# A Hybrid Method for a Countable Family of Multivalued Maps, Equilibrium Problems, and Variational Inequality Problems

Authors: Watcharaporn Cholamjiak; Suthep Suantai
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We introduce a new monotone hybrid iterative scheme for
finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions of variational inequality problem, and the set of the solutions of the equilibrium problem in a Hilbert space. Strong convergence theorems of the purposed iteration are established.


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## Body

## 1. Introduction

LetD be  a nonempty convex subset of a  Banach spaces E. Let F be a bifunction from D×D to ℝ, where ℝ is the set of all real numbers. The equilibrium problem for F is to find x∈D such that F(x,y)≥0 for all y∈D. The set of such solutions is denoted by EP(F). The set D is called proximal if for each x∈E, there exists an element y∈D such that ∥x-y∥=d(x,D), where d(x,D)=inf{∥x-z∥:z∈D}. Let CB(D), K(D), and P(D) denote the families of nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by(1.1)H(A,B)=max{supx∈Ad(x,B),supy∈Bd(y,A)}
for A,B∈CB(D). A single-valued map T:D→D is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈D. A multivalued map T:D→CB(D) is said to benonexpansive if H(Tx,Ty)≤∥x-y∥ for all x,y∈D. An element p∈D is called a fixed point of T:D→D (resp.,  T:D→CB(D))  if p=Tp (resp.,  p∈Tp). The set of fixed points of T is denoted by F(T). The mapping T:D→CB(D) is called quasi-nonexpansive[1] if F(T)≠∅ and H(Tx,Tp)≤∥x-p∥ for all x∈D and all p∈F(T). It is clear that every nonexpansive multivalued map T with F(T)≠∅ is quasi-nonexpansive. But there exist quasi-nonexpansive mappings that are not nonexpansive; see [2].The mappingT:D→CB(D) is called hemicompact if, for any sequence {xn} in D such that d(xn,Txn)→0 as n→∞, there exists a subsequence {xnk}  of  {xn} such that xnk→p∈D. We note that if D is compact, then every multivalued mapping T:D→CB(D) is hemicompact.A mappingT:D→CB(D) is said to satisfy Condition (I) if there is a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0, f(r)>0 for r∈(0,∞) such that(1.2)d(x,Tx)≥f(d(x,F(T)))
for all x∈D.In 1953, Mann [3] introduced the following iterative procedure to approximate a fixed point of a nonexpansive mapping T in a Hilbert space H:(1.3)xn+1=αnxn+(1-αn)Txn,∀n∈ℕ,
where the initial point x0 is taken in C arbitrarily and {αn} is a sequence in [0,1].However, we note that Mann's iteration process (1.3) has only weak convergence, in general; for instance, see [4–6].In 2003, Nakajo and Takahashi [7] introduced the method which is the so-called CQ method to modify the process (1.3) so that strong convergence is guaranteed. They also proved a strong convergence theorem for a nonexpansive mapping in a Hilbert space.Recently, Tada and Takahashi [8] proposed a new iteration for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H.In 2005, Sastry and Babu [9] proved that the Mann and Ishikawa iteration schemes for multivalued map T with a fixed point p converge to a fixed point q of T under certain conditions. They also claimed that the fixed point q may be different from p. More precisely, they proved the following result for nonexpansive multivalued map with compact domain.In 2007, Panyanak [10] extended the above result of Sastry and Babu [9] to uniformly convex Banach spaces but the  domain of T remains compact.Later, Song and Wang [11] noted that there was a gap in the proofs of Theorem 3.1 [10] and Theorem 5 [9]. They further solved/revised the gap and also gave the affirmative answer to Panyanak [10] question using the following Ishikawa iteration scheme. In the main results, domain of T is still compact, which is a strong condition (see [11, Theorem 1]) and T satisfies condition (I)  (see [11, Theorem 1]).In 2009, Shahzad and Zegeye [2] extended and improved the results of Panyanak [10], Sastry and Babu [9], and Song and Wang [11] to quasi-nonexpansive multivalued maps. They also relaxed compactness of the domain of T and constructed an iteration scheme which removes the restriction of T, namely, Tp={p} for any p∈F(T). The results provided an affirmative answer to Panyanak [10] question in a more general setting. In the main results, T satisfies Condition (I) (see [2, Theorem 2.3]) and T is hemicompact and continuous (see [2, Theorem 2.5]).A mappingA:D→H is called α-inverse-strongly monotone [12]  if there exists a positive real number α such that(1.4)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈D.Remark 1.1.
It is easy to see that ifA:D→H is α-inverse-strongly monotone, then it is a (1/α)-Lipschitzian mapping.
LetA:D→H be a mapping. The classical variational inequality problem is to find a u∈D such that
(1.5)〈Au,v-u〉≥0,∀v∈D.
The set of solutions of variational inequality (3.9) is denoted by VI(D,A).Question 1.
How can we construct an iteration process for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of a variational inequality problem, and the set of common fixed points of nonexpansive multivalued maps ?In the recent years, the problem of finding a common element of the set of solutions of equilibrium problems and the set of fixed points of single-valued nonexpansive mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; for instance, see [8, 13–20] and the references cited theorems.In this paper, we introduce a monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a countable family of nonexpansive multivalued maps, the set of variational inequality, and the set of solutions of an equilibrium problem in a Hilbert space.


## 2. Preliminaries

The following lemmas give some characterizations and a useful property of the metric projectionPD in a Hilbert space.LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let D be a closed and convex subset of H. For every point x∈H, there exists a unique nearest point in D, denoted by PDx, such that(2.1)∥x-PDx∥≤∥x-y∥,∀y∈D.PD is called the metric projection of H onto D. We know that PD is a nonexpansive mapping of H onto D.Lemma 2.1 (see [21]).
LetD be a closed and convex subset of a real Hilbert space H and let PD be the metric projection from H onto D. Given x∈H and z∈D, then z=PDx if and only if the following holds:
(2.2)〈x-z,y-z〉≤0,∀y∈D.Lemma 2.2 (see [7]).
LetD be a nonempty, closed and convex subset of a real Hilbert space H and PD:H→D   the metric projection from H onto D. Then the following inequality holds:
(2.3)∥y-PDx∥2+∥x-PDx∥2≤∥x-y∥2,∀x∈H,∀y∈D.Lemma 2.3 (see [21]).
LetH be a real Hilbert space. Then the following equations hold:   (i)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉,   for all  x,y∈H;(ii)
∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2,  for all t∈[0,1]    and x,y∈H.Lemma 2.4 (see [22]).
LetD be a nonempty, closed and convex subset of a real Hilbert space H. Given x,y,z∈H and also given a∈ℝ, the set
(2.4){v∈D:∥y-v∥2≤∥x-v∥2+〈z,v〉+a}
is convex and closed.For solving the equilibrium problem, we assume that the bifunctionF:D×D→ℝ satisfies the following conditions:(A1)
F(x,x)=0 for all x∈D;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0  for all x,y∈D;(A3)
for eachx,y,z∈D,lim supt↓0F(tz+(1-t)x,y)≤F(x,y);(A4)
F(x,·) is convex and lower semicontinuous for each x∈D.Lemma 2.5 (see [13]).
LetD be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bifunction from D×D to ℝ satisfying (A1)–(A4) and let r>0 and x∈H. Then, there exists z∈D such  that
(2.5)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈D.Lemma 2.6 (see [18]).
Forr>0, x∈H, defined a mapping Tr:H→D as follows:(2.6)Tr(x)={z∈D:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈D}.
Then the following  holds:(1)
Tr is a single  value;(2)
Tr is firmly nonexpansive, that is, for any x,y∈H,
(2.7)∥Trx-Try∥2≤〈Trx-Try,x-y〉;(3)
F(Tr)=EP(F);(4)
EP(F) is closed and convex.In the context of the variational inequality problem,(2.8)u∈VI(D,A)⇔u=PD(u-λAu),∀λ>0.
A set-valued mapping T:H→2H is said to be monotone  if for all  x,y∈H,  f∈Tx, and g∈Ty imply that 〈f-g,x-y〉≥0. A monotone mapping T:H→H is said to be maximal [23]  if the graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal  if and only if for (x,f)∈H×H, 〈f-g,x-y〉≥0, ∀(y,g)∈G(T) imply that f∈Tx. Let A:D→H be an inverse strongly monotone mapping and let NDv be the normal cone to D at v∈D, that is,(2.9)NDv={w∈H:〈v-u,w〉≥0,∀u∈D},
and define(2.10)Tv={Av+NDv,v∈D,∅,v∉D.
Then T is maximal monotone and 0∈Tv if and only if v∈VI(D,A) (see, e.g., [24]).In general, the fixed point set of a nonexpansive multivalued mapT is not necessary to be closed and convex (see [25, Example 3.2]). In the next Lemma, we show that F(T) is closed and convex under the assumption that Tp={p} for all p∈F(T).Lemma 2.7.
LetD be a closed and convex subset of a real Hilbert space H. Let T:D→CB(D) be a nonexpansive multivalued map with F(T)≠∅ and Tp={p} for each p∈F(T). Then F(T) is a closed and convex subset of D.Proof.
First, we will show thatF(T) is closed. Let {xn} be a sequence in F(T) such that xn→x as n→∞. We have
(2.11)d(x,Tx)≤d(x,xn)+d(xn,Tx)≤d(x,xn)+H(Txn,Tx)≤2d(x,xn).
It follows that d(x,Tx)=0, so x∈F(T). Next, we show that F(T) is convex. Let p=tp1+(1-t)p2 where  p1,p2∈F(T) and t∈(0,1). Let z∈Tp; by Lemma 2.3, we have
(2.12)∥p-z∥2=∥t(z-p1)+(1-t)(z-p2)∥2=t∥z-p1∥2+(1-t)∥z-p2∥2-t(1-t)∥p1-p2∥2=td(z,Tp1)2+(1-t)d(z,Tp2)2-t(1-t)∥p1-p2∥2≤tH(Tp,Tp1)2+(1-t)H(Tp,Tp2)2-t(1-t)∥p1-p2∥2≤t∥p-p1∥2+(1-t)∥p-p2∥2-t(1-t)∥p1-p2∥2=t(1-t)2∥p1-p2∥2+(1-t)t2∥p1-p2∥2-t(1-t)∥p1-p2∥2=0.
Hence p=z. Therefore, p∈F(T).


## 3. Main Results

In the following theorem, we introduce a new monotone hybrid iterative scheme for finding a common element of the set of a common fixed points of a countable family of nonexpansive multivalued maps, the set of variational inequality, and the set of solutions of an equilibrium problem in a Hilbert space, and we prove strong convergence theorem without the condition (I).Theorem 3.1.
LetD be a nonempty, closed and convex subset of a real Hilbert space H. Let F be a bifunction from D×D to ℝ satisfying (A1)–(A4), let A:D→H be an α-inverse strongly monotone mapping, and let Ti:D→CB(D) be nonexpansive multivalued maps for all i∈ℕ with Ω:=⋂i=1∞F(Ti)∩EP(F)∩VI(D,A)≠∅ and Tip={p}, ∀p∈⋂i=1∞F(Ti). Assume that αi,n∈[0,1) with lim supn→∞αi,n<1 for all i∈ℕ, {rn}⊂[b,∞) for some b∈(0,∞), and {λn}⊂[c,d] for some c,d∈(0,2α). For an initial point x0∈H with C1=D and x1=PC1x0, let {xn}, {yn}, {si,n}, and {un} be sequences generated by
(3.1)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈D,yn=PD(un-λnAun),si,n=αi,nyn+(1-αi,n)zi,n,Ci,n+1={z∈Ci,n:∥si,n-z∥≤∥yn-z∥≤∥xn-z∥},Cn+1=⋂i=1∞Ci,n+1,xn+1=PCn+1x0,∀n∈ℕ,
where zi,n∈Tiyn. Then,  {xn}, {yn},  and {un} converge strongly to z0=PΩx0.Proof.
We split the proof into six steps.
Step 1.
Show thatPCn+1x0 is well defined for every x0∈H.
Since0<c≤λn≤d<2α for all n∈ℕ, we get that PC(I-λnA) is nonexpansive for all n∈ℕ. Hence, ⋂n=1∞F(PC(I-λnA))=VI(D,A) is closed and convex. By Lemma 2.6(4), we know that EP(F) is closed and convex. By Lemma 2.7, we also know that ⋂i=1∞F(Ti) is closed and convex. Hence, Ω:=⋂i=1∞F(Ti)∩EP(F)∩VI(D,A) is a nonempty, closed and convex set. By Lemma 2.4, we see that Ci,n+1 is closed and convex for all i,n∈ℕ. This implies that Cn+1 is also closed and convex. Therefore, PCn+1x0 is well defined. Let p∈Ω and i∈ℕ. From un=Trnxn, we have
(3.2)∥un-p∥=∥Trnxn-Trnp∥≤∥xn-p∥
for every n≥0. From this, we have
(3.3)∥si,n-p∥=∥αi,nyn+(1-αi,n)zi,n-p∥≤αi,n∥yn-p∥+(1-αi,n)∥zi,n-p∥≤αi,n∥yn-p∥+(1-αi,n)d(zi,n,Tip)≤αi,n∥yn-p∥+(1-αi,n)H(Tiyn,Tip)≤∥yn-p∥=∥PD(un-λnAun)-PD(p-λnAp)∥≤∥un-p∥≤∥xn-p∥.
So, we have p∈Ci,n+1, hence Ω⊂Ci,n+1, ∀i∈ℕ. This shows that  Ω⊂Cn+1⊂Cn.
Step 2.
Show thatlimn→∞∥xn-x0∥ exists.
SinceΩ is a nonempty closed convex subset of H, there exists a unique v∈Ω such that
(3.4)z0=PΩx0.
From xn=PCnx0, Cn+1⊂Cn and xn+1∈Cn, ∀n≥0, we get
(3.5)∥xn-x0∥≤∥xn+1-x0∥,∀n≥0.
On the other hand, as Ω⊂Cn, we obtain
(3.6)∥xn-x0∥≤∥z0-x0∥,∀n≥0.
It follows that the sequence {xn} is bounded and nondecreasing. Therefore, limn→∞∥xn-x0∥ exists.
Step 3.
Show thatxn→q∈D as n→∞.
Form>n, by the definition of Cn, we see that xm=PCmx0∈Cm⊂Cn. By Lemma 2.2, we get
(3.7)∥xm-xn∥2≤∥xm-x0∥2-∥xn-x0∥2.
From Step 2, we obtain that  {xn} is Cauchy. Hence, there exists q∈D such that xn→q as n→∞.
Step 4.
Show thatq∈F.
From Step3, we get
(3.8)∥xn+1-xn∥→0
as n→∞. Since xn+1∈Cn+1⊂Cn, we have
(3.9)∥si,n-xn∥≤∥si,n-xn+1∥+∥xn+1-xn∥≤2∥xn+1-xn∥→0
as n→∞ for all i∈ℕ,
(3.10)∥yn-xn∥≤∥yn-xn+1∥+∥xn+1-xn∥≤2∥xn+1-xn∥→0
as n→∞. Hence, yn→q as n→∞. It follows from (3.9) and (3.10) that
(3.11)∥zi,n-yn∥=11-αi,n∥si,n-yn∥→0
as n→∞ for all i∈ℕ. For each i∈ℕ, we have
(3.12)d(q,Tiq)≤∥q-yn∥+∥yn-zi,n∥+d(zi,n,Tiq)≤∥q-yn∥+∥yn-zi,n∥+H(Tiyn,Tiq)≤∥q-yn∥+∥yn-zi,n∥+∥yn-q∥.
From (3.11), we obtain d(q,Tiq)=0. Hence q∈F.
Step 5.
Show thatq∈EP(F).
By the nonexpansiveness ofPD and the inverse strongly monotonicity of A, we obtain
(3.13)∥yn-p∥2≤∥un-λnAun-(p-λnAp)∥2≤∥un-p∥2+λn(λn-2α)∥Aun-Ap∥2=∥Trnxn-Trnp∥2+λn(λn-2α)∥Aun-Ap∥2≤∥xn-p∥2+c(d-2α)∥Aun-Ap∥2.
This implies
(3.14)c(2α-d)∥Aun-Ap∥2≤∥xn-p∥2-∥yn-p∥2≤∥xn-yn∥(∥xn-p∥+∥yn-p∥).
It follows from (3.10) that
(3.15)limn→∞∥Aun-Ap∥=0.
Since PD is firmly nonexpansive, we have
(3.16)∥yn-p∥2=∥PD(un-λnAun)-PD(p-λnAp)∥2≤〈un-λnAun)-(p-λnAp),yn-p〉=12(∥(un-λnAun)-(p-λnAp)∥2+∥yn-p∥2-∥(un-λnAun)-(p-λnAp)-(yn-p)∥2)≤12(∥un-p∥2+∥yn-p∥2-∥(un-yn)-λn(Aun-Ap)∥2)≤12(∥xn-p∥2+∥yn-p∥2-∥un-yn∥2+2λn〈un-yn,Aun-Ap〉)≤12(∥xn-p∥2+∥yn-p∥2-∥un-yn∥2+2λn∥un-yn∥∥Aun-Ap∥).
This implies that
(3.17)∥yn-p∥2≤∥xn-p∥2-∥un-yn∥2+2λn∥un-yn∥∥Aun-Ap∥.
It follows that
(3.18)∥un-yn∥2≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2d∥un-yn∥∥Aun-Ap∥.
From (3.10) and (3.15), we get
(3.19)limn→∞∥un-yn∥=0.
It follows from (3.10) and (3.19) that
(3.20)limn→∞∥un-xn∥=0.
Since  un=Trnxn, we have
(3.21)F(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈D.
From the monotonicity of F, we have
(3.22)1rn〈y-un,un-xn〉≥F(y,un),∀y∈D,
hence
(3.23)〈y-un,un-xnrn〉≥F(y,un),∀y∈D.
From (3.20) and condition (A4), we have
(3.24)0≥F(y,q),∀y∈D.
For t with 0<t≤1 and y∈D, let yt=ty+(1-t)q. Since y,q∈D and D is convex, then yt∈D and hence F(yt,q)≤0. So, we have
(3.25)0=F(yt,yt)≤tF(yt,y)+(1-t)F(yt,q)≤tF(yt,y).
Dividing by t, we obtain
(3.26)F(yt,y)≥0,∀y∈D.
Letting t↓0 and from (A3), we get
(3.27)F(q,y)≥0,∀y∈D.
Therefore, we obtain q∈EP(F).
Step 6.
Show thatq∈VI(D,A).
SinceT is the maximal monotone mapping defined by (2.10),
(3.28)Tx={Ax+NDx,x∈D,∅,x∉D.
For any given (x,u)∈G(T), hence u-Ax∈NDx. It follows that
(3.29)〈x-yn,u-Ax〉≥0.
On the other hand, since yn=PD(un-λnAun), we have
(3.30)〈x-yn,yn-(un-λnAun)〉≥0,
and so
(3.31)〈x-yn,yn-unλn+Aun〉≥0.
From (3.29), (3.31),  and the α-inverse monotonicity of A, we have
(3.32)〈x-yn,u〉≥〈x-yn,Ax〉≥〈x-yn,Ax〉-〈x-yn,yn-unλn+Aun〉=〈x-yn,Ax-Ayn〉+〈x-yn,Ayn-Aun〉-〈x-yn,yn-unλn〉≥〈x-yn,Ayn-Aun〉-〈x-yn,yn-unλn〉.
It follows that
(3.33)limn→∞〈x-yn,u〉=〈x-q,u〉≥0.
Again since T is maximal monotone, hence 0∈Tq. This shows that q∈VI(D,A).
Step 7.
Show thatq=z0=PΩx0.
Sincexn=PCnx0 and Ω⊂Cn, we obtain
(3.34)〈x0-xn,xn-p〉≥0∀p∈Ω.
By taking the limit in (3.34), we obtain
(3.35)〈x0-q,q-p〉≥0∀p∈Ω.
This shows that q=PΩx0=z0.
From Steps3 to  5, we obtain that {xn}, {yn},  and {un} converge strongly to z0=PΩx0. This completes the proof.Theorem 3.2.
LetD be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti:D→CB(D) be nonexpansive multivalued maps for all i∈ℕ with Ω:=⋂i=1∞F(Ti)∩VI(D,A)≠∅ and  Tip={p}, for all p∈⋂i=1∞F(Ti). Assume that αi,n∈[0,1) with lim supn→∞αi,n<1 and {λn}⊂[c,d] for some c,d∈(0,2α). For an initial point x0∈H with C1=D and x1=PC1x0, let {xn},  {yn}, and {si,n} be sequences generated by
(3.36)yn=PD(xn-λnAxn),si,n=αi,nyn+(1-αi,n)zi,n,Ci,n+1={z∈Ci,n:∥si,n-z∥≤∥yn-z∥≤∥xn-z∥},Cn+1=⋂i=1∞Ci,n+1,xn+1=PCn+1x0,∀n∈ℕ,
where zi,n∈Tiyn. Then, {xn} and {yn} converge strongly to z0=PΩx0.Proof.
PuttingF(x,y)=0 for all x,y∈D in Theorem 3.1, we obtain the desired result directly from Theorem 3.1.Theorem 3.3.
LetD be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti:D→CB(D) be nonexpansive multivalued maps for all i∈ℕ with Ω:=⋂i=1∞F(Ti)≠∅ and Tip={p}, for all p∈⋂i=1∞F(Ti). Assume that αi,n∈[0,1) with lim supn→∞αi,n<1. For an initial point x0∈H with C1=D and x1=PC1x0, let {xn} and {si,n} be sequences generated by
(3.37)si,n=αi,nxn+(1-αi,n)zi,n,Ci,n+1={z∈Ci,n:∥sn-z∥≤∥xn-z∥},Cn+1=⋂i=1∞Ci,n+1,xn+1=PCn+1x0,∀n∈ℕ,
where zi,n∈Tiyn. Then, {xn} converge strongly to z0=PΩx0.Proof.
PuttingA=0 in Theorem 3.2, we obtain the desired result directly from Theorem 3.2.



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*Source: 349158-2010-06-15.xml*