# Fixed Points Approximation and Solutions of Some Equilibrium and Variational Inequalities Problems

Authors: Bashir Ali
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.


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## Body

## 1. Introduction

LetH be a real Hilbert space; a mapping B:D(B)→H is said to be monotone if for all x,y∈D(B)(1.1)〈Bx-By,x-y〉≥0.
For some λ>0, the mapping B is said to be  λ-inverse strongly monotone if(1.2)〈Bx-By,x-y〉≥λ∥Bx-By∥2.
A λ-inverse strongly monotone map is some time called  λ-cocoercive. A map B is said to be relaxedλ-cocoercive if there exists a constant λ>0 such that(1.3)〈Bx-By,x-y〉≥-λ∥Bx-By∥2∀x,y∈K.B is said to be relaxed  (λ,γ)-cocoercive, if there exist λ,γ>0 such that(1.4)〈Bx-By,x-y〉≥-λ∥Bx-By∥2+γ∥x-y∥2.
A map B:H→H is said to be  λ-Lipschitzian if there exists a real number λ≥0 such that(1.5)∥Bx-By∥≤λ∥x-y∥∀x,y∈H.B  is a contraction map, if in the above inequality λ∈[0,1) and nonexpansive if λ=1.LetK be a nonempty, closed, and convex subset of a real Hilbert space H. A variational inequality problem is searched for x*∈K such that(1.6)〈Bx*,y-x*〉≥0∀y∈K,
where B is some nonlinear mapping of K into H. Inequality (1.6) is called the variational inequality.Recall that for eachx∈H there exists a unique nearest point in K to x denoted by PKx. That is, ∥x-PKx∥≤∥x-y∥ for all y∈K.  PK is called a metric projection of H onto K. The mapping PK is nonexpansive in this setting, that is, ∥PKx-PKy∥≤∥x-y∥ for all  x,y∈H. It is also known that PK satisfies the following inequality ∥PKx-PKy∥2≤〈x-y,PKx-PKy〉.The solution set of the problem (1.6) is denoted by  VI(K,B). It is well known (see [1]) that x*∈VI(K,B) if and only if(1.7)x*=PK(x*-λBx*),∀λ>0.
A monotone map B is said to be maximal if the graph Γ(B) of B is not properly contained in the graph of any other monotone map, where  Γ(B)={(x,y)∈H×H:y∈Bx}  for a multivalued map B. It is also known that B is maximal monotone if and only if for (x,f)∈H×H,〈x-y,f-g〉≥0 for every (y,g)∈Γ(B) implies f∈Bx. Let B be a monotone mapping defined from K into H and let NKq be a normal cone to K at q∈K, that is, NKq={p∈H:〈q-u,p〉≥,forallu∈K}. Define a map M by(1.8)Mq={Bq+NKq,q∈K,∅,q∉K.
Then, M is maximal monotone and x*∈M-1(0)⇔x*∈VI(K,B), see, for example, [2].LetG:K×K→ℝ be a bifunction on a closed convex nonempty subset K of a real Hilbert space H. An equilibrium problem is searched for x*∈K such that(1.9)G(x*,y)≥0∀y∈K.
The set of solutions of the equilibrium problem above is denoted by EP(G).Several physical problems (such as the theories of lubrications, filterations and flows, moving boundary problems, see, e.g., [1, 3]) can be reduced to variational inequality or equilibrium problems. Consequently, these problems have solutions as the solutions of these resultant variational inequality or equilibrium problems.Maingé [4] introduced a Halpern-type scheme and proved a strong convergence theorem for family of nonexpansive mappings in Hilbert space.Recently,  S. Takahashi and W.  Takahashi [5] introduced an iterative scheme which they used to study the problem of approximating a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.More recently, Kumam and Katchang [6], W. Kumam and P. Kumam [7], Li and Su [8] and many others (see, e.g. [9–12] and the references contained in them) studied the problem of fixed point approximations and solutions of some equilibrium and, or solutions of some variational inequalities problems.In this paper we introduce a new iterative scheme for approximation of a common element in the intersection of the set of fixed points of some countable family of nonexpansive mappings, the set of solutions of some equilibrium problem, and the set of solutions of some variational inequality problem and prove a new theorem. Our theorem generalizes and improves some recent results.


## 2. Preliminaries

For a sequence{xn} the notation xn→x* and xn⇀x* means that the sequence {xn} converges strongly and weakly to x*, repectively.  A Banach space E is said to satisfy an Opial's condition (or in other words is an Opial's space) if for a sequence {xn} in E with xn⇀x*, then(2.1)lim infn→∞∥xn-x*∥<lim infn→∞∥xn-y∥for anyy∈Ey≠x*.
It is well known that Hilbert spaces are Opial’s spaces (see [13]).In the sequel we shall make use of the following results.Lemma 2.1 (see [14]).
LetK be a nonempty closed convex subset of H and let G be a bifunction of K×K into ℝ satisfying(A1)
G(x,x)=0forallx∈K;(A2)
G is  monotone, i.e. G(x,y)+G(y,x)≤0forallx,y∈K;(A3)
forallx,y,z∈K,limsupt→0+G(tz+(1+t)x,y)≤G(x,y);(A4)
forallx∈K,G(x,.) is convex and lower semicontinuous.Let r>0 and x∈H. Then there exists z∈K such that
(2.2)G(z,y)+1r〈y-z,z-x〉≥0,∀y∈K.Lemma 2.2 (see [15]).
LetK be a nonempty closed convex subset of H and let G be a bifunction of K×K into ℝ satisfying (A1)–(A4). For r>0 and x∈H define a map Tr:H→K by
(2.3)Trx={z∈K:G(z,y)+1r〈y-z,z-x〉≥0,∀y∈K}.
Then, the following holds:(1)
Tr is single-valued;(2)
Tr is firmly nonexpansive, that is, for any x,y∈H(2.4)∥Trx-Try∥2≤〈Trx-Try,x-y〉;(3)
Fix(Tr)=EP(G);(4)
EP(G) is closed and convex.Lemma 2.3 (see [16]).
LetH be a real inner product space. Then, the following inequality holds:
(2.5)∥x+y∥2≤∥x∥2+2〈y,x+y〉∀x,y∈H.Lemma 2.4 (see [17]).
Let{xn} and {yn} be bounded sequences in a Banach space E and let  {βn}  be a sequence in [0,1] with 0<lim infβn≤lim supβn<1. Suppose xn+1=βnyn+(1-βn)xn for all integers n≥0 and lim sup(∥yn+1-yn∥-∥xn+1-xn∥)≤0. Then, lim∥yn-xn∥=0.Lemma 2.5 (see [18]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(2.6)an+1≤(1-αn)an+αnσn+γn,n≥0,
Where (i){αn}⊂[0,1], ∑αn=∞;  (ii)lim supσn≤0;  (iii)γn≥0;(n≥0),∑γn<∞. Then, an→0 as n→∞.


## 3. Main Results

In the sequel we assume that the sequences{αn},{σi,n}i⊂(0,1) satisfy ∑i≥1σi,n=1-αn.Theorem 3.1.
LetK be a nonempty, closed, and convex subset of a real Hilbert space H. Let G be a bifunction from K×K to ℝ which satisfies conditions (A1)–(A4). Let {Ti,i=1,2,3,…} be family of nonexpansive mappings of K into H and let B be a μ-Lipschitzian, relaxed (λ,γ)-cocoercive map of K into H such that F:=⋂i=1∞Fix(Ti)∩EP(G)∩VI(K,B)≠∅. For an arbitrary but fixed δ∈(0,1), let {xn} and {yn} be sequences generated by
(3.1)x1,u∈H,G(yn,η)+1rn〈η-yn,yn-xn〉≥0,∀η∈K,xn+1=αnu+(1-δ)(1-αn)xn+δ∑i≥1σinTiPK(I-snB)yn,n≥1,
where {αn} and {σin} are sequences in [0,1] and {rn} and {sn} are sequences in [0,∞) satisfying (C1)
limn→∞αn=0,(C2)
∑n=1∞αn=∞,(C3)
{sn}⊂[a,b] for some a,b satisfying  0≤a≤b≤2(γ-λμ2)/μ2,(C4)
limn→∞|sn+1-sn|=0, limn→∞|rn+1-rn|=0,  limn→∞∑i≥1|σi,n+1-σi,n|=0,(C5)
lim infn→∞rn>0.Then, both {xn} and {yn} converge strongly to PFu.Proof.
First, we show that(I-snB) is nonexpansive, actually, using the property of B we have for x,y∈H,
(3.2)∥(I-snB)x-(I-snB)y∥2=∥x-y-sn(Bx-By)∥2=∥x-y∥2-2sn〈x-y,Bx-By〉+sn2∥Bx-By∥2≤∥x-y∥2-2sn[-λ∥Bx-By∥2+γ∥x-y∥2]+sn2∥Bx-By∥2≤∥x-y∥2+2snμ2λ∥x-y∥2-2snγ∥x-y∥2+μ2sn2∥x-y∥2=(1+2snμ2λ-2snγ+μ2sn2)∥x-y∥2≤∥x-y∥2,
and thus (I-snB) is nonexpansive.
Letx*∈F; since yn=Trnxn, n∈ℕ and the fact that  Trn is firmly nonexpansive (and hence nonexpansive) we have the following:
(3.3)∥yn-x*∥=∥Trnxn-Trnx*∥≤∥xn-x*∥.
We claim that {xn} satisfies
(3.4)∥xn-x*∥≤max{∥u-x*∥,∥x1-x*∥}∀n≥1.
We prove this by induction. Cearly the result is true for n=1. Assume that the result holds for n=k for some k∈ℕ. Then, for n=k+1 we have
(3.5)∥xk+1-x*∥=∥αk(u-x*)+(1-δ)(1-αk)(xk-x*)+δ∑i≥1σikTiPK(I-skB)(yk-x*)∥≤αk∥u-x*∥+(1-δ)(1-αk)∥xk-x*∥+δ∑i≥1σik∥TiPK(I-skB)yk-TiPK(I-skB)x*∥≤αk∥u-x*∥+(1-αk)∥xk-x*∥≤max{∥u-x*∥,∥xk-x*∥}.
Hence the result, and so {xn} is bounded. Furthermore, {yn}, {TiPK(yn-snByn)} and {Byn} are each bounded.
We now show thatlimn→∞∥xn+1-xn∥=0. Note that yn=Trnxn, yn+1=Trn+1xn+1, so that
(3.6)G(yn,η)+1rn〈η-yn,yn-xn〉≥0∀η∈K,G(yn+1,η)+1rn+1〈η-yn+1,yn+1-xn+1〉≥0∀η∈K.
Using (3.6)  and (A2), we have
(3.7)〈yn+1-yn,yn-xnrn-yn+1-xn+1rn+1〉≥0.
which implies that
(3.8)〈yn+1-yn,yn-yn+1+yn+1-xn-rnrn+1(yn+1-xn+1)〉≥0.
from which we get
(3.9)∥yn+1-yn∥2≤∥yn+1-yn∥[∥xn+1-xn∥+|1-rnrn+1|∥yn+1-xn+1∥].
If, without loss of generality M, m are real numbers such that rn>m>0 for all  n and M:=supn,i{∥yn-xn∥,∥TiPK(I-snB)yn∥,∥Byn∥}  we then have
(3.10)∥yn+1-yn∥≤∥xn+1-xn∥+|1-rnrn+1|∥yn+1-xn+1∥≤∥xn+1-xn∥+Mm|rn+1-rn|.
Now, define two sequences {βn} and {zn} by βn:=(1-δ)αn+δ and zn:=(xn+1-xn+βnxn)/βn. Then,
(3.11)zn=αnu+δ∑i≥1σi,nTiPK(I-snB)ynβn.
Observe that {zn} is bounded and that
(3.12)∥zn+1-zn∥-∥xn+1-xn∥≤|αn+1βn+1-αnβn|∥u∥+δ(1-αn+1)βn+1∥yn+1-yn∥-∥xn+1-xn∥+δβn+1∑i≥1|σi,n+1-σi,n|∥TiPK(I-sn+1B)yn∥+δβn+1βn|βn-βn+1|∑i≥1σi,n+1∥TiPK(I-sn+1B)yn∥+δ(1-αn)∥Byn∥βn|sn-sn+1|.
Using (3.10) we have that
(3.13)∥zn+1-zn∥-∥xn+1-xn∥≤|αn+1βn+1-αnβn|∥u∥+|δ(1-αn+1)βn+1-1|∥xn+1-xn∥+δ(1-αn+1)Mβn+1m|rn+1-rn|+δMβn+1∑i≥1|σi,n+1-σi,n|‍+δMβn+1βn|βn-βn+1|+δ(1-αn)Mβn|sn-sn+1|.
This implies
(3.14)lim supn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0,
and by Lemma 2.4, limn→∞∥zn-xn∥=0. Hence,
(3.15)∥xn+1-xn∥=βn∥zn-xn∥→0asn→∞.
Using (3.10) we have
(3.16)∥yn+1-yn∥→0asn→∞.
We now have
(3.17)∥yn-x*∥2=∥Trnxn-Trnx*∥2≤〈Trnxn-Trnx*,xn-x*〉=〈yn-x*,xn-x*〉=12(∥yn-x*∥2+∥xn-x*∥2-∥xn-yn∥2),
so that
(3.18)∥yn-x*∥2≤∥xn-x*∥2-∥xn-yn∥2.
But,
(3.19)∥xn+1-x*∥2=∥∑i≥1αn(u-x*)+(1-δ)(1-αn)(xn-x*)+δ[∑i≥1σinTiPK(I-snB)yn-TiPK(I-snB)x*]∥2≤αn∥u-x*∥2+(1-δ)(1-αn)∥xn-x*∥2+δ∑i≥1σin∥TiPK(I-snB)yn-TiPK(I-snB)x*∥2≤αn∥u-x*∥2+(1-δ)(1-αn)∥xn-x*∥2+δ(1-αn)∥yn-x*∥2.
Putting (3.18) in (3.19), we have
(3.20)δ(1-αn)∥xn-yn∥2≤αn∥u-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2=αn∥u-x*∥2+(∥xn-x*∥-∥xn+1-x*∥)(∥xn-x*∥+∥xn+1-x*∥)≤αn∥u-x*∥2+(∥xn-xn+1∥)(∥xn-x*∥+∥xn+1-x*∥),
and hence,
(3.21)limn→∞∥xn-yn∥=0.
Observe also that if wn:=PK(I-snB)yn, then
(3.22)∥wn-x*∥2=∥PK(I-snB)yn-PK(I-snB)x*∥2≤∥yn-x*-sn(Byn-Bx*)∥2=∥yn-x*∥2-2sn〈yn-x*,Byn-Bx*〉+sn2∥Byn-Bx*∥2≤∥yn-x*∥2-2sn[-λ∥Byn-Bx*∥2+γ∥yn-x*∥2]+sn2∥Byn-Bx*∥2=∥yn-x*∥2+2snλ∥Byn-Bx*∥2-2snγ∥yn-x*∥2+sn2∥Byn-Bx*∥2≤∥yn-x*∥2+[2snλ+sn2-2snγμ2]∥Byn-Bx*∥2.
Also,
(3.23)∥xn+1-x*∥2=∥∑i≥1αn(u-x*)+(1-δ)(1-αn)(xn-x*)+δ[∑i≥1σinTiPK(I-snB)yn-TiPK(I-snB)x*]∥2≤αn∥u-x*∥2+(1-δ)(1-αn)∥xn-x*∥2+δ∑i≥1σin∥TiPK(I-snB)yn-TiPK(I-snB)x*∥2=αn∥u-x*∥2+(1-δ)(1-αn)∥xn-x*∥2+δ∑i≥1σin∥Tiwn-x*∥2≤αn∥u-x*∥2+(1-δ)(1-αn)∥xn-x*∥2+δ(1-αn)∥wn-x*∥2≤αn∥u-x*∥2+(1-δ)(1-αn)∥xn-x*∥2+δ(1-αn)[∥yn-x*∥2+[2snλ+sn2-2snγμ2]∥Byn-Bx*∥2]≤αn∥u-x*∥2+(1-αn)∥xn-x*∥2+δ(1-αn)[2snλ+sn2-2snγμ2]∥Byn-Bx*∥2,
which implies
(3.24)-δ(1-αn)[2snλ+sn2-2snγμ2]∥Byn-Bx*∥2≤αn∥u-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2
so that
(3.25)limn→∞∥Byn-Bx*∥=0.
We go further to prove that for eachi∈ℕ,(3.26)limn→∞∥TiPK(I-sn)Byn-yn∥=0.
Consider the following estimates:
(3.27)∥(TiPK(I-snB))yn-yn∥2=∥(TiPK(I-snB))yn-x*+x*-yn∥2=∥x*-yn∥2+2〈(TiPK(I-snB))yn-x*,x*-yn〉+∥(TiPK(I-snB))yn-x*∥2≤2∥x*-yn∥2+2〈(TiPK(I-snB))yn-yn+yn-x*,x*-yn〉=2∥yn-x*∥2+2〈(TiPK(I-snB))yn-yn,x*-yn〉-2∥yn-x*∥2=2〈(TiPK(I-snB))yn-yn,x*-yn〉.
Using (3.1), we have
(3.28)〈xn+1-x*,yn-x*〉=αn〈u-x*,yn-x*〉+(1-αn)(1-δ)〈xn-x*,yn-x*〉+δ∑i≥1σi,n〈(TiPK(I-snB))yn-yn+yn-x*,yn-x*〉=αn〈u-x*,yn-x*〉+(1-αn)(1-δ)〈xn-yn+yn-x*,yn-x*〉+δ∑i≥1σi,n〈(TiPK(I-snB))yn-yn,yn-x*〉+δ(1-αn)∥yn-x*∥2=αn〈u-x*,yn-x*〉+(1-αn)(1-δ)〈xn-yn,yn-x*〉+δ∑i≥1σi,n〈(TiPK(I-snB))yn-yn,yn-x*〉+(1-αn)(1-δ)∥yn-x*∥2+δ(1-αn)∥yn-x*∥2,
which implies
(3.29)δ∑i≥1σi,n〈(TiPK(I-snB))yn-yn,x*-yn〉=αn〈u-x*,yn-x*〉+(1-αn)(1-δ)〈xn-yn,yn-x*〉+(1-αn)〈yn-x*,yn-x*〉-〈xn+1-x*,yn-x*〉=αn〈u-xn+1,yn-x*〉+(1-αn)(1-δ)〈xn-yn,yn-x*〉+(1-αn)〈yn-xn+1,yn-x*〉.
Using this and (3.27), we get
(3.30)δ2∑i≥1σi,n∥(TiPK(I-snB))yn-yn∥2≤αn〈u-xn+1,yn-x*〉+(1-αn)(1-δ)〈xn-yn,yn-x*〉+(1-αn)〈yn-xn+1,yn-x*〉.
Since {xn}, {yn} are each bounded and using (3.21), we have that
(3.31)limn→∞∥(TiPK(I-snB))yn-yn∥=0∀i∈ℕ.
Using this and (3.21), we also have
(3.32)limn→∞∥(TiPK(I-snB))yn-xn∥=0∀i∈ℕ.
Next we show that  lim supn→∞〈u-w,xn-w〉≤0, where  w=P⋂i=1∞Fix(Ti)∩EP(G)∩VI(K,B)u. Let {xnj} be a subsequence of {xn} such that lim supn→∞〈u-w,xn-w〉=limj→∞〈u-w,xnj-w〉.
Since{yn} is bounded, there exists a subsequence {ynj} of {yn} such that ynj⇀z. Then, z∈⋂i=1∞Fix(TiPK(I-snB)),otherwise, for i∈ℕ, we have(3.33)lim infj→∞∥ynj-z∥<lim infj→∞∥ynj-TiPK(I-snjB)z∥≤lim infj→∞[∥ynj-TiPK(I-snjB)ynj∥+∥TiPK(I-snjB)ynj-TiPK(I-snjB)z∥]≤lim infj→∞∥ynj-z∥,
which is a contradiction, so z∈⋂i=1∞Fix(TiPK(I-snB)).
Next we show thatz∈EP(G). Since yn=Trnxn, we have
(3.34)G(yn,η)+1rn〈η-yn,yn-xn〉≥0,∀η∈K.
It follows from (A2) that
(3.35)〈η-yn,yn-xnrn〉≥G(η,yn),
and so
(3.36)〈η-yni,yni-xnirni〉≥G(η,yni).
Since (yni-xni)/rni→0, yni⇀z and using (A4), we have G(η,z)≤0 for all  η∈K. For a real number t, 0<t≤1 and η∈K, let ηt=tη+(1-t)z. Clearly ηt∈K, so that using (A1) and (A4), we have
(3.37)0=G(ηt,ηt)≤tG(ηt,η)+(1-t)G(ηt,z)≤tG(ηt,η).
This implies G(ηt,η)≥0, and using this and (A3) we have that G(z,η)≥0 for all  η∈K and hence z∈EP(G).
Next we show thatz∈VI(K,B).
Letx*∈F then, we have the following:
(3.38)∥wn-x*∥2=∥PK(I-snB)yn-PK(I-snB)x*∥2≤〈yn-snByn-(x*-snBx*),wn-x*〉=〈yn-x*,wn-x*〉-sn〈Byn-Bx*,wn-x*〉≤12[∥wn-x*∥2+∥yn-x*∥2-∥wn-yn∥2]-sn〈Byn-Bx*,wn-x*〉,
so that
(3.39)∥wn-x*∥2≤∥yn-x*∥2-∥wn-yn∥2-2sn〈Byn-Bx*,wn-x*〉.
We then have
(3.40)∥xn+1-x*∥2=∥αn(u-x*)+(1-αn)(1-δ)(xn-x*)+δ∑i≥1σi,n[Tiwn-x*]∥2≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+(1-αn)δ∥wn-x*∥2≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+(1-αn)δ[∥yn-x*∥2-∥wn-yn∥2-2sn〈Byn-Bx*,wn-x*〉]≤αn∥u-x*∥2+(1-αn)(1-δ)∥xn-x*∥2+(1-αn)δ[∥xn-x*∥2-∥wn-yn∥2-2sn〈Byn-Bx*,wn-x*〉]=αn∥u-x*∥2+(1-αn)∥xn-x*∥2-δ(1-αn)∥wn-yn∥2-2snδ(1-αn)〈Byn-Bx*,wn-x*〉,
so that
(3.41)δ(1-αn)∥wn-yn∥2≤αn∥u-x*∥2+∥xn-x*∥2-∥xn+1-x*∥2-2δsn(1-αn)〈Byn-Bx*,wn-x*〉,
and as αn→0, ∥xn-xn+1∥→0 and ∥Byn-Bx*∥→0 as n→∞, we get
(3.42)limn→∞∥yn-wn∥=0.
Using this and (3.21) we also have
(3.43)limn→∞∥xn-wn∥=0.
As B is a relaxed (λ,γ)-cocoercive and using condition (C3) we have for x,y∈H(3.44)〈Bx-By,x-y〉≥(-λ)∥Bx-By∥2+γ∥x-y∥2≥(γ-λμ2)∥x-y∥2≥0,
and so B is monotone. If
(3.45)Mq={Bq+NKq,q∈K,∅,q∉K,
then M is maximal monotone. Let Γ(M) denote the graph of M.
Let(q,p)∈Γ(M). Since p-Bq∈NKq and wn∈K, we have 〈p-Bq,q-wn〉≥0 by definition of NKq. Also, as wn=PK(yn-snByn) (using property of the projection PK), we have
(3.46)〈wn-(yn-snByn),q-wn〉≥0,
and hence
(3.47)〈wn-ynsn+Byn,q-wn〉≥0.
Using this, we obtain the following estimates:
(3.48)〈p,q-wni〉≥〈Bq,q-wni〉≥〈Bq,q-wni〉-〈wni-ynisni+Byni,q-wni〉=〈Bq-wni-ynisn-Byni,q-wni〉=〈Bq-Bwni,q-wni〉+〈Bwni-Byni,q-wni〉-〈wni-ynisn,q-wni〉≥〈Bwni-Byni,q-wni〉-〈wni-ynisn,q-wni〉,
which implies 〈p,q-z〉≥0 (letting i→∞).
SinceM is maximal monotone, we obtained that z∈M-1(0) and hence z∈VI(K,B).
We mention here that since we proved thatz∈⋂i=1∞Fix(TiPK(I-snB)) and z∈VI(K,B), then z∈⋂i=1∞Fix(Ti). So, clearly z∈⋂i=1∞Fix(Ti)∩EP(G)∩VI(K,B).
Sincew=P⋂i=1∞Fix(Ti)∩EP(G)∩VI(K,B)u, we have
(3.49)lim supn→∞〈u-w,xn-w〉=limi→∞〈u-w,xni-w〉=〈u-w,z-w〉≤0.
Hence, lim supn→∞〈u-w,xn-w〉≤0.
From the recursion formula (3.1) and Lemma 2.3, we have
(3.50)∥xn+1-w∥2=∥∑i≥1αn(u-w)+(1-αn)(1-δ)(xn-w)+δ∑i≥1σi,n[TiPK(I-snB)yn-w]∥2≤∥(1-αn)(1-δ)(xn-w)+δ∑i≥1σi,n[TiPK(I-snB)yn-w]∥2+2αn〈u-w,xn+1-w〉≤[(1-αn)(1-δ)∥xn-w∥+δ∑i≥1σi,n∥TiPK(I-snB)yn-w∥]2+2αn〈u-w,xn+1-w〉≤(1-αn)∥xn-w∥2+2αn〈u-w,xn+1-w〉.
Using Lemma 2.5, we have that {xn} and consequently {yn} converge to w and the proof is complete.The following corollaries follow from Theorem3.1.Corollary 3.2.
LetK,G and B be as in Theorem 3.1 and let {Ti,i=1,2,3,…,N} be finite family of nonexpansive mappings of K into H. Let F:=⋂i=1NFix(Ti)∩EP(G)∩VI(K,B)≠∅. For an arbitrary but fixed δ∈(0,1), let {xn}  and {yn} be sequences generated by
(3.51)x1,u∈H,G(yn,η)+1rn〈η-yn,yn-xn〉≥0,∀η∈K,xn+1=αnu+(1-δ)(1-αn)xn+δ∑i=1NσinTiPK(I-snB)yn,n≥1,
where {αn} and {σin} are sequences in [0,1] and {rn} and {sn} are sequences in [0,∞) satisfying (C1)
limn→∞αn=0, ∑i=1Nσin=(1-αn),(C2)
∑n=1∞αn=∞,(C3)
{sn}⊂[a,b] for some a,b satisfying 0≤a≤b≤2(γ-λμ2)/μ2,(C4)
limn→∞|sn+1-sn|=0, limn→∞|rn+1-rn|=0, limn→∞∑i=1N|σi,n+1-σi,n|=0,(C5)
lim infn→∞rn>0.Then, both {xn} and {yn} converge strongly to PFu.Corollary 3.3.
LetK and G be as in Theorem 3.1. Let T be a nonexpansive map of K into H. Let F:=Fix(T)∩EP(G)≠∅. For an arbitrary but fixed δ∈(0,1), let {xn} and {yn} be sequences generated by
(3.52)x1,u∈H,G(yn,η)+1rn〈η-yn,yn-xn〉≥0,∀η∈K,xn+1=αnu+(1-δ)(1-αn)xn+δ(1-αn)Tyn,n≥1,
where {αn} is a sequence in [0,1] and {rn} is a sequences in [0,∞) satisfying (C1)
limn→∞αn=0(C2)
∑n=1∞αn=∞,(C3)
limn→∞|rn+1-rn|=0,(C4)
lim infn→∞rn>0.Then, both {xn} and {yn} converge strongly to PFu.Remark 3.4.
Prototypes of the sequences{αn} and {σi,n} in our theorem are the following:
(3.53)αn:=1n+1,σi,n:=n2i(n+1)∀i∈ℕ.



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*Source: 656534-2010-01-27.xml*