Adv. Fixed Point Theory, 4 (2014), No. 1, 125-139 ISSN: 1927-6303
CONVERGENCE OF ALGORITHMS FOR AN INFINITE FAMILY NONEXPANSIVE MAPPINGS AND RELAXED COCOERCIVE MAPPINGS IN
HILBERT SPACES
C. WU
School of Business and Administration, Kaifeng 475000, China
Copyright c2014 C. Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract.In this article, the problem of finding a common element of the set of solutions of a variational problem and the set of fixed points of nonexpansive mappings. Our results improve and extend the recent ones announced by many others.
Keywords: solution; variational inequality; fixed point; nonexpansive mapping. 2010 AMS Subject Classification:47H09, 90C33.
1. Introduction
Optimization theory has emerged as a powerful and effective tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [1-25] and the references therein. The computation of solutions of variational inequalities (fixed points of nonexpansive mappings) is important in the study of many real world problems. The aim of this paper is to investigate a solution problem
E-mail address: [email protected] Received June 20, 2013
of a family of nonexpansive mappings and relaxed cocoercive mappings. In Section 2, strong convergence theorems of common solutions are established in a Hilbert space.
2. Preliminaries
LetH be a real Hilbert space, whose inner product and norm are denoted byh·,·iandk · k, respectively. LetCbe a closed convex subset ofH and letA:C→H be a nonlinear map. Let PC be the projection ofH onto the convex subsetC. The classical variational inequality which denoted by V I(C,A) is to find u∈C such that hAu,v−ui ≥0, v∈C. One can see that the variational inequality is equivalent to a fixed point problem. The functionu∈Cis a solution of the variational inequality if and only if u∈Csatisfies the relationu=PC(u−λAu), where λ >0 is a constant.
Recall that Ais said to be inverse-strongly monotone if there exists a constants u>0 such that
hAx−Ay,x−yi ≥ukAx−Ayk2, ∀x,y∈C.
A mappingS:C→Cis said to be nonexpansive ifkSx−Syk ≤ kx−ykfor allx,y∈C.Next, we denote byF(S)the set of fixed points ofS. A mapping f :C→Cis said to be a contraction if there exists a coefficientα (0<α <1)such that
kf(x)−f(y)k ≤αkx−yk,
for∀x,y∈C. A linear bounded operator Bis strongly positive if there exists a constant ¯γ >0
with the propertyhBx,xi ≥γ¯kxk2,x∈H.A set-valued mappingT :H→2H is called monotone if for allx,y∈H, f ∈T xandg∈Tyimplyhx−y,f−gi ≥0.A monotone mappingT:H→2His maximal if the graph ofG(T)ofT is not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingT is maximal if and only if for(x,f)∈H×H, hx−y,f−gi ≥0 for every(y,g)∈G(T)implies f ∈T x.LetAbe a monotone map ofCintoH and letNCvbe the normal cone toCatv∈C, i.e.,NCv={w∈H:hv−u,wi ≥0, ∀u∈C}and define
T v=
Av+NCv, v∈C
ThenT is maximal monotone and 0∈T vif and only ifv∈V I(C,A); see [1] and the references therein.
Recently iterative methods for nonexpansive mappings have been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert spaceH:
min
x∈C
1
2hBx,xi − hx,bi, (2.1) whereBis a linear bounded operator,Cis the fixed point set of a nonexpansive mappingSandb is a given point inH. In [11], it is proved that the sequence{xn}defined by the iterative method
below, with the initial guessx0∈H chosen arbitrarily,
xn+1= (I−αnB)Sxn+αnb, n≥0,
converges strongly to the unique solution of the minimization problem (2.1) provided the se-quence{αn}satisfies certain conditions. More recently, Marino and Xu [12] introduced a new
iterative scheme by the viscosity approximation method:
xn+1= (I−αnB)Sxn+αnγf(xn), n≥0.
They proved the sequence{xn}generated by above iterative scheme converges strongly to the
unique solution of the variational inequality
h(B−γf)x∗,x−x∗i ≥0, x∈C,
which is the optimality condition for the minimization problem minx∈C12hBx,xi −h(x), where
C is the fixed point set of a nonexpansive mapping S, h is a potential function for δf (i.e., h0(x) =δf(x)forx∈H.)
of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation.
Recently Yaoet al.[13] considered a general iterative algorithm for an infinite family of non-expansive mapping in the framework of Hilbert spaces. To be more precisely, they introduced the following general iterative algorithm.
xn+1=λnγf(xn) +βnxn+ ((1−βn)I−λnA)Wnx,
where f is a contraction onH,Ais a stronglyy positive bounded linear operator,Wnare nonex-pansive mappings which are generated by an finite family of nonexnonex-pansive mappingT1,T2, . . .. To be more precisely,
Un,n+1=I,
Un,n=γnTnUn,n+1+ (1−γn)I,
.. .
Un,k=γkTkUn,k+1+ (1−γk)I,
un,k−1=γk−1Tk−1Un,k+ (1−γk−1)I, ..
.
Un,2=γ2T2Uu,3+ (1−γ2)I,
Wn=Un,1=γ1T1Un,2+ (1−γ1)I,
(2.2)
where {γ1},{γ2}, . . . are real numbers such that 0≤γ ≤1, T1,T2, . . . be an infinite family of mappings ofCinto itself. Nonexpansivity of eachTiensures the nonexpansivity ofWn.
ConcerningWnwe have the following lemmas which are important to prove our main results.
Lemma 2.1 [14]Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T1,T2, . . .be nonexpansive mappings of C into itself such that∩∞
n=1F(Tn) is nonempty,
and letγ1,γ2, . . .be real numbers such that0<γn≤η<1for any n≥1. Then, for every x∈C
Using Lemma 2.1, one can define the mappingW ofCinto itself as follows.W x=limn→∞Wnx=
limn→∞Un,1x,for every x∈C. Such aW is called theW-mapping generated byT1,T2, . . .and
γ1,γ2, . . .. Throughout this paper, we will assume that 0<γn≤η <1 for alln≥1.
Lemma 2.2[14]Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T1,T2, . . .be nonexpansive mappings of C into itself such that∩∞
n=1F(Tn)is nonempty, and
letγ1,γ2, . . .be real numbers such that0<γn≤η<1for any n≥1. Then, F(W) =∩∞n=1F(Tn).
In this paper, we introduce a composite iterative process as following:
x1∈C
yn=PC βnγf(xn) + (I−βnB)WnPC(I−rnA)xn
,
xn+1=PC αnxn+ (1−αn)yn+en
, n≥1,
(2.3)
where A is an inverse-strongly monotone mapping, B is a strongly positive linear bounded operator, f is a contraction onCandWnis a mapping generated by (2.2).
We prove the sequence{xn}generated by the above iterative scheme converges strongly to a common element of the set of common fixed points of an infinite nonexpansive mappings and the set of solutions of the variational inequalities for an inverse-strongly mapping, which solves another variation inequality hγf(q)−Bq,q−pi ≤0, p∈ ∩∞i=1F(Ti)∩V I(C,A) and is also
the optimality condition for the minimization problem minx∈C12hBx,xi −h(x),whereC is the
intersection of the common fixed points set of a nonexpansive mappings and the set of solutions of the variational inequalities for relaxed (γ,r)-cocoercive maps, h is a potential function for δf (i.e.,h0(x) =δf(x)forx∈H.)
In order to prove our main results, we need the following lemmas.
Lemma 2.3[11]Assume that{αn}is a sequence of nonnegative real numbers such that
αn+1≤(1−γn)αn+δn,
whereγnis a sequence in (0,1) and{δn}is a sequence such that
(i) ∑∞n=1γn=∞;
(ii) lim supn→∞δn/γn≤0or∑∞n=1|δn|<∞.
Lemma 2.4[12]Assume B is a strongly positive linear bounded operator on a Hilbert space H with coefficientγ¯>0and0<ρ≤ kBk−1. Then
kI−ρBk ≤1−ργ¯.
Lemma 2.5[15]Let{xn}and{yn}be bounded sequences in a Banach space X and letβnbe a
sequence in [0,1] with
0<lim inf
n→∞ βn≤lim supn→∞ βn<1.
Suppose xn+1= (1−βn)yn+βnxnfor all integers n≥0and
lim sup
n→∞
(kyn+1−ynk − kxn+1−xnk)≤0.
Thenlimn→∞kyn−xnk=0.
Lemma 2.6. In a real Hilbert space H, there holds the the following inequality
kx+yk2≤ kxk2+2hy,x+yi,
for all x,y∈H.
3. Main results
Theorem 3.1. Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let A:C→H be an u-inverse-strongly monotone mapping. Let f :C→C be a contraction with the coefficient α (0<α <1)and {Ti}∞i=1 be an infinite nonexpansive mappings from C into
itself generated by (2.2) such that F=T∞
i=1F(Ti)∩V I(C,A)6= /0. Let B be a strongly positive
linear bounded self-adjoint operator of C into itself with coefficient γ¯>0such that kBk ≤1. Assume that0<γ<γ¯/α. Assume that x1∈C and{xn}is generated by
x1∈C
yn=PC βnγf(xn) + (I−βnB)WnPC(I−rnA)xn
,
where enis a bounded sequence in H,{αn}and{βn}are sequences in(0,1). If{αn},{βn}and
{rn}are chosen such that
(a) 0<lim infn→∞αn≤lim supn→∞αn<1;
(b) limn→∞βn=0, ∑∞n=1βn=∞;
(c) limn→∞|rn+1−rn|=0,∑∞n=1kenk<∞;
(d) {rn} ⊂[a,b] f or some a,b with0<a<b<2u.
Then{xn}converges strongly to q∈F, where q=PF(γf+ (I−B))(q),which solves the
varia-tion inequalityhγf(q)−Bq,p−qi ≤0, ∀p∈F.
Proof. From the definition of inverse-strongly monotone mappings, we find that I−rnA is nonexpansive. Since the condition (a), we may assume, with no loss of generality, thatβn<
kBk−1 for all n.From Lemma 2.4, we know that if 0<ρ ≤ kBk−1, thenkI−ρBk ≤1−ργ¯. Letting p∈F, we have
kyn−pk ≤βnkγf(xn)−Bpk+ (1−βnγ¯)kWnPC(I−rnA)xn−pk
≤βnγkf(xn)− f(p)k+βnkγf(p)−Bpk+ (1−βnγ¯)kxn−pk
= [1−βn(γ¯−γ α)]kxn−pk+βnkγf(p)−Bpk.
On the other hand, we have
kxn+1−pk ≤αnkxn−pk+ (1−αn)kyn−pk+kenk
≤αnkxn−pk+ (1−αn)[(1−βn(γ¯−γ α))kxn−pk
+βnkγf(p)−Bpk] +kenk.
By simple inductions, we have that the sequence{xn}is bounded. Puttingρn=PC(I−rnA)xn,
we have
kρn−ρn+1k ≤k(I−rnA)xn−(I−rn+1A)xn+1k
=k(xn−rnAxn)−(xn+1−rnAxn+1) + (rn+1−rn)Axn+1k
≤kxn−xn+1k+|rn+1−rn|M1,
whereM1is an appropriate constant. It follows that
kyn−yn+1k ≤(1−βn+1γ¯)(kρn+1−ρnk+kWn+1ρn−Wnρnk)
+|βn+1−βn|M2+γ βn+1αkxn+1−xnk,
(3.2)
where M2 is an appropriate constant. Since both Ti andUn,i are nonexpansive, we find from
(2.2) that
kWn+1ρn−Wnρnk ≤γ1kUn+1,2ρ−Un,2ρnk
=γ1kγ2T2Uu+1,3ρn−γ2T2Un,3ρnk
≤γ1γ2kUu+1,3ρn−Un,3ρnk
≤ · · ·
≤γ1γ2· · ·γnkUn+1,n+1ρn−Un,n+1ρnk
≤M3
n
∏
i=1
γi,
(3.3)
whereM3≥0 is an appropriate. Therefore, we have
kyn−yn+1k ≤[1−βn+1(γ¯−α γ)]kxn+1−xnk
+M4(|rn+1−rn|+|βn+1−βn|+ n
∏
i=1
γi),
whereM4is an appropriate appropriate constant. It follows that
lim sup
n→∞
{kyn+1−ynk − |xn+1−xnk} ≤0. (3.4)
Using Lemma 2.6, we obtain that
lim
n→∞
kyn−xnk=0. (3.5)
It follows that
lim
n→∞
kxn+1−xnk=0 (3.6)
and
lim
n→∞kWnρn−
Forp∈F, we have kρn−pk2
=kPC(I−rnA)xn−PC(I−rnA)pk2
≤ k(xn−p)−rn(Axn−Ap)k2
=kxn−pk2−2rnhxn−p,Axn−Api+r2nkAxn−Apk2
≤ kxn−pk2−2rn[−γkAxn−Apk2+rkxn−pk2] +r2nkAxn−Apk2
≤ kxn−pk2+2rnγkAxn−Apk2−2rnrkxn−pk2+rn2kAxn−Apk2
≤ kxn−pk2+ (2rnγ+r2n−2rnr
µ2 )kAxn−Apk
2.
(3.8)
Since
kyn−pk2=kβn(γf(xn)−Bp) + (I−βnB)(Wnρn−p)k2
≤(βnkγf(xn)−Bpk+ (1−βnγ¯)kρn−pk)2
≤βnkγf(xn)−Bpk2+kρn−pk2+2βnkγf(xn)−Bpkkρn−pk,
(3.9)
we see that
kxn+1−pk2≤2kαn(xn−p) + (1−αn)(yn−p)k2+2kenk2
≤2αnkxn−pk2+2(1−αn)kyn−pk2+2kenk2
≤2αnkxn−pk2+2(1−αn)[βnkγf(xn)−Bpk2+2kρn−pk2
+4βnkγf(xn)−Bpkkρn−pk] +2kenk2.
(3.10)
Substituting (3.8) into (3.10), we have kxn+1−pk2
≤ kxn−pk2+βnkγf(xn)−Bpk2+ (2rnγ+rn2−
2rnr
µ2 )kAxn−Apk
2
+2βnkγf(xn)−Bpkkρn−pk+kenk2.
(3.11)
This obtains that
lim
On the other hand, we have
kρn−pk2=kPC(I−rnA)xn−PC(I−rnA)pk2
≤h(I−rnA)xn−(I−rnA)p,ρn−pi
=1
2{k(I−rnA)xn−(I−rnA)pk
2+k
ρn−pk2
− k(I−rnA)xn−(I−rnA)p−(ρn−p)k2}
≤1
2{kxn−pk
2+k
ρn−pk2− k(xn−ρn)−rn(Axn−Ap)k2}
=1
2{kxn−pk
2+k
ρn−pk2− kxn−ρnk2−r2nkAxn−Apk2
+2rnhxn−ρn,Axn−Api},
which yields that
kρn−pk2≤kxn−pk2− kρn−xnk2+2rnkρn−xnkkAxn−Apk. (3.13)
Therefore, we have
(1−αn)kρn−xnk2
≤ kxn−pk2− kxn+1−pk2+βnkγf(xn)−Bpk2+2rnkρn−xnkkAxn−Apk
+2βnkγf(xn)−Bpkkρn−pk
≤(kxn−pk+kxn+1−pk)kxn−xn+1k+βnkγf(xn)−Bpk2
+2rnkρn−xnkkAxn−Apk+2βnkγf(xn)−Bpkkρn−pk.
From the conditions (i), (ii), (3.6) and (3.12), we have lim
n→∞kρn−
xnk=0. (3.14)
On the other hand, we havekρn−Wnρnk ≤ kxn−ρnk+kxn−ynk+kyn−Wnρnk.Therefore, we
have limn→∞kWρn−Wnρnk=0. Since
kWρn−ρnk ≤ kWnρn−ρnk+kWnρn−Wρnk,
we have
lim
SincePF(γf+ (I−B))is a contraction, we find thatPF(γf+ (I−B))has a unique fixed point,
sayq∈H. That is,q=PF(γf+ (I−B))(q).To show it, we choose a subsequence{xni}of{xn}
such that lim supn→∞hγf(q)−Bq,xn−qi=limi→∞hγf(q)−Bq,xni−qi.As{xni}is bounded,
we have that there is a subsequence {xni j} of {xni}converges weakly to p. We may assume
that without loss of generality thatxni*p.Hence we havep∈F. Indeed, let us first show that
p∈V I(C,A).Put
Tw1=
Aw1+NCw1, w1∈C
/0, w1∈/C.
SinceAis inverse-strongly monotone, we haveT is maximal monotone. Let(w1,w2)∈G(T). Sincew2−Aw1∈NCw1andρn∈C,we havehw1−ρn,w2−Aw1i ≥0.On the other hand, from
ρn=PC(I−rnA)xn,we havehw1−ρn,ρn−(I−rnA)xni ≥0 and hencehw1−ρni,w2i ≥ hw1−
ρni,Aρni−Axnii − hw1−ρni,ρnir−xni
ni i,which implies thathw1−p,w2i ≥0. We have p∈T −10 and hence p∈V I(C,A). Next, let us show p∈T∞
i=1F(Ti). Since Hilbert spaces are Opial’s
spaces, from (3.15), we have
lim inf
i→∞
kρni−pk<lim inf i→∞
kρni−W pk
=lim inf
i→∞
kρni−Wρni+Wnρni−W pk
≤lim inf
i→∞
kWρni−W pk
≤lim inf
i→∞
kρni−pk,
which derives a contradiction. Thus, we have p∈F(W) =T∞
i=1F(Ti). On the other hand, we
have
lim sup
n→∞
hγf(q)−Bq,xn−qi=lim n→∞hγ
f(q)−Bq,xni−qi
=hγf(q)−Bq,p−qi ≤0.
It follows from Lemma 2.5 that kyn−qk2≤
(1−βnγ¯)2+βnγ α
1−βnγ α kxn−qk
2+ 2βn
1−αnγ αhγ
f(q)−Bq,yn−qi
=(1−2βnγ¯+βnα γ) 1−βnγ α
kxn−qk2+
βn2γ¯2
1−βnγ α
kxn−qk2
+ 2βn 1−βnγ α
hγf(q)−Bq,yn−qi
≤[1−2βn(γ¯−α γ) 1−βnγ α
]kxn−qk2
+2βn(γ¯−α γ) 1−βnγ α
[ 1 ¯
γ−α γhγf(q)−Bq,yn−qi+
αnγ¯2
2(γ¯−α γ)M5],
whereM5is an appropriate constant. On the other hand, we have
kxn+1−pk2≤ kαn(xn−p) + (1−αn)(yn−p)k2+kenk2
≤2αnkxn−pk2+2(1−αn)kyn−pk2+2kenk2.
Hence, we have
kxn+1−pk2≤[1−(1−αn)
2βn(γ¯−α γ)
1−βnγ α
]kxn−qk2
+ (1−αn)
2βn(γ¯−α γ)
1−βnγ α
[ 1 ¯
γ−α γhγf(q)−Aq,yn−qi+
βnγ¯2
2(γ¯−α γ)M5].
Using Lemma 2.3, we find thatxn→qasn→∞.This completes the proof.
Corollary 3.2. Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let A:C→H be an u-inverse-strongly monotone mapping. Let f :C→C be a contraction with the coefficientα (0<α<1)and{Ti}∞i=1be an infinite nonexpansive mappings from C into itself generated by (2.2) such that F=T∞
i=1F(Ti)∩V I(C,A)6= /0. Let B be a strongly positive
linear bounded self-adjoint operator of C into itself with coefficient γ¯>0such that kBk ≤1. Assume that0<γ<γ¯/α. Assume that x1∈C and{xn}is generated by
x1∈C
yn=PC βnγf(xn) + (I−βnB)WnPC(I−rnA)xn
,
xn+1=αnxn+ (1−α)yn, n≥1,
where ,{αn}and{βn}are sequences in(0,1). If{αn},{βn}and{rn}are chosen such that
(b) limn→∞βn=0, ∑∞n=1βn=∞;
(c) limn→∞|rn+1−rn|=0;
(d) {rn} ⊂[a,b] f or some a,b with0<a<b<2u.
Then{xn}converges strongly to q∈F, where q=PF(γf+ (I−B))(q),which solves the
varia-tion inequalityhγf(q)−Bq,p−qi ≤0, ∀p∈F.
Corollary 3.3. Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let A:C→H be an u-inverse-strongly monotone mapping. Let f :C→C be a contraction with the coefficientα (0<α <1) and{Ti}∞i=1 be an infinite nonexpansive mappings from C
into itself generated by (2.2) such that F =T∞
i=1F(Ti)∩V I(C,A)6= /0. Assume that x1∈C and {xn}is generated by
x1∈C
yn=βnf(xn) + (1−βn)WnPC(I−rnA)xn
,
xn+1=PC αnxn+ (1−α)yn+en
, n≥1,
where enis a bounded sequence in H,{αn}and{βn}are sequences in(0,1). If{αn},{βn}and
{rn}are chosen such that
(a) 0<lim infn→∞αn≤lim supn→∞αn<1;
(b) limn→∞βn=0, ∑∞n=1βn=∞;
(c) limn→∞|rn+1−rn|=0,∑∞n=1kenk<∞;
(d) {rn} ⊂[a,b] f or some a,b with0<a<b<2u.
Then{xn}converges strongly to q∈F, where q=PFf(q),which solves the variation inequality
hf(q)−q,p−qi ≤0, ∀p∈F.
Conflict of Interests
The author declares that there is no conflict of interests.
Acknowledgements
REFERENCES
[1] R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970) 75-88.
[2] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems. Math. Stud, 63 (1994) 123-145.
[3] S. Kamimura, W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000) 226-240.
[4] S.Y. Cho, S.M. Kang, Zero point theorems form-accretive operators in a Banach space, Fixed Point Theory 13 (2012) 49-58.
[5] H.K. Xu, A regularization method for the proximal point algorithm, J. Global Optim. 36 (2006) 115-125. [6] J.W. Chen, Z. Wan, Y. Zou, Strong convergence theorems for firmly nonexpansive-type mappings and
equi-librium problems in Banach spaces, Optim. 62 (2013) 483-497.
[7] X. Qin, S.Y. Cho, L. Wang, Iterative algorithms with errors for zero points of m-accretive operators, Fixed Point Theory Appl. 2013, 2013:148.
[8] Q. Yuan, M. Shang, Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings, Fixed Point Theory Appl. 2013, 2013: 67.
[9] J. Ye, J. Huang, Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces, J. Math. Comput. Sci. 1 (2011) 1-18. [10] B.C. Dhage, H.K. Nashine, V.S. Patil, Common fixed points for some variants of weakly contraction
map-pings in partially ordered metric spaces, Adv. Fixed Point Theory, 3 (2013) 29-48.
[11] H.K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003) 659-678. [12] G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal.
Appl. 318 (2006) 43-52.
[13] Y. Yao, A general iterative method for a finite family of nonexpansive mappings, Nonlinear Anal. 66 (2007) 2676-2687.
[14] K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001) 387-404.
[15] T. Suzuki, Strong convergence of Krasnoselskii and Manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005) 227-239.
[16] A.Y. Al-Bayati, R.Z. Al-Kawaz, A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization, J. Math. Comput. Sci. 2 (2012) 937-966.
[18] H. Zegeye, N. Shahzad, Strong convergence theorem for a common point of solution of variational inequality and fixed point problem, Adv. Fixed Point Theory, 2 (2012) 374-397.
[19] J.M. Chen, L.J. Zhang, T.G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2007) 1450-1461.
[20] Y. Hao, M. Shang, Convergence theorems for nonexpansive and inverse-strongly monotone mappings, Fixed Point Theory 11 (2010) 273-288.
[21] Z.M. Wang, W.D. Lou, A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems, J. Math. Comput. Sci. 3 (2013) 57-72.
[22] B.O. Osu, O.U. Solomon, A stochastic algorithm for the valuation of financial derivatives using the hyperbolic distributional variates, Math. Fianc. Lett. 1 (2012) 43-56.
[23] V.A. Khan, K. Ebadullah, I-convergent difference sequence spaces defined by a sequence of moduli, J. Math. Comput. Sci. 2 (2012) 265-273.
[24] H. Iiduka, W. Takahashi, M. Toyoda, Approximation of solutions ofvariational inequalities for monotone mappings, Pan Amer. Math. J. 14 (2004) 49-61.
