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Convergence theorems for uniformly quasiϕasymptotically nonexpansive mappings, generalized equilibrium problems, and variational inequalities
Journal of Inequalities and Applications volume 2011, Article number: 96 (2011)
Abstract
In this article, we introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings, the set of the variational inequality for an αinversestrongly monotone operator, and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a 2uniformly convex and uniformly smooth Banach space. The results presented in this article improve and extend the recent results of Zegeye [Nonlinear Anal. 72, 21362146 (2010)], Wattanawitoon and Kumam [Nonlinear Anal. Hybrid Syst. 3(1), 1120 (2009)] and many others.
2000 MSC: 47H05, 47H09, 47H10.
1 Introduction and preliminaries
Let C be a nonempty closed convex subset of a real Banach space E with  ·  and E* the dual space of E. Recall that a mapping T : C → C is said to be LLipschitz 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}. Let A : C → E* be a mapping. Then, A is called

(i)
monotone if
$$\u3008AxAy,xy\u3009\ge 0,\phantom{\rule{1em}{0ex}}\forall x,y\in C,$$ 
(ii)
αinversestrongly monotone if there exists a constant α > 0 such that
$$\u3008AxAy,xy\u3009\ge \alpha \parallel AxAy{\parallel}^{2},\phantom{\rule{1em}{0ex}}\forall x,y\in C.$$
Remark 1.1. It is easy to see that an αinversestrongly monotone is monotone and $\frac{1}{\alpha}$Lipschitz continuous.
Let f be a bifunction of C × C into ℝ and B : C → E* be a monotone mapping. The generalized equilibrium problem, denoted by GEP, is to find x ∈ C such that
The set of solutions for the problem (1.1) is denoted by GEP(f, B), that is,
If B ≡ 0, the problem (1.1) reduce into the equilibrium problem for f, denoted by EP(f), is to find x ∈ C such that
If f ≡ 0, the problem (1.1) reduce into the classical variational inequality problem, denoted by V I(B, C), is to find x* ∈ C such that
The above formulation (1.1) is more general than equilibrium problem (1.2) and cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, vector equilibrium problems, and 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 GEP(f, B). 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 articles 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 GEP(f, B) and EP(f); see, for example, [1, 3–13] and references therein.
Consider the functional defined by
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
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[14] 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, ${\Pi}_{C}x=\stackrel{\u0304}{x}$, where $\stackrel{\u0304}{x}$ is the solution to the minimization problem
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, [14–18]).
Recall that a point p in C is said to be an asymptotic fixed point of T[19] 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 $\stackrel{\u0303}{F\left(T\right)}$. A mapping T is said to be ϕnonexpansive, if ϕ(Tx, Ty) ≤ ϕ(x, y) for x, y ∈ C.
A mapping T from C into itself is said to be relatively nonexpansive mapping[20–22] if
(R1) F(T) is nonempty;
(R2) ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T);
(R3) $\stackrel{\u0303}{F\left(T\right)}=F\left(T\right)$.
A mapping T is said to be relatively quasinonexpansive (or quasi ϕnonexpansive) if the conditions (R1) and (R2) are satisfied. The asymptotic behavior of a relatively nonexpansive mapping was studied in [23–25].
A mapping T is said to be quasi ϕasymptotically nonexpansive if F(T) ≠ ∅ and there exists a real sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 such that ϕ(p, T^{n}x) ≤ k_{ nϕ }(p, x) for all n ≥ 1 x ∈ C and p ∈ F(T). We note that the class of relatively quasinonexpansive mappings is more general than the class of relatively nonexpansive mappings [23–27] which requires the strong restriction: $F\left(T\right)=\stackrel{\u0303}{F\left(T\right)}$.
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.
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 ${lim}_{t\to 0}\frac{\parallel x+ty\parallel \parallel x\parallel}{t}$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 $\delta \left(\epsilon \right)=inf\left\{1\parallel \frac{x+y}{2}\parallel :x,y\in E,\parallel x\parallel =\parallel y\parallel =1,\parallel xy\parallel \ge \epsilon \right\}$. 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 puniformly convex if there exists a constant c > 0 such that δ(ε) ≥ cε ^{p} for all ε ∈ [0, 2]; see [28, 29] for more details. Observe that every puniform convex is uniformly convex. One should note that no a Banach space is puniform convex for 1 < p < 2. It is well known that a Hilbert space is 2uniformly convex, uniformly smooth. For each p > 1, the generalized duality mapping${J}_{p}:E\to {2}^{{E}^{*}}$ is defined by J_{ p }>(x) = {x* ∈ E* : 〈x, x*〉 = x^{p}, x* = x^{p1}} for all x ∈ E. In particular, J = J_{2} is called the normalized duality mapping. If E is a Hilbert space, then J = I, where I is the identity mapping.
Remark 1.2. 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, Jy〉 = x^{2} = Jy^{2}. From the definition of J, one has Jx = Jy. Therefore, we have x = y; see [16, 18] for more details.
Remark 1.3. The following basic properties can be found in Cioranescu [16].

(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 normweak*continuous.

(iii)
If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping $J:E\to {2}^{{E}^{*}}$ is singlevalued, onetoone, 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 KadecKlee property, that is, for any sequence {x_{ n } } ⊂ E, if x_{ n } ⇀ x ∈ E and x_{ n }  → x, then x_{ n } → x.
In 2004, Matsushita and Takahashi [30] introduced the following iteration: a sequence {x_{ n } } is defined by
where the initial guess element x_{0} ∈ 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 . Later, in year 2005, Matsushita and Takahashi [26] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping T in a Banach space E:
They proved that {x_{ n } } converges strongly to Π_{F(T)}x_{0}, where Π_{F(T)}is the generalized projection from C onto F(T).
In 2008, Iiduka and Takahashi [31] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inversestrongly monotone operator A in a 2uniformly convex and uniformly smooth Banach space E : x_{1} = x ∈ C and
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 (1.9) converges weakly to some element of V I(A, C).
In [32, 33], Takahashi and Zembayashi 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. Wattanawitoon and Kumam [34] using the idea of Takahashi and Zembayashi [32] extend the notion from relatively nonexpansive mappings or ϕnonexpansive mappings to two relatively quasinonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasinonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces.
On the other hand, the block iterative method is a method which often used by many authors to solve the convex feasibility problem (see, [11, 35, 36], etc.). In 2008, Plubtieng and Ungchittrakool [37] 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. In 2010, Chang et al. [38] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mapping, they obtain the strong convergence theorems in a Banach space.
In this article, motivated and inspired by the study of Chang et al. [38], Qin et al. [9], Takahashi and Zembayashi [32], Wattanawitoon and Kumam [34], and Zegeye [39], we introduce a new modified block hybrid projection algorithm for finding a common element of the set of the variational inequality for an αinversestrongly monotone operator, the set of solutions of the generalized equilibrium problems, and the set of common fixed points of an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings which more general than closed quasiϕnonexpansive mappings in the framework Banach spaces. The results presented in this article improve and generalize the main results of Chang et al. [38], Zegeye [39], Wattanawitoon and Kumam [34], and some wellknown results in the literature.
2 Basic results
We also need the following lemmas for the proof of our main results.
Lemma 2.1. (Beauzamy[40] and Xu[41]). If E be a 2uniformly convex Banach space. Then, for all x, y ∈ E, we have
where J is the normalized duality mapping of E and 0 < c ≤ 1.
The best constant $\frac{1}{c}$ in lemma is called the puniformly convex constant of E.
Lemma 2.2. (Beauzamy[40] and Zalinescu[42]). If E be a puniformly 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)
where J_{ p } is the generalized duality mapping of E, and$\frac{1}{c}$is the puniformly convexity constant of E.
Lemma 2.3. (Kamimura and Takahashi[17]). 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 2.4. (Alber[14]). Let C be a nonempty closed convex subset of a smooth Banach space E and × ∈ E. Then, x_{0} = Π _{ C }x if and only if
Lemma 2.5. (Alber[14]). Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E and let × ∈ E. Then,
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,
(A4) for each x ∈ C, y α f(x, y) is convex and lower semicontinuous.
Lemma 2.6. (Blum and Oettli[1]). 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 ℝ satisfying (A1)(A4), and let r > 0 and × ∈ E. Then, there exists z ∈ C such that
Replacing x with J^{1}(Jx  rBx), where B is a monotone mapping from C into E*, then there exists z ∈ C such that
Lemma 2.7. (Zegeye[39]). 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 ℝ satisfying (A1)(A4), and let B be a monotone mapping from C into E*. For r > 0 and × ∈ E, define a mapping T_{ r } : C → C as follows:
for all × ∈ C. Then, the following hold:

(1)
T _{ r } is singlevalued;

(2)
T_{ r } is a firmly nonexpansivetype mapping, for all x, y ∈ E,
$$\u3008{T}_{r}x{T}_{r}y,J{T}_{r}xJ{T}_{r}y\u3009\le \u3008{T}_{r}x{T}_{r}y,JxJy\u3009;$$ 
(3)
F(T_{ r } ) = GEP(f, B);

(4)
GEP(f, B) is closed and convex.
Lemma 2.8. (Zegeye[39]). 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 ℝ satisfying (A1)(A4), and let B be a monotone mapping from C into E*. For r > 0, × ∈ E, and q ∈ F(T_{ r } ), we have that
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, onetoone, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [14]
for all x ∈ E and x* ∈ E*, that is, V(x, x*) = ϕ(x, J^{1}(x*)).
Lemma 2.9. (Alber[14]). Let E be a reflexive, strictly convex, smooth Banach space and let V be as in (2.1). Then,
for all × ∈ E and x*, y* ∈ E*.
An operator $\mathcal{M}\subset E\times {E}^{*}$ is said to be monotone if 〈x  y, x*  y*〉 ≥ 0 whenever (x, x*), (y, y*) ∈ T. We denote the set {x ∈ E : 0 ∈ Tx} by ${\mathcal{M}}^{1}0$. A monotone $\mathcal{M}$ is said to be maximal if its graph $G\left(\mathcal{M}\right)=\left\{\left(x,y\right):y\in \mathcal{M}x\right\}$ is not property contained in the graph of any other monotone operator. If $\mathcal{M}$ is maximal monotone, then the solution set ${\mathcal{M}}^{1}0$ is closed and convex. Let E be a reflexive, strictly convex, and smooth Banach space, it is known that $\mathcal{M}$ is a maximal monotone if and only if $R\left(J+r\mathcal{M}\right)={E}^{*}$ for all r > 0. Define the resolvent of $\mathcal{M}$ by J_{ r }x = x_{ r }. In other words, ${J}_{r}={(J+r\mathcal{M})}^{1}J$ for all r > 0. J_{ r }is a singlevalued mapping from E to $D\left(\mathcal{M}\right)$. Also, ${\mathcal{M}}^{1}\left(0\right)=F\left({J}_{r}\right)$ for all r > 0, where F(J_{ r }) is the set of all fixed points of J_{ r }. Define, for r > 0, the Yosida approximation of $\mathcal{M}$ by ${\mathcal{M}}_{r}=\left(JJ{J}_{r}\right)\u2215r$. We know that ${\mathcal{M}}_{r}x\in \mathcal{M}\left({J}_{r}x\right)$ for all r > 0 and x ∈ E.
Let A be an inversestrongly 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(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,
Lemma 2.10. (Rockafellar[43]). 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$\mathcal{M}\subset E\times {E}^{*}$be an operator defined as follows:
Then, $\mathcal{M}$is maximal monotone and${\mathcal{M}}^{1}0=VI\left(A,C\right)$.
Lemma 2.11. (Chang et al. [38]). 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,
Lemma 2.12. (Chang et al. [38]). 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 ϕasymptotically nonexpansive mapping with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1. Then, F (T ) is a closed convex subset of C.
3 Main results
Definition 3.1. (Chang et al. [38]) (1) Let ${\left\{{S}_{i}\right\}}_{i=1}^{\infty}:C\to C$ be a sequence of mapping. ${\left\{{S}_{i}\right\}}_{i=1}^{\infty}$ is said to be a family of uniformly quasi ϕasymptotically nonexpansive mappings, if $\mathcal{F}:={\cap}_{n=1}^{\infty}F\left({S}_{n}\right)\ne \varnothing $, and there exists a sequence {k_{ n } } ⊂ [1, ∞) with k_{ n } → 1 such that for each i ≥ 1
(2) A mapping S : C → C is said to be uniformly LLipschitz continuous, if there exists a constant L > 0 such that
In this section, we prove the new convergence theorems for finding the set of solutions of a general equilibrium problems, the common fixed point set of a family of closed and uniformly quasiϕasymptotically nonexpansive mappings, and the solution set of variational inequalities for an αinverse strongly monotone mapping in a 2uniformly convex and uniformly smooth Banach space.
Theorem 3.2. Let C be a nonempty closed and convex subset of a 2uniformly convex and uniformly smooth Banach space E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ V I(A, C) ≠ ∅. Let f be a bifunction from C × C to ℝ satisfying (A1)(A4) and B be a continuous monotone mapping of C into E*. Let${\left\{{S}_{i}\right\}}_{i=1}^{\infty}:C\to C$be an infinite family of closed uniformly L_{ i }Lipschitz continuous and uniformly quasi ϕasymptotically nonexpansive mappings with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 such that$F:={\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\cap GEP\left(f,B\right)\cap VI\left(A,C\right)$is a nonempty and bounded subset in C. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ n }= sup_{q∈F}(k_{ n }1)ϕ(q, x_{ n }), {α_{n,i}} is sequence in [0, 1], {r_{ n }} ⊂ [d, ∞) for some d > 0 and {λ_{ n }} ⊂ [a, b] for some a,b with 0 < a < b < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If${\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 }x_{0}.
Proof. We first show that C_{n+1}is closed and convex for each n ≥ 0. Clearly, C_{1} = 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 that ϕ(z, u_{ n } ) ≤ ϕ(z, x_{ n } ) + ζ_{ n } is equivalent to 2〈z, Jx_{ n }  Ju_{ n } 〉 ≤ x_{ n } ^{2}  u_{ n } ^{2} + ζ_{ n } . Hence, C_{n+1}is closed and convex. Next, we show that F ⊂ C_{ n } for all n ≥ 0. Indeed, put ${u}_{n}={T}_{{r}_{n}}{y}_{n}$ for all n ≥ 0. On the other hand, from Lemma 2.7, one has ${T}_{{r}_{n}}$is relatively quasinonexpansive mappings and F ⊂ C_{1} = C. Suppose F ⊂ C_{ n } for n ∈ ℕ, by the convexity of  · ^{2}, property of ϕ, Lemma 2.11 and by uniformly quasiϕasymptotically nonexpansive of S_{ n } for each q ∈ F ⊂ C_{ n } , we have
It follows from Lemma 2.9 that
Since q ∈ V I(A, C) and A is an αinversestrongly monotone mapping, we have
From Lemma 2.1 and A is an αinversestrongly monotone mapping, we also have
Substituting (3.6) and (3.7) into (3.5), we obtain
Substituting (3.8) into (3.4), we also have
This shows 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
By Lemma 2.5, we get
From (3.10) and (3.11), then {ϕ(x_{ n } , x_{0})} are nondecreasing and bounded. Hence, we obtain that lim_{n→∞}ϕ(x_{ n } , x_{0}) exists. In particular, by (1.5), the sequence {(x_{ n }   x_{0})^{2}} is bounded. This implies {x_{ n } } is also bounded. Denote
Moreover, by the definition of {ζ_{ n } } and (3.12), it follows that
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 2.5, we have
Since lim_{n→∞}ϕ(x_{ n } , x_{0}) exists and we taking m, n → ∞, then we get ϕ(x_{ m } , x_{ n } ) → 0. From Lemma 2.3, 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 } → p as n → ∞.
Now, we claim that Ju_{ n }  Jx_{ n }  → 0, as n → ∞. By definition of ${\Pi}_{{C}_{n}}{x}_{0}$, we have
Since lim_{n→∞}ϕ(x_{ n } , x_{0}) exists, we also have
Again from Lemma 2.3 that
From J is uniformly normtonorm continuous on bounded subsets of E, we obtain
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
By (3.14) and (3.13) that
Again applying Lemma 2.3, we have
Since
It follows that
Since J is uniformly normtonorm continuous on bounded subsets of E, we also have
Next, we will show that $p\in F:=GEP\left(f,B\right)\cap \left({\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\right)\cap VI\left(A,C\right).$
(a) First, we show that p ∈ GEP(f, B). From (3.4) and (3.8), we get ϕ(p, y_{ n } ) ≤ ϕ(p, x_{ n } ). By Lemma 2.8 and ${u}_{n}={T}_{{r}_{n}}{y}_{n}$, we observe that
From (3.19), (3.20), and Lemma 2.3, we have
Again since J is uniformly normtonorm continuous, we also have
From (A2), we note that
and hence
For t with 0 < t < 1 and y ∈ C, let y_{ t } = ty + (1 t)p. Then, y_{ t } ∈ C and hence
It follows that
By the continuity of B, J, and from (3.22) and (3.23), we obtain that Bu_{ n }  By_{ n } → 0 as n → ∞. From r_{ n } > 0 then $\frac{\parallel J{u}_{n}J{y}_{n}\parallel}{{r}_{n}}\to 0$ as n → ∞. Since B is monotone, we know that 〈By_{ t }  Bu_{ n } , y_{ t }  u_{ n } 〉 ≥ 0. Thus, it follows from (A4) that
From the conditions (A1) and (A4) we have
and hence
Letting t → 0, we have
This implies that p ∈ GEP(f, B).
(b) We show that $p\in {\cap}_{i=1}^{\infty}F\left({S}_{i}\right)$. From (3.4) and (3.8), for q ∈ F, we have
We note that
From Lemma 2.3, we get
By using the triangle inequality, we have
It follows from (3.19) and (3.26) that
and J is uniformly normtonorm continuous, we also have
By using the triangle inequality, we obtain
By (3.18) and (3.22), we get
Since J is uniformly normtonorm continuous, we obtain
From (3.3), we note that
and hence
From (3.16), (3.31), and ${liminf}_{n\to \infty}{\sum}_{i=1}^{\infty}{\alpha}_{n,i}0$, for each i ≥ 1, we obtain that
Since J^{1} is uniformly normtonorm continuous on bounded sets, we have
Again by using the triangle inequality, for each i ≥ 1, we get
From (3.15), (3.27), and (3.34), for each i ≥ 1, it follows that
Since lim_{n→∞}x_{ n }  z_{ n }  = 0 and x_{ n } → p as n → ∞, imply that z_{ n } → p as n → ∞. By using the triangle inequality, for each i ≥ 1
For each i ≥ 1, we have
Moreover, by the assumption that for each i ≥ 1, S_{ i } is uniformly L_{ i } Lipschitz continuous, hence we have
By (3.15) and (3.35), it yields that $\parallel {S}_{i}^{n+1}{z}_{n}{S}_{i}^{n}{z}_{n}\parallel \to 0$. From ${S}_{i}^{n}{z}_{n}\to p$, we have ${S}_{i}^{n+1}{z}_{n}\to p$, that is, ${S}_{i}{S}_{i}^{n}{z}_{n}\to p$. In view of closeness of S_{ i } , we have S_{ i }p = p, for all i ≥ 1. This imply that $p\in {\cap}_{i=1}^{\infty}F\left({S}_{i}\right)$
(c) We show that p ∈ V I(A, C). Indeed, define $\mathcal{M}\subset E\times {E}^{*}$ by
By Lemma 2.10, $\mathcal{M}$ is maximal monotone and ${\mathcal{M}}^{1}0=VI\left(A,C\right)$. Let $\left(v,w\right)\in G\left(\mathcal{M}\right)$. Since $w\in \mathcal{M}v=Av+{N}_{C}\left(v\right)$, we get w  Av ∈ N_{ C } (v).
From z_{ n } ∈ C, we have
On the other hand, since z_{ n } = Π _{ C }J^{1}(Jx_{ n }  λ_{ n }Ax_{ n } ). Then, by Lemma 2.4, we have
and thus
It follows from (3.39) and (3.40) that
where M = sup_{n≥1}v  z_{ n } . Take the limit as n → ∞ and (3.28), we obtain 〈v  p, w〉 ≥ 0. By the maximality of $\mathcal{M}$, we have $p\in {\mathcal{M}}^{1}0$, that is, p ∈ V I(A, C).
Finally, we show that p = Π _{ F }x_{0}. From ${x}_{n}={\Pi}_{{C}_{n}}{x}_{0}$, we have 〈Jx_{0}  Jx_{ n } , x_{ n }  z〉 ≥ 0, ∀z ∈ C_{ n } . Since F ⊂ C_{ n } , we also have
Taking limit n → ∞, we obtain
By Lemma 2.4, we can conclude that p = Π _{ F }x_{0} and x_{ n } → p as n → ∞. This completes the proof. □
If S_{ i } = S for each i ∈ ℕ, then Theorem 3.2 is reduced to the following corollary.
Corollary 3.3. Let C be a nonempty closed and convex subset of a 2uniformly convex and uniformly smooth Banach space E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ V I(A, C) ≠ ∅. Let f be a bifunction from C × C to ℝ satisfying (A1)(A4) and B be a continuous monotone mapping of C into E*. Let S : C → C be a closed uniformly LLipschitz continuous and quasi ϕasymptotically nonexpansive mappings with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 such that F := F(S)∩GEP(f, B)∩V I(A, C) is a nonempty and bounded subset in C. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)ϕ(q, x_{ n } ), {α_{ n } } 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 < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If lim inf_{n→∞}α_{ n }(1  α_{ n } ) > 0, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F } x_{0}.
For a special case that i = 1, 2, we can obtain the following results on a pair of quasiϕasymptotically nonexpansive mappings immediately from Theorem 3.2.
Corollary 3.4. Let C be a nonempty closed and convex subset of a 2uniformly convex and uniformly smooth Banach space E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ V I(A, C) ≠ ∅. Let f be a bifunction from C × C to ℝ satisfying (A1)(A4) and B be a continuous monotone mapping of C into E*. Let S, T : C → C be two closed quasi ϕasymptotically nonexpansive mappings and uniformly L_{ S } , L_{ T } Lipschitz continuous, respectively, with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 such that F := F(S) ∩ F (T ) ∩ GEP(f, B) ∩ V I(A, C) is a nonempty and bounded subset in C. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)ϕ(q, x_{ n } ), {α_{ n } }, {β_{ n } } and {γ_{ n } } are sequences in [0, 1], {r_{ n } } ⊂ [d, ∞) for some d > 0 and {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly 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 and lim inf_{n→∞}α_{ n }γ_{ n }> 0, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Corollary 3.5. Let C be a nonempty closed and convex subset of a 2uniformly convex and uniformly smooth Banach space E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ V I(A, C) ≠ ∅. Let f be a bifunction from C × C to ℝ satisfying (A1)(A4) and 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 quasi ϕnonexpansive mappings such that$F:={\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\cap GEP\left(f,B\right)\cap VI\left(A,C\right)\ne \varnothing .$For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n }} as follows:
where {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 < c^{2}α/2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. 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 }x_{0}.
Proof Since ${\left\{{S}_{i}\right\}}_{i=1}^{\infty}:C\to C$ is an infinite family of closed quasiϕnonexpansive mappings, it is an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings with sequence k_{ n } = 1. Hence, the conditions appearing in Theorem 3.2 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 the conditions in Theorem 3.2 are satisfied. The conclusion of Corollary 3.5 is obtained from Theorem 3.2 immediately. □
Corollary 3.6. [39, Theorem 3.2] Let C be a nonempty closed and convex subset of a 2uniformly convex and uniformly smooth Banach space E. Let A be an αinversestrongly monotone mapping of C into E* satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ V I(A, C) ≠ ∅. Let f be a bifunction from C × C to ℝ satisfying (A1)(A4) and B be a continuous monotone mapping of C into E*. Let${\left\{{S}_{i}\right\}}_{i=1}^{N}:C\to C$be a finite family of closed quasi ϕnonexpansive mappings such that$F:={\cap}_{i=1}^{N}F\left({S}_{i}\right)\cap GEP\left(f,B\right)\cap VI\left(A,C\right)\ne \varnothing $. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C 1 = C, we define the sequence {x_{ n } } as follows:
where {α_{n,i}} is sequence in [0, 1], {r_{ n } } ⊂ [d, ∞) for some d > 0 and {λ_{ n } } ⊂ [a, b] for some a, b with 0 < a < b < c^{2}α/ 2, where$\frac{1}{c}$is the 2uniformly convexity constant of E. If α_{ i } ∈ (0, 1) such that${\sum}_{i=0}^{N}{\alpha}_{i}=1$, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Remark 3.7. Theorem 3.2, Corollaries 3.5 and 3.6 improve and extend the corresponding results in Wattanawitoon and Kumam [34] and Zegeye [39] in the following senses:

from a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family of quasiϕasymptotically mappings;

for the mappings, we extend the mappings from nonexpansive mappings, relatively quasinonexpansive mappings or quasiϕnonexpansive mappings and a finite family of closed relatively quasinonexpansive mappings to an infinite family of quasiϕasymptotically nonexpansive mappings.
4 Deduced theorems
Corollary 4.1. 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). 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 ϕasymptotically nonexpansive mappings with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 and uniformly L_{ i }Lipschitz continuous such that$F:={\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\cap GEP\left(f,B\right)$is a nonempty and bounded subset in C. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ 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 }x_{0}.
Proof Put A ≡ 0 in Theorem 3.2. Then, we get that z_{ n } = x_{ n } . Thus, the method of proof of Theorem 3.2 gives the required assertion without the requirement that E be 2uniformly convex. □
If setting B ≡ 0 in Corollary 4.1, then we have the following corollary.
Corollary 4.2. 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). Let${\left\{{S}_{i}\right\}}_{i=1}^{\infty}:C\to C$be an infinite family of closed and uniformly quasi ϕasymptotically nonexpansive mappings with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 and uniformly L_{ i }Lipschitz continuous such that$F:={\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\cap EP\left(f\right)$is a nonempty and bounded subset in C. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ 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 }x_{0}.
If setting f ≡ 0 in Corollary 4.1, then we obtain the following corollary.
Corollary 4.3. Let C be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space E. 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 ϕasymptotically nonexpansive mappings with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 and uniformly L_{ i }Lipschitz continuous such that$F:={\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\cap VI\left(B,C\right)$is a nonempty and bounded subset in C. For an initial point x_{0} ∈ E with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ 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 }x_{0}.
Remark 4.4. Corollaries 4.1{4.3 improve and extend the corresponding results in Zegeye [39] and Wattanawitoon and Kumam [34] in the sense from a finite family of closed relatively quasinonexpansive mappings and closed relatively quasinonexpansive mappings to more general than an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings.
5 Application to Hilbert spaces
If E = H, a Hilbert space, then E is 2uniformly convex (we can choose c = 1) and uniformly smooth real Banach space and closed relatively quasinonexpansive map reduces to closed quasinonexpansive 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 5.1. Let C be a nonempty closed and convex subset of a Hilbert space H. Let f be a bifunction from C × C to ℝ satisfying (A1)(A4). Let A be an αinversestrongly monotone mapping of C into H satisfying Ay ≤ Ay  Au, ∀y ∈ C and u ∈ V I(A, C) ≠ ∅ and B be a continuous monotone mapping of C into H. Let${\left\{{S}_{i}\right\}}_{i=1}^{\infty}:C\to C$be an infinite family of closed and uniformly quasi ϕasymptotically nonexpansive mappings with a sequence {k_{ n } } ⊂ [1, ∞), k_{ n } → 1 and uniformly L_{ i }Lipschitz continuous such that$F:={\cap}_{i=1}^{\infty}F\left({S}_{i}\right)\cap GEP\left(f,B\right)\cap VI\left(A,C\right)$is a nonempty and bounded subset in C. For an initial point x_{0} ∈ H with${x}_{1}={\Pi}_{{C}_{1}}{x}_{0}$and C_{1} = C, we define the sequence {x_{ n } } as follows:
where ζ_{ n } = sup_{q∈F}(k_{ n }  1)q  x_{ n } , {α_{n,i}} is sequence in [0, 1], {r_{ 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,0}α_{n,i}> 0 for all i ≥ 1, then {x_{ n } } converges strongly to p ∈ F, where p = Π _{ F }x_{0}.
Remark 5.2. Theorem 5.1 improve and extend the Corollary 3.7 in Zegeye [39] in the aspect for the mappings, we extend the mappings from a finite family of closed relatively quasinonexpansive mappings to more general an infinite family of closed and uniformly quasiϕasymptotically nonexpansive mappings.
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