# A New Hybrid Projection Algorithm for System of Equilibrium Problems and Variational Inequality Problems and Two Finite Families of Quasi-ϕ-Nonexpansive Mappings

Authors: Pongrus Phuangphoo; Poom Kumam
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce a modified Mann’s iterative procedure by using the hybrid projection method for solving the common solution of the system of equilibrium problems for a finite family of bifunctions satisfying certain condition, the common solution of fixed point problems for two finite families of quasi-ϕ-nonexpansive mappings, and the common solution of variational inequality problems for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper improves and generalizes some well-known results in the literature.


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## Body

## 1. Introduction

Throughout this paper, we denote byℝ and ℕ the set of all real numbers and the set of all positive numbers, respectively. We also assume that E is a real Banach space and E* is the dual space of E. Let C be a nonempty, closed and convex subset of a real Banach space E with the dual E*. We recall the following definitions.A mappingA:C→C is said to be nonexpansive if
(1)∥Ax-Ay∥≤∥x-y∥,∀x,y∈C.A mappingA:C→E* is said to be monotone if for each x,y∈C, such that
(2)〈x-y,Ax-Ay〉≥0.A mappingA:C→E* is said to be δ-strongly monotone if there exists a constant δ>0 such that
(3)〈x-y,Ax-Ay〉≥δ∥x-y∥2,∀x,y∈C.A mappingA:C→E* is said to be δ-inverse strongly monotone if there exists a constant δ>0 such that
(4)〈x-y,Ax-Ay〉≥δ∥Ax-Ay∥2,∀x,y∈C.IfA is δ-inverse strongly monotone, then it is Lipschitz continuouswith constant 1/δ, that is, for all x,y∈C, ∥Ax-Ay∥≤(1/δ)∥x-y∥.Clearly, the class of monotone mappings include the class ofδ-inverse strongly monotone mappings.LetA:C→E* be a monotone mapping. The variational inequality problem is to find a point z∈C such that
(5)〈y-z,Az〉≥0,∀y∈C.The set of the solution of the variational inequality problem is denoted byVI(C,A).LetC be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space E, let T:C→C be a mapping, and F(T) be the set of fixed points of T.A pointx∈C is said to be afixed pointof T if Tx=x. The set of the solution of the fixed point of T is denoted by F(T):={x∈C:Tx=x}.A pointp∈C is said to be an asymptotic fixed point of T if there exists a sequence {xn}⊂C such that xn⇀p and ∥xn-Txn∥→0. We denoted the set of all asymptotic fixed points of T by F^(T).A pointp∈C is said to be a strong asymptotic fixed pointof T if there exists a sequence {xn}⊂C such that xn→p and ∥xn-Txn∥→0. We denoted the set of all strong asymptotic fixed points of T by F~(T).LetF:C×C→ℝ be a bifunction. The equilibrium problem is to find a point z∈C such that
(6)F(z,y)≥0,∀y∈C.
The set of the solution of equilibrium problem is denoted by EP(F). Numerous problems in sciences, mathematics, optimizations, and economics reduced to find a solution of equilibrium problems. The equilibrium problems include variational inequality problems and fixed point problem, and optimization problems as special cases (see, e.g., [1–3]). Recently, many authors have considered the problem for finding the common solution of fixed point problems, the common solution of equilibrium problems, and the common solution of variational inequality problems.In 1953, Mann [4] introduced the iterative sequence {xn}n∈ℕ which is defined by
(7)xn+1=αnxn+(1-αn)Txn,
where the initial element x0∈C is arbitrary, T is a nonexpansive mapping, and {αn} is the sequence in [0,1] such that limn→∞αn=0 and ∑n=1∞‍αn=∞. The sequence of (7) is generally referred to as the Mann iteration.In 2009, Takahashi and Zembayashi [5] introduced the following iterative scheme by the shrinking projection method, and they proved that {xn}n∈ℕ converges strongly to q=ΠEP(F)∩F(S), under appropriate conditions.Theorem TZ.  LetEbe a uniformly smooth and uniformly convex real Banach space, and letCbe a nonempty, closed and convex subset ofE. LetFbe a bifunction fromC×Ctoℝsatisfying (A1)–(A4) and letTbe a relatively nonexpansive mapping fromCinto itself such thatF(S)∩EP(F)≠∅. Let{xn}be a sequence generated by(8)x0∈C0=Cchosenarbitrary,yn=J-1(αnJxn+(1-αn)JTxn),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+1(x0),∀n≥0,for everyn∈ℕ∪{0},whereJis the duality mapping onE, the sequenceαn⊂[0,1]satisfiesliminfn→∞αn(1-αn)>0, and{rn}n∈ℕ⊂(0,∞)satisfiesliminfn→∞rn>0. Then, the sequence{xn}converges strongly toΠF(S)∩EP(f)x0, whereΠF(S)∩EP(F)x0is the generalized projection ofEontoF(S)∩EP(F).In 2009, Qin et al. [6] extended the iterative process (8) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings. In 2011, Zegeye and Shahzad [7] introduced an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems, and common solutions of the finite family of variational inequality problems for monotone mappings in Banach spaces.Theorem ZS.  LetCbe a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach spaceEwith the dualE*. Letfk:C×C→ℝ,k=1,2,3,…,L, be a finite family of bifunctions. LetSj:C→C,j=1,2,3,…,Dbe a finite family relatively quasi-nonexpansive mappings andAi:C→E*,i=1,2,3,…,Nbe a finite family of continuous monotone mappings.For
x
∈
E,
they define mappingsFrn,Trn:E→Cby(9)Frnx:={1rn〈y-z,Jz-Jx〉z∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn},Trnx:={1rn〈y-z,Jz-Jx〉z∈C:fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn},whereSn≡Sn(modD),An≡An(modN),fn(·,·)≡fn(modL)(·,·), and{rn}n∈ℕ⊂[c1,∞)for somec1>0. Assume thatℱ:=(⋂j=1D‍F(Sj))⋂‍(⋂i=1N‍VI(C,Ai))⋂‍(⋂k=1L‍EP(fk))≠∅. Let{xn}be a sequence generated by(10)x0∈C0=Cchosenarbitrary,zn=Frnxn,un=Trnxn,yn=J-1(α0Jxn+α1Jzn+α2JSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,where the real numbersα0,α1,α2∈(0,1)such thatα0+α1+α2=1. 					Then,{xn}n∈ℕconverges strongly to an element ofℱ.In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative procedure for solving the common solution of system of equilibrium problems for a finite family of bifunctions satisfying certain conditions and the common solution of fixed point problems for two countable families of quasi-ϕ-nonexpansive mappings and the common solution of variational inequality problems for a finite family of monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by the conditions. The results obtained in this paper extend and improve several recent results in this area.


## 2. Preliminaries

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. It is said to be uniformly convexif limn→∞∥xn-yn∥=0 for any two sequences {xn} and {yn} in E such that ∥xn∥≤1, ∥yn∥≤1 and
(11)limn→∞∥xn+yn2∥=1.LetUE = {x∈E:∥x∥=1} be the unit sphere of E. Then the Banach space E is said to be smooth if
(12)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈UE. It is said to be uniformly smooth if the limit (12) is attained uniformly for all x,y∈UE.LetE be a Banach space. Then a function ρE:ℝ+→ℝ+ is said to be the modulus of smoothness of E if
(13)ρE(t)=sup{∥x+y∥+∥x-y∥2-1:∥x∥=1,∥y∥=t}.The spaceE is said to be smoothif ρE(t)>0, ∀t>0 and is said to be uniformly smoothif and only if limt→0+ρE(t)/t=0.Themodulus of convexity of E is the function δE:[0,2]→[0,1] defined by
(14)δE(ϵ)=inf{1-∥x+y2∥:∥x∥≤1,∥y∥≤1;∥x-y∥≥ϵ}.A Banach spaceE is said to be uniformly convexif and only if δE(ϵ)>0 for all ϵ∈(0,2].We recall the following definitions.Definition 1.
LetC be a nonempty set.(1)
A mappingT:C→C is said to be closed if for each {xn}⊂C,xn→x and Txn→y imply Tx=y.(2)
A mappingT:C→C is said to be quasi-ϕ-nonexpansive(relatively quasi-nonexpansive) if F(T)≠∅, and
(15)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).(3)
A mappingT:C→C is said to be relatively nonexpansive [8, 9] if F(T)≠∅,  F(T)=F^(T) and
(16)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).(4)
A mappingT:C→C is said to be weak relatively nonexpansive [10] if F(T)≠∅,  F(T)=F~(T) and
(17)ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T).Remark 2.
We here the following  basic properties.(1)
Each relatively nonexpansive mapping is closed.(2)
The class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse may be not true.(3)
The class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(4)
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.(5)
IfE is a real uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.(6)
IfE is a strictly convex reflexive Banach space, then J-1 is hemicontinuous, that is, J-1 is norm-to-weak* continuous.(7)
IfE is a smooth and strictly convex reflexive Banach space, then J is single-valued, one-to-one, and onto.(8)
A Banach spaceE is uniformly smooth if and only if E* is uniformly convex.(9)
Each uniformly convex Banach spaceE has the Kadec-Klee property, that is, for any sequence {xn}⊂E, if {xn}⇀x∈E and ∥xn∥→∥x∥, then xn→x.(10)
A Banach spaceE is strictly convex if and only if J is strictly monotone, that is,
(18)〈x-y,x*-y*〉>0,wheneverx,y∈E,x≠y,x*∈Jx,y*∈Jy.(11)
Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.(12)
E* is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E.
LetE be a real Banach space and {xn} be a sequence in E. We denote by xn→x and xn⇀x the strong convergence and weak convergence of {xn}, respectively. The normalized duality mappingJ from E to 2E* is defined by
(19)Jx={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},∀x∈E,
where 〈·,·〉 denotes the duality pairing. It is well known that if E is smooth, then J is single-valued and demicontinuous, and if E is uniformly smooth, then J is uniformly continuous on bounded subset of E. Moreover, if E is reflexive and strictly convex Banach space with a strictly convex dual, then J-1 is single-valued, one-to-one, surjective, and it is the duality mapping from E* to E and so JJ-1=IE* and J-1J=IE (see [11, 12]). We note that in a Hilbert space H, the mapping J is the identity operator.
Now, letE be a smooth and strictly convex reflexive Banach space. As Alber (see [13]) and Kamimura and Takahashi (see [14]) did, the Lyapunov functionalϕ:E×E→ℝ+ is defined by
(20)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
It follows from Kohsaka and Takahashi (see [15]) that ϕ(x,y)=0 if and only if x=y and that
(21)(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2.
Further suppose thatC is nonempty, closed and convex subset of E. The generalized, projection (Alber see [13]) ΠC:E→C is defined by for each x∈E,
(22)ΠC(x)=argminy∈Cϕ(x,y).Remark 3.
IfE is a real Hilbert space H, then ϕ(x,y)=∥x-y∥2 and ΠC=PC (the metric projection of H onto C).Lemma 4 (Alber [13]).
LetC be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space E, and let x∈E. Then
(23)ϕ(x,ΠC(y))+ϕ(ΠC(y),y)≤ϕ(x,y),∀x∈C,∀y∈E.Lemma 5 (Kamimura and Takahashi [14]).
LetC be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space E, and let x∈E and p^∈C. Then,
(24)p^=ΠC(x)⟺〈p^-y,Jx-Jp^〉≥0,∀y∈C.Lemma 6 (Qin et al. [6] and Kohsaka and Takahashi [16]).
LetE be a smooth, strictly convex and reflexive Banach space, and A⊂E×E* is a continuous monotone mapping with A-1(0)≠ϕ. Then, it is proved in [16] that the resolvent Jr:=(J+rA)-1J, for r>0 is quasi-ϕ-nonexpansive. Moreover, if T:E→E is quasi-ϕ-nonexpansive, then using the definition of ϕ one can show that F(T) is closed and convex (see [6]).Lemma 7 (Kamimura and Takahashi [14]).
LetE be a uniformly convex and smooth real 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 8.
LetE be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let {xn} and {yn} be two sequences of E, and p∈E. If xn→p and ϕ(xn,yn)→0, then yn→p.For solving the equilibrium problem, let us assume that the bifunctionF:C×C→ℝ satisfies the following conditions: (A1)
F(x,x)=0, for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0,∀x,y∈C;(A3)
limt→0F(tz+(1-t)x,y)≤F(x,y),∀x,y,z∈C;(A4)
for anyx∈C, y↦F(x,y) is convex and lower semicontinuous.Lemma 9 (Blum and Oettli [1]).
LetC be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space E and let F:C×C→ℝ be a bifunction satisfying the following conditions (A1)–(A4). Let r>0 be any given number and x∈E be any point. Then, there exists a point z∈C such that
(25)F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 10 (Takahashi and Zembayashi [5]).
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E, and F:C×C→ℝ be a bifunction satisfying the following conditions(A1)–(A4). Let r>0 be any given number and x∈E be any point defined a mapping Tr:E→C as follows. Then, there exists a point z∈C such that
(26)Tr(x)={1r〈y-z,Jz-Jx〉z∈C:F(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following conclusions hold:(1)
Tr is single-valued;(2)
Tr is a firmly nonexpansive type mapping, that is,
(27)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉,∀x,y∈E;(3)
F(Tr)=EP(F);(4)
EP(F) is a closed and convex subset of C;(5)
ϕ(p,Trx)+ϕ(Trx,x)≤ϕ(p,x),∀p∈F(Tr).Lemma 11 (Zegeye and Shahzad [7]).
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E, and A:C×E*→ℝ be a continuous monotone mapping. Let r>0 be any given number and x∈E be any point defined a mapping Kr:E→C as follows:
(28)Kr(x)={1r〈y-z,Jz-Jx〉z∈C:〈y-z,Az〉+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then, the following conclusions hold:(1)
Kr is single valued;(2)
F(Kr) is a firmly nonexpansive type mapping, that is,
(29)〈Krx-Kry,JKrx-JKry〉≤〈Krx-Kry,Jx-Jy〉,∀x,y∈E;(3)
F(Kr)=VI(C,A);(4)
VI(C,A) is a closed and convex subset of C;(5)
ϕ(p,Krx)+ϕ(Krx,x)≤ϕ(p,x),∀p∈F(Kr).Lemma 12 (Xu [17]).
LetE be a uniformly convex Banach space, let Br(0) be a closed ball of E, where r>0. Then, there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(30)∥αx+βy∥2≤α∥x∥2+β∥y∥2-αβg(∥x-y∥),
for all x,y∈Br(0) and all α,β∈[0,1] with α+β=1.


## 3. Main Results

In this section, we prove a strong convergence theorem which solves the problem for finding a common solution of the system of equilibrium problems and variational inequality problems and fixed point problems in Banach spaces.Theorem 13.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are two finite families of quasi-ϕ-nonexpansive mappings from C into E, where i=1,2,3,…,D;(3)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:E→C by
(31)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn},Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂k=1MSEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C. Let {xn} be a sequence generated by
(32)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJTnzn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ and α,β,γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).Proof.
We will complete this proof by seven steps below.
Step  1. We will show that
C
n
is closed and convex for each
n
≥
0
.
From the definition ofCn, it is obvious that Cn is closed. Moreover, since
(33)ϕ(z,yn)≤ϕ(z,xn)isequivalentto2〈z,Jxn-Jyn〉-∥xn∥2+∥yn∥2≤0.
It follows that Cn is convex for each n≥0. Therefore, Cn is closed and convex for each n≥0.
Step  2. We will show that
Ω
⊂
C
n
for each
n
≥
0
.
From the assumption, we see thatF⊂C0=C. Suppose that F⊂Ck for some k≥1. Now, for p∈Ω, Since Sk and Tk are quasi-ϕ-nonexpansive and by Lemmas 10 and 11, we compute
(34)ϕ(p,yk)=ϕ(p,J-1(αJxk+βJTkzk+γJSkuk))=∥p∥2-2〈p,αJxk+βJTkzk+γJSkuk〉+∥αJxk+βJTkzk+γJSkuk∥2≤∥p∥2-2α〈p,Jxk〉-2β〈p,JTkzk〉-2γ〈p,JSkuk〉+α∥Jxk∥2+β∥JTkzk∥2+γ∥JSkuk∥2=αϕ(p,xk)+βϕ(p,Tkzk)+γϕ(p,Skuk)≤αϕ(p,xk)+βϕ(p,zk)+γϕ(p,uk)=αϕ(p,xk)+βϕ(p,Krkxk)+γϕ(p,Trkxk)≤αϕ(p,xk)+βϕ(p,xk)+γϕ(p,xk)=(α+β+γ)ϕ(p,xk)=ϕ(p,xk).
Therefore, p∈Ck+1. By the induction, this implies that Ω⊂Cn for each n≥0.
Hence, the sequence{xn} is well defined.
Step  3. We will show that the sequence
{
x
n
}
is bounded.
LetG:=⋂n=0∞‍Cn. From Ω⊂Cn for each n≥0 and {xn} is well defined. From the assumption of G, we see that G is closed and convex subset of C.
Letp^=ΠG(x0), where p^ is the unique element that satisfies infx∈Gϕ(x,x0)=ϕ(p^,x0). Now, we will show that ∥xn-p^∥→0 as n→∞.
From the assumption ofCn, we know that C⊃C1⊃C2⊃C3⊃⋯ and since xn=ΠCn(x0) and xn+1=ΠCn+1(x0)∈Cn+1⊂Cn. Using Lemma 4, we get
(35)ϕ(xn,x0)≤ϕ(xn+1,x0)≤⋯≤ϕ(p^,x0),
for all p^∈Ω⊂Cn, where n≥1. Then, the sequence {ϕ(xn,x0)} is bounded. Hence, the sequence {xn} is also bounded.
Step  4. We will show that there exists
p
^
∈
C
such that
x
n
→
p
^, asn→∞.
Sincexn=ΠCn(x0) and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have
(36)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
Therefore, the sequence {ϕ(xn,x0)} is nondecreasing. Hence limn→∞ϕ(xn,x0) exists. By the definition of Cn, one has that Cm⊂Cn and xm=ΠCmx0∈Cn for any positive integer m≥n.
It follows that(37)ϕ(xm,xn)=ϕ(xm,ΠCnx0)≤ϕ(xm,x0)-ϕ(ΠCnx0,x0)=ϕ(xm,x0)-ϕ(xn,x0).
Since limn→∞ϕ(xn,x0) exists, by taking m,n→∞ in (37), we have ϕ(xm,xn)→0.
From Lemma7, it follows that ∥xm-xn∥→0 as m,n→∞. Thus {xn} is a Cauchy sequence.
Without a loss of generalization, we can assume thatxn⇀p0∈E. Since {xn} is bounded and E is reflexive. Since G:=⋂n=0∞‍Cn is closed and convex, it follows that p0∈G, ∀n≥0. Moreover, by using the weak lower semicontinuous of the norm on E and (35), we obtain
(38)ϕ(p^,x0)≤ϕ(p0,x0)≤liminfn→∞ϕ(xn,x0)≤limsupn→∞ϕ(xn,x0)≤infx∈Gϕ(x,x0)=ϕ(p^,x0).
This implies that
(39)limn→∞ϕ(xn,x0)=ϕ(p^,x0)=ϕ(p0,x0)=infx∈Gϕ(x,x0).
By using Lemma 5, we have 〈p^-p0,Jp^-Jp0〉=0 and hence, p^=p0. By the definition of ϕ, we get
(40)liminfn→∞ϕ(xn,x0)=liminfn→∞(∥xn∥2-2〈xn,Jx0〉+∥x0∥2)≥∥p^∥2-2〈p^,Jx0〉+∥x0∥2=ϕ(p^,x0).
Therefore, ∥xn∥→∥p^∥. Since xn⇀p^, by the Kadec-Klee property of E, we obtain
(41)limn→∞xn=p^.
From J is uniformly continuous, we also have
(42)limn→∞Jxn=Jp^.Step  5. We will show thatyn→p^,zn→p^andun→p^asn→∞.
Sincexn+1∈Cn+1, we have ϕ(xn+1,yn)≤ϕ(xn+1,xn)→0, as n→∞. Thus, from (21), we obtain
(43)∥yn∥→∥p^∥,asn→∞,
and so
(44)∥Jyn∥→∥Jp^∥,asn→∞.
This implies that {Jyn} is bounded. Note that reflexivity of E implies reflexivity of E*. Thus, we assume that Jyn⇀y∈E*. Furthermore, the reflexivity of E implies there exists x∈E such that y=Jx. Then, it follows that
(45)ϕ(xn+1,yn)=∥xn+1∥2-2〈xn+1,Jyn〉+∥yn∥2=∥xn+1∥2-2〈xn+1,Jyn〉+∥Jyn∥2.
Taking the liminfn→∞ on both sides of (45) and using weak lower semicontinuous of norm to get that
(46)0≥∥p^∥2-2〈p^,y〉+∥y∥2=∥p^∥2-2〈p^,Jx〉+∥Jx∥2=∥p^∥2-2〈p^,Jx〉+∥x∥2=ϕ(p^,x).
Thus p^=x, and so y=Jx=Jp^. It follows that Jyn⇀Jp^. Now, from (44) and the Kadec-Klee property of E*, we obtain
(47)Jyn→Jp^,asn→∞.
Thus the demicontinuity of J-1 implies that yn⇀p^. Now, from (43) and the fact that E has the Kadec-Klee property, we obtain
(48)limn→∞yn=p^.
In the fact that xn→p^ and yn→p^ as n→∞, we get
(49)limn→∞ϕ(p,zn)=ϕ(p,p^).
Since zn=Krnxn, it follows from Lemma 11 that
(50)ϕ(zn,xn)=ϕ(Krnxn,xn)≤ϕ(p,xn)-ϕ(p,Krnxn)≤ϕ(p,xn)-ϕ(p,zn)→0,asn→∞.
From (21), we obtain
(51)∥zn∥→∥p^∥,
and so {zn} is bounded. Since E is reflexive, we assume that zn⇀z∈E. It follows that
(52)ϕ(zn,xn)=∥zn∥2-2〈zn,Jxn〉+∥Jxn∥2.
Taking the liminfn→∞ on both sides of (52) and using the continuity of J, we get that
(53)0≥∥z∥2-2〈z,Jp^〉+∥Jp^∥2=ϕ(z,p^).
This implies that p^=z and hence zn⇀p^.
Now, from (51) and the Kadec-Klee property of E, we obtain
(54)limn→∞zn=p^.
In the fact that xn→p^, yn→p^ and zn→p^ as n→∞, we get
(55)limn→∞ϕ(p,un)=ϕ(p,p^).
Since un=Trnxn, it follows from Lemma 10 that
(56)ϕ(un,xn)=ϕ(Trnxn,xn)≤ϕ(p,xn)-ϕ(p,Trnxn)≤ϕ(p,xn)-ϕ(p,un)→0,asn→∞.
From (21), we obtain
(57)∥un∥→∥p^∥,
and so {un} is bounded. Since E is reflexive, we assume that un⇀u∈E. It follows that
(58)ϕ(un,xn)=∥un∥2-2〈un,Jxn〉+∥Jxn∥2.
Taking the liminfn→∞ on both sides of (58) and using the continuity of J, we get that
(59)0≥∥u∥2-2〈u,Jp^〉+∥Jp^∥2=ϕ(u,p^).
This implies that p^=u and hence un⇀p^.
Now, from (57) and the Kadec-Klee property of E, we obtain
(60)limn→∞un=p^.Step  6. We will show thatp^∈Ω.
Substep 1. We will show that
p
^
∈
⋂
n
=
1
N
‍
VI
(
C
,
A
n
).
From the definition ofKrn of algorithm (32), we have
(61)〈y-zn,Anzn〉+1rn〈y-zn,Jzn-Jxn〉≥0,∀y∈C.
Let {nk}k∈ℕ⊂ℕ be such that Ank=A1, for all k∈ℕ. Then from (61), we obtain
(62)〈y-znk,A1znk〉+1rnk〈y-znk,Jznk-Jxnk〉≥0,∀y∈C,
and that is
(63)〈y-znk,A1znk〉+〈y-znk,Jznk-Jxnkrnk〉≥0,∀y∈C.
Now, we set vt=tv+(1-t)p^, for all t∈(0,1] and v∈C.
Therefore, we getvt∈C. From (63), it follows that
(64)〈vt-znk,A1vt〉≥〈vt-znk,A1vt〉-〈vt-znk,A1znk〉-〈vt-znk,Jznk-Jxnkrnk〉=〈vt-znk,A1vt-A1znk〉-〈vt-znk,Jznk-Jxnkrnk〉.
From the continuity of J and (41) and (54), we have xnk→p^, znk→p^, as k→∞, we obtain
(65)Jznk-Jxnkrnk→0,ask→∞.
Since A1 is a monotone mapping, we also have 〈vt-znk,A1vt-A1znk〉≥0.
Thus, it follows that(66)0≤limn→∞〈vt-znk,A1vt〉=〈vt-p^,A1vt〉,
and hence
(67)〈y-p^,A1vt〉≥0,∀y∈C.
If t→0, we obtain
(68)〈y-p^,A1p^〉≥0,∀y∈C.
This implies that p^∈VI(C,A1).
Similarly, let{nk}k∈ℕ⊂ℕ be such that Ank=A2, for all k∈ℕ.
Then, we have again thatp^∈VI(C,A2).
Continuing in the same way, we obtain thatp^∈VI(C,An), where n=3,4,5,…,N.
Hence,p^∈⋂n=1N‍VI(C,An).
Substep 2. We will show that
p
^
∈
⋂
k
=
1
M
‍
SEP
(
F
k
)
.
From the definition ofTrn of algorithm (32) and (A2), we have
(69)1rn〈y-un,Jun-Jxn〉≥-Fn(un,y)≥F(y,un),∀y∈C.
Let {nk}k∈ℕ⊂ℕ be such that Fnk=F1, for all k∈ℕ. Then from (69), we obtain
(70)1rnk〈y-unk,Junk-Jxnk〉=〈y-unk,Junk-Jxnkrnk〉≥F1(y,unk),∀y∈C.
From the continuity of J and (41) and (60), we have xnk→p^, unk→p^, as k→∞ and we obtain
(71)Junk-Jxnkrnk→0,ask→∞.
Therefore, F1(y,unk)≤0,∀y∈C.
Now, we setwt=tw+(1-t)p^, for all t∈(0,1] and w∈C.
Consequently, we getwt∈C. And so F1(wt,p^)≤0.
Therefore, from (A1), we obtain(72)0=F1(wt,wt)≤tF1(wt,y)+(1-t)F1(wt,p^)≤tF1(wt,y).
Thus, F1(wt,y)≥0,∀y∈C. From (A3), if t→0, then we get F1(p^,y)≥0,∀y∈C. This implies that p^∈EP(F1).
Similarly, let{nk}k∈ℕ⊂ℕ be such that Fnk=F2, for all k∈ℕ. Then, we have again that p^∈EP(F2).
Continuing in the same way, we obtain thatp^∈EP(Fk), where k=3,4,5,…,M. Hence, p^∈⋂k=1M‍SEP(Fk).
Substep 3. We will show that
p
^
∈
⋂
i
=
1
D
‍
F
(
S
i
).
From algorithm (32) and Lemma 12, we compute
(73)ϕ(p,yn)=ϕ(p,J-1(αJxn+βJTnzn+γJSnun))=∥p∥2-2〈p,αJxn+βJTnzn+γJSnun〉+∥αJxn+βJTnzn+γJSnun∥2≤∥p∥2-2α〈p,Jxn〉-2β〈p,JTnzn〉-2γ〈p,JSnun〉+α∥xn∥2+β∥Tnzn∥2+γ∥Snun∥2-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)=αϕ(p,xn)+βϕ(p,Tnzn)+γϕ(p,Snun)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)≤αϕ(p,xn)+βϕ(p,zn)+γϕ(p,un)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)=αϕ(p,xn)+βϕ(p,Krnxn)+γϕ(p,Trnxn)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)≤αϕ(p,xn)+βϕ(p,xn)+γϕ(p,xn)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)=(α+β+γ)ϕ(p,xn)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)=ϕ(p,xn)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)≤ϕ(p,xn).
From (73), we have
(74)ϕ(p,yn)≤ϕ(p,xn)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)≤ϕ(p,xn)-αγg(∥Jxn-JSnun∥).
From xn→p^, yn→p^, and α,γ>0, we obtain
(75)αγg(∥Jxn-JSnun∥)≤ϕ(p,xn)-ϕ(p,yn)→0,asn→∞.
Therefore,
(76)g(∥Jxn-JSnun∥)→0,asn→∞.
It follows from the property of g that
(77)∥Jxn-JSnun∥→0,asn→∞.
From (42), we have Jxn→Jp^,  as  n→∞. Then,
(78)JSnun→Jp^,asn→∞,
and so
(79)∥Snun∥→∥p^∥,asn→∞.
Moreover, the demicontinuity of J-1 implies that Snun⇀p^ as n→∞. Thus, the Kadec-Klee property of E, we obtain
(80)Snun→p^,asn→∞.
Let {nk}k∈ℕ⊂ℕ be such that Snk=S1, for all k∈ℕ.
Then, from (60), we have unk→p^, as n→∞. It follows from (80) and the closedness of S1 that
(81)p^=limk→∞Snkunk=limk→∞S1unk=S1p^.
This implies that p^∈F(S1).
Similarly, let{nk}k∈ℕ⊂ℕ be such that Snk=S2, for all k∈ℕ. Then, we have again that p^∈F(S2).
Continuing in the same way, we obtain thatp^∈F(Si), where i=3,4,5,…,D.
Hence,p^∈⋂i=1D‍F(Si).
Substep 4. We will show that
p
^
∈
⋂
i
=
1
D
‍
F
(
T
i
).
From (73), we have
(82)ϕ(p,yn)≤ϕ(p,xn)-αβg(∥Jxn-JTnzn∥)-αγg(∥Jxn-JSnun∥)≤ϕ(p,xn)-αβg(∥Jxn-JTnzn∥).
From xn→p^, yn→p^, and α,β>0, we obtain
(83)αβg(∥Jxn-JTnzn∥)≤ϕ(p,xn)-ϕ(p,yn)→0,asn→∞.
Therefore,
(84)g(∥Jxn-JTnzn∥)→0,asn→∞.
It follows from the property of g that
(85)∥Jxn-JTnzn∥→0,asn→∞.
From (42), we have Jxn→Jp^, as  n→∞. Then,
(86)JTnzn→Jp^,asn→∞,
and so
(87)∥Tnzn∥→∥p^∥,asn→∞.
Moreover, the demicontinuity of J-1 implies that Tnzn⇀p^ as n→∞. Thus, the Kadec-Klee property of E, we obtain
(88)Tnzn→p^,asn→∞.
Let {nk}k∈ℕ⊂ℕ be such that Tnk=T1, for all k∈ℕ.
Then, from (54), we have znk→p^, as n→∞. It follows from (88) and the closedness of T1 that
(89)p^=limk→∞Tnkznk=limk→∞T1znk=T1p^.
This implies that p^∈F(T1).
Similarly, let{nk}k∈ℕ⊂ℕ be such that Tnk=T2, for all k∈ℕ. Then, we have again that p^∈F(T2).
Continuing in the same way, we obtain thatp^∈F(Ti), where i=3,4,5,…,D.
Hence,p^∈⋂i=1D‍F(Ti).
From Substeps (6.1)–(6.4), we can conclude that(90)p^∈Ω:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂‍(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)).
Step  7. Finally, we will show that
p
^
=
Π
Ω
(
x
0
)
.
Fromxn=ΠCn(x0), we have
(91)〈Jx0-Jxn,xn-p〉≥0,∀p∈Ω.
Taking n→∞ in (91), one has
(92)〈Jx0-Jp^,p^-p〉≥0,∀p∈Ω.
Now, we have p^∈Ω and by Lemma 5, we get
(93)p^=ΠΩ(x0).
This completes the proof of Theorem 13.If we setM=N=D=1 in Theorem 13, then we obtain the following result.Corollary 14.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
F:C×C→ℝ is a bifunction satisfying conditions (A1)–(A4);(2)
T and S are two quasi-ϕ-nonexpansive mappings from C into E;(3)
A:C→E* is a continuous monotone mapping.(4)
Forx∈E, one defines the mappings Krn,Trn:E→C as follows:
(94)Krn(x):={1rnz∈C:〈y-z,Az〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn};Trn(x):={1rnz∈C:F(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=F(T)∩F(S)∩EP(F)∩VI(C,A) is a nonempty and bounded in C and let {xn} be a sequence generated by
(95)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJTzn+γJSun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).If we setTi=I, for any i=1,2,3,…,D in Theorem 13, then we obtain the following result which extends and improves the result’s Zegeye and Shahzad [7].Corollary 15 (Zegeye and Shahzad [7]).
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Si}i=1D:C→E is a finite family of quasi-ϕ-nonexpansive mappings, where i=1,2,3,…,D;(3)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:E→C by
(96)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn};Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Si))⋂‍(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(97)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJzn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).


## 4. Deduced Theorems

If we setFk≡0, for any k=1,2,3,…,M in Theorem 13, then we obtain the following result.Corollary 16.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Ti}i=1D and {Si}i=1D are finite families of quasi-ϕ-nonexpansive mappings from C into E, where i=1,2,3,…,D;(2)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(3)
Forx∈E, we define the mappings Krn:E→C by
(98)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(99)x0∈C0=C,chosenarbitrary,zn=Krnxn,yn=J-1(αJxn+βJTnzn+γJSnxn),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).If we setAn≡0, for any n=1,2,3,…,N in Theorem 13, then we obtain the following result.Corollary 17.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of quasi-ϕ-nonexpansive mappings from C into E, where i=1,2,3,…,D;(3)
Forx∈E, one defines the mappings Trn:E→C by
(100)Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂k=1M‍SEP(Fk)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(101)x0∈C0=C,chosenarbitrary,un=Trnxn,yn=J-1(αJxn+βJTnxn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).If we setFk≡0, for any k=1,2,3,…,M and An≡0, for any n=1,2,3,…,N in Theorem 13, then we obtain the following result.Corollary 18.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that {Ti}i=1D and {Si}i=1D are finite families of quasi-ϕ-nonexpansive mappings from C into E, where i=1,2,3,…,D.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(102)x0∈C0=C,chosenarbitrary,yn=J-1(αJxn+βJTnxn+γJSnxn),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).


## 5. Some Applications

### 5.1. Application to Weak Relatively Nonexpansive Mappings

If we change the condition(2) in Theorem 13 as follows: {Ti}i=1D and {Si}i=1D are finite families of weak relatively nonexpansive mappings. From Remark 2(2), every weak relatively nonexpansive mappings is quasi-ϕ-nonexpansive mappings. Then, we obtain the following result.Corollary 19.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of weak relatively nonexpansive mappings from C into E, where i=1,2,3,…,D;(3)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:E→C by
(103)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn};Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(104)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJTnzn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).


### 5.2. Application to Relatively Nonexpansive Mappings

If we change the condition(2) in Theorem 13 as follows: {Ti}i=1D and {Si}i=1D are finite families of relatively nonexpansive mappings. From Remark 2(3) and (2) every relatively nonexpansive mappings is weak relatively nonexpansive mappings and every weak relatively nonexpansive mappings is quasi-ϕ-nonexpansive mappings. Then, we obtain the following result.Corollary 20.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of relatively nonexpansive mappings from C into E, where i=1,2,3,…,D;(3)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:E→C by
(105)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn};Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(106)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJTnzn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).


### 5.3. Application to Hilbert Spaces

IfE=H, a real Hilbert space, then E is uniformly smooth and strictly convex real Banach space. In this case, J=I and ΠC=PC. Then, we obtain the following result.Corollary 21.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of nonexpansive mappings from C into C, where i=1,2,3,…,D;(3)
{An}n=1N:C→H is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:H→C by
(107)Krn′(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,z-x〉≥0,∀y∈C1rn};Trn′(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,z-x〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂‍(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(108)x0∈C0=C,chosenarbitrary,zn=Krn′xn,un=Trn′xn,yn=αxn+βTnzn+γSnun,Cn+1={z∈Cn:∥z-yn∥≤∥z-xn∥},xn+1=PCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α,β,γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to an element of Ω.Remark 22.
Our theorem extends and improves the corresponding results in [5–7] in the following aspect.(a)
For the mapping, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, and weak relatively nonexpansive mappings to more general than quasi-ϕ-nonexpansive mappings.(b)
For the common solution, we extend the common solution of a single finite family of quasi-ϕ-nonexpansive mappings to two finite families of quasi-ϕ-nonexpansive mappings.


## 5.1. Application to Weak Relatively Nonexpansive Mappings

If we change the condition(2) in Theorem 13 as follows: {Ti}i=1D and {Si}i=1D are finite families of weak relatively nonexpansive mappings. From Remark 2(2), every weak relatively nonexpansive mappings is quasi-ϕ-nonexpansive mappings. Then, we obtain the following result.Corollary 19.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of weak relatively nonexpansive mappings from C into E, where i=1,2,3,…,D;(3)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:E→C by
(103)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn};Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(104)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJTnzn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).


## 5.2. Application to Relatively Nonexpansive Mappings

If we change the condition(2) in Theorem 13 as follows: {Ti}i=1D and {Si}i=1D are finite families of relatively nonexpansive mappings. From Remark 2(3) and (2) every relatively nonexpansive mappings is weak relatively nonexpansive mappings and every weak relatively nonexpansive mappings is quasi-ϕ-nonexpansive mappings. Then, we obtain the following result.Corollary 20.
LetC be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space E which has the Kadec-Klee property. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of relatively nonexpansive mappings from C into E, where i=1,2,3,…,D;(3)
{An}n=1N:C→E* is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:E→C by
(105)Krn(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn};Trn(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(106)x0∈C0=C,chosenarbitrary,zn=Krnxn,un=Trnxn,yn=J-1(αJxn+βJTnzn+γJSnun),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α, β, γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to p^=ΠΩ(x0).


## 5.3. Application to Hilbert Spaces

IfE=H, a real Hilbert space, then E is uniformly smooth and strictly convex real Banach space. In this case, J=I and ΠC=PC. Then, we obtain the following result.Corollary 21.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Suppose that(1)
{Fk}k=1M:C×C→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where k=1,2,3,…,M;(2)
{Ti}i=1D and {Si}i=1D are finite families of nonexpansive mappings from C into C, where i=1,2,3,…,D;(3)
{An}n=1N:C→H is a finite family of continuous monotone mappings, where n=1,2,3,…,N;(4)
Forx∈E, one defines the mappings Krn,Trn:H→C by
(107)Krn′(x):={1rnz∈C:〈y-z,Anz〉+1rn〈y-z,z-x〉≥0,∀y∈C1rn};Trn′(x):={1rnz∈C:Fn(z,y)+1rn〈y-z,z-x〉≥0,∀y∈C1rn}.
Assume thatΩ:=(⋂i=1D‍F(Ti))⋂‍(⋂i=1D‍F(Si))⋂‍(⋂k=1M‍SEP(Fk))⋂‍(⋂n=1N‍VI(C,An)) is a nonempty and bounded in C and let {xn} be a sequence generated by
(108)x0∈C0=C,chosenarbitrary,zn=Krn′xn,un=Trn′xn,yn=αxn+βTnzn+γSnun,Cn+1={z∈Cn:∥z-yn∥≤∥z-xn∥},xn+1=PCn+1(x0),∀n≥0,
where An≡An(modN), Fn≡Fn(modM), Tn≡Tn(modD), Sn≡Sn(modD), {rn}⊂[a,∞) for some a>0 and n∈ℕ, and α,β,γ are real numbers in (0,1) such that α+β+γ=1.
Then, the sequence{xn}n=0∞ converges strongly to an element of Ω.Remark 22.
Our theorem extends and improves the corresponding results in [5–7] in the following aspect.(a)
For the mapping, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, and weak relatively nonexpansive mappings to more general than quasi-ϕ-nonexpansive mappings.(b)
For the common solution, we extend the common solution of a single finite family of quasi-ϕ-nonexpansive mappings to two finite families of quasi-ϕ-nonexpansive mappings.



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*Source: 107296-2013-02-11.xml*