# Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems

Authors: H. Zegeye; N. Shahzad
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We introduce an iterative process which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces.


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## Body

## 1. Introduction

LetE be a real Banach space with dual E*. The function ϕ:E×E→ℝ+, defined by
(1.1)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,forx,y∈E,
is studied by Alber [1] and Reich [2], where J is the normalized duality mapping from E to 2E* defined by  Jx:={f*∈E*:〈x,f*〉=||x||2=||f*||2}, where 〈·,·〉 denotes the generalized duality pairing. It is well known that E is smooth if and only if J is single valued and if E is uniformly smooth then J is uniformly continuous on bounded subsets of E. We note that in a Hilbert space H,  J is the identity operator.LetC be a nonempty closed convex subset of a Hilbert space H. It is well known that the metric projection of H onto C,  PC:H→C, is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this direction, Alber [1] introduced a generalized projection operator ΠC in a Banach space E which is an analogue of metric projection in Hilbert spaces.LetC be a nonempty closed and convex subset of a reflexive, strictly convex and smooth Banach space E. The generalized projection mapping, introduced by Alber [1], is a mapping ΠC:E→C, that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(y,x), that is, ΠCx=x-, where x- is the solution to the minimization problem
(1.2)ϕ(x-,x)=min{ϕ(y,x),y∈C}.
Let C be a nonempty closed convex subset of a Banach space E. Let T:C→C be a single-valued mapping. An element p∈C is called a fixed point of T if T(p)=p. The set of fixed points of T is denoted by F(T). A point p in C is said to be an asymptotic fixed pointof  T (see [2]) 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). T is said to be nonexpansive if ||Tx-Ty||≤||x-y|| for each x,y∈C and is called relatively nonexpansive if (A1) F(T)≠∅; (A2) ϕ(p,Tx)≤ϕ(p,x) for x∈C and p∈F(T) and (A3) F(T)=F^(T).LetC be a nonempty closed convex subset of a Banach space E and let N(C) and CB(C) denote the family of nonempty subsets and nonempty closed bounded subsets of C, respectively. Let H be the Hausdorff metric on CB(C) defined by
(1.3)H(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)},
for all A,B∈CB(C), where d(a,B)=inf{||a-b||:b∈B} is the distance from the point a to the subset B.LetT:C→CB(C) be a multivalued mapping. T is said to be a nonexpansiveif H(Tx,Ty)≤||x-y||, for x,y∈C. An element p∈C is called a fixed point of T, if p∈F(T), where F(T):={p∈C:p∈T(p)}. A point p∈C called an asymptotic fixed point of T, if there exists a sequence {xn} in C which converges weakly to p such that limn→∞d(xn,Txn)=0. T is said to be relatively nonexpansiveif (B1) F(T)≠∅; (B2) ϕ(p,u)≤ϕ(p,x) for x∈C, u∈Tx, p∈F(T) and (B3) F(T)=F^(T), where F^(T) is the set of asymptotic fixed points of T.We remark that the class of relatively nonexpansive single-valued mappings is contained in a class of relatively nonexpansive multi-valued mappings. An example of relatively nonexpansive multi-valued mapping by Homaeipour and Razani [3] is given below.Example 1.1.
LetI=[0,1], X=Lp(I), 1<p<∞ and C={f∈X:f(x)≥0,forallx∈I}. Let T:C→CB(C) be defined by
(1.4)T(f)={{g∈C:f(x)-34≤g(x)≤f(x)-14,∀x∈I},iff(x)>1,x∈I,{0},otherwise.
It is shown in [3] that T is relatively nonexpansive multi-valued mapping which is not nonexpansive.The study of fixed points for multi-valued nonexpansive mappings in relation to Hausdorff metric was introduced by Markin [4] (see also [5]). Since then a lot of activity in this area and fixed point theory for multi-valued nonexpansive mappings has been developed which has some nontrivial applications in pure and applied sciences including control theory, convex optimization, differential inclusion, and economics (see, e.g., [6] and references therein). Later, Lim [7] established the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces.It is well known that the normalMann’s iterative [8] algorithm has only weak convergence in an infinite-dimensional Hilbert space even for nonexpansive single-valued mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann’s iteration algorithm, the so called hybrid projection iteration method is such a modification. The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [9] in 1968. For 40 years, (HPIA) has received rapid developments. For details, the readers are referred to papers [10–12] and the references therein.In 2003, Nakajo and Takahashi [12] proposed the following modification of the Mann iteration method for a nonexpansive single-valued mapping T in a Hilbert space H:
(1.5)x0∈C,chosen  arbitrary,yn=αnxn+(1-αn)Txn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qn(x0),n≥0,
where C is a closed convex subset of H, PC denotes the metric projection from H onto C. They proved that if the sequence {αn} is bounded above from one then the sequence {xn} generated by (1.5) converges strongly to PF(T)(x0).In spaces more general than Hilbert spaces, Matsushita and Takahashi [11] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive single-valued mapping T in a Banach space E:
(1.6)x0∈C,chosen  arbitrary,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∩Qn(x0),n≥0.
They proved the following convergence theorem.Theorem MT.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive single-valued mapping from C into itself, and let {αn} be a sequence of real numbers such that 0≤αn<1 and limsupn→∞αn<1. Suppose that {xn} is given by (1.6), where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to ΠF(T)x0, where ΠF(T)(·) is the generalized projection from E onto F(T).Letf:C×C→ℝ be a bifunction, where ℝ is the set of real numbers. The equilibrium problem for f is
(1.7)findingx*∈Csuch  thatf(x*,y)≥0,∀y∈C.
The solution set of (1.7) is denoted by EP(f).Iff(x,y)=〈Ax,y-x〉, where A:C→C is a monotone mapping, then the problem (1.7) reduces to the system of variational inequality problem
(1.8)find  an  elementx*∈Csuch  that〈Ax*,y-x*〉≥0,∀y∈C.
That is, the problem (1.8) is a special case of (1.7). The set of solutions of inequality (1.8) is denoted by VI(C,A).For solving the equilibrium problem for a bifunctionf:C×C→R, we assume that f 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, for all x,y∈C,(A3) for eachx,y,z∈C, limt→0+f(tz+(1-t)x,y)≤f(x,y),(A4) for eachx∈C,  y→f(x,y) is convex and lower semicontinuous.Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive single-valued mapping and the set of solutions of an equilibrium problems in the frame work of Hilbert spaces and Banach spaces, respectively: see, for instance, [2, 13–21] and the references therein.In [22], Kumam introduced the following iterative scheme in a Hilbert space:
(1.9)x0∈H,un∈Csuch  thatf(un,y)+1rn〈y-un,un-yn〉≥0,∀y∈C,wn=αnxn+(1-αn)Tun,Cn={z∈H:∥wn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn+1∩Qn(x0),n≥0,
for finding a common element of the set of fixed point of nonexpansive single-valued mapping T and set of solution of equilibrium problems.In the case thatE is a Banach space, Takahashi and Zembayashi [16] introduced the following iterative scheme which is called the shrinking projection method:
(1.10)x0∈C,chosen  arbitrary,yn=J-1(αnJxn+(1-αn)JTxn),un∈Csuch  thatf(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,
where J is the duality mapping on E, ΠC is the generalized projection from E onto C and T is relatively nonexpansive single-valued mapping. They proved that the sequence {xn} converges strongly to a common element of the set of fixed point of relatively nonexpansive single-valued mapping and set of solution of equilibrium problem under appropriate conditions.We remark that the computation ofxn+1 in (1.9) and (1.10) is not simple because of the involvement of computation of Cn+1 from Cn for each n≥0.More recently, Homaeipour and Razani [3] studied the following iterative scheme for a fixed point of relatively nonexpansive multi-valued mapping in uniformly convex and uniformly smooth Banach space E:
(1.11)x0∈C,chosen  arbitrary,xn+1=ΠCJ-1(αnJxn+(1-αn)Jzn),zn∈Txn,n≥0,
where {αn}⊂(0,1) for all n≥0 and  liminfn→∞αn(1-αn)>0. They proved that if J is weakly sequentially continuous then the sequence {xn} converges weakly to a fixed point of T. Furthermore, it is shown that the scheme converges strongly to a fixed point of T if interior ofF(T) isnonempty.But it is worth mentioning that the convergence of the scheme is eitherweak or it requires that the interiorof  F(T) is nonempty.In this paper, motivated by Kumam [22], Takahashi and Zembayashi [16], and Homaeipour and Razani [3], we construct an iterative scheme which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces. Our scheme does not involve computation of Cn and Qn, for each n≥0, and the requirement that the interior of F is nonempty is dispensed with. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.


## 2. Preliminaries

LetE be a normed linear space with  dimE≥2. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by
(2.1)ρE(τ):=sup{∥x+y∥+∥x-y∥2-1:∥x∥=1;∥y∥=τ}.
The space E is said to be smooth if  ρE(τ)>0,  for all  τ>0 and E is called uniformly smooth if and only if limt→0+(ρE(t)/t)=0.Themodulus of convexity of E is the function δE:(0,2]→[0,1] defined by
(2.2)δE(ϵ):=inf{1-∥x+y2∥:∥x∥=∥y∥=1;ϵ=∥x-y∥}.E is called uniformly convex if and only if δE(ϵ)>0, for every ϵ∈(0,2].In the sequel, we will need the following results.Lemma 2.1   (see [1]).
LetK be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E and let x∈E. Then for all  y∈K,
(2.3)ϕ(y,ΠKx)+ϕ(ΠKx,x)≤ϕ(y,x).We make use of the functionV:E×E*→ℝ, defined by
(2.4)V(x,x*)=∥x∥2-2〈x,x*〉+∥x*∥2,∀x∈E,x*∈E*,
studied by Alber [1]. That is, V(x,x*)=ϕ(x,J-1x*) for all x∈E and x*∈E*. We know the following lemma.Lemma 2.2 (see [1]).
LetE be reflexive strictly convex and smooth Banach space with E* as its dual. Then
(2.5)V(x,x*)+2〈J-1x*-x,y*〉≤V(x,x*+y*),
for all x∈E and x*,y*∈E*.Lemma 2.3 (see [1]).
LetC be a convex subset of a real smooth Banach space E. Let x∈E. Then x0=ΠCx if and only if
(2.6)〈z-x0,Jx-Jx0〉≤0,∀z∈C.Lemma 2.4 (see [23]).
LetE be a uniformly convex Banach space and BR(0) be a closed ball of E. Then, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.7)∥α1x1+α2x2+⋯+αNxN∥2≤∑i=1N‍αi∥xi∥2-αiαjg(∥xi-xj∥),
for i,j∈{1,…,N}, αi∈(0,1) such that ∑i=1N‍αi=1, and xi∈BR(0):={x∈E:||x||≤R}, for i=1,2,…,N.Lemma 2.5 (see [24]).
LetE be a real smooth and uniformly convex Banach space and let {xn} and {yn} be two sequences of E. If either {xn} or {yn} is bounded and ϕ(xn,yn)→0 as n→∞, then xn-yn→0, as n→∞.Proposition 2.6 (see [3]).
LetE be a strictly convex and smooth Banach space and C be a nonempty closed convex subset of E. Let T:C→N(C) be a relatively nonexpansive multi-valued mapping. Then F(T) is closed and convex.Lemma 2.7 (see [16]).
LetC be a nonempty, closed and convex subset of a uniformly smooth, strictly convex and reflexive real Banach space E. Let f be a bifunction from C×C to ℝ which satisfies conditions (A1)–(A4). For r>0 and x∈E, define the mapping Fr:E→C as follows:
(2.8)Frx:={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C}.
Then the following statements hold:(1)
Fr is single-valued,(2)
F(Fr)=EP(f),(3)
ϕ(q,Frx)+ϕ(Frx,x)≤ϕ(q,x), for q∈F(Fr),(4)
EP(f) is closed and convex.Lemma 2.8 (see [25]).
Let{an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈ℕ. Then there exists a nondecreasing sequence {mk}⊂ℕ such that mk→∞ and the following properties are satisfied by all (sufficiently large) numbers k∈ℕ:
(2.9)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.Lemma 2.9 (see [26]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.10)an+1≤(1-βn)an+βnδn,n≥n0,forsomen0∈ℕ,
where {βn}⊂(0,1) and {δn}⊂R satisfying the following conditions: limn→∞βn=0,∑n=1∞‍βn=∞, and limsupn→∞δn≤0. Then, limn→∞an=0.


## 3. Main Result

LetC be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive real Banach space E with dual E*. Let f:C×C→ℝ be a bifunction. For the rest of this paper, Frnx is a mapping defined as follows. For x∈E, let Frn:E→C be given by
(3.1)Frnx:={z∈C:f(z,y)+1rn〈y-z,Jz-Jx〉≥0,∀y∈C},
where {rn}n∈ℕ⊂[c1,∞), for some c1>0.Theorem 3.1.
LetC be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let f:C×C→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let Ti:C→CB(C), for i=1,2,…,N, be a finite family of relatively nonexpansive multi-valued mappings. Assume that F:=∩i=1NF(Ti)∩EP(f) is nonempty. Let {xn} be a sequence generated by
(3.2)x0=w∈C,chosenarbitrarily,wn=Frnxn,yn=ΠCJ-1(αnJw+(1-αn)Jwn),xn+1=J-1(βn,0Jwn+∑i=1N‍βn,iJun,i),un,i∈Tiyn,n≥0,
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞‍αn=∞, {βn,i}⊂[a,b]⊂(0,1),  for i=1,2,…,N, satisfying βn,0+βn,1+⋯+βn,N=1, for each n≥0. Then {xn} converges strongly to an element of F.Proof.
SinceF is nonempty closed and convex, put x*:=ΠFw. Now from (3.2), Lemma 2.7(3) and property of ϕ, we get that
(3.3)ϕ(x*,yn)=ϕ(x*,ΠCJ-1(αnJw+(1-αn)Jwn))≤ϕ(x*,J-1(αnJw+(1-αn)Jwn))=∥x*∥2-2〈x*,αnJw+(1-αn)Jwn〉+∥αnJw+(1-αn)Jwn∥2≤∥x*∥2-2αn〈x*,Jw〉-2(1-αn)〈x*,Jwn〉+αn∥w∥2+(1-αn)∥wn∥2≤αnϕ(x*,w)+(1-αn)ϕ(x*,wn)=αnϕ(x*,w)+(1-αn)ϕ(x*,Frnxn)≤αnϕ(x*,w)+(1-αn)ϕ(x*,xn).
Now, from (3.2), Lemma 2.7(3), relatively nonexpansiveness of Ti, property of ϕ and (3.3), we have that
(3.4)ϕ(x*,xn+1)=ϕ(x*,J-1(βn,0Jwn+∑i=1N‍βn,iJun,i))≤βn,0ϕ(x*,wn)+∑i=1N‍βn,iϕ(x*,un,i)=βn,0ϕ(x*,Frnxn)+∑i=1N‍βn,iϕ(x*,un,i)≤βn,0ϕ(x*,xn)+(1-βn,0)ϕ(x*,yn)≤βn,0ϕ(x*,xn)+(1-βn,0)[αnϕ(x*,w)+(1-αn)ϕ(x*,xn)]≤δnϕ(x*,w)+(1-δn)ϕ(x*,xn),
where δn=(1-βn,0)αn. Thus, by induction,
(3.5)ϕ(x*,xn+1)≤max{ϕ(x*,x0),ϕ(x*,w)},∀n≥0,
which implies that {xn} is bounded and hence {yn} and {wn} are bounded. Now let zn=J-1(αnJw+(1-αn)Jwn). Then we have that yn=ΠCzn. Using Lemma 2.2 and property of ϕ, we obtain that
(3.6)ϕ(x*,yn)≤ϕ(x*,zn)=V(x*,Jzn)≤V(x*,Jzn-αn(Jw-Jx*))-2〈zn-x*,-αn(Jw-Jx*)〉=ϕ(x*,J-1(αnJx*+(1-αn)Jwn)+2αn〈zn-x*,Jw-Jx*〉)≤αnϕ(x*,x*)+(1-αn)ϕ(x*,wn)+2αn〈zn-x*,Jw-Jx*〉=(1-αn)ϕ(x*,wn)+2αn〈zn-x*,Jw-Jx*〉≤(1-αn)ϕ(x*,xn)+2αn〈zn-x*,Jw-Jx*〉.
Furthermore, from (3.2), Lemma 2.4, relatively nonexpansiveness of Ti, for each i=1,2,…,N, Lemma 2.7(3), and (3.6) we have that
(3.7)ϕ(x*,xn+1)=ϕ(x*,J-1(βn,0Jwn+∑i=1N‍βn,iJun,i))≤βn,0ϕ(x*,wn)+∑i=1N‍βn,iϕ(x*,un,i)-βn,0βn,ig(∥Jwn-Jun,i∥)=βn,0ϕ(x*,Frnxn)+∑i=1N‍βn,iϕ(x*,un,i)-βn,0βn,ig(∥Jwn-Jun,i∥)≤βn,0(ϕ(x*,xn)-ϕ(xn,wn))+(1-βn,0)ϕ(x*,yn)-βn,0βn,ig(∥Jwn-Jun,i∥)≤βn,0ϕ(x*,xn)-βn,0ϕ(xn,wn)+(1-βn,0)×[(1-αn)ϕ(x*,xn)+2αn〈zn-x*,Jw-Jx*〉]-βn,0βn,ig(∥Jwn-Jun,i∥)=(1-δn)ϕ(x*,xn)+2δn〈zn-x*,Jw-Jx*〉-βn,0ϕ(xn,wn)-βn,0βn,ig(∥Jwn-Jun,i∥),
and hence
(3.8)ϕ(x*,xn+1)≤(1-δn)ϕ(x*,xn)+2δn〈zn-x*,Jw-Jx*〉,
where δn:=αn(1-βn,0), for all n∈ℕ. Note that δn satisfies limnδn=0 and ∑n=1∞‍δn=∞.Now, we consider two cases.Case 1.
Suppose that there existsn0∈ℕ such that {ϕ(x*,xn)} is nonincreasing for all n≥n0. In this situation, {ϕ(x*,xn)} is then convergent. Then from (3.7), we have that ϕ(xn,wn)→0 and hence Lemma 2.5 implies that
(3.9)xn-wn→0,asn→∞.
Moreover, from (3.7), we have that βn,0βn,ig(||Jwn-Jun,i||)→0, as n→∞, which implies by the property of g that Jwn-Jun,i→0, as n→∞, for each i∈{1,2,…,N}, and hence, since J-1 uniformly continuous on bounded sets, we obtain that
(3.10)wn-un,i→0,asn→∞.
Furthermore, by Lemma 2.1, property of ϕ and the fact that αn→0, as n→∞, imply that
(3.11)ϕ(wn,yn)=ϕ(wn,ΠCzn)≤ϕ(wn,zn)=ϕ(wn,J-1(αnJw+(1-αn)Jwn))≤αnϕ(wn,w)+(1-αn)ϕ(wn,wn)→0,asn→∞,
and hence
(3.12)wn-yn→0,wn-zn→0,asn→∞.
Therefore, from (3.9), (3.10), and (3.12), we obtain that
(3.13)xn-zn→0,yn-xn→0,asn→∞,(3.14)d(yn,Tiyn)≤∥yn-un,i∥≤∥yn-wn∥+∥wn-un,i∥→0,
as n→∞, for each i∈{1,2,…,N}.Let{zni} be a subsequence of {zn} such that zni⇀z  and limsupn→∞〈zn-x*,Jw-Jx*〉=limi→∞〈zni-x*,Jw-Jx*〉. Then, from (3.12), (3.13), and the uniform continuity of J, we get that
(3.15)xni,wni,yni⇀z,Jxn-Jwn→0,asn→∞.
Now, we show that z∈EP(f). But, from the definition of wn and (A2) we note that
(3.16)1rni〈y-wni,Jwni-Jxni〉≥-f(wni,y)≥f(y,wni),∀y∈C.
Letting i→∞, we have from (3.15) and (A4) that f(y,z)≤0, for all y∈C. Now, for 0<t≤1 and y∈C, let yt=ty+(1-t)z. Since y∈C and z∈C, we have yt∈C and hence f(yt,z)≤0. So, from the convexity of the equilibrium bifunction f(x,y) on the second variable y, we have
(3.17)0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,z)≤tf(yt,y),
and hence f(yt,y)≥0. Now, letting t→0 and condition (A3), we obtain that f(z,y)≥0, for all y∈C, and hence z∈EP(f).Next, we show thatz∈∩i=1NF(Ti). But, since each Ti satisfies condition (B3) we obtain from (3.13) and (3.15) that z∈F(Ti), for each i=1,2,…,N, and hence z∈∩i=1NF(Ti). Thus, from the above discussions we obtain that z∈F:=∩i=1NF(Ti)∩EP(f). Therefore, by Lemma 2.3, we immediately obtain that limsupn→∞〈zn-x*,Jw-Jx*〉=limi→∞〈zni-x*,Jw-Jx*〉=〈z-x*,Jw-Jx*〉≤0. It follows from (3.8) and Lemma 2.9 that ϕ(x*,xn)→0, as n→∞. Consequently, xn→x* by Lemma 2.5.Case 2.
Suppose that there exists a subsequence{ni} of {n} such that
(3.18)ϕ(x*,xni)<ϕ(x*,xni+1)
for all i∈ℕ. Then, by Lemma 2.8, there exist a nondecreasing sequence {mk}⊂ℕ such that mk→∞, ϕ(x*,xmk)≤ϕ(x*,xmk+1) and ϕ(x*,xk)≤ϕ(x*,xmk+1), for all k∈ℕ. Now, from (3.7) and the fact that δn→0, we have
(3.19)βmk,0ϕ(xmk,wmk)+βmk,0βmk,ig(∥Jwmk-Jumk,i∥)≤(ϕ(x*,xmk)-ϕ(x*,xmk+1))-δmkϕ(x*,xmk)+2δmk〈zmk-x*,Jw-Jx*〉,
as k→∞. Thus, using the same proof of Case 1, we obtain that xmk-wmk→0 and wmk-umk,i→0, as k→∞, for each i=1,2,…,N and hence
(3.20)limsupn→∞〈zmk-x*,Jw-Jx*〉≤0.
Then from (3.8), we have that
(3.21)ϕ(x*,xmk+1)≤(1-δmk)ϕ(x*,xmk)+2δmk〈zmk-x*,Jw-Jx*〉.
Since ϕ(x*,xmk)≤ϕ(x*,xmk+1), (3.21) implies that
(3.22)δmkϕ(x*,xmk)≤ϕ(x*,xmk)-ϕ(x*,xmk+1)+2δmk〈zmk-x*,Jw-Jx*〉≤2δmk〈zmk-x*,Jw-Jx*〉.
In particular, since δmk>0, we get
(3.23)ϕ(x*,xmk)≤2〈zmk-x*,Jw-Jx*〉.
Then, from (3.20), we obtain that ϕ(x*,xmk)→0, as k→∞. This together with (3.21) gives ϕ(x*,xmk+1)→0, as k→∞. But ϕ(x*,xk)≤ϕ(x*,xmk+1) for all k∈ℕ, thus we obtain that xk→x*. Therefore, from the above two cases, we can conclude that {xn} converges strongly to x* and the proof is complete.If in Theorem3.1, we assume that f(x,y)=〈Ax,y-x〉, for A continuous monotone mapping, then we obtain the following corollary.Corollary 3.2.
LetC be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let A:C→E* be a continuous monotone mapping. Let Ti:C→CB(C), for i=1,2,…,N, be a finite family of relatively nonexpansive multi-valued mappings. Assume that F:=∩i=1NF(Ti)∩VI(C,A) is nonempty. Let {xn} be a sequence generated by
(3.24)x0=w∈C,chosenarbitrarily,wn∈Csuchthat〈Awn,y-wn〉+1rn〈y-wn,Jwn-Jxn〉≥0,∀y∈C,yn=ΠCJ-1(αnJw+(1-αn)Jwn),xn+1=J-1(βn,0Jwn+∑i=1N‍βn,iJun,i),un,i∈Tiyn,n≥0,
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞‍αn=∞, {βn,i}⊂[a,b]⊂(0,1), for i=1,2,…,N, satisfying βn,0+βn,1+⋯+βn,N=1, for each n≥0. Then {xn} converges strongly to an element of F.Proof.
Letf(x,y)=〈Ax,y-x〉. Since A is monotone and continuous, we get that a bifunction f satisfies conditions (A1)–(A4). Thus, the conclusion follows from Theorem 3.1.If in Theorem3.1, we assume that N=1, then we get the following theorem.Corollary 3.3.
LetC be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let f:C×C→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let T:C→CB(C) be a relatively nonexpansive multi-valued mapping. Assume that F:=F(T)∩EP(f) is nonempty. Let {xn} be a sequence generated by
(3.25)x0=w∈C,chosenarbitrarily,wn=Frnxn,yn=ΠCJ-1(αnJw+(1-αn)Jwn),xn+1=J-1(βnJwn+(1-βn)Jun),un∈Tyn,n≥0,
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞‍αn=∞, {βn}⊂[a,b]⊂(0,1), for each n≥0. Then {xn} converges strongly to an element of F.Proof.
The proof follows from Theorem3.1 with N=1.If in Theorem3.1, we assume that f≡0, we get the following corollary.Corollary 3.4.
LetC be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let Ti:C→CB(C), for i=1,2,…,N, be a finite family of relatively nonexpansive multi-valued mappings. Assume that F:=∩i=1NF(Ti) is nonempty. Let {xn} be a sequence generated by
(3.26)x0=w∈C,chosenarbitrarily,yn=ΠCJ-1(αnJw+(1-αn)Jxn),xn+1=J-1(βn,0Jxn+∑i=1N‍βn,iJun,i),un,i∈Tiyn,n≥0,
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞‍αn=∞, {βn,i}⊂[a,b]⊂(0,1), for i=1,2,…,N satisfying βn,0+βn,1+⋯+βn,N=1, for each n≥0. Then {xn} converges strongly to an element of F.If in Theorem3.1, we assume that each Ti,  i=1,2,…,N is single valued, we get the following corollary.Corollary 3.5.
LetC be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let f:C×C→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let Ti:C→C, for i=1,2,…,N, be a finite family of relatively nonexpansive single-valued mappings. Assume that F:=∩i=1NF(Ti)∩EP(f) is nonempty. Let {xn} be a sequence generated by
(3.27)x0=w∈C,chosenarbitrarily,wn=Frnxn,yn=ΠCJ-1(αnJw+(1-αn)Jwn),xn+1=J-1(βn,0Jwn+∑i=1N‍βn,iJTiyn),n≥0,
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞‍αn=∞, {βn,i}⊂[a,b]⊂(0,1), for i=1,2,…,N, satisfying βn,0+βn,1+⋯+βn,N=1, for each n≥0. Then {xn} converges strongly to an element of F.IfE=H, a real Hilbert space, then E is uniformly convex and uniformly smooth real Banach space. In this case, J=I, identity map on H and ΠC=PC, projection mapping from H onto C. Thus, the following corollary holds.Corollary 3.6.
LetC be a nonempty, closed, and convex subset of a Hilbert space H. Let f:C×C→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let Ti:C→CB(C), for i=1,2,…,N, be a finite family of relatively nonexpansive multi-valued mappings. Assume that F:=∩i=1NF(Ti)∩EP(f) is nonempty. Let {xn} be a sequence generated by
(3.28)x0=w∈C,chosenarbitrarily,wn=Frnxn,yn=PC(αnw+(1-αn)wn),xn+1=βn,0wn+∑i=1N‍βn,iun,i,un,i∈Tiyn,n≥0,
where αn∈(0,1) such that limn→∞αn=0, ∑n=1∞‍αn=∞, {βn,i}⊂[a,b]⊂(0,1), for i=1,2,…,N, satisfying βn,0+βn,1+⋯+βn,N=1, for each n≥0. Then {xn} converges strongly to an element of F.Remark 3.7.
(1) Theorem3.1 improves and extends the corresponding results of Kumanm [22] and Takahashi and Zembayashi [16] in the sense that either our scheme does not require computation of Cn+1, for each n≥1, or the space considered is more general.
(2) Theorem3.1 improves the corresponding results of Homaeipour and Razani [3] in the sense that our convergence is strong and the requirement that the interior of F is nonempty is dispensed with.



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*Source: 859598-2012-11-20.xml*