# Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi-ϕ-Asymptotically Nonexpansive Mappings

Authors: Siwaporn Saewan; Poom Kumam
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings 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 uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.


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## Body

## 1. Introduction and Preliminaries

The convex feasibility problem (CFP) is the problem of computing points laying in the intersection of a finite family of closed convex subsetsCj,j=1,2,…,N, of a Banach space E. This problem appears in various fields of applied mathematics. The theory of optimization [1], Image Reconstruction from projections [2], and Game Theory [3] are some examples. There is a considerable investigation on (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [4]. The advantage of a Hilbert space H is that the projection PC onto a closed convex subset C of H is nonexpansive. So projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [5] deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if C is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection (see, Alber [6] or Kamimura and Takahashi [7]) from E onto C is relatively nonexpansive, whereas the metric projection from H onto C is not generally nonexpansive.We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem (CFP) (see, [8, 9], etc.). In 2008, Plubtieng and Ungchittrakool [10] 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.LetC be a nonempty closed convex subset of a real Banach space E with ∥·∥ and E* being the dual space of E. Let f be a bifunction of C×C into ℝ and B:C→E* a monotone mapping. The generalized equilibrium problem, denoted by GEP, is to find x∈C such that(1.1)f(x,y)+〈Bx,y-x〉≥0,∀y∈C.
The set of solutions for the problem (1.1) is denoted by GEP(f,B), that is (1.2)GEP(f,B):={x∈C:f(x,y)+〈Bx,y-x〉≥0,∀y∈C}.
If B≡0, the problem (1.1) reducing into the equilibrium problem forf, denoted by EP(f), is to find x∈C such that(1.3)f(x,y)≥0,∀y∈C.
If f≡0, the problem (1.1) reducing into the classical variational inequality, denoted by VI(B,C), is to find x*∈C such that(1.4)〈Bx*,y-x*〉≥0,∀y∈C.
The above formulation (1.3) was shown in [11] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality 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 EP(f). In other words, the EP(f) is a unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(f); see, for example [11] and references therein. Some solution methods have been proposed to solve the EP(f); see, for example, [12–29] and references therein.Consider the functional defined by(1.5)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E,
where J is the duality mapping from E into E*. It is well known that if C is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC 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(1.6)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,∀x,y∈E.
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 (Alber [6]) ΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y), that is, ΠCx=x¯, where x¯ is the solution to the minimization problem(1.7)ϕ(x¯,x)=infy∈Cϕ(y,x),
and existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, for example, [6, 7, 30–32]).Remark 1.1.
IfE 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 (1.5), 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 [31, 32] for more details.
LetC be a closed convex subset of E; a mapping T:C→C is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥,forallx,y∈C. A point x∈C is a fixed point of T provided Tx=x. Denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}. Recall that a point p in C is said to be an asymptotic fixed point of T [33] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F(T)̃.
A mappingT from C into itself is said to be relatively nonexpansive [34–36] if F(T)̃=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in [37–39]. T is said to be ϕ-nonexpansive, if ϕ(Tx,Ty)≤ϕ(x,y) for x,y∈C. T is said to be relatively quasi-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). T is said to be quasi-ϕ-asymptotically nonexpansive if F(T)≠∅ and there exists a real sequence {kn}⊂[1,∞) with kn→1 such that ϕ(p,Tnx)≤knϕ(p,x) for all n≥1x∈C and p∈F(T). A mapping T is said to be closed if for any sequence {xn}⊂C with xn→x and Txn→y, Tx=y. It is easy to know that each relatively nonexpansive mapping is closed. The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass and the class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true (see more details [37–41]).
A Banach spaceE is said to be strictly convex if ∥(x+y)/2∥<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 limt→0(∥x+ty∥-∥x∥)/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 δ(ε)=inf{1-∥(x+y)/2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥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)≥cεp for all ε∈[0,2]; see [42] for more details. Observe that every p-uniform convex is uniformly convex. One should note that no Banach space is p-uniform convex for 1<p<2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each p>1, the generalized duality mapping  Jp:E→2E* is defined by Jp(x)={x*∈E*:〈x,x*〉=∥x∥p,∥x*∥=∥x∥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. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.
The following basic properties can be found in Cioranescu [31].(i)
IfE is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.(ii)
IfE is a reflexive and strictly convex Banach space, then J-1 is norm-weak*-continuous.(iii)
IfE is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J:E→2E* is single-valued, one-to-one, and onto.(iv)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex.(v)
Each uniformly convex Banach spaceE has the Kadec-Klee property, that is, for any sequence {xn}⊂E, if xn⇀x∈E and ∥xn∥→∥x∥,xn→x.In 2005, Matsushita and Takahashi [40] 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:(1.8)x0∈Cchosenarbitrarily,yn=J-1(αnJxn+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0.
They proved that {xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection from C onto F(T). In 2007, Plubtieng and Ungchittrakool [43] generalized the processes (1.8) to the new general processes of two relatively nonexpansive mappings in a Banach space. Let C be a closed convex subset of a Banach space E and S,T:C→C relatively nonexpansive mappings such that F:=F(S)∩F(T)≠∅. Define {xn} in the following way:(1.9)x0=x∈C,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={z∈C:ϕ(z,yn)⩽ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉⩾0},xn+1=PHn∩Wnx,n=0,1,2,…,
where {αn}, {βn(1)}, {βn(2)}, and {βn(3)} are sequences in [0,1] with βn(1)+βn(2)+βn(3)=1 for all n∈ℕ∪{0}.In 2009, Qin et al. [26] introduced a hybrid projection algorithm to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two quasi-ϕ-nonexpansive mappings in the framework of Banach spaces:(1.10)x0=xchosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αnJxn+βnJTxn+γnJSxn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,
where ΠCn+1 is the generalized projection from E onto Cn+1. They proved that the sequence {xn} converges strongly to ΠF(T)∩F(S)∩EP(f)x0. In the same year, Wattanawitoon and Kumam [44] and Petrot et al. [45] using the idea of Takahashi and Zembayashi [46, 47] and Plubtieng and Ungchittrakool [43] extend the notion from relatively nonexpansive mappings or quasi-ϕ-nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2010, Chang et al. [48] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mapping; they obtain the strong convergence theorems in a Banach space. Recently, many authors considered the iterative methods for finding a common element of the set of solutions to the problem (1.3) and of the set of fixed points of nonexpansive mappings; see, for instance, [12–27] and the references therein.Motivated by Chang et al. [48], Qin et al. [26, 49], Wattanawitoon and Kumam [44], Petrot et al. [45], Zegeye [50], and other recent works, in this paper we introduce a new modified block hybrid projection algorithm for finding a common element of 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 which is more general than closed quasi-ϕ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. The results presented in this paper improve and generalize some well-known results in the literature.


## 2. Basic Results

We also need the following lemmas for the proof of our main results.Lemma 2.1 (Kamimura and Takahashi [7]).
LetE be a uniformly convex and smooth Banach space and let  {xn} and  {yn} be two sequences of  E. If  ϕ(xn,yn)→0 and either  {xn} or  {yn} is bounded, then  ∥xn-yn∥→0.Lemma 2.2 (Alber [6]).
LetC be a nonempty closed convex subset of a smooth Banach space  E and  x∈E. Then  x0=ΠCx if and only if
(2.1)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.3 (Alber [6]).
LetE be a reflexive, strictly convex and smooth Banach space, let  C be a nonempty closed convex subset of  E, and let  x∈E. Then
(2.2)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.For solving the equilibrium problem for a bifunctionf:C×C→ℝ, let us assume that f satisfies the following conditions:  (A1)
f(x,x)=0for all x∈C;(A2)
f is monotone, that is, f(x,y)+f(y,x)≤0 for all x,y∈C;(A3)
for eachx,y,z∈C,
(2.3)limt↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
for eachx∈C, y↦f(x,y) is convex and lower semicontinuous.Lemma 2.4 (Blum and Oettli [11]).
LetC 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  x∈E. Then there exists  z∈C such that
(2.4)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.5 (Zegeye [50]).
LetC 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  x∈E, define a mapping  Tr:C→C as follows:
(2.5)Trx={z∈C:f(z,y)+〈Bx,y-z〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C}
for all x∈C. Then the following hold:  (1)
Tr is single-valued;(2)
Tr is a firmly nonexpansive-type mapping, for all x,y∈E,
(2.6)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(3)
F(Tr)=GEP(f,B);(4)
GEP(f,B) is closed and convex.Lemma 2.6 (Zegeye [50]).
(LetC 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, x∈E, and q∈F(Tr), we have that
(2.7)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 2.7 (Chang et al. [48]).
LetE be a uniformly convex Banach space, r>0 a positive number, and Br(0) a closed ball of E. Then, for any given sequence {xi}i=1∞⊂Br(0) and for any given sequence {λi}i=1∞ of positive number with ∑n=1∞λn=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,
(2.8)∥∑n=1∞λnxn∥2≤∑n=1∞λn∥xn∥2-λiλjg(∥xi-xj∥).Lemma 2.8 (Chang et al. [48]).
LetE be a real uniformly smooth and strictly convex Banach space, and C a nonempty closed convex subset of E. Let T:C→C be a closed and quasi-ϕ-asymptotically nonexpansive mapping with a sequence {kn}⊂[1,∞), kn→1. Then F(T) is a closed convex subset of C.Definition 2.9 (Chang et al. [48]).
(1) Let{Si}i=1∞:C→C be a sequence of mapping. {Si}i=1∞ is said to be a family of uniformly quasi-ϕ-asymptotically nonexpansive mappings, if ∩n=1∞F(Sn)≠∅, and there exists a sequence {kn}⊂[1,∞) with kn→1 such that for each i≥1,(2.9)ϕ(p,Sinx)≤knϕ(p,x),∀p∈⋂n=1∞F(Sn),x∈C,∀n≥1.
(2) A mappingS:C→C is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that
(2.10)∥Snx-Sny∥≤L∥x-y∥,∀x,y∈C.


## 3. Main Results

In this section, we prove the new convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property.Theorem 3.1.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let B be a continuous monotone mapping of C into E*. Let {Si}i=1∞:C→C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1 such that F:=∩i=1∞F(Si)∩GEP(f,B) is a nonempty and bounded subset in C. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.1)yn=J-1(βnJxn+(1-βn)Jzn),zn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),un∈Csuchthatf(un,y)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E,θn=supq∈F(kn-1)ϕ(q,xn), {αn,i},{βn} are sequences in [0,1] and {rn}⊂[a,∞) for some a>0. If ∑i=0∞αn,i=1 for all n≥0 and liminfn→∞αn,0αn,i>0 for all i≥1, then {xn} converges strongly to p∈F, where p=ΠFx0.Proof.
We first show thatCn+1 is closed and convex for each n≥0. Clearly C1=C is closed and convex. Suppose that Cn is closed and convex for each n∈ℕ. Since for any z∈Cn, we know
(3.2)ϕ(z,un)≤ϕ(z,xn)+θn⇔2〈z,Jxn-Jun〉≤∥xn∥2-∥un∥2+θn.
So, Cn+1 is closed and convex. Therefore, ΠFx0 and ΠCnx0 are well defined. Next, we show that F⊂Cn for all n≥0. Indeed, put un=Trnyn for all n≥0. It is clear that F1⊂C1=C. Suppose F⊂Cn for n∈ℕ, by the convexity of ∥·∥2, property of ϕ, Lemma 2.7, and uniformly quasi-ϕ-asymptotically nonexpansive of Sn for each q∈F⊂Cn, we observed that
(3.3)ϕ(q,un)=ϕ(q,Trnyn)≤ϕ(q,yn)=ϕ(q,J-1(βnJxn+(1-βn)Jzn)=∥q∥2-2〈q,βnJxn+(1-βn)Jzn〉+∥βnJxn+(1-βn)Jzn∥2≤∥q∥2-2βn〈q,Jxn〉-2(1-βn)〈q,Jzn〉+βn∥xn∥2+(1-βn)∥zn∥2=βnϕ(q,xn)+(1-βn)ϕ(q,zn)
and
(3.4)ϕ(q,zn)=ϕ(q,J-1(αn,0Jxn+∑i=1∞αn,iJSinxn))=∥q∥2-2〈q,αn,0Jxn+∑i=1∞αn,iJSinxn〉+∥αn,0Jxn+∑i=1∞αn,iJSinxn∥2≤∥q∥2-2αn,0〈q,Jxn〉-2∑i=1∞αn,i〈q,JSinxn〉+∥αn,0Jxn+∑i=1∞αn,iJSinxn∥2≤∥q∥2-2αn,0〈q,Jxn〉-2∑i=1∞αn,i〈q,JSinxn〉+αn,0∥Jxn∥2+∑i=1∞αn,i∥JSinxn∥2-αn,0αn,jg∥Jxn-JSjnxn∥=∥q∥2-2αn,0〈q,Jxn〉+αn,0∥Jxn∥2-2∑i=1∞αn,i〈q,JSinxn〉+∑i=1∞αn,i∥JSinxn∥2-αn,0αn,jg∥Jxn-JSjnxn∥=αn,0ϕ(q,xn)+∑i=1∞αn,iϕ(q,Sinxn)-αn,0αn,jg∥Jxn-JSjnxn∥≤αn,0knϕ(q,xn)+∑i=1∞αn,iknϕ(q,xn)-αn,0αn,jg∥Jxn-JSjnxn∥≤knϕ(q,xn).
Substituting (3.4) into (3.3), we get
(3.5)ϕ(q,un)≤βnϕ(q,xn)+(1-βn)ϕ(q,zn)≤βnϕ(q,xn)+(1-βn)knϕ(q,xn)≤βnϕ(q,xn)+(1-βn)[ϕ(q,xn)+supq∈F(kn-1)ϕ(q,xn)]≤ϕ(q,xn)+(1-βn)supq∈F(kn-1)ϕ(q,xn)≤ϕ(q,xn)+θn.
This show that q∈Cn+1 implies that F⊂Cn+1 and hence, F⊂Cn for all n≥0. Since F is nonempty, Cn is a nonempty closed convex subset of E, and hence ΠCn exist for all n≥0. This implies that the sequence {xn} is well defined. From definition of Cn+1 that xn=ΠCnx0 and xn+1=ΠCn+1x0, we have
(3.6)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
By Lemma 2.3, we also have
(3.7)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(q,x0)-ϕ(q,xn)≤ϕ(q,x0),∀q∈F⊂Cn,∀n≥0.
From (3.6) and (3.7), then {ϕ(xn,x0)} are nondecreasing and bounded. So, we obtain that limn→∞ϕ(xn,x0) exists. In particular, by (1.6), the sequence {(∥xn∥-∥x0∥)2} is bounded. This implies that {xn} is also bounded. Denote
(3.8)K=supn≥0{∥xn∥}<∞.
Moreover, by the definition of {θn} and (3.8), it follows that
(3.9)θn→0,n→∞.
Next, we show that{xn} is a Cauchy sequence in C. Since xm=ΠCmx0∈Cm⊂Cn, for m>n, by Lemma 2.3, we have (3.10)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
Since limn→∞ϕ(xn,x0) exists and we take m,n→∞, then, we get ϕ(xm,xn)→0. From Lemma 2.1, we have limn→∞∥xm-xn∥=0. Thus {xn} is a Cauchy sequence and by the completeness of E, and there exists a point p∈C such that xn→p as n→∞.
Now, we claim that∥Jun-Jxn∥→0, as n→∞. By definition of ΠCnx0, one has (3.11)ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0).
Since limn→∞ϕ(xn,x0) exists, we have
(3.12)limn→∞ϕ(xn+1,xn)=0.
By Lemma 2.1, we obtain that
(3.13)limn→∞∥xn+1-xn∥=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we get
(3.14)limn→∞∥Jxn+1-Jxn∥=0.
From xn+1=ΠCn+1x0∈Cn+1⊂Cn and the definition of Cn+1, we have
(3.15)ϕ(xn+1,un)≤ϕ(xn+1,xn)+θn.
By (3.9) and (3.12), we also have
(3.16)limn→∞ϕ(xn+1,un)=0.
Applying Lemma 2.1, we obtain
(3.17)limn→∞∥xn+1-un∥=0.
Since ∥un-xn∥≤∥un-xn+1∥+∥xn+1-xn∥, we get
(3.18)limn→∞∥un-xn∥=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we have
(3.19)limn→∞∥Jun-Jxn∥=0.
Next, we will show thatxn→p∈F:=GEP(f,B)∩(∩i=1∞F(Si)).
(i) First, we show thatxn→p∈GEP(f,B). It follows from (3.3) and (3.4) we observe that ϕ(p,yn)≤ϕ(p,xn)+θn. By Lemma 2.6 and un=Trnyn, we obtain
(3.20)ϕ(un,yn)=ϕ(Trnyn,yn)≤ϕ(p,yn)-ϕ(p,Trnyn)≤ϕ(p,xn)-ϕ(p,Trnyn)+θn=ϕ(p,xn)-ϕ(p,un)+θn=∥p∥2-2〈p,Jxn〉+∥xn∥2-(∥p∥2-2〈p,Jun〉+∥un∥2)+θn=∥xn∥2-∥un∥2-2〈p,Jxn-Jun〉+θn≤∥xn-un∥(∥xn∥+∥un∥)+2∥p∥∥Jxn-Jun∥+θn.
By Lemma 2.1, (3.9), (3.18), and (3.19), we have
(3.21)limn→∞∥un-yn∥=0.
Again since J is uniformly norm-to-norm continuous, we also have
(3.22)limn→∞∥Jun-Jyn∥=0.
From (A2), we note that
(3.23)〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥-f(un,y)≥f(y,un),∀y∈C,
and hence
(3.24)〈Byn,y-un〉+〈y-un,Jun-Jynrn〉≥f(y,un),∀y∈C.
For t with 0<t<1 and y∈C, let yt=ty+(1-t)p. Then yt∈C and hence
(3.25)0≥-〈Byn,yt-un〉-〈yt-un,Jun-Jynrn〉+f(yt,un),∀y∈C.
It follows that
(3.26)〈Byt,yt-un〉≥〈Byt,yt-un〉-〈Byn,yt-un〉-〈yt-un,Jun-Jynrn〉+f(yt,un),∀yt∈C=〈Byt,yt-un〉-〈Bun,yt-un〉+〈Bun,yt-un〉-〈Byn,yt-un〉-〈yt-un,Jun-Jynrn〉+f(yt,un),∀yt∈C=〈Byt-Bun,yt-un〉+〈Bun-Byn,yt-un〉-〈yt-un,Jun-Jynrn〉+f(yt,un),∀yt∈C.
Since xn→p as n→∞ from (3.18) and (3.21), we can get un→p and yn→p as n→∞. Furthermore, it follows from the continuity of B that Bun-Byn→0 as n→∞. From rn>0 and (3.22), we have ∥Jun-Jyn∥/rn→0 as n→∞. Since B is monotone, we know that 〈Byt-Bun,yt-un〉≥0. Thus, it follows from (A4) that
(3.27)f(yt,p)≤liminfn→∞f(yt,un)≤limn→∞〈Byt,yt-un〉=〈Byt,yt-p〉.
From the conditions (A1) and (A4), we have
(3.28)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,p)≤tf(yt,y)+(1-t)〈Byt,yt-p〉≤tf(yt,y)+(1-t)t〈Byt,y-p〉,
and hence
(3.29)0≤f(yt,y)+(1-t)〈Byt,y-p〉.
Letting t→0, we get
(3.30)0≤f(p,y)+〈Bp,y-p〉,∀y∈C.
This implies that p∈GEP(f,B).
(ii) We show that xn→p∈∩i=1∞F(Si). From definition of Cn+1 and since xn+1=ΠCn+1x0∈Cn+1, we have (3.31)ϕ(xn+1,zn)≤ϕ(xn+1,xn)+θn.
Form Lemma 2.1 and (3.9), we obtain that
(3.32)limn→∞∥xn+1-zn∥=0.
Since J is uniformly norm-to-norm continuous, we also have
(3.33)limn→∞∥Jxn+1-Jzn∥=0.
From (3.1), we compute
(3.34)∥Jxn+1-Jzn∥=∥Jxn+1-(αn,0Jxn+∑i=1∞αn,iJSinxn)∥=∥αn,0Jxn+1-αn,0Jxn+∑i=1∞αn,iJxn+1-∑i=1∞αn,iJSinxn∥=∥αn,0(Jxn+1-Jxn)+∑i=1∞αn,i(Jxn+1-JSinxn)∥=∥∑i=1∞αn,i(Jxn+1-JSinxn)-αn,0(Jxn-Jxn+1)∥≥∑i=1∞αn,i∥Jxn+1-JSinxn∥-αn,0∥Jxn-Jxn+1∥,
and hence
(3.35)∥Jxn+1-JSinxn∥≤1∑i=1∞αn,i(∥Jxn+1-Jzn∥+αn,0∥Jxn-Jxn+1∥).
From (3.14), (3.33), and liminfn→∞∑i=1∞αn,i>0, we get
(3.36)limn→∞∥Jxn+1-JSinxn∥=0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we obtain that
(3.37)limn→∞∥xn+1-Sinxn∥=0.
By the triangle inequality,
(3.38)∥xn-Sinxn∥=∥xn-xn+1+xn+1-Sinxn∥≤∥xn-xn+1∥+∥xn+1-Sinxn∥.
Hence from (3.13) and (3.37), we have
(3.39)limn→∞∥xn-Sinxn∥=0.
Since J is uniformly continuous on any bounded subset of E, we obtain
(3.40)limn→∞∥Jxn-JSinxn∥=0,∀i≥1.
Since xn→p and J is uniformly continuous, it yields Jxn→Jp. Thus from (3.40), we get
(3.41)JSinxn→Jp,∀i≥1.
Since J-1:E*→E is norm-weak*-continuous, we have
(3.42)Sinxn⇀p,∀i≥1.
On the other hand, for each i≥1, we observe that
(3.43)|∥Sinxn∥-∥p∥|=|∥J(Sinxn)∥-∥Jp∥|≤∥J(Sinxn)-Jp∥.
In view of (3.41), we obtain ∥Sinxn∥→∥p∥  for each i≥1. Since E has the Kadec-Klee property, we get
(3.44)Sinxn→p,foreachi≥1.
By the assumption that for each i≥1,Si is uniformly Li-Lipschitz continuous, we have
(3.45)∥Sin+1xn-Sinxn∥≤∥Sin+1xn-Sin+1xn+1∥+∥Sin+1xn+1-xn+1∥+∥xn+1-xn∥+∥xn-Sinxn∥≤(Li+1)∥xn+1-xn∥+∥Sin+1xn+1-xn+1∥+∥xn-Sinxn∥.
From (3.13) and (3.39), it yields that ∥Sin+1xn-Sinxn∥→0. From Sinxn→p, we get Sin+1xn→p, that is SiSinxn→p. In view of closeness of Si, we have Sip=p, for all i≥1. This implies that p∈∩i=1∞F(Si).
Finally, we show thatxn→p=ΠFx0. Let u=ΠFx0. From xn=ΠCnx0 and u∈F⊂Cn, we have(3.46)ϕ(xn,x0)≤ϕ(u,x0),∀n≥0.
This implies that
(3.47)ϕ(p,x0)=limn→∞ϕ(xn,x0)≤ϕ(u,x0).
By definition of p=ΠFx0, we have p=u. Therefore, xn→p=ΠFx0. This completes the proof.If Si=S for each i∈ℕ, then Theorem 3.1 is reduced to the following corollary.Corollary 3.2.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let B be a continuous monotone mapping of C into E*. Let S:C→C be a closed uniformly L-Lipschitz continuous and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1, such that F:=F(S)∩GEP(f,B) is a nonempty and bounded subset in C. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.48)yn=J-1(βnJxn+(1-βn)Jzn),zn=J-1(αnJxn+(1-αn)JSnxn),un∈Csuchthatf(un,y)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, θn=supq∈F(kn-1)ϕ(q,xn), {αn},{βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0. If liminfn→∞αn(1-αn)>0, then {xn} converges strongly to p∈F, where p=ΠFx0.For a special case thati=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.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let B be a continuous monotone mapping of C into E*. Let S,T:C→C be two closed quasi-ϕ-asymptotically nonexpansive mappings and uniformly LS,LT-Lipschitz continuous, respectively with a sequence {kn}⊂[1,∞), kn→1 such that F:=F(S)∩F(T)∩GEP(f,B) is a nonempty and bounded subset in C. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.49)yn=J-1(βnJxn+(1-βn)Jzn),zn=J-1(αnJxn+βnJSnxn+γnJTnxn),un∈Csuch thatf(un,y)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, θn=supq∈F(kn-1)ϕ(q,xn), {αn,i},{βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0. If αn+βn+γn=1 for all n≥0 and liminfn→∞αnβn>0 and liminfn→∞αnγn>0, then {xn} converges strongly to p∈F, where p=ΠFx0.Remark 3.4.
Corollary3.3 improves and extends [44, Theorem 3.1] in the following senses:  (i)
for the mappings, we extend the mappings from two closed relatively quasi-nonexpansive mappings to an infinite family of closed and uniformly quasi-ϕ-asymptotically mappings,(ii)
from a solution of the classical equilibrium problem to the generalized equilibrium problem,(iii)
for the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.Corollary 3.5.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let B be a continuous monotone mapping of C into E*. Let {Si}i=1∞:C→C be an infinite family of closed quasi-ϕ-nonexpansive mappings such that F:=∩i=1∞F(Si)∩GEP(f,B)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.50)yn=J-1(βnJxn+(1-βn)Jzn),zn=J-1(αn,0Jxn+∑i=1∞αn,iJSixn),un∈Csuch  thatf(un,y)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, {αn,i},{βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0. If ∑i=0∞αn,i=1 for all n≥0 and liminfn→∞αn,0αn,i>0 for all i≥1, then {xn} converges strongly to p∈F, where p=ΠFx0.Proof.
Since{Si}i=1∞:C→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 kn=1. Hence the conditions appearing in Theorem 3.1F is a bounded subset in C and for each i≥1,Si is uniformly Li-Lipschitz continuous are of no use here. By virtue of the closeness of mapping Si for each i≥1, it yields that p∈F(Si) for each i≥1, that is, p∈∩i=1∞F(Si). Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Corollary 3.5 is obtained from Theorem 3.1 immediately.Corollary 3.6.
LetC be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let B be a continuous monotone mapping of C into E*. Let {Si}i=1∞:C→C be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1 and uniformly Li-Lipschitz continuous such that F:=∩i=1∞F(Si)∩GEP(f,B) is a nonempty and bounded subset in C. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
(3.51)yn=J-1(αn,0Jxn+∑i=1∞αn,iJSinxn),un∈Csuch  thatf(un,y)+〈Byn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n≥0,
where J is the duality mapping on E, θn=supq∈F(kn-1)ϕ(q,xn), {αn,i} is a sequence in [0,1], and {rn}⊂[a,∞) for some a>0. If ∑i=0∞αn,i=1 for all n≥0 and liminfn→∞αn,0αn,i>0 for all i≥1, then {xn} converges strongly to p∈F, where p=ΠFx0.Proof.
Settingβn≡0 in Theorem 3.1, then, we get that yn=zn. Thus, the method of proof of Theorem 3.1 we obtain Corollary 3.6 immediately.Remark 3.7.
Theorem3.1, Corollary 3.3, and Corollary 3.5, improve and extend the corresponding results in Qin et al. [49] and Zegeye [50] in the following senses:  (i)
from a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family of quasi-ϕ-asymptotically mappings,(ii)
for the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or quasi-ϕ-nonexpansive mappings and a finite family of closed relatively quasi-nonexpansive mappings to an infinite family of quasi-ϕ-asymptotically nonexpansive mappings,(iii)
for the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.


## 4. Applications

IfE=H, a Hilbert space, then H is uniformly smooth and strictly convex Banach space E with Kadec-Klee property and closed relatively quasi-nonexpansive mappings reducing to closed quasi-nonexpansive mappings. Moreover, J=I, identity operator on H and ΠC=PC, projection mapping from H into C. Thus, the following corollaries hold.Theorem 4.1.
LetC 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 B be a continuous monotone mapping of C into H. Let {Si}i=1∞:C→C be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1 and uniformly Li-Lipschitz continuous such that F:=∩i=1∞F(Si)∩GEP(f,B) is a nonempty and bounded subset in C. For an initial point x0∈H with x1=PC1x0 and C1=C, we define the sequence {xn} as follows:
(4.1)yn=βnxn+(1-βn)zn,zn=αn,0xn+∑i=1∞αn,iSinxn,un∈Csuch thatf(un,y)+〈Byn,y-un〉+1rn〈y-un,un-yn〉≥0,∀y∈C,Cn+1={z∈Cn:∥z-un∥≤αn∥z-xn∥+(1-αn)∥z-zn∥≤∥z-xn∥+θn},xn+1=PCn+1x0,∀n≥0,
where θn=supq∈F(kn-1)∥q-xn∥, {αn,i},{βn} are sequences in [0,1], and {rn}⊂[a,∞) for some a>0. If ∑i=0∞αn,i=1 for all n≥0 and liminfn→∞αn,0αn,i>0 for all i≥1, then {xn} converges strongly to p∈F, where p=ΠFx0.Corollary 4.2.
LetC 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 B be a continuous monotone mapping of C into H. Let {Si}i=1∞:C→C be an infinite family of closed and quasi-ϕ-nonexpansive mappings with a sequence {kn}⊂[1,∞), kn→1 and uniformly Li-Lipschitz continuous such that F:=∩i=1∞F(Si)∩GEP(f,B) is a nonempty and bounded subset in C. For an initial point x0∈H with x1=PC1x0 and C1=C, we define the sequence {xn} as follows:
(4.2)yn=βnxn+(1-βn)zn,zn=αn,0xn+∑i=1∞αn,iSinxn,un∈Csuch  thatf(un,y)+〈Byn,y-un〉+1rn〈y-un,un-yn〉≥0,∀y∈C,Cn+1={z∈Cn:∥z-un∥≤αn∥z-xn∥+(1-αn)∥z-zn∥≤∥z-xn∥},xn+1=PCn+1x0,∀n≥0,
where {αn,i},{βn} are sequences in [0,1] and {rn}⊂[a,∞) for some a>0. If ∑i=0∞αn,i=1 for all n≥0 and liminfn→∞αn,0αn,i>0 for all i≥1, then {xn} converges strongly to p∈F, where p=ΠFx0.Remark 4.3.
Theorem4.1 improves and extends the Corollary  3.7 in Zegeye [50] in the aspect for the mappings; we extend the mappings from a finite family of closed relatively quasi-nonexpansive mappings to more general a infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings.



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