R E S E A R C H
Open Access
A new explicit iterative algorithm for solving
a class of variational inequalities over the
common fixed points set of a finite family of
nonexpansive mappings
Cuijie Zhang
*and Caiping Yang
*Correspondence:
[email protected] College of Science, Civil Aviation University of China, Tianjin, 300300, P.R. China
Abstract
In this paper, we introduce a new explicit iterative algorithm for finding a solution for a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the variational inequality. Our result improves and extends the
corresponding results announced by many others. At the end of the paper, we extend our result to the more broad family of
λ
-strictly pseudo-contractive mappings.Keywords: nonexpansive mapping; strong convergence; variational inequalities; common fixed points
1 Introduction
LetH be a real Hilbert space with inner product·,·and norm · . Throughout this paper, we always assume thatTis a nonexpansive operator onH. The fixed point set ofT is denoted byFix(T),i.e.,Fix(T) ={x∈H:Tx=x}. The typical problem is to minimize a quadratic function on a real Hilbert spaceH:
min x∈C
Ax,x–x,u, (.)
whereCis a nonempty closed convex subset ofH,uis a given point inHandAis a strongly positive bounded linear operator onH.
In , Xu [] introduced the following iterative scheme:
xn+=αnu+ (I–αnA)Txn, (.)
whereuis some point ofHand{αn}is a sequence in (, ). He proved that the sequence
{xn}converges strongly to the unique solution of the minimization problem (.) withC= Fix(T).
In , Marino and Xu [] considered the viscosity method on the iterative scheme (.), and they gave the following general iterative method:
xn+=αnγf(xn) + (I–αnA)Txn, (.)
wheref is a contraction onH. They proved the above sequence{xn}converges strongly
to the unique solution of the variational inequality
(A–γf)x∗,x–x∗≥, ∀x∈Fix(T),
which is the optimality condition for the minimization problem
min x∈Fix(T)
Ax,x–h(x),
wherehis a potential function forγf (i.e.,h(x) =γf(x) forx∈H). In , Yamada [] considered the following hybrid iterative method:
xn+=Txn–μλnF(Txn), (.)
whereFisL-Lipschitzian continuous andη-strongly monotone operator withL> ,η> and <μ< η/L. Under some appropriate conditions, the sequence{x
n}generated by
(.) converges strongly to the unique solution of the variational inequality
Fx∗,x–x∗≥, ∀x∈Fix(T).
Combining (.) and (.), Tian [] considered the following general viscosity type iterative method:
xn+=αnγf(xn) + (I–μαnF)Txn. (.)
Improving and extending the corresponding results given by Marinoet al., he proved that the sequence{xn}generated by (.) converges strongly to the unique solutionx∗∈Fix(T)
of the variational inequality
(γf–μF)x,˜ x–x˜≤, ∀x∈Fix(T).
In [], Tian generalized the iterative method (.) replacing the contraction operatorf with a Lipschitzian continuous operatorVto solve the following variational inequality:
(γV–μF)x,˜ x–x˜≤, ∀x∈Fix(T). (.) On the other hand, let{Ti}Ni= be a finite family of nonexpansive self-mappings ofH. AssumeNi=Fix(Ti)=∅. In [], Xu also defined the following sequence{xn}:
xn+=αnu+ (I–αnA)Tn+xn, n≥, (.)
whereTn=TnmodN and themodfunction takes values in{, , . . . ,N}. He found that the
sequence{xn}generated by (.) converges strongly to the unique solution of the
mini-mization problem (.) withC=Ni=Fix(Ti) under suitable conditions on{αn}and the
following additional condition on{Tn}:
F(TN· · ·TT) =F(TTN· · ·TT) =· · ·=F(TN–· · ·TTN). (.)
In , Atsushiba and Takahashi [] defined theWn-mappings generated byT,T, . . . , TNand{γn,},{γn,}, . . . ,{γn,N} ⊂[, ] as follows:
Un,=I,
Un,=γn,TUn,+ ( –γn,)I,
Un,=γn,TUn,+ ( –γn,)I,
.. .
Un,N–=γn,N–TN–Un,N–+ ( –γn,N–)I,
Wn=Un,N=γn,NTNUn,N–+ ( –γn,N)I.
From [, Lemma .], we know thatF(Wn) =
N i=F(Ti).
In , Yao [] introduced the following iterative method:
xn+=αnγf(xn) +βxn+
( –β)I–αnA
Wnxn. (.)
Without the condition (.), he proved that the sequence{xn}generated by (.) converges
strongly to the unique solution of the following variational inequality:
(A–γf)x∗,x∗–x≤, ∀x∈
N
i=
Fix(Ti), (.)
which is the optimality condition for the minimization problem
min x∈C
Ax,x–h(x), (.)
whereC=Ni=Fix(Ti) andhis a potential function forγf (i.e.,h(x) =γf(x)).
Shanget al.[] introduced the following scheme:
⎧ ⎨ ⎩
yn=βnxn+ ( –βn)Wnxn,
xn+=αnγf(xn) + (I–αnA)yn.
(.)
Under certain appropriate conditions, without (.), they proved that{xn}defined by (.)
converges strongly to the unique solution of (.) which is also the optimality condition for (.).
Recently, combining the Krasnoselskii-Mann type algorithm and the steepest-descent method, Buong and Duong [] introduced a new explicit iterative algorithm:
xk+=
–βkxk+βkTkTNk· · ·Tkxk, (.)
whereTik= ( –βi
k)I+βkiTifori= , , . . . ,N,T k
=I–λkμF, andFis anL-Lipschitz
variational inequality:
Fx∗,x–x∗≥, ∀x∈
N
i=
Fix(Ti). (.)
Very recently, Zhou and Wang [] proposed a simpler iterative algorithm than the it-erative algorithm (.) given by Buong and Duong:
xk+= (I–λkμF)TNk· · ·Tkxk. (.)
They proved that the sequence{xk}defined by (.) converges strongly to the unique
solution of the variational inequality (.) in a faster rate of convergence.
Motivated and inspired by the results of Zhouet al., in this paper, we consider a new it-erative algorithm to solve the class of variational inequalities (.). The itit-erative algorithm improves and extends the results of Yaoet al., and the corresponding results announced by many others. At the end of this paper, we extend our iterative algorithm to the more broad family ofλ-strictly pseudo-contractive mappings.
2 Preliminaries
Throughout this paper, we writexnxandxn→xto indicate that{xn}converges weakly
toxand converges strongly tox, respectively.
An operatorT:H→His said to be nonexpansive ifTx–Ty ≤ x–yfor allx,y∈H. It is well known thatFix(T) is closed and convex.Ais called strongly positive if there exists a constantγ > such thatAx,x ≥γxfor allx∈H. The operatorFis calledη-strongly monotone if there exists a constantη> such that
x–y,Fx–Fy ≥ηx–y
for allx,y∈H.
In order to prove our results, we collect some necessary conceptions and lemmas in this section.
Definition . A mappingT :H→H is said to be an averaged mapping if there exists
some numberα∈(, ) such that
T= ( –α)I+αS, (.)
whereI:H→His the identity mapping andS:H→His nonexpansive. More precisely, when (.) holds, we say thatTisα-averaged.
Lemma .([]) (i)The composite of finitely many averaged mappings is averaged.That
is,if each of the mappings{Ti}Ni=is averaged,then so is the composite T· · ·TN.In
partic-ular,if Tisα-averaged and Tisα-averaged,whereα,α∈(, ),then both TTand TTareα-averaged,whereα=α+α–αα.
(ii)If the mappings{Ti}Ni=are averaged and have a common fixed point,then
N
i=
Fix(Ti) =Fix(T· · ·TN).
Lemma .([]) Let C be a closed convex subset of a real Hilbert space H.Given x∈H and y∈C.Then y=PCx if and only if the following inequality holds:
x–y,z–y ≤
for every z∈C.
Lemma .([]) Assume V is a contraction on a Hilbert space H with coefficientα> , and F :H→H is an L-Lipschitzian continuous andη-strongly monotone operator with L> ,η> .Then,for <γ <μηα,μF–γV is strongly monotone with coefficientμη–γ α.
Lemma .([]) Let H be a Hilbert space,C a closed convex subset of H,and T:C→C a nonexpansive mapping withFix(T)=∅.If{xn}is a sequence in C weakly converging to
x∈C and{(I–T)xn}converges strongly to y∈C,then(I–T)x=y.In particular,if y= ,
then x∈Fix(T).
Lemma .([]) Let{xn}and{zn}be bounded sequences in a Banach space X and{βn}
be a sequence in[, ]which satisfies the following condition:
<lim inf
n→∞ βn≤lim supn→∞ βn< .
Suppose xn+= ( –βn)zn+βnxnfor all integers n≥and lim sup
n→∞
zn+–zn–xn+–xn
≤.
Thenlimn→∞zn–xn= .
Lemma .([]) Assume{an}is a sequence of nonnegative real numbers such that
an+≤( –γn)an+δn, n≥,
where{γn}is a sequence in(, )and{δn}is a sequence such that
(i) ∞n=γn=∞,
(ii) lim supn→∞δn
γn ≤or
∞
n=|δn|<∞.
Thenlimn→∞an= .
Lemma . ([]) Assume S is a λ-strictly pseudo-contractive mapping on a Hilbert
space H.Define a mapping T by Tx=αx+ ( –α)Sx for all x∈H andα∈[λ, ).Then T is a nonexpansive mapping such thatFix(T) =Fix(S).
3 Main results
Now we state and prove our main results in this paper.
Theorem . Let{Ti}Ni=be N nonexpansive mappings of a real Hilbert space H such that C=Ni=Fix(Ti)=∅,F be an L-Lipschitzian continuous andη-strongly monotone operator
on H with L> andη> ,V be anα-Lipschitzian on H withα> .Suppose x∈H and <μ<η
L.Define a sequence{xk}as follows:
where <γ <ταwithτ=μ(η–μL)and Tk
i = ( –βki)I+βkiTifor i= , , . . . ,N.Suppose
αk∈(, )andβki∈(ξ,ζ)for someξ,ζ ∈(, ).If the following conditions are satisfied:
(i) limk→∞αk= ;
(ii) ∞k=αk=∞;
(iii) limk→∞|βki+–βki|= fori= , , . . . ,N.
Then the sequence{xk}converges strongly to the unique solution x∗of the variational
in-equality:
(μF–γV)x∗,x–x∗≥, ∀x∈
N
i=
Fix(Ti). (.)
Equivalently,we have PC(I–μF+γV)x∗=x∗.
Proof Since our methods easily deduce the general case, we prove Theorem . forN= . First, we show{xk}is bounded. In fact, for some pointp∈C, by (.) we have
xk+–p=αkγVxk+ (I–μαkF)TkTkxk–p
=(I–μαkF)TkTkxk– (I–μαkF)p+αk(γVxk–μFp)
≤( –αkτ)TkTkxk–TkTkp+αk
γVxk–γVp+γVp–μFp
≤( –αkτ)xk–p+αkγ αxk–p+αkγVp–μFp
= –αk(τ–γ α)
xk–p+αk(τ–γ α)
γVp–μFp τ–γ α
≤max
xk–p,
τ–γ αγVp–μFp
≤ · · · ≤max
x–p,
τ–γ αγVp–μFp
.
Therefore,{xk}is bounded. Hence we also see that{TkTkxk},{FTkTkxk}, and{Vxk}are all
bounded. From (.), it follows that
lim
k→∞xk+–T
k
Tkxk= . (.)
We next show thatlimk→∞xk+–xk= . Noting thatTkandTkareβk-averaged and
βk-averaged, respectively, by Lemma ., we find thatTk
Tk istk-averaged for everyk,
where tk=βk+βk–βkβk. Set ξ∗= ξ –ξ andζ∗= ζ –ζ. It is easy to deduce that
<ξ∗≤tk≤ζ∗< for allkand
lim
k→∞tk+–tk= . (.)
Since for everyk,Tk
Tk istk-averaged, we can find a family of nonexpansive mappings
{Sk}k≥onHsuch that
Substituting (.) into (.) yields
xk+=αkγVxk+ (I–μαkF)
( –tk)xk+tkSkxk
= ( –tk)xk+tk
Skxk+
αk
tk
γVxk–μFTkTkxk
.
Define a sequence{zk}byzk=Skxk+αtkk(γVxk–μFTkTkxk), so
xk+= ( –tk)xk+tkzk. (.)
Now, we claim that
lim sup k→∞
zk+–zk–xk+–xk
≤.
To this end, we observe that
zk+–zk ≤ Sk+xk+–Skxk+
αk+ tk+
γVxk+–μFTk+Tk+xk+
+αk tk
γVxk–μFTkTkxk
≤ Sk+xk+–Sk+xk+Sk+xk–Skxk
+αk+ tk+
γVxk++μFTk+Tk+xk+
+αk tk
γVxk+μFTkTkxk
≤ xk+–xk+Sk+xk–Skxk
+αk+ tk+
γVxk++μFTk+Tk+xk+
+αk tk
γVxk+μFTkTkxk (.)
and
Sk+xk–Skxk=
tk+Tk+Tk+xk–
tk
TkTkxk–
–tk+ tk+
xk+
–tk
tk
xk
≤tk+–tk
tk+tk
Tk+Tk+xk+xk
+ tk
Tk+Tk+xk–TkTkxk
≤tk+–tk
tk+tk
M+
tk
Tk+
Tk+xk–Tk+Tkxk
+Tk+Tkxk–TkTkxk
≤tk+–tk
tk+tk
M+
ξ∗T
k+
xk–Tkxk
+Tk+Tkxk–TkTkxk, (.)
whereMis a fixed constant satisfying
M≥sup
k≥
Tk+Tk+xk+xk
Note that
Tk+
xk–Tkxk= –βk+
xk+βk+Txk–
–βkxk–βkTxk
≤βk+–βkxk+Txk
.
Sincelimk→∞|βki+–βki|= fori= , , and{xk}and{Txk}are bounded, we easily obtain lim
k→∞T
k+
xk–Tkxk= . (.)
Similarly,
Tk+Tkxk–TkTkxk≤βk+–βkTkxk+TTkxk,
from which it follows that
lim k→∞T
k+
Tkxk–TkTkxk= . (.)
Using (.), (.), and (.), from (.) we have
lim
k→∞Sk+xk–Skxk= . (.)
Sincelimk→∞αk= and <ξ∗<tk<ζ∗< , combining (.) and (.) we get
lim sup k→∞
zk+–zk–xk+–xk
≤.
By Lemma ., we conclude thatlimk→∞zk–xk= , which implies thatlimk→∞xk+– xk= by (.). Thus from (.), it is true that
lim k→∞xk–T
k
Tkxk= . (.)
From [, Theorem .], we know that the solution of the variational inequality (.) is unique. We usex∗to denote the unique solution of (.). Since{xk}k≥is bounded, there exists a subsequence{xkj}j≥of{xk}k≥such thatxkjxˆasj→ ∞and
lim sup k→∞
(μF–γV)x∗,x∗–xk
=lim j→∞
(μF–γV)x∗,x∗–xkj
.
Since{βki}is bounded fori= , , we can assume thatβki j→β
i
∞asj→ ∞, where <ξ≤ β∞i ≤ζ < fori= , . DefineTi∞= ( –β∞i )I+β∞i Ti(i= , ). Then we haveFix(Ti∞) = Fix(Ti) fori= , . Note that
Tikjx–Ti∞x≤βki j–β
i
∞x+Tix
.
Hence, we deduce that
lim j→∞supx∈D
Tikjx–Ti∞x= , (.)
SinceFix(T∞)∩Fix(T∞) =Fix(T)∩Fix(T) =C=∅andTi∞isβ∞i -averaged fori= , ,
by Lemma ., we know thatFix(T∞T∞) =Fix(T∞)∩Fix(T∞) =C. Combining (.) and (.), we obtain
xkj–T∞T∞xkj≤xkj–T
kj
T
kj
xkj+T
kj
T
kj
xkj–T∞T
kj
xkj
+T∞Tkjxkj–T ∞
T∞xkj
≤xkj–T
kj
T
kj
xkj+T
kj
T
kj
xkj–T∞T
kj
xkj
+Tkjxkj–T ∞
xkj
≤xkj–T
kj
T
kj
xkj+sup
x∈D
Tkjx–T∞x
+sup x∈D
Tkjx–T∞x,
where D is a bounded subset including{Tkjxkj}andD is a bounded subset including
{xkj}. Hencelimj→∞xkj–T∞T∞xkj= . From Lemma ., we havexˆ∈Fix(T∞T∞) =C. It follows that
lim sup k→∞
(μF–γV)x∗,x∗–TkTkxk
=lim sup k→∞
(μF–γV)x∗,x∗–xk
=lim j→∞
(μF–γV)x∗,x∗–xkj
=(μF–γV)x∗,x∗–xˆ≤. (.) Finally, we show thatxk→x∗ask→ ∞. From (.), we have
xk+–x∗ =αkγVxk+ (I–μαkF)TkTkxk–x∗
=(I–μαkF)TkTkxk– (I–μαkF)x∗+αk
γVxk–μFx∗
=(I–μαkF)TkTkxk– (I–μαkF)x∗
+αkγVxk–μFx∗
+ αk
(I–μαkF)TkTkxk– (I–μαkF)x∗,γVxk–μFx∗
≤( –αkτ)xk–x∗
+αkγVxk–μFx∗
+ αk
TkTkxk–x∗,γVxk–μFx∗
– μαkFTk
Tkxk–Fx∗,γVxk–μFx∗
≤( –αkτ)xk–x∗
+αkγVxk–μFx∗
+ αkγ
TkTkxk–x∗,Vxk–Vx∗
+ αk
TkTkxk–x∗,γVx∗–μFx∗
– μαkFTkTkxk–Fx∗,γVxk–μFx∗
≤( –αkτ)+ ααkγxk–x∗+αk
TkTkxk–x∗, (γV–μF)x∗
+αkγVxk–μFx∗+ αkLTkTkxk–x∗γVxk–μFx∗
= – αk(τ–αγ)xk–x∗
+αk
TkTkxk–x∗, (γV–μF)x∗
+αkγVxk–μFx∗
+ Lxk–x∗γVxk–μFx∗+τxk–x∗
≤ – αk(τ–αγ)xk–x∗
+αk
TkTkxk–x∗, (γV–μF)x∗
+αkM
,
whereM is a constant satisfying
M ≥sup
k≥
γVxk–μFx∗+ LTkTkxk–x∗γVxk–μFx∗+τxk–x∗
.
Consequently, according to the conditions (i) and (ii), (.), and Lemma ., we conclude thatxk→x∗ask→ ∞. This completes the proof. 4 An extension of our result
In this section, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings. Now let us recall that a mappingS:H→His said to beλ-strictly pseudo-contractive if there exists a constantλ∈[, ) such that
Sx–Sy≤ x–y+λ(I–S)x– (I–S)y, ∀x,y∈H.
Let {Si}Ni= be a family ofλi-strictly pseudo-contractive self-mappings of H with ≤
λi< . Fori= , , . . . ,N, define
ˆ
Ti=ωiI+ ( –ωi)Si, (.)
where ≤λi≤ωi< . By virtue of Lemma ., we know that{ ˆTi}Ni=is a family of non-expansive mappings. Thus we extend Theorem . to the family of λi-strictly
pseudo-contractions.
Theorem . Let H be a real Hilbert space,F:H→H be an L-Lipschitizian continuous
andη-strongly monotone operator on H with L> andη> , V be anα-Lipschitzian continuous on H withα> .Let{Si}Ni= be N λi-strictly pseudo-contractive mappings on
H such that C =Ni=Fix(Si)= ∅. Suppose <μ< τα, <γ < τ
α withτ =μ(η–
μL
),
αk∈(, ),βki ∈(ξ,ζ)for some ξ,ζ ∈(, )and≤λi≤ωi< for i= , , . . . ,N.If the
conditions(i)-(iii)of Theorem.are satisfied,the sequence{xk}k≥defined by(.)with Ti
replaced by(.),converges strongly to the unique solution x∗ of the following variational inequality:
(μF–γV)x∗,x–x∗≥, ∀x∈
N
i=
Fix(Si).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. 3122013k004).
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10.1186/1687-1812-2014-60
