The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems

Abstract

In this paper, we construct the hybrid 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 solutions of the variational inequality for an α-inverse-strongly monotone operator, and the set of solutions of a system of equilibrium problems. Moreover, we obtain a strong convergence theorem for the sequence generated by this process in the framework Banach spaces. The results presented in this paper improve and generalize some well-known results in the literature.

Introduction

In the theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. Equilibrium theory represents an important area of mathematical sciences such as optimization, operations research, game theory, financial mathematics and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, fixed point problems, and complementarity problems as special cases.

Let C be a nonempty closed convex subset of a real Banach space E with ∥·∥ and E^∗ the dual space of E and A : C→E^∗ be an operator. The classical variational inequality problem for an operator A is to find x^∗ ∈ C such that

$$〈A{x}^{\ast },y-x^{\ast }〉\ge 0,\forall y\in C.$$

(1.1)

The set of solution of (1.1) is denoted by V I(A, C). Recall that let A : C → E^∗ be a mapping. Then A is called

• (i) monotone if

  $$〈\mathit{\text{Ax}}-\mathit{\text{Ay}},x-y〉\ge 0,\forall x,y\in C,$$

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

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

Such a problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point x^∗ ∈ E satisfying A x^∗ = 0.

Let {f _i }_i∈Γ : $C\times C\to \mathbb{R}$ be a bifunction, {φ _i }_i∈Γ : $C\to \mathbb{R}$ be a real-valued function, where Γ is an arbitrary index set. The system of equilibrium problems, is to find x ∈ C such that

$$f_i(x,y)\ge 0,i\in \mathrm{\Gamma },\forall y\in C.$$

(1.2)

The set of solution of (1.2) is denoted by SEP. If Γ is a singleton, then problem (1.2) reduces to the equilibrium problem, is to find x ∈ C such that

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

(1.3)

The set of solution of (1.3) is denoted by E P(f). The above formulation (1.3) was shown in (Blum and Oettli 1994) to cover monotone inclusion problems, saddle point problems, minimization problems, optimization problems, variational inequality 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 E P(f). In other words, the E P(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 E P(f); see, for example (Blum and Oettli 1994; Combettes and Hirstoaga 2005) and references therein. Some solution methods have been proposed to solve the E P(f); see, for example, (Blum and Oettli 1994; Combettes and Hirstoaga 2005; Jaiboon and Kumam 2010; Katchang and Kumam 2010; Kumam 2009; Moudafi 2003; Qin et al. 2009a, 2009b, 2009c; Saewan and Kumam 2010b, 2011a, 2011b, 2011c, 2011d, 2011e, 2011f, 2011g, 2012b; Zegeye et al. 2010) and references therein.

For each p > 1, the generalized duality mapping J _p : $E\to 2^{E^{\ast }}$ is defined by

$$J_p\left(x\right)=\{x^{\ast }\in E^{\ast }:〈x,x^{\ast }〉=\parallel x{\parallel }^p,\parallel x^{\ast }\parallel =\parallel x{\parallel }^{p-1}\}$$

for all x ∈ E. In particular, J = J₂ is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identity mapping. Consider the functional defined by

$$\varphi (x,y)=\parallel x{\parallel }^2-2〈x,\mathit{\text{Jy}}〉+\parallel y{\parallel }^2,\forall x,y\in E.$$

(1.4)

As well know 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

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

(1.5)

If E is a Hilbert space, then ϕ(x, y) = ∥x - y∥², for all x, y ∈ E. On the author hand, the generalized projection (Alber 1996) 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

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

(1.6)

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, Alber 1996); Alber and Reich 1994; Cioranescu 1990; Kamimura and Takahashi 2002; Takahashi 2000).

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 (1.4), we have ∥x∥ = ∥y∥. This implies that 〈x, J y〉 = ∥x∥² = ∥J y∥². From the definition of J, one has J x = J y. Therefore, we have x = y; see (Cioranescu 1990; Takahashi 2000) 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 ∥T x - T y ∥ ≤ L∥x - y∥, ∀x, y ∈ C and a mapping T is said to be nonexpansive if ∥T x - T y∥ ≤ ∥x - y∥,∀x, y ∈ C. A point x ∈ C is a fixed point of T provided T x = x. Denote by F(T) the set of fixed points of T; that is, F(T) = {x ∈ C : T x = x}. Recall that a point p in C is said to be an asymptotic fixed point of T(Reich 1996) if C contains a sequence {x _n } which converges weakly to p such that limn→∞∥x _n  - T x _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 (Nilsrakoo and Saejung 2008; Su et al. 2008; Zegeye and Shahzad 2009) if $\widetilde{F\left(T\right)}=F\left(T\right)$ and ϕ(p, T x) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in (Butnariu et al. 2001, 2003; Censor and Reich 1996). T is said to be ϕ-nonexpansive, if ϕ(T x, T y) ≤ ϕ(x, y) for x, y ∈ C. T is said to be relatively quasi-nonexpansive if F(T) ≠ ∅ and ϕ(p, T x) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T). 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).

We note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings (Butnariu et al. 2001, 2003; Censor and Reich 1996; Matsushita and Takahashi 2005; Saewan et al. 2010) 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 T x _n  → y, then T x = y. It is easy to know that each relatively nonexpansive mapping is closed.

Definition 1.2

(Chang et al. 2010 (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 ${\cap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$, and there exists a sequence {k _n } ⊂[1, ∞) with k _n  → 1 such that for each i ≥ 1

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

(1.7)

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

$$\begin{array}{c}\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y\parallel ,\forall x,y\in C.\hfill \end{array}$$

(1.8)

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 (2004) introduced the following iteration: a sequence {x _n } defined by

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

(1.9)

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 (2005) 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:

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

(1.10)

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 (2008) 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

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

(1.11)

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 (1.11) converges weakly to some element of V I(A, C). Takahashi and Zembayashi (2008, 2009), 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 (2009) using the idea of Takahashi and Zembayashi (2009) extend 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 problen in the framework of Banach spaces. Cholamjiak (2009), proved the following iteration:

$$\left\{\begin{array}{c}{z}_n={\pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_n\mathit{\text{JT}}{x}_n+{\gamma }_n\mathit{\text{JS}}{z}_n),\hfill \\ u_n\in C\mathit{\text{such}}\mathit{\text{that}}f(u_n,y)+\frac{1}{r_n}〈y-u_n,J{u}_n-J{y}_n〉\ge 0,\forall y\in C,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,x_n),\hfill \\ x_{n+1}={\pi }_{C_{n+1}}{x}_0,\hfill \end{array}\right.$$

(1.12)

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

In 2010, Saewan et al. (2010) introduced a new hybrid projection iterative scheme which is difference from the algorithm (1.12) of Cholamjiak in (2009, Theorem 3.1) for two relatively quasi-nonexpansive mappings in a Banach space. Motivated by the results of Takahashi and Zembayashi (2008); Cholamjiak and Suantai (2010) proved the strong convergence theorem by the hybrid iterative scheme for approximation of a 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

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

(1.13)

Then, they proved that under certain appropriate conditions imposed on {α _n }, and {r_n,i}, the sequence {x _n } converges strongly to ${\pi }_{C_{n+1}}{x}_0$.

We note that the block iterative method is a method which often used by many authors to solve the convex feasibility problem (see, Kohsaka and Takahashi 2007; Kikkawa and Takahashi 2004, etc.). In 2008, Plubtieng and Ungchittrakool (2008) 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(2010) proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi- ϕ-asymptotically nonexpansive mappings, they obtained the strong convergence theorems in a Banach space. In 2010, Saewan and Kumam (2010a) obtained the 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.

Very recently, Qin, Cho and Kang (2009a) 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. (2010) 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 the 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. (2010); Qin et al. (2009c); Takahashi and Zembayashi (2009); Wattanawitoon and Kumam (2009); Zegeye (2010); Saewan and Kumam (2010a, 2012a), we introduce a modified hybrid block projection algorithm for finding a common element of the set of the solution of the variational inequality for an α-inverse-strongly monotone operator, and the set of solutions of the system of equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi- ϕ-asymptotically nonexpansive mappings in a 2-uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and generalize some well-known results in the literature.

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

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

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

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

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) ≥ c ϵ^p for all ϵ ∈ [0, 2]; see (Ball et al.1994; Takahashi et al. 2002) for more details. Observe that every p-uniformly convex is uniformly convex. 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 (1990).

• (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 reflexive and strictly convex Banach space, then J ^-1 is norm-weak ^∗-continuous.

• (iii) If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping $J:E\to 2^{E^{\ast }}$ is single-valued, one-to-one, and onto.

• (iv) A Banach space E is uniformly smooth if and only if E ^∗ is uniformly convex.

• (v) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence {x _n } ⊂ E, if $x_n\rightharpoonup x\in E$ 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 (1985); Xu (1991)). If E be a 2-uniformly convex Banach space. Then for all x, y ∈ E, we have

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

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 (1985); Zalinescu (1983)). 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 (2002)). 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 (1996)). Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E. Then x₀ = π _C x if and only if

$$〈x_0-y,\mathit{\text{Jx}}-J{x}_0〉\ge 0,\forall y\in C.$$

Lemma 2.6

(Alber (1996, Lemma 2.4)). Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and let x ∈ E. Then

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

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 (1996)

$$V(x,x^{\ast })=\parallel x{\parallel }^2-2〈x,x^{\ast }〉+\parallel x^{\ast }{\parallel }^2,$$

(2.1)

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

Lemma 2.7

(Alber (1996)). Let E be a reflexive, strictly convex smooth Banach space and let V be as in ( 2.1). Then

$$V(x,x^{\ast })+2〈J^{-1}\left(x^{\ast }\right)-x,y^{\ast }〉\le V(x,x^{\ast }+y^{\ast }),$$

for all x ∈ E and x^∗, y^∗ ∈ 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, the mapping F of [0,1] into E^∗, defined by F(t) = A(t x+(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,

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

(2.2)

Lemma 2.8

(Rockafellar (1970)). 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 B ⊂ E × E^∗be an operator defined as follows:

$$\mathit{\text{Bv}}=\left\{\begin{array}{c}\mathit{\text{Av}}+N_C\left(v\right),v\in C;\hfill \\ \varnothing ,\text{otherwise.}\hfill \end{array}\right.$$

(2.3)

Then B is maximal monotone and B^-10 = V I(A, C).

Lemma 2.9

(Chang et al. (2010)). 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,

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

(2.4)

Lemma 2.10

(Chang et al. (2010)). 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 $f:C\times C\to \mathbb{R}$, 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,

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

• (A4) for each x ∈ C, y ↦ f(x, y) is convex and lower semi-continuous.

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

$$f(x,y)=〈\mathit{\text{Ax}},y-x〉,\forall x,y\in C.$$

Then, f satisfies (A1)-A4). The following result is in Blum and Oettli (1994.

Lemma 2.11

(Blum and Oettli (1994)). Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from C × C to$\mathbb{R}$satisfying (A 1)- (A 4), and let r > 0 and x ∈ E. Then, there exists z ∈ C such that

$$f(z,y)+\frac{1}{r}〈y-z,\mathit{\text{Jz}}-\mathit{\text{Jx}}〉\ge 0,\forall y\in C.$$

Lemma 2.12

(Takahashi and Zembayashi (2009)). Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let f be a bifunction from C × C to$\mathbb{R}$satisfying conditions (A 1)- (A 4). For all r > 0 and x ∈ E, define a mapping$T_r^f:E\to C$as follows:

$$T_r^{f}x=\{z\in C:f(z,y)+\frac{1}{r}〈y-z,\mathit{\text{Jz}}-\mathit{\text{Jx}}〉\ge 0,\forall y\in C\}.$$

Then the following hold:

• (1) $T_r^f$ is single-valued;

• (2) $T_r^f$ is a firmly nonexpansive-type mapping (Kohsaka and Takahashi 2008), that is, for all x, y ∈ E,

  $$〈T_r^{f}x-T_r^{f}y,J{T}_r^{f}x-J{T}_r^{f}y〉\le 〈T_r^{f}x-T_r^{f}y,\mathit{\text{Jx}}-\mathit{\text{Jy}}〉;$$

• (3) $F\left(T_r^f\right)=\mathit{\text{EP}}\left(f\right);$

• (4) E P(f) is closed and convex.

Lemma 2.13

(Takahashi and Zembayashi (2009)). Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C × C to$\mathbb{R}$satisfying (A 1)- (A 4) and let r > 0. Then, for x ∈ E and$q\in F\left(T_r^f\right)$,

$$\varphi (q,T_r^{f}x)+\varphi (T_r^{f}x,x)\le \varphi (q,x).$$

Strong convergence theorems

In this section, we prove the new convergence theorems for finding the set of solutions of system of 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$\mathbb{R}$which satisfies conditions (A1)-(A4). Let A be an α -inverse-strongly monotone mapping of C into E^∗satisfying ∥A y ∥ ≤∥A y - A u∥, ∀y ∈ C and u ∈ V I(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$F:=\left({\cap }_{i=1}^{\infty }F\right(S_i\left)\right)\cap \left({\cap }_{j=1}^m\mathit{\text{EP}}\right(f_j\left)\right)(\cap \mathit{\text{VI}}(A,C\left)\right)$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, define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}{v}_n={\pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n\left)J{z}_n\right),\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,z_n)\le \varphi (z,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\pi }_{C_{n+1}}{x}_0,\forall n\ge 1,\hfill \end{array}\right.$$

(3.1)

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→∞β _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+1 is closed and convex for each n ≥ 0. Clearly C₁ = C is closed and convex. Suppose that C _n is closed and convex for each $n\in \mathbb{N}$. Since for any z ∈ C _n , we known that

$$\varphi (z,u_n)\le \varphi (z,x_n)+{\theta }_n$$

is equivalent to

$$2〈z,J{x}_n-J{u}_n〉\le \parallel x_n{\parallel }^2-\parallel u_n{\parallel }^2+{\theta }_n.$$

Hence, C_n+1 is closed and convex.

Next, we show that F ⊂ C _n for all n ≥ 0. Since 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

$$\begin{array}{ll}\varphi (q,u_n)\hfill & =\varphi (q,T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n)\hfill \\ \le \varphi (q,y_n)\hfill \\ =\varphi (q,J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n)\hfill \\ =\parallel q{\parallel }^2-2〈q,{\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n〉\hfill \\ +\parallel {\beta }_{n}J{x}_n+(1-{\beta }_n)J{z}_n{\parallel }^2\hfill \\ \le \parallel q{\parallel }^2-2{\beta }_n〈q,J{x}_n〉-2(1-{\beta }_n)〈q,J{z}_n〉\hfill \\ +{\beta }_n\parallel x_n{\parallel }^2+(1-{\beta }_n)\parallel z_n{\parallel }^2\hfill \\ ={\beta }_n\varphi (q,x_n)+(1-{\beta }_n)\varphi (q,z_n),\hfill \end{array}$$

(3.2)

and

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

(3.3)

It follows from Lemma 2.7, that

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

(3.4)

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

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

(3.5)

By Lemma 2.2 and ∥A x _n ∥ ≤ ∥A x _n  - A q∥, ∀q ∈ V I(A, C), we also have

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

(3.6)

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

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

(3.7)

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

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

(3.8)

and substituting (3.8) into (3.2), we obtain

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

(3.9)

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

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

(3.10)

Form Lemma 2.6, it follows that

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

(3.11)

By (3.10) and (3.11), then {ϕ(x _n , x₀)} are nondecreasing and bounded. So, we obtain that $\underset{n\to \infty }{\text{lim}}\varphi (x_n,x_0)$ exists. In particular, by (1.5), the sequence {(∥x _n ∥ - ∥x₀∥)²} is bounded. This implies {x _n } is also bounded. We denote

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

(3.12)

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

$$\begin{array}{c}{\theta }_n\to 0\text{as}n\to \mathrm{\infty .}\hfill \end{array}$$

(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

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

Since lim_n→∞ϕ(x _n , x₀) exists and we taking m, n → ∞ then, 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 and there exist a point p ∈ C such that

$$x_n\to p\text{as}n\to \mathrm{\infty .}$$

(3.14)

Now, we claim that ∥J u _n  - J x _n ∥ → 0, as n → ∞. By definition of $x_n={\pi }_{C_n}{x}_0$, we have

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

Since $\underset{n\to \infty }{\text{lim}}\varphi (x_n,x_0)$ exists, we also have

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

(3.15)

Again form Lemma 2.4, that

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel x_{n+1}-x_n\parallel =0.\hfill \end{array}$$

(3.16)

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

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel J{x}_{n+1}-J{x}_n\parallel =0.\hfill \end{array}$$

(3.17)

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

$$\begin{array}{c}\varphi (x_{n+1},u_n)\le \varphi (x_{n+1},x_n)+{\theta }_n.\hfill \end{array}$$

By (3.13) and (3.15), that

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

(3.18)

Applying Lemma 2.4, we have

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel x_{n+1}-u_n\parallel =0.\hfill \end{array}$$

(3.19)

Since

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

It follows from (3.23) and (3.19), that

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

(3.20)

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

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel J{u}_n-J{x}_n\parallel =0.\hfill \end{array}$$

(3.21)

Next, we will show that$x_n\to p\in F:={\cap }_{j=1}^m\mathit{\text{EP}}\left(f_j\right)\cap \left({\cap }_{i=1}^{\infty }F\right(S_i\left)\right)\cap \mathit{\text{VI}}(A,C).$

(i) We show that $x_n\to p\in {\cap }_{i=1}^{\infty }F\left(S_i\right)$. It follows from definition of $x_{n+1}={\pi }_{C_{n+1}}{x}_0\in C_{n+1}\subset C_n$, we have

$$\varphi (x_{n+1},z_n)\le \varphi (x_{n+1},x_n)+{\theta }_n,\forall n\ge 1.$$

By (3.13) and (3.15), that

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

(3.22)

Form Lemma 2.4, that

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel x_{n+1}-z_n\parallel =0.\hfill \end{array}$$

(3.23)

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

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

(3.24)

From (3.45), we note that

$$\begin{array}{ccc}\parallel J{x}_{n+1}-J{z}_n\parallel \hfill & =\hfill & \parallel J{x}_{n+1}-({\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n)\parallel \hfill \\ =\hfill & \parallel {\alpha }_{n,0}J{x}_{n+1}-{\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}J{x}_{n+1}\hfill \\ -\sum _{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n\parallel \hfill \\ =\hfill & \parallel {\alpha }_{n,0}(J{x}_{n+1}-J{x}_n)\hfill \\ +\sum _{i=1}^{\infty }{\alpha }_{n,i}(J{x}_{n+1}-J{S}_i^n{v}_n)\parallel \hfill \\ =\hfill & \parallel \sum _{i=1}^{\infty }{\alpha }_{n,i}(J{x}_{n+1}-J{S}_i^n{v}_n)\hfill \\ -{\alpha }_{n,0}(J{x}_n-J{x}_{n+1})\parallel \hfill \\ \ge \hfill & \sum _{i=1}^{\infty }{\alpha }_{n,i}\parallel J{x}_{n+1}-J{S}_i^n{v}_n\parallel \hfill \\ -{\alpha }_{n,0}\parallel J{x}_n-J{x}_{n+1}\parallel ,\hfill \end{array}$$

and hence

$$\begin{array}{c}\parallel J{x}_{n+1}-J{S}_i^n{v}_n\parallel \le \frac{1}{\sum _{i=1}^{\infty }{\alpha }_{n,i}}\hfill \\ \times (\parallel J{x}_{n+1}-J{z}_n\parallel +{\alpha }_{n,0}\parallel J{x}_n-J{x}_{n+1}\parallel ).\hfill \end{array}$$

(3.25)

From (3.17), (3.24) and $\underset{n\to \infty }{liminf}\sum _{i=1}^{\infty }{\alpha }_{n,i}>0,$ we get

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel J{x}_{n+1}-J{S}_i^n{v}_n\parallel =0.\hfill \end{array}$$

(3.26)

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

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

(3.27)

Using the triangle inequality, that

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

From (3.23) and (3.27), we have

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

(3.28)

On the other hand, we observe that

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

It follows from θ _n  → 0, ∥x _n  - u _n ∥ → 0 and ∥J x _n  - J u _n ∥ → 0, that

$$\begin{array}{c}\varphi (q,x_n)-\varphi (q,u_n)+{\theta }_n\to 0\text{as}n\to \mathrm{\infty .}\hfill \end{array}$$

(3.29)

From (3.2), (3.3) and (3.7), we compute

$$\begin{array}{ccc}\varphi (q,u_n)\hfill & \le \hfill & \varphi (q,y_n)\hfill \\ \le \hfill & {\beta }_n\varphi (q,x_n)+(1-{\beta }_n)\varphi (q,z_n)\hfill \\ \le \hfill & {\beta }_n\varphi (q,x_n)+(1-{\beta }_n)\left[{\alpha }_{n,0}\varphi \right(q,x_n)\hfill \\ +\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi (q,v_n)\hfill \\ -{\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel ]\hfill \\ =\hfill & {\beta }_n\varphi (q,x_n)+(1-{\beta }_n){\alpha }_{n,0}\varphi (q,x_n)\hfill \\ +(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi (q,v_n)\hfill \\ -(1-{\beta }_n){\alpha }_{n,0}{\alpha }_{n,j}g\parallel J{v}_n-J{S}_j^n{v}_n\parallel \hfill \\ \le \hfill & {\beta }_n\varphi (q,x_n)+(1-{\beta }_n){\alpha }_{n,0}\varphi (q,x_n)\hfill \\ +(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi (q,v_n)\hfill \\ \le \hfill & {\beta }_n\varphi (q,x_n)+(1-{\beta }_n){\alpha }_{n,0}\varphi (q,x_n)\hfill \\ +(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_n\left[\varphi \right(q,x_n)\hfill \\ -2{\lambda }_n(\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-\mathit{\text{Aq}}{\parallel }^2]\hfill \\ \le \hfill & {\beta }_n\varphi (q,x_n)+(1-{\beta }_n){\alpha }_{n,0}{k}_n\varphi (q,x_n)\hfill \\ +(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_n\varphi (q,x_n)\hfill \\ -(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_{n}2{\lambda }_n\hfill \\ \times (\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-\mathit{\text{Aq}}{\parallel }^2\hfill \\ =\hfill & {\beta }_n{k}_n\varphi (q,x_n)+(1-{\beta }_n)k_n\varphi (q,x_n)\hfill \\ -(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_{n}2{\lambda }_n\hfill \\ \times (\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-\mathit{\text{Aq}}{\parallel }^2\hfill \\ \le \hfill & k_n\varphi (q,x_n)-(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_{n}2{\lambda }_n\hfill \\ \times (\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-\mathit{\text{Aq}}{\parallel }^2]\hfill \\ \le \hfill & \varphi (q,x_n)+\underset{q\in F}{sup}(k_n-1)\varphi (q,x_n)\hfill \\ -(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_{n}2{\lambda }_n\hfill \\ \times (\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-\mathit{\text{Aq}}{\parallel }^2\hfill \\ \le \hfill & \varphi (q,x_n)+{\theta }_n-(1-{\beta }_n)\sum _{i=1}^{\infty }{\alpha }_{n,i}{k}_{n}2{\lambda }_n\hfill \\ \times (\alpha -\frac{2}{c^2}{\lambda }_n)\parallel A{x}_n-\mathit{\text{Aq}}{\parallel }^2\hfill \end{array}$$

and hence

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

(3.30)

From (3.29), {λ _n } ⊂[a, b] for some a, b with 0 < a < b < c²α / 2, liminf_n→∞(1 - β _n ) > 0 and liminf_n→∞α_n,0α_n,i > 0, for i ≥ 0 and k _n  → 1 as n → ∞, we obtain that

$$\begin{array}{c}\underset{n\to \infty }{\text{lim}}\parallel A{x}_n-\mathit{\text{Aq}}\parallel =0.\hfill \end{array}$$

(3.31)

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

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

Applying Lemma 2.4 and (3.31) that

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

(3.32)

and we also obtain

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

(3.33)

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

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

(3.34)

Again by the triangle inequality, we get

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

From (3.28) and (3.34), we have

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

(3.35)

By using the triangle inequality, we have

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

That is

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

(3.36)

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

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

(3.37)

By (3.23) and (3.36), it follows 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 imply that$p\in {\cap }_{i=1}^{\infty }F\left(S_i\right).$

(i i) We show that $x_n\to p\in {\cap }_{j=1}^m\mathrm{EP}\left(f_j\right).$ From Lemma 2.13 and $u_n={\Omega }_n^m{y}_n$, when ${\Omega }_n^j=T_{r_{j,n}}^{f_j}{T}_{r_{j-1,n}}^{f_{j-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}$, j = 1, 2, 3, ..., m, Ω n 0 = I, for q ∈ F, we observe that

$$\begin{array}{ccc}\varphi (u_n,{\Omega }_n^j{y}_n)\hfill & \le \hfill & \varphi (q,{\Omega }_n^j{y}_n)-\varphi (q,u_n)\hfill \\ \le \hfill & \varphi (q,x_n)-\varphi (q,u_n)+{\theta }_n\hfill \\ =\hfill & \parallel q{\parallel }^2-2〈q,J{x}_n〉+\parallel x_n{\parallel }^2\hfill \\ -(\parallel q{\parallel }^2-2〈q,J{u}_n〉+\parallel u_n{\parallel }^2)+{\theta }_n\hfill \\ =\hfill & \parallel x_n{\parallel }^2-\parallel u_n{\parallel }^2-2〈q,J{x}_n-J{u}_n〉+{\theta }_n\hfill \\ \le \hfill & \parallel x_n-u_n\parallel (\parallel x_n+u_n\parallel )\hfill \\ +2\parallel q\parallel \parallel J{x}_n-J{u}_n\parallel +{\theta }_n.\hfill \end{array}$$

(3.38)

From (3.20), (3.21), θ _n  → 0 as n → ∞ and Lemma 2.4, we get

$$\underset{n\to \infty }{\text{lim}}\parallel u_n-{\Omega }_n^j{y}_n\parallel =0,\forall j=1,2,3,\mathrm{...},\mathrm{m.}$$

(3.39)

By using triangle inequality, we have

$$\parallel x_n-{\Omega }_n^j{y}_n\parallel \le \parallel x_n-u_n\parallel +\parallel u_n-{\Omega }_n^j{y}_n\parallel .$$

From (3.20) and (3.39), we have

$$\underset{n\to \infty }{\text{lim}}\parallel x_n-{\Omega }_n^j{y}_n\parallel =0,\forall j=1,2,3,\mathrm{...},\mathrm{m.}$$

(3.40)

Again by using triangle inequality, we have

$$\parallel {\Omega }_n^j{y}_n-{\Omega }_n^{j-1}{y}_n\parallel \le \parallel {\Omega }_n^j{y}_n-x_n\parallel +\parallel x_n-{\Omega }_n^{j-1}{y}_n\parallel .$$

From (3.40),we also have

$$\underset{n\to \infty }{\text{lim}}\parallel {\Omega }_n^j{y}_n-{\Omega }_n^{j-1}{y}_n\parallel =0,\forall j=1,2,3,\mathrm{...},\mathrm{m.}$$

(3.41)

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

$$\underset{n\to \infty }{\text{lim}}\parallel J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n\parallel =0,\forall j=1,2,3,\mathrm{...},\mathrm{m.}$$

From r_j,n > 0 we have $\frac{\parallel J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n\parallel }{r_{j,n}}\to 0$ as n → ∞, ∀j = 1, 2, 3, ..., m, and

$$\begin{array}{c}{f}_j({\Omega }_n^j{y}_n,y)+\frac{1}{r_{j,n}}〈y-{\Omega }_n^j{y}_n,J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n〉\ge 0,\forall y\in C.\hfill \end{array}$$

By (A2), that

$$\begin{array}{c}\parallel y-{\Omega }_n^j{y}_n\parallel \frac{\parallel J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n\parallel }{r_n}\hfill \\ \ge \frac{1}{r_{j,n}}〈y-{\Omega }_n^j{y}_n,J{\Omega }_n^j{y}_n-J{\Omega }_n^{j-1}{y}_n〉\hfill \\ \ge -f_j({\Omega }_n^j{y}_n,y)\hfill \\ \ge f_j(y,{\Omega }_n^j{y}_n),\forall y\in C,\hfill \end{array}$$

and ${\Omega }_n^j{y}_n\to p$ we get f(y, p) ≤ 0 for all y ∈ C. For 0 < t < 1, define y _t  = t y + (1 - t)p. Then y _t  ∈ C which imply that f _j (y _t , p) ≤ 0. From (A1), we obtain that

$$0=f_j(y_t,y_t)\le t{f}_j(y_t,y)+(1-t)f_j(y_t,p)\le t{f}_j(y_t,y).$$

Thus f _j (y _t , y) ≥ 0. From (A3), we have f _j (p, y) ≥ 0 for all y ∈ C and j = 1, 2, 3, ..., m. Hence p ∈ E P(f _j ), ∀j = 1, 2, 3, ..., m. This imply that $p\in {\cap }_{j=1}^m\mathrm{EP}\left(f_j\right)$.

(i i i) We show that x _n  → p ∈ V I(A, C). Indeed, define B ⊂ E × E^∗ by

$$\mathit{\text{Bv}}=\left\{\begin{array}{c}\mathit{\text{Av}}+N_C\left(v\right),v\in C;\hfill \\ \varnothing ,v\notin C.\hfill \end{array}\right.$$

(3.42)

By Lemma 2.8, B is maximal monotone and B^-10 = V I(A, C). Let (v, w) ∈ G(B). Since w ∈ B v = A v + N _C (v), we get w - A v∈N _C (v). From v _n  ∈ C, we have

$$\begin{array}{c}〈v-v_n,w-\mathit{\text{Av}}〉\ge 0.\hfill \end{array}$$

(3.43)

On the other hand, since v _n  = π _C J^-1(J x _n  - λ _n A x _n ). Then by Lemma 2.5, we have

$$\begin{array}{c}〈v-v_n,J{v}_n-(J{x}_n-{\lambda }_{n}A{x}_n)〉\ge 0,\hfill \end{array}$$

and thus

$$\begin{array}{c}〈v-v_n,\frac{J{x}_n-J{v}_n}{{\lambda }_n}-A{x}_n〉\le 0.\hfill \end{array}$$

(3.44)

It follows from (3.43), (3.44) and A is monotone and $\frac{1}{\alpha }$-Lipschitz continuous, that

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

where H = sup_n≥1∥v - v _n ∥. Take the limit as n → ∞, (3.32) and (3.33), we obtain 〈v - p, w〉 ≥ 0. By the maximality of B we have p ∈ B^-10, that is p ∈ V I(A, C).

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

$$〈J{x}_0-J{x}_n,x_n-y〉\ge 0,\forall y\in \mathrm{F.}$$

Taking limit n → ∞, we obtain

$$〈J{x}_0-\mathit{\text{Jp}},p-y〉\ge 0,\forall y\in \mathrm{F.}$$

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 $i\in \mathbb{N}$, 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$\mathbb{R}$which satisfies conditions (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C into E^∗satisfying ∥A y∥ ≤ ∥A y - A u∥, ∀y∈C and u ∈ V I(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$F:=\left(F\right(S\left)\right)\cap \left({\cap }_{j=1}^m\mathrm{EP}\right(f_j\left)\right)\cap \left(\mathrm{VI}\right(A,C\left)\right)$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:

$$\left\{\begin{array}{c}{v}_n={\pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n}J{x}_n+(1-{\alpha }_n\left)J{S}^n{v}_n\right),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n\left)J{z}_n\right),\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,z_n)\le \varphi (z,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\pi }_{C_{n+1}}{x}_0,\forall n\ge 1,\hfill \end{array}\right.$$

(3.45)

where J is the duality mapping on E, 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 $\frac{1}{c}$is the 2-uniformly convexity constant of E. If lim inf_n→∞(1 - β _n ) > 0 and lim inf_n→∞α _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- ϕ-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 $\mathbb{R}$ which satisfies conditions (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C into E^∗ satisfying ∥A y∥ ≤ ∥A y - A u∥, ∀y ∈ C and u ∈ V I(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 $F:=F\left(S\right)\cap F\left(T\right)\cap \left({\cap }_{j=1}^m\mathrm{EP}\right(f_j\left)\right)\cap \mathrm{VI}(A,C)$ 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:

$$\left\{\begin{array}{c}{v}_n={\pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{S}^n{v}_n+{\gamma }_{n}J{T}^n{v}_n),\hfill \\ y_n=J^{-1}({\delta }_{n}J{x}_n+(1-{\delta }_n\left)J{z}_n\right),\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,z_n)\le \varphi (z,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.46)

where J is the duality mapping on E, 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→∞δ _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 $\mathbb{R}$ which satisfies conditions (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C into E^∗ satisfying ∥A y∥ ≤ ∥A y - A u∥, ∀y ∈ C and u ∈ V I(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 $F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathit{\text{EP}}\right(f_j\left)\right)\cap \mathit{\text{VI}}(A,C)\ne \mathrm{\varnothing .}$ 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:

$$\left\{\begin{array}{c}{v}_n={\pi }_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}J{S}_i{v}_n),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n\left)J{z}_n\right),\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,z_n)\le \varphi (z,x_n),\hfill \\ x_{n+1}={\pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.47)

where J is the duality mapping on E, {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

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 3.1F 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. □

Corollary 3.5

(Zegeye 2010, 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$\mathbb{R}$satisfying (A 1)- (A 4). Let A be an α-inverse-strongly monotone mapping of C into E^∗satisfying ∥A y∥ ≤ ∥A y - A u∥, ∀y ∈ C and u ∈ V I(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$F:={\cap }_{i=1}^{N}F\left(S_i\right)\cap \mathit{\text{EP}}\left(f\right)\cap \mathit{\text{VI}}(A,C)\ne \mathrm{\varnothing .}$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:

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

(3.48)

where J is the duality mapping on E, {α_n,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$\frac{1}{c}$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.

Corollary 3.6

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$\mathbb{R}$satisfying (A 1)- (A 4). 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$F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \mathit{\text{EP}}\left(f\right)$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:

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

(3.49)

where J is the duality mapping on E, 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,0α_n,i > 0 for all i ≥ 1, then {x _n } converges strongly to p ∈ F, where p = π _F x₀.

Deduced 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  = P _C , projection mapping from H into C. Thus, the following corollaries hold.

Theorem 4.1

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$\mathbb{R}$which satisfies conditions (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C into H satisfying ∥A y∥ ≤ ∥A y - A u ∥ , ∀y ∈ C and u ∈ V I(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$F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathit{\text{EP}}\right(f_j\left)\right)\cap \mathit{\text{VI}}(A,C)$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, define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}{z}_n=P_C(x_n-{\lambda }_{n}A{x}_n),\hfill \\ y_n={\alpha }_{n,0}{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}{S}_i^n{z}_n,\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ C_{n+1}=\{z\in C_n:\parallel z-u_n\parallel \le \parallel z-y_n\parallel \le \parallel z-x_n\parallel +{\theta }_n\},\hfill \\ x_{n+1}=P_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(4.1)

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 4.2

Theorem 4.1 improve and extend the Corollary 3.7 in Zegeye (2010) in the aspect for the mappings, we extend the mappings from a finite family of closed relatively quasi-nonexpansive mappings to more general an infinite family of closed and uniformly quasi- ϕ-asymptotically nonexpansive mappings.

Zero points of an inverse-strongly monotone operator

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 : A u = 0} ≠ ∅.

Theorem 5.1

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$\mathbb{R}$which satisfies conditions (A1)-(A4). 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

$$F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathit{\text{EP}}\right(f_j\left)\right)\cap A^{-1}0$$

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, define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}{z}_n=J^{-1}({\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n\left)J{z}_n\right),\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ C_{n+1}=\{z\in C_n:\varphi (z,u_n)\le \varphi (z,z_n)\le \varphi (z,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\pi }_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(5.1)

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,0α_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 V I(A, C) = V I(A, E) = {x ∈ E : A x = 0} ≠ ∅ and then the condition ∥A y∥ ≤ ∥A y - A u∥ holds for all y ∈ E and u ∈ A^-10. So, we obtain the result. □

Complementarity problems

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

$$\begin{array}{c}{K}^{\ast }=\{y^{\ast }\in E^{\ast }:〈x,y^{\ast }〉\ge 0,\forall x\in K\}.\hfill \end{array}$$

(6.1)

If A : K → E^∗ is an operator, then an element u ∈ K is called a solution of the complementarity problem (Takahashi 2000) if

$$\begin{array}{c}\mathit{\text{Au}}\in K^{\ast },\mathit{\text{and}}〈u,\mathit{\text{Au}}〉=0.\hfill \end{array}$$

(6.2)

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

Theorem 6.1

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$\mathbb{R}$which satisfies conditions (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of K into E^∗satisfying ∥A y∥ ≤ ∥A y - A u∥, ∀y ∈ K and u ∈ C P(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$F:={\cap }_{i=1}^{\infty }F\left(S_i\right)\cap \left({\cap }_{j=1}^m\mathit{\text{EP}}\right(f_j\left)\right)\cap \mathit{\text{CP}}(A,K)$is a nonempty and bounded subset in K. For an initial point x₀ ∈ E with$x_1={\pi }_{K_1}{x}_0$and K₁ = K, we define the sequence {x _n } as follows:

$$\left\{\begin{array}{c}{v}_n={\pi }_K{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ z_n=J^{-1}({\alpha }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\alpha }_{n,i}J{S}_i^n{v}_n),\hfill \\ y_n=J^{-1}({\beta }_{n}J{x}_n+(1-{\beta }_n\left)J{z}_n\right),\hfill \\ u_n=T_{r_{m,n}}^{f_m}{T}_{r_{m-1,n}}^{f_{m-1}}\mathrm{...}{T}_{r_{2,n}}^{f_2}{T}_{r_{1,n}}^{f_1}{y}_n,\hfill \\ K_{n+1}=\{z\in K_n:\varphi (z,u_n)\le \varphi (z,z_n)\le \varphi (z,x_n)+{\theta }_n\},\hfill \\ x_{n+1}={\pi }_{K_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(6.3)

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,0α_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 (Takahashi 2000, Lemma 7.11), we get that V I(A, K) = C P(A, K). So, we obtain the result. □