain the result.   □

5.3 Application to Hilbert spaces

If E = H, a Hilbert space, then E is 2-uniformly convex (we can choose c = 1) and uniformly smooth real Banach space and closed relatively quasi-nonexpansive map reduces to closed quasi-nonexpansive map. Moreover, J = I, identity operator on H and Π _C = PC, projection mapping from H into C: Thus, the following corollaries hold.

Theorem 5.3. Let C be a nonempty closed and convex subset of a Hilbert space H. For each j = 1, 2, ..., m let f _j be a bifunction from C × C to ℝ which satisfies conditions (A1)-(A4), B _j : C → E* be a continuous and monotone mapping and φ _j : C → ℝ be a lower semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C into H satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let${\left\{S_i\right\}}_{i=1}^{\infty }:C\to C$be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k _n } ⊂ [1, ∞), k _n → 1 and uniformly L _i -Lipschitz continuous such that $F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathsf{\text{GMEP}}\left(f_j,B_j,{\phi }_j\right)\right)\cap \mathsf{\text{VI}}\left(A,C\right)$ is a nonempty and bounded subset in C. For an initial point x₀ ∈ H with $x_1=P_{C_1}{x}_0$ and C₁ = C, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}\hfill z_n=P_C\left(x_n-{\lambda }_{n}A{x}_n\right),\hfill \\ \hfill y_n={\alpha }_{n,0}{x}_n+{\sum }_{i=1}^{\infty }{\alpha }_{n,i}{S}_i^n{z}_n,\hfill \\ \hfill u_n=T_{r_{m,n}}^{Q_m}{T}_{r_{m-1,n}}^{Q_{m-1}}\cdots T_{r_{2,n}}^{Q_2}{T}_{r_{1,n}}^{Q_1}{y}_n,\hfill \\ \hfill C_{n+1}=\left\{z\in C_n:\parallel z-u_n\parallel \le \parallel z-x_n\parallel +{\theta }_n\right\},\hfill \\ \hfill x_{n+1}=P_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(5.5)

where θ _n = sup_q∈F( k _n - 1)||| q - x _n ||, { α _{n, i} } is sequence in [0, 1], { r _{j, n} } ⊂ [a, ∞) for some a > 0 and {λ _n } ⊂ [a, b] for some a; b with 0 < a < b < α/ 2. If${\sum }_{i=0}^{\infty }{\alpha }_{n,i}=1$for all n ≥ 0 and lim inf_{n →∞}α_n,0α _{n, i} > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = Π _F x₀.

Remark 5.4. Theorem 5.3 improves and extends the Corollary 3.7 in Zegeye [41] in the aspect for the mappings, and we extend the mappings from a finite family of closed relatively quasi-nonexpansive mappings to a more general infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings.