R E S E A R C H
Open Access
Convergence theorems of solutions of a
generalized variational inequality
Li Yu
1*and Ma Liang
2* Correspondence: [email protected]
1School of Business Administration, Henan University, Kaifeng 475000, Henan Province, China Full list of author information is available at the end of the article
Abstract
The convex feasibility problem (CFP) of finding a point in the nonempty intersection r
m=1Cmis considered, wherer≥1 is an integer and eachCmis assumed to be the
solution set of a generalized variational inequality. LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. LetAm,Bm: C®Hbe relaxed cocoercive
mappings for each 1≤m≤r. It is proved that the sequence {xn} generated in the
following algorithm:
x1∈C, xn+1=αnu+βnxn+γn r
m=1
δ(m,n)PC(τmBmxn−λmAmxn), n≥1,
whereuÎCis a fixed point, {an}, {bn}, {gn}, {δ(1,n)}, ..., and {δ(r,n)} are sequences in (0, 1) and{τm}rm=1,{λm}rm=1are positive sequences, converges strongly to a solution of CFP
provided that the control sequences satisfies certain restrictions. 2000 AMS Subject Classification: 47H05; 47H09; 47H10.
Keywords:nonexpansive mapping, fixed point, relaxed cocoercive mapping, varia-tional inequality
1. Introduction and Preliminaries
Many problems in mathematics, in physical sciences and in real-world applications of various technological innovations can be modeled as a convex feasibility problem (CFP). This is the problem of finding a point in the intersection of finitely many closed convex sets in a real Hilbert spacesH. That is,
finding anx∈ r
m=1
Cm, (1:1)
wherer≥1 is an integer and eachCmis a nonempty closed and convex subset ofH. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [1,2], computer tomography [3] and radiation therapy treatment planning [4].
Throughout this paper, we always assume thatHis a real Hilbert space, whose inner product and norm are denoted by〈·, ·〉and ||·||. LetCbe a nonempty closed and con-vex subset ofHandA:C® Ha nonlinear mapping. Recall the following definitions:
(a)Ais said to be monotone if
Ax−Ay,x−y ≥0, ∀x,y∈C.
(b)Ais said to ber-strongly monotone if there exists a positive real numberr>0 such that
Ax−Ay,x−y ≥ρ||x−y||2, ∀x,y∈C.
(c)Ais said to beh-cocoercive if there exists a positive real numberh>0 such that
Ax−Ay,x−y ≥η||Ax−Ay||2, ∀x,y∈C.
(d)Ais said to be relaxed h-cocoercive if there exists a positive real numberh>0 such that
Ax−Ay,x−y ≥(−η)||Ax−Ay||2, ∀x,y∈C.
(e)Ais said to be relaxed (h,r)-cocoercive if there exist positive real numbersh, r
>0 such that
Ax−Ay,x−y ≥(−η)||Ax−Ay||2+ρ||x−y||2, ∀x,y∈C.
The main purpose of this paper is to consider the following generalized variational inequality. Given nonlinear mappings A:C ®H andB: C® H, find au ÎC such that
u−τBu+λAu,v−u ≥0, ∀v∈C, (1:2)
where land τ are two positive constants. In this paper, we use GV I(C, B, A) to denote the set of solutions of the generalized variational inequality (1.2).
It is easy to see that an element uÎCis a solution to the variational inequality (1.2) if and only ifuÎ Cis a fixed point of the mappingPC(τB- lA), wherePCdenotes the metric projection fromHontoC. Indeed, we have the following relations:
u=PC(τB−λA)u⇔ u−τBu+λAu,v−u ≥0, ∀v∈C. (1:3)
Next, we consider a special case of (1.2). If B=I, the identity mapping and τ = 1, then the generalized variational inequality (1.1) is reduced to the following. FinduÎ C
such that
Au,v−u ≥0, ∀v∈C. (1:4)
The variational inequality (1.4) emerging as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences was introduced by Stam-pacchia [5]. In this paper, we useV I(C, A) to denote the set of solutions of the variational inequality (1.4).
Let S :C ®C be a mapping. We useF(S) to denote the set of fixed points of the mapping S. Recall thatSis said to be nonexpansive if
It is well known that if Cis nonempty bounded closed and convex subset ofH, then the fixed point set of the nonexpansive mappingS is nonempty, see [6] more details. Recently, fixed point problems of nonexpansive mappings have been considered by many authors; see, for example, [7-16].
Recall thatSis said to be demi-closed at the origin if for each sequence {xn} inC,xn⇀
x0andSxn®0 implySx0= 0, where⇀and®stand for weak convergence and strong convergence.
Recently, many authors considered the variational inequality (1.4) based on iterative methods; see [17-32]. For finding solutions to a variational inequality for a cocoercive mapping, Iiduka et al. [22] proved the following theorem.
Theorem ITT.Let C be a nonempty closed convex subset of a real Hilbert space H and let A be an a-cocoercive operator of H into H with V I(C, A)≠∅. Let{xn}be a
sequence defined as follows.x1=xÎ C and
xn+1=PC(αnxn+ (1−αn)PC(xn−λnAxn))
for every n = 1, 2, ..., where C is the metric projection from H onto C, {an} is a
sequence in[-1, 1],and{ln}is a sequence in[0, 2a].If{an}and{ln}are chosen so that {an}Î [a, b]for some a,b with-1< a < b <1and{ln}Î[c, d]for some c,d with0 <
c < d <2(1 +a)a, then{xn}converges weakly to some element of V I(C,A).
Subsequently, Iiduka and Takahashi [23] further studied the problem of finding solu-tions of the classical variational inequality (1.4) for cocoercive mappings (inverse-strongly monotone mappings) and nonexpansive mappings. They obtained a strong convergence theorem. More precisely, they proved the following theorem.
Theorem IT.Let C be a closed convex subset of a real Hilbert space H. Let S:C®
C be a nonexpanisve mapping and A an a-cocoercive mapping of C into H such that F
(S)∩V I(C,A)≠∅.Suppose x1=uÎC and{xn}is given by
xn+1=αnu+ (1−αn)SPC(xn−λnAxn)
for every n = 1, 2, ...,where{an}is a sequence in [0, 1)and{ln}is a sequence in[a,
b].
If{an}and{ln}are chosen so that{ln}Î[a,b]for some a,b with0< a < b <2a,
lim
n→∞αn= 0, ∞
n=1
αn=∞, ∞
n=1
|αn+1−αn| <∞ and ∞
n=1
|λn+1−λn| <∞,
then{xn}converges strongly to PF(S)∩V I(C,A)x.
In this paper, motivated by research work going on in this direction, we study the CFP in the case that each Cm is a solution set of generalized variational inequality (1.2). Strong convergence theorems of solutions are established in the framework of real Hilbert spaces.
In order to prove our main results, we need the following lemmas.
Lemma 1.1 [33]. Let{xn}and {yn} be bounded sequences in a Hilbert space H and {bn}a sequence in(0, 1)with
0<lim inf
Suppose that xn+1= (1 -bn)yn+bnxnfor all integers n≥0and
lim sup
n→∞ (||yn+1−yn|| − ||xn+1−xn||)≤0.
Thenlimn®∞||yn-xn|| = 0.
Lemma 1.2 [34]. Let C be a nonempty closed and convex subset of a real Hilbert
space H. Let S1 : C ® C and S2 : C® C be nonexpansive mappings on C. Suppose
that F(S1)∩F(S2)is nonempty. Define a mapping S:C® C by
Sx=aS1x+ (1−a)S2x, ∀x∈C,
where a is a constant in(0, 1).Then S is nonexpansive with F(S) =F(S1)∩F(S2).
Lemma 1.3 [35]. Let C be a nonempty closed and convex subset of a real Hilbert
space H and S:C®C a nonexpansive mapping. Then I- S is demi-closed at zero.
Lemma 1.4 [36]. Assume that{an}is a sequence of nonnegative real numbers such that
αn+1≤(1−γn)αn+δn,
where {gn}is a sequence in(0, 1)and{δn}is a sequence such that (a)∞n=1γn=∞;
(b) lim supn®∞δn/gn≤0 or
∞
n=1|δn|<∞.
Thenlimn®∞an= 0.
2. Main results
Theorem 2.1.Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Am:C® H be a relaxed(hm,rm)-cocoercive andμm-Lipschitz continuous mapping
and Bm: C® H a relaxed(ηm,ρm)-cocoercive and μm-Lipschitz continuous mapping for each1≤m≤r. Assume thatrm=1GVI(C,Bm,Am)=∅.Let{xn}be a sequence
gener-ated in the following manner:
x1∈C, xn+1=αnu+βnxn+γn r
m=1
δ(m,n)PC(τmBmxn−λmAmxn), n≥1, (ϒ)
where u ÎC is a fixed point, {an}, {bn}, {gn}, {δ(1,n)}, ...,and{δ(r,n)}are sequences in(0, 1)satisfying the following restrictions:
(a)αn+βn+γn=
r
m=1δ(m,n)= 1,∀n≥1; (b) 0<lim infn®∞bn≤lim supn®∞bn<1; (c) limn®∞an= 0and∞n=1αn=∞; (d) limn®∞δ(m,n)=δmÎ(0, 1),∀1≤m≤r,
And{τm}rm=1,{λm}rm=1are two positive sequences such that
(e)1−2λmρm+λm2μ2m+ 2λmηmμ2m+
1−2λmρm+λ2mμ2m+ 2λmηmμ2m≤1, ∀1≤m≤r.
common element x¯∈rm=1GVI(C,Bm,Am),which uniquely solves the following varia-tional inequality.
u− ¯x,x¯−x∗ ≥0, ∀x∗∈r
m=1GVI(C,Bm,Am).
Proof. First, we prove that the mappingPC(τmBm-lmAm) is nonexpansive for each 1
≤m≤r. For eachx,yÎ C, we have
||PC(τmBm−λmAm)x−PC(τmBm−λmAm)y||
≤ ||(τmBm−λmAm)x−(τmBm−λmAm)y||
≤ ||(x−y)−λm(Amx−Amy)||+||(x−y)−τm(Bmx−Bmy)||.
(2:1)
It follows from the assumption that each Amis relaxed (hm,rm)-cocoercive andμm -Lipschitz continuous that
||x−y−λm(Amx−Amy)||2
= ||x−y||2−2λmAmx−Amy,x−y+λ2m||Amx−Amy||2
≤ ||x−y||2−2λm[(−ηm)||Amx−Amy||2+ρm||x−y||2] +λm2μ2m||x−y||2 = (1−2λmρm+λ2mμ2m)||x−y||2+ 2λmηm||Amx−Amy||2
= (1−2λmρm+λ2mμ2m)||x−y||2+ 2λmηmμ2m||Amx−Amy||2
=ξm2||x−y||2,
whereξm=
1−2λmρm+λ2mμ2m+ 2λmηmμ2m. This shows that
||x−y−λm(Amx−Amy)|| ≤ξm||x−y||. (2:2)
In a similar way, we can obtain that
||x−y−τm(Bmx−Bmy)|| ≤ζm||x−y||, (2:3)
whereζm=
1−2λmρm+λ2mμ2m+ 2λmηmμ2m. Substituting (2.2) and (2.3) into (2.1),
we from the condition (e) see that PC(τmBm- lmAm) is nonexpansive for each 1≤m≤
r. Put
yn= r
m=1
δ(m,n)PC(τmBmxn−λmAmxn), ∀n≥1.
Fixing p∈rm=1GVI(C,Bm,Am), we see that
||yn−p|| ≤ ||xn−p||.
It follows that
||xn+1−p||=||αnu+βnxn+γnyn−p||
≤αn||u−p||+βn||xn−p||+γn||yn−p||
≤αn||u−p||+βn||xn−p||+γn||xn−p|| =αn||u−p||+ (1−αn)||xn−p||.
By mathematical inductions we arrive at
Since the mappingPC(τmBm-lmAm) is nonexpansive for each 1≤m≤r, we see that ||yn+1−yn||
=|| r
m=1
δ(m,(n+1))PC(τmBmxn+1−λmAmxn+1)− r
m=1
δ(m,n)PC(τmBmxn−λmAmxn)||
≤ ||xn+1−xn||+M r
m=1
|δ(m,(n+1))−δ(m,n)|,
(2:4)
whereMis an appropriate constant such that
M= max{sup n≥1||
PC(τmBmxn−λmAmxn)||,∀1≤m≤r}.
Putln= xn+1−βnxn
1−βn , for alln≥1. That is,
xn+1= (1−βn)ln+βnxn, ∀n≥1. (2:5)
Now, we estimate ||ln+1- ln||. Note that
ln+1−ln= αn+1
u+γn+1yn+1
1−βn+1 −
αnu+γnyn 1−βn
= αn+1 1−βn+1
(u−yn+1) + αn 1−βn
(yn−u) +yn+1−yn,
which combines with (2.4) yields that
||ln+1−ln|| − ||xn+1−xn||
≤ αn+1 1−βn+1||
u−yn+1||+ α n 1−βn||
yn−u||+M r
m=1
|δ(m,(n+1))−δ(m,n)|.
It follows from the conditions (b), (c) and (d) that
lim sup n→∞
(||ln+1−ln|| − ||xn+1−xn+1||)≤0.
It follows from Lemma 1.1 that limn®∞||ln-xn|| = 0. In view of (2.5), we see thatxn +1xn= (1 -bn)(ln-xn). It follows that
lim
n→∞||xn+1−xn||= 0. (2:6)
On the other hand, from the iterative algorithm (ϒ), we see thatxn+1 -xn=an(u
-xn) + gn(yn- xn). It follows from (2.6) and the conditions (b), (c) that
lim
n→∞||yn−xn||= 0. (2:7)
Next, we show thatlim supn→∞u− ¯x,xn− ¯x ≤0. To show it, we can choose a sub-sequence{xni}of {xn} such that
lim sup n→∞
u− ¯x,xn− ¯x= lim
i→∞u− ¯x,xni− ¯x. (2:8)
Rx= r
m=1
δmPC(τmBm−λmAm)x, ∀x∈C,
whereδm= limn®∞δ(m,n). From Lemma 1.2, we see thatRis nonexpansive with
F(R) = r
m=1
F(PC(τmBm−λmAm)) = r
m=1
GVI(C,Bm,Am).
Now, we show thatRxn-xn®0 asn®∞. Note that
||Rxn−xn||
=|| r
m=1
δmPC(τmBm−λmAm)xn− r
m=1
δ(m,n)PC(τmBmxn−λmAmxn)||+||yn−xn||
≤M r
m=1
|δ(m,n)−δm|+||yn−xn||.
From the condition (d) and (2.7), we obtain that limn®∞ ||Rxn - xn|| = 0. From Lemma 1.3, we see that
q∈F(R) = r
m=1
F(PC(τmBm−λmAm)) = r
m=1
GVI(C,Bm,Am).
In view of (2.8), we arrive at
lim sup n→∞
u− ¯x,xn− ¯x=u− ¯x,q− ¯x ≤0. (2:9)
Finally, we show that xn→ ¯xasn-∞. Note that
||xn+1− ¯x||2
=αnu+βnxn+γnyn− ¯x,xn+1− ¯x
=αnu− ¯x,xn+1− ¯x+βnxn− ¯x,xn+1− ¯x+γnyn− ¯x,xn+1− ¯x
≤αnu− ¯x,xn+1− ¯x+βn||xn− ¯x||||xn+1− ¯x||+γn||yn− ¯x|| ||xn+1− ¯x||
≤αnu− ¯x,xn+1− ¯x+ (1−αn)||xn− ¯x|| ||xn+1− ¯x||
≤ 1−αn
2 (||xn− ¯x|| 2+||xn
+1− ¯x||2) +αnu− ¯x,xn+1− ¯x,
which implies that
||xn+1− ¯x||2≤(1−αn)||xn− ¯x||2+ 2αnu− ¯x,xn+1− ¯x. (2:10)
From the condition (c), (2.9) and applying Lemma 1.4 to (2.10), we obtain that
lim
n→∞||xn− ¯x||= 0.
This completes the proof.
If Bm≡I, the identity mapping andτm≡1, then Theorem 2.1 is reduced to the fol-lowing result on the classical variational inequality (1.4).
Corollary 2.2. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Am:C®H be a relaxed(hm,rm)-cocoercive andμm-Lipschitz continuous
map-ping for each1≤m≤r. Assume thatrm=1VI(C,Am)=∅.Let{xn}be a sequence
x1∈C, xn+1=αnu+βnxn+γn r
m=1
δ(m,n)PC(xn−λmAmxn), n≥1,
where u ÎC is a fixed point, {an}, {bn}, {gn}, {δ(1,n)}, ...,and{δ(r,n)}are sequences in(0, 1)satisfying the following restrictions.
(a)αn+βn+γn=mr=1δ(m,n)= 1,∀n≥1; (b) 0<lim infn®∞bn≤lim supn®∞bn< 1; (c) limn®∞an= 0and∞n=1α=∞;
(d) limn®∞ δ(m,n)=δmÎ(0, 1),∀1 ≤m≤r,and{λm}rm=1is a positive sequence such that
(e)λm≤ 2ρm−2ηmμ
2
m
μ2
m ,∀1≤m≤r.
Then the sequence {xn}converges strongly to a common elementx¯∈ r
m=1VI(C,Am), which uniquely solves the following variational inequality
u− ¯x,x¯−x∗ ≥0, ∀x∗∈r
m=1VI(C,Am).
Ifr = 1, then Theorem 2.1 is reduced to the following.
Corollary 2.3. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A :C® H be a relaxed(h, r)-cocoercive andμ-Lipschitz continuous mapping
and B : C ® H a relaxed(η,ρ)-cocoercive and μ-Lipschitz continuous mapping.
Assume that GV I(C, B, A)is not empty. Let{xn}be a sequence generated in the
follow-ing manner:
x1∈C, xn+1=αnu+βnxn+γnPC(τBxn−λAxn), n≥1,
where u ÎC is a fixed point, {an}, {bn}and{gn}are sequences in(0, 1)satisfying the following restrictions.
(a)an+bn+gn= 1,∀n≥1;
(b) 0<lim infn®∞bn≤lim supn®∞bn<1; (c) limn®∞an= 0and∞n=1αn=∞
(d)1−2λρ+λ2μ2+ 2λημ2+1−2λρ+λ2μ2+ 2λημ2≤1.
Then the sequence {xn} converges strongly to a common element x¯∈GVI(C,B,A),
which uniquely solves the following variational inequality
u− ¯x,x¯−x∗ ≥0, ∀x∗∈GVI(C,B,A).
For the variational inequality (1.4), we can obtain from Corollary 2.3 the following immediately.
Corollary 2.4. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A: C®H be a relaxed(h,r)-cocoercive andμ-Lipschitz continuous mapping. Assume that V I(C,A)is not empty. Let {xn}be a sequence generated in the following
manner:
where u ÎC is a fixed point, {an}, {bn}and{gn}are sequences in(0, 1)satisfying the
following restrictions.
(a)an+bn+gn= 1,∀n≥1;
(b) 0<lim infn®∞bn≤lim supn®∞bn< 1; (c) limn®∞an= 0and
∞
n=1αn=∞; (d)λ≤ 2ρ−μ22ημ2.
Then the sequence {xn}converges strongly to a common element x¯∈VI(C,A), which
uniquely solves the following variational inequality
u− ¯x,x¯−x∗ ≥0, ∀x∗∈VI(C,A).
Remark 2.5. In this paper, the generalized variational inequality (1.2), which includes the classical variational inequality (1.4) as a special case, is considered based on itera-tive methods. Strong convergence theorems are established under mild restrictions imposed on the parameters. It is of interest to extend the main results presented in this paper to the framework of Banach spaces.
Abbreviation
CFP: convex feasibility problem.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant no. 70871081 and Important Science and Technology Research Project of Henan province, China (102102210022).
Author details 1
School of Business Administration, Henan University, Kaifeng 475000, Henan Province, China2School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China
Authors’contributions
LYdesigned and performed all the steps of proof in this research and also wrote the paper.MLparticipated in the design of the study. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 14 November 2010 Accepted: 25 July 2011 Published: 25 July 2011
References
1. Combettes PL:The convex feasibility problem: in image recovery.InAdvances in Imaging and Electron Physics. Volume 95.Edited by: Hawkes P. Academic Press, Orlando; 1996:155-270.
2. Kotzer T, Cohen N, Shamir J:Images to ration by a novel method of parallel projection onto constraint sets.Opt Lett 1995,20:1172-1174.
3. Sezan MI, Stark H:Application of convex projection theory to image recovery in tomograph and related areas.In Image Recovery: Theory and Application.Edited by: Stark H. Academic Press, Orlando; 1987:155-270.
4. Censor Y, Zenios SA:Parallel Optimization. Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation.Oxford University Press, New York; 1997.
5. Stampacchia G:Formes bilineaires coercitives sur les ensembles convexes.CR Acad Sci Paris1964,258:4413-4416. 6. Baillon JB:Quelques aspects de la theorie des points fixes dans les espaces de Banach. I, II, Séminaire d’Analyse
Fonctionnelle (1978-1979).Exp. No. 7-8, Ecole Polytech., (in French). Palaiseau1979, 45.
7. Qin X, Su Y:Approximation of a zero point of accretive operator in Banach spaces.J Math Anal Appl2007,
329:415-424.
8. Qin X, Su Y:Strong convergence theorems for relatively nonexpansive mappings in a Banach space.Nonlinear Anal 2007,67:1958-1965.
9. Cho YJ, Kang SM, Qin X:Approximation of common fixed points of an infinite family of nonex-pansive mappings in Banach spaces.Comput Math Appl2008,56:2058-2064.
10. Park S:Fixed point theorems in locally G-convex spaces.Nonlinear Anal2002,48:869-879.
12. Qin X, Cho YJ, Kang JI, Kang SM:Strong convergence theorems for an infinite family of nonex-pansive mappings in Banach spaces.J Comput Appl Math2009,230:121-127.
13. Kim JK, Nam YM, Sim JY:Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings.Nonlinear Anal2009,71:e2839-e2848.
14. Qin X, Cho YJ, Kang SM, Zho H:Convergence of a modified Halpern-type iteration algorithm for quasi-ϕ -nonexpansive mappings.Appl Math Lett2009,22:1051-1055.
15. Qin X, Cho SY, Zhou H:Common fixed points of a pair of non-expansive mappings with applications to convex feasibility problems.Glasgow Math J2010,52:241-252.
16. Wu C, Cho SY, Shang M:Moudafi’s viscosity approximations with demi-continuous and strong pseudo-contractions for non-expansive semigroups.J Inequal Appl2010,2010:Article ID 645498.
17. Cho SY:Approximation of solutions of a generalized variational inequality problem based on iterative methods.
Commun Korean Math Soc2010,25:207-214.
18. Kim JK, Cho SY, Qin X:Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings.J Inequal Appl2010,2010:Article ID 312602.
19. Hao Y:Strong convergence of an iterative method for inverse strongly accretive operators.J Inequal Appl2008,
2008:Article ID 420989.
20. Qin X, Cho YJ, Kang SM:Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.J Comput Appl Math2009,225:20-30.
21. Qin X, Shang M, Su Y:Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems.Math Comput Model2008,48:1033-1046.
22. Iiduka H, Takahashi W, Toyoda M:Approximation of solutions of variational inequalities for monotone mappings.
PanAmer Math J2004,14:49-61.
23. Iiduka H, Takahashi W:Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.Nonlinear Anal2005,61:341-350.
24. Iiduka H, Takahashi W:Weak convergence of a projection algorithm for variational inequalities in a Banach space.J Math Anal Appl2008,339:668-679.
25. Qin X, Cho SY, Kang SM:Some results on generalized equilibrium problems involving a family of nonexpansive mappings.Appl Math Comput2010,217:3113-3126.
26. Park S, Kang BG:Generalized variational inequalities and fixed point theorems.Nonlinear Anal1998,31:207-216. 27. Park S, Kum S:An application of a Browder-type fixed point theorem to generalized variational inequalities.J Math
Anal Appl1998,218:519-526.
28. Park S:Fixed points, intersection theorems, variational inequalities, and equilibrium theorems.Int J Math Math Sci 2000,24(2):73-93.
29. Park S, Chen MP:Generalized variational inequalities of the Hartman-Stampacchia-Browder type.J Inequal Appl 1998,2:71-87.
30. Qin X, Chang SS, Cho YJ:Iterative methods for generalized equilibrium problems and fixed point problems with applications.Nonlinear Anal2010,11:2963-2972.
31. Qin X, Cho YJ, Kang SM:Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications.Nonlinear Anal2010,72:99-112.
32. Takahashi W, Toyoda M:Weak convergence theorems for nonexpansive mappings and monotone mappings.J Optim Theory Appl2003,118:417-428.
33. Suzuki T:Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-expansive semigroups without Bochne integrals.J Math Anal Appl2005,305:227-239.
34. Bruck RE:Properties of fixed point sets of nonexpansive mappings in Banach spaces.Trans Am Math Soc1973,
179:251-262.
35. Browder FE:Nonlinear operators and nonlinear equations of evolution in Banach spaces.Proc Symp Pure Math1976,
18:78-81.
36. Liu LS:Ishikawa and Mann iterative processes with errors for nonlinear strongly acretive mappings in Banach spaces.J Math Anal Appl1995,194:114-125.
doi:10.1186/1687-1812-2011-19
Cite this article as:Yu and Liang:Convergence theorems of solutions of a generalized variational inequality. Fixed Point Theory and Applications20112011:19.
Submit your manuscript to a
journal and benefi t from:
7Convenient online submission
7Rigorous peer review
7Immediate publication on acceptance
7Open access: articles freely available online
7High visibility within the fi eld
7Retaining the copyright to your article
