# Modified Hybrid Block Iterative Algorithm for Uniformly Quasi-ϕ-Asymptotically Nonexpansive Mappings

Authors: Qiansheng Feng; Yongfu Su; Fangfang Yan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Saewan and Kumam (2010) have proved the 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. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subspace of a real Banach space E. A mapping A:D(A)⊂E→E* is said to be monotone if for each x,y∈D(A), the following inequality holds:
(1.1)〈x-y,Ax-Ay〉≥0.A mappingA:C→E* is called α-inverse-strongly monotone if there exists α>0 such that
(1.2)〈x-y,Ax-Ay〉≥α∥Ax-Ay∥2.A monotone mappingT is said to be maximal monotone if R(J+rT)=E*, for all r>0, where J is the normalized duality mapping. We denote by T-1(0)={x∈E:0∈Tx} the set of zero points of T.Remark 1.1.
It is well know that ifA:C→E* is an α-inverse-strongly monotone mapping, then it is (1/α)-Lipschitzian, and hence uniformly continuous. Clearly, the class of monotone mappings includes the class of α-inverse strongly monotone mappings.
LetC be a nonempty closed convex subspace of a real Banach space E with dual E* and 〈·,·〉 is the pairing between E and E*. Let f:C×C→R be a bifunction and A:C→E* be a monotone mapping. The generalized equilibrium problem means that finding a u∈C such that
(1.3)f(u,y)+〈Au,y-u〉≥0,∀y∈C.The set of solutions of (1.3) is denoted by GEP(f,A), that is,
(1.4)GEP(f,A)={u∈C:f(u,y)+〈Au,y-u〉≥0,∀y∈C}.IfA=0, then the problem (1.3) is equivalent to that of finding a u∈C such that
(1.5)f(u,y)≥0,∀y∈C,
which is called the equilibrium problem. The solution of (1.5) is denoted by EP(f). If f=0, then the problem (1.3) is equivalent to that of finding a u∈C such that
(1.6)〈Au,y-u〉≥0,∀y∈C,
which is called the variational inequality of Browder type. The solution of (1.6) is denoted by VI(C,A).The problem (1.3) was shown in [1] to cover variational inequality problems, monotone inclusion problems, vector equilibrium problems, numerous problems in physics, minimization problems, saddle point problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, fixed point problem, the complementarity 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 on. In the past two decades, Some methods have been modified for solving the generalized equilibrium problem and the equilibrium problem in Hilbert space and Banach space, see [2–9].The convex feasibility problem (CFP) is the problem for computing points that lay in the intersection of a finite family of closed convex subsetsCj,j=1,2,…,N, of a Banach space E. This problem appears in many fields of applied mathematics, such as the theory of optimization [1], Image Reconstruction from projections [10], and Game Theory [11] and plays an important role in these domains. There is a considerable investigation of (CFP) in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [12]. Also the projection methods have dominated in the iterative approaches to (CFP) in Hilbert spaces. In 1993, Kitahara and Takahashi [13] deal with the convex feasibility problem by convex and sunny nonexpansive retractions in a uniformly convex Banach space.We note that the block iterative method is a common method by many authors to solve (CFP) [14]. In 2008, Plubtieng and Ungchittrakool [15] established block iterative methods for a finite family of relatively nonexpansive mappings and got some strong convergence theorems in a Banach space by using the hybrid method.In 2009, Takahashi and Zembayashi [16] introduced the following iterative scheme in the case that E is uniformly smooth and uniformly convex Banach space:
(1.7)x0=x∈C,arbitrarily,yn=J-1(αnJxn+(1-αn)Jxn);un∈Cs.t.f(un,y)+1rn〈y-un,Jun-Jyn〉,∀y∈CCn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)};xn+1=ΠCn+1x,n=1,2,…,
where T is a relatively nonexpansive mapping and f is a bifunction from C×C into R. They prove that the sequence {xn} converges strongly to q=ΠF(T)∩EP(f) under appropriate conditions.In the same year, Qin et al. [7] introduced a hybrid projection algorithm to two quasi-ϕ-nonexpansive mappings in Banach spaces as follows:
(1.8)x0=x∈C,arbitrarily,C1=C;x1=ΠC1x0;yn=J-1(αnJxn+βnJTxn+γnJSxn);un∈Cs.t.f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C;Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)};xn+1=ΠCn+1x,n=1,2,…,
where ΠCn+1 is the generalized projection from E onto Cn+1. They proved that the sequence {xn} converges strongly to ΠF(S)∩F(T)∩EP(f)x0. Then Petrot et al. [17] improved the notion from a relatively nonexpansive mapping or a quasi-ϕ-nonexpansive mapping to two relatively quasi-nonexpansive mappings; they also proved some strong convergence theorems to find a common element of the set of fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces.In 2010, Saewan and Kumam [18] introduced the following iterative method to find a common element of the set of solutions of an equilibrium problem 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 with Kadec-Klee property. 						(1.9)x0=x∈E,arbitrarily,C1=C;x1=ΠC1x0;yn=J-1(βnJxn+(1-βn)Jzn);zn=J-1(αn,0Jxn+∑i=1∞‍αn,iJSinxn);un∈Cs.t.f(un,y)+〈Ayn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0;Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn};xn+1=ΠCn+1x,n=1,2,….They proved that{xn} converges strongly to Π∩i=1∞F(Si)∩GEP(f,A) under the proper conditions. The same year, Chang et al. [19] 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.Motivated by Saewan and Kumam [18], in this paper we use some new conditions to prove strong convergence theorems for 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. In (1.9) we find iterative step yn is not essential, so we combine yn with zn of (1.9), and use an equally continuous mapping that is more weak than uniformly L-Lipschitz mapping in a uniformly smooth and strictly convex Banach space E, but the Banach space E does not have Kadec-Klee property, under the circumstances we prove strong convergence theorems and get some results same as the results of Saewan and Kumam [18]. The results presented in this paper improve some well-known results in the literature.


## 2. Preliminaries

The spaceE is said to be smooth if the limit
(2.1)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈U={z∈E:∥z∥=1}, and E is said to be uniformly smooth if the limit (2.1) exists uniformly for all x,y∈U. Then a Banach space E is said to be strictly convex if ∥x+y∥/2≤1 for all x,y∈U and x≠y. It is said to be uniformly convex if for each ɛ∈(0,2], there exists δ>0 such that ∥x+y∥/2≤1-δ for all x,y∈U with ∥x-y∥≥ɛ.LetE be a Banach space and let E* be the topological dual of E. For all x∈E and x*∈E*, we denote the value of x* at x by 〈x,x*〉. Then, the duality mapping J:E→2E* is defined by
(2.2)J(x)={f∈E*:〈x,f〉=∥x∥2=∥f∥2},
for every x∈E. By the Hahn-Banach theorem, J(x) is nonempty.The following basic properties can be found in Cioranescu [20].(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.LetE be a smooth, strictly convex, and reflexive Banach space and let C be a nonempty closed convex subset of E. Throughout this paper, we denote by ϕ the function defined by
(2.3)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
Following Alber [21], the generalized projection ΠC from E onto C is defined by
(2.4)ΠC(x)=argminu∈Cϕ(u,x),∀x∈E.IfE is a Hilbert space, then ϕ(x,y)=∥x-y∥2 and ΠC is the metric projection of H onto C. We know the following lemmas for generalized projections.Lemma 2.1 (see Alber [21] and Kamimura and Takahashi [22]).
LetC be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Then
(2.5)ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y),∀x∈C,y∈E.Lemma 2.2 (see Alber [21], Kamimura and Takahashi [22]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space and let x∈E and z∈C. Then
(2.6)z=ΠCx⟺〈y-z,Jx-Jz〉≤0,∀y∈C.Lemma 2.3 (see Kamimura and Takahashi [22]).
LetE be a smooth and uniformly convex Banach space and let {xn} and {yn} be sequences in E such that either {xn} or {yn} is bounded. If limn→∞ϕ(xn,yn)=0, then limn→∞∥xn-yn∥=0.LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E and let T be a mapping from C into itself. We denoted by F(T) the set of fixed points of T, that is F(T)={x:Tx=x}. A point p∈C is said to be an asymptotic fixed point of T if there exists {xn} in C which converges weakly to p and limn→∞∥xn-Txn∥=0. We denote the set of all asymptotic fixed points of T by F^(T).A mappingT from C into itself is said to be relatively nonexpansive [23] if the following conditions are satisfied:(1)
F(T) is nonempty,(2)
ϕ(u,Tx)≤ϕ(u,x),forallu∈F(T),x∈E,(3)
F(T)=F^(T).A mappingT from C into itself is said to be relatively quasi-nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty,(2)
ϕ(u,Tx)≤ϕ(u,x),forallu∈F(T),x∈E,The asymptotic behavior of a relatively nonexpansive mapping was studied in [24]. T is said to be ϕ-nonexpansive, if ϕ(Tx,Ty)≤ϕ(x,y) for x,y∈C. T is said to be quasi-ϕ-asymptotically nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty,(2)
ϕ(u,Tnx)≤knϕ(u,x),forallu∈F(T),x∈E and n≥1,where {kn}  is a real sequence within [1,∞) and kn→1 as n→∞.A mappingT is said to be closed if for any sequence {xn}⊂C with xn→x and Txn→y, then 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 [24, 25]).By using the similar method as in Su et al. [26], the following Lemma is not hard to prove.Lemma 2.4.
LetE be a strictly convex and uniformly smooth real Banach space, let C be a closed convex subset of E, and let T be a closed and quasi-ϕ-asymptotically nonexpansive mapping from C into itself with a sequence {kn}⊂[1,∞) and kn→1 as n→∞. Then F(T) is a closed and convex subset of C.For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:(A1)
f(x,x)=0,forallx∈E,(A2)
f is monotone, that is, f(x,y)+f(y,x)≤0,forallx,y∈E,(A3)
for allx,y,z∈E, limsupt↓0f(tz+(1-t)x,y)≤f(x,y),(A4)
for allx∈C,   y↦f(x,y) is convex and lower semicontinuous.Lemma 2.5 (see Blum and Oettli [1]).
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
(2.7)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 2.6 (see Kumam [5]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E. Let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4) and let A be a monotone mapping from C into E*. For r>0, define a mapping Tr:C→C as follows:
(2.8)Tr(x)={z∈C:f(z,y)+〈Ax,y-z〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈E. Then, the following hold:(1)
Tr is single-valued,(2)
Tr is a firmly nonexpansive-type mapping [6], that is, for all x,y∈E,
(2.9)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉,(3)
F(Tr)=GEP(f,A),(4)
GEP(f,A) is closed and convex.Lemma 2.7 (see Kumam [5]).
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 A be a monotone mapping from C into E*. For x∈E, q∈F(Tr), then the following holds:
(2.10)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 2.8 (see Chang et al. [19]).
Let E 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,
(2.11)∥∑n=1∞‍λnxn∥2≤∑n=1∞‍λn∥xn∥2-λiλig(∥xi-xj∥).Definition 2.9.
A mappingS from C into itself is said to be equally continuous if it is follows that
(2.12)limn→∞∥xn-yn∥=0⇒limn→∞∥Snxn-Snyn∥=0,∀xn,yn∈C.
A mappingS from C into itself is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that
(2.13)∥Snx-Sny∥≤L∥x-y∥,∀x,y∈C.
It is easy to know that eachL-Lipschitz continuous mapping is equally continuous, but the converse may be not true.Definition 2.10.
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 ∩i=1∞F(Si)≠∅, and there exists a sequence {kn}⊂[1,∞) with kn→1 such that for each i≥1,
(2.14)ϕ(p,Sinx)≤knϕ(p,x),∀p∈∩i=1∞F(Si),x∈C,∀n≥1.


## 3. Main Results

Theorem 3.1.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4) and let A be a continuous monotone mapping of C into E*. Let {Si}i=1∞:C→C be an infinite family of closed equally continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞),kn→1 such that F:=∩i=1∞F(Si)∩GEP(f,A) is a nonempty and bounded subset in C. Let {xn} be a sequence generated by
(3.1)x0∈Echosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αn,0Jxn+∑i=1∞‍αn,iJSinxn),un=Trnyn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n=1,2,3,…,
where J is the duality mapping on E, {αn,i}i=0∞ are sequences in [0,1] which satisfies ∑i=0∞‍αn,i=1, θn=supp∈F(kn-1)ϕ(p,xn), and rn∈[a,+∞) for some a>0. If liminfn→∞αn,0αn,i>0 for all n≥0, then {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.
We first show thatCn is closed and convex. It is obvious that Cn is closed. In addition, since
(3.2)ϕ(z,un)≤ϕ(z,xn)+θn⟺∥un∥2-∥xn∥2-2〈z,Jun-Jxn〉-θn≤0,
soCn is convex, therefore, Cn is a closed convex subset of E for all n≥0.
Next, we show thatF⊂Cn for all n≥1. It is clear that F⊂C1=C. Suppose F∈Cn for n>1, by the property of ϕ, ∑i=0∞‍αn,i=1, Lemmas 2.6 and 2.8, and uniformly quasi-ϕ-asymptotically nonexpansive of Sn for each u∈F⊂Cn, then we have(3.3)ϕ(u,un)=ϕ(u,Trnyn)≤ϕ(u,yn)=ϕ(u,J-1(αn,0Jxn+∑i=1∞‍αn,iJSinxn))=∥u∥2-2〈u,(αn,0Jxn+∑i=1∞‍αn,iJSinxn)〉+∥αn,0Jxn+∑i=1∞‍αn,iJSinxn∥2≤∥p∥2-2αn,0〈p,Jxn〉-2∑i=1∞‍αn,i〈u,JSinxn〉+αn,0∥xn∥2+∑i=1∞‍αn,i∥Sinxn∥2-αn,0αn,jg(∥Jxn-JSjnxn∥)≤αn,0ϕ(u,xn)+∑i=1∞‍αn,iϕ(u,Sinxn)≤αn,0ϕ(u,xn)+∑i=1∞‍αn,iknϕ(u,xn)≤knϕ(u,xn)≤ϕ(u,xn)+θn.
This shows thatu∈Cn+1 implies that F⊂Cn for all n≥1 by induction. On the one hand, since xn+1=ΠCnx0 and Cn+1⊂Cn for all n≥1, we have
(3.4)ϕ(xn,x0)≤ϕ(xn+1,x0).
Therefore{ϕ(xn,x0)} is nondecreasing. In the other hand, by Lemma 2.1, we have
(3.5)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(u,x0)-ϕ(u,xn)≤ϕ(u,x0),
for each u∈F(T)⊂Cn for all n≥0. Therefore, {ϕ(xn,x0)} is bounded; this together with (3.4) implies that the limit of {ϕ(xn,x0)} exists.
Since{ϕ(xn,x0)} is bounded, so {xn} is bounded by (1.7), together with limn→∞kn=1, we have that
(3.6)limn→∞θn=0.
From Lemma2.1, we have, for any positive integers n,m, that
(3.7)ϕ(xn+m,xn)=ϕ(xn+m,ΠCnx0)≤ϕ(xn+m,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+m,x0)-ϕ(xn,x0).
Because the limit of{ϕ(xn,x0)} exists, then we have
(3.8)limn→∞ϕ(xn+m,xn)=0
uniformly for positive integers m>1. Since {xn} is a bounded sequence, by using Lemma 2.4, we have
(3.9)limn→∞∥xn+m-xn∥=0
uniformly for positive integers m>1. Hence {xn} is a Cauchy sequence, therefore, there exists a point p∈C such that {xn} converges strongly to p.
In addition, from (3.7) we have limn→∞ϕ(xn+1,xn)=0, this together with the fact xn+1∈Cn implies that
(3.10)ϕ(xn+1,un)≤ϕ(xn+1,xn)+θn.
Taking limit on both side of (3.10) and from (3.6),we get that
(3.11)limn→∞ϕ(xn+1,un)=0.
By using Lemma2.4, we have
(3.12)limn→∞∥xn+1-un∥=0,
which implies that {un} converges strongly to p.
From (3.3), we have ϕ(u,yn)≤(u,xn)+θn, together with un=Trnyn and Lemma 2.7, we have
(3.13)ϕ(un,yn)=ϕ(Trnyn,yn)≤ϕ(u,yn)-ϕ(u,Trnyn)≤ϕ(u,xn)-ϕ(u,Trnyn)+θn=ϕ(u,xn)-ϕ(u,un)+θn
for anyu∈F. This implies that
(3.14)limn→∞ϕ(un,yn)=0.
Therefore, we have(3.15)limn→∞∥un-yn∥=0,
which implies that {yn} converges strongly to p. Thus we have proved that
(3.16)xn→p,un→p,yn→p,
as  n→∞, where p∈C. From (3.1)
(3.17)∥Jxn-Jyn∥=∥Jxn-(αn,0Jxn+∑i=1∞‍αn,iJSinxn)∥=∥∑i=1∞‍αn,i(Jxn-JSinxn)∥≤∑i=1∞‍αn,i∥Jxn-JSinxn∥,
and hence
(3.18)∥Jxn-JSinxn∥≤1∑i=1∞‍αn,i∥Jxn-Jyn∥.
Taking limit on both side of above inequality, byliminfn→∞∑i=1∞‍αn,i>0 and from (3.16), we have
(3.19)limn→∞∥Jxn-JSinxn∥=0.
SinceJ-1 is uniformly norm-to-norm continuous on bounded sets, we have
(3.20)limn→∞∥xn-Sinxn∥=0,
for each i≥1, together with (3.16), we get that
(3.21)limn→∞Sinxn=p,
for each i≥1. Since Si is equally continuous, we have
(3.22)∥Sin+1xn-Sinxn∥=∥Sin+1xn-Sin+1xn+1∥+∥Sin+1xn+1-xn+1∥+∥xn+1-xn∥+∥xn-Sin+1xn∥≤(Li+1)∥xn+1-xn∥+∥Sin+1xn+1-xn+1∥+∥xn-Sin+1xn∥.
Together with (3.16) and (3.20), we have limn→∞∥Sin+1xn-Sinxn∥=0. From (3.21), we have 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).
Next we showp∈GEP(f,A). By un=Trnyn, we have
(3.23)f(un,y)+〈Ayn,y-un〉+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
From (A2), we get that(3.24)〈Ayn,y-un〉+1rn〈y-un,Jun-Jyn〉≥-f(un,y)≥f(y,un),∀y∈C,
and hence
(3.25)0≥-〈Ayn,y-un〉-〈y-un,Jun-Jynrn〉+f(y,un),∀y∈C.
Fort with 0<t<1 and y∈C, let yt=ty+(1-t)p, then yt∈C, from (3.25) we have
(3.26)〈Ayt,yt-un〉≥〈Ayt,yt-un〉-〈Ayn,yt-un〉-〈yt-un,Jun-Jynrn〉+f(yt,un)=〈Ayt-Aun,yt-un〉+〈Aun-Ayn,yt-un〉-〈yt-un,Jun-Jynrn〉+f(yt,un).
Since J is uniformly norm-to-norm continuous on bounded sets, A is monotone and (3.16), we have
(3.27)〈Ayt,yt-un〉≥0.
It follows from (A4) that(3.28)f(yt,p)≤liminfn→∞f(yt,un)≤limn→∞〈Ayt,yt-un〉=〈Ayt,yt-p〉=t〈Ayt,y-p〉.
From the conditions (A1) and (A4), we have(3.29)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,p)≤tf(yt,y)+(1-t)t〈Ayt,y-p〉≤f(yt,y)+(1-t)〈Ayt,y-p〉.
Lettingt→0, we get
(3.30)f(p,y)+〈Ap,y-p〉≥0,∀y∈C.
This implies thatp∈GEP(f,A).
Finally, we show thatp=ΠFx0. Let w=ΠFx0, from p∈F, we have
(3.31)ϕ(p,x0)≥ϕ(w,x0).
Sincexn=ΠCnx0 and w∈F⊂Cn,
(3.32)ϕ(xn,x0)≤ϕ(w,x0),
together with above two hands andlimn→∞xn=p, we obtain
(3.33)ϕ(p,x0)=ϕ(w,x0).
that isp=w=ΠFx0. The proof is completed.By using the similar method of proof as in Theorem3.1, the following theorem is not hard to prove.Theorem 3.2.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4). Let {Si}i=1∞:C→C be an infinite family of closed equally continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞),kn→1 such that F:=∩i=1∞F(Si)∩EP(f) is a nonempty and bounded subset in C. Let {xn} be a sequence generated by
(3.34)x0∈Echosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αn,0Jxn+∑i=1∞‍αn,iJSinxn),un∈Cs.t.f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n=1,2,3,…,
where J is the duality mapping on E, {αn,i}i=0∞ are sequences in [0,1] which satisfies ∑i=0∞‍αn,i=1, θn=supp∈F(kn-1)ϕ(p,xn), and rn∈[a,+∞) for some a>0. If liminfn→∞αn,0αn,i>0 for all n≥0, then {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.
In Theorem3.1, put A=0 we can obtain the conclusion of Theorem 3.2.Theorem 3.3.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4) and let A be a continuous monotone mapping of C into E*. Let S:C→C be an infinite family of closed equally continuous and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞),kn→1 such that F:=F(S)∩GEP(f,A) is a nonempty and bounded subset in C. Let {xn} be a sequence generated by
(3.35)x0∈Echosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αnJxn+(1-αn)JSnxn),un=Trnyn,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0,∀n=1,2,3,…,
where J is the duality mapping on E, {αn} are sequences in [0,1] which satisfies liminfn→∞αn(1-αn)>0 for all n≥0, θn=supp∈F(kn-1)ϕ(p,xn) and rn∈[a,+∞) for some a>0. Then {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.
In Theorem3.1, put Si=S, for i=1,2,…, we can obtain the conclusion of Theorem 3.3.


## 4. Application for Optimization Problem

In this section, we study a kind of optimization problem by using the result of this paper. that is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:(4.1)maxh(x),x∈C,
where h(x) is a convex and lower semicontinuous functional defined on a closed convex subset C of a Banach space H. We denoted by S the set of solutions of (4.1). Let F be a bifunction from C×C to R defined by f(x,y)=h(x)-h(y). We consider the following equilibrium problem, that is, to find x∈C such that
(4.2)f(x,y)≥0,∀y∈C.It is obvious thatEP(F)=S, where EP(F) denote the set of solutions of equilibrium problem (4.2). In addition, it is easy to see that f(x,y) satisfies the conditions (A1)–(A4) in the Section 2. Therefore, from the Theorem 3.1, we can obtain the following theorem.Theorem 4.1.
LetE be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4) and let A be a continuous monotone mapping of C into E*. Let {Si}i=1∞:C→C be an infinite family of closed equally continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {kn}⊂[1,∞),kn→1 such that F:=∩i=1∞F(Si)∩S is a nonempty and bounded subset in C. Let {xn} be a sequence generated by
(4.3)x0∈Echosenarbitrarily,C1=C,x1=ΠC1x0,yn=J-1(αn,0Jxn+∑i=1∞‍αn,iJSinxn),un∈C,s.t.h(un)-h(y)+1rn〈y-un,un-yn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)+θn},xn+1=ΠCn+1x0∀n=1,2,3,…,
where J is the duality mapping on E, {αn,i}i=0∞ are sequences in [0,1] which satisfies ∑i=0∞‍αn,i=1, θn=supp∈F(kn-1)ϕ(p,xn), and rn∈[a,+∞) for some a>0. If liminfn→∞αn,0αn,i>0 for all n≥0, then {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Proof.
By the proof of Theorem3.2, we can obtain Theorem 4.1.It is easy to see that, this paper has some new methods and conditions than the conditions of Takahashi and Zembayashi [16]. In this paper, we prove the convergence theorems for uniformly quasi-ϕ-asymptotically nonexpansive mappings and do not need the Kadec-Klee property of Banach space and use the condition of equally continuous that is more weak different from the condition of uniformly L-Lipscitz.



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*Source: 215261-2012-09-11.xml*