Volume 2011, Article ID 276859,10pages doi:10.1155/2011/276859
Research Article
An Implicit Iteration Method for Variational
Inequalities over the Set of Common Fixed Points
for a Finite Family of Nonexpansive Mappings in
Hilbert Spaces
Nguyen Buong
1and Nguyen Thi Quynh Anh
21Vietnamese Academy of Science and Technology, Institute of Information Technology, 18,
Hoang Quoc Viet, Cau Giay, Ha Noi 122100, Vietnam
2Department of Information Technology, Thai Nguyen National University,
Thainguye 842803, Vietnam
Correspondence should be addressed to Nguyen Buong,[email protected]
Received 17 December 2010; Accepted 7 March 2011 Academic Editor: Jong Kim
Copyrightq2011 N. Buong and N. T. Quynh Anh. 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.
We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.
1. Introduction
LetCbe a nonempty closed and convex subset of a real Hilbert spaceHwith inner product ·,·and norm · , and letF :H → Hbe a nonlinear mapping. The variational inequality problem is formulated as finding a pointp∗∈Csuch that
Fp∗, p−p∗≥0, ∀p∈C. 1.1
It is well known that ifFis anL-Lipschitz continuous andη-strongly monotone, that is,Fsatisfies the following conditions:
Fx−F
y≤Lx−y,
Fx−Fy, x−y≥ηx−y2,
1.2
whereLandηare fixed positive numbers, then1.1has a unique solution. It is also known that1.1is equivalent to the fixed-point equation
pPC
p−μFp, 1.3
where PC denotes the metric projection from x ∈ H onto C and μ is an arbitrarily fixed
positive constant.
Let {Ti}Ni1 be a finite family of nonexpansive self-mappings of C. For finding an elementp∈ ∩N
i1FixTi, Xu and Ori introduced in4the following implicit iteration process. Forx0∈Cand{βk}∞k1⊂0,1, the sequence{xk}is generated as follows:
x1β1x0
1−β1
T1x1,
x2β2x1
1−β2
T2x2,
.. .
xNβNxN−1
1−βN
TNxN,
xN1βN1xN
1−βN1
T1xN1,
.. .
1.4
The compact expression of the method is the form
xkβkxk−1
1−βk
Tkxk, k≥1, 1.5
where Tn TnmodN, for integer n ≥ 1, with the mod function taking values in the set
{1,2, . . . , N}. They proved the following result.
Theorem 1.1. LetHbe a real Hilbert space andCa nonempty closed convex subset ofH. Let{Ti}Ni1 beNnonexpansive self-maps ofCsuch that∩N
i1FixTi/∅, whereFixTi {x∈C:Tixx}. Let x0∈Cand{βk}∞k1be a sequence in0,1such thatlimk→ ∞βk0. Then, the sequence{xk}defined
Further, Zeng and Yao introduced in 5 the following implicit method. For an arbitrary initial pointx0∈H, the sequence{xk}∞k1is generated as follows:
x1β1x0
1−β1
T1x1−λ1μFT1x1
,
x2β2x1
1−β2
T2x2−λ2μFT2x2
,
.. .
xNβNxN−1
1−βN
TNxN−λNμFTNxN
,
xN1βN1xN
1−βN1
T1xN1−λN1μFT1xN1
,
.. .
1.6
The scheme is written in a compact form as
xkβkxk−1
1−βk
Tkxk−λkμF
Tkxk
, k≥1. 1.7
They proved the following result.
Theorem 1.2. LetHbe a real Hilbert space andF:H → Ha mapping such that for some constants
L, η >0,F isL-Lipschitz continuous andη-strongly monotone. Let{Ti}Ni1beNnonexpansive self-maps ofHsuch thatC∩N
i1FixTi/∅. Letμ∈0,2η/L2, and letx0∈H, with{λk}∞k1⊂0,1 and {βk}∞k1 ⊂ 0,1 satisfying the conditions: k∞1λk < ∞ and α ≤ βk ≤ β, k ≥ 1, for some α, β ∈0,1. Then, the sequence{xk}defined by1.7converges weakly to a common fixed point of
the mappings{Ti}Ni1. The convergence is strong if and only iflim infk→ ∞dxk, C 0.
Recently, Ceng et al.6extended the above result to a finite family of asymptotically self-maps.
Clearly, from ∞k1λk < ∞ we have that λk → 0 as k → ∞. To obtain strong
convergence without the condition ∞k1λk < ∞, in this paper we propose the following
implicit algorithm:
xtTtxt, Tt:T0tTNt · · ·T1t, t∈0,1, 1.8
whereTt
i are defined by
Titx1−βitxβitTix, i1, . . . , N, T0ty
I−λtμF
y, x, y∈H, 1.9
I denotes the identity mapping ofH, and the parameters{λt},{βit} ⊂0,1for allt ∈ 0,1
2. Main Result
We formulate the following facts for the proof of our results.
Lemma 2.1 see 7. ixy2 ≤ x2 2y, x yand for any fixed t ∈ 0,1,
ii 1−txty2 1−tx2ty2−1−ttx−y2, for allx, y∈H.
PutTλx Tx−λμFTx, x ∈H, λ ∈ 0,1; for any nonexpansive mappingT ofH,
we have the following lemma.
Lemma 2.2see8. Tλx−Tλy ≤ 1−λτx−y, for allx, y ∈H and for a fixed number μ∈0,2η/L2, whereτ1−1−μ2η−μL2∈0,1.
Lemma 2.3Demiclosedness Principle9. Assume thatT is a nonexpansive self-mapping of a closed convex subsetKof a Hibert spaceH. IfT has a fixed point, thenI−T is demiclosed; that is, whenever{xk}is a sequence inKweakly converging to somex ∈Kand the sequence{I−Txk}
strongly converges to somey, it follows thatI−Txy.
Now, we are in a position to prove the following result.
Theorem 2.4. LetHbe a real Hilbert space andF:H → Ha mapping such that for some constants
L, η >0,F isL-Lipschitz continuous andη-strongly monotone. Let{Ti}Ni1beNnonexpansive self-maps ofHsuch thatC∩Ni1FixTi/∅. Letμ∈0,2η/L2and lett∈0,1, {λt},{βit} ⊂0,1,
such that
λt−→0, as t−→0, 0<lim inf t→0β
i
t≤lim sup t→0
βit<1, i1, . . . , N. 2.1
Then, the net{xt}defined by1.8-1.9converges strongly to the unique elementp∗in1.1.
Proof. By usingLemma 2.2withTλTt
0, that is,T I, we have that
Ttx−Tty≤1−λ
tτTNt · · ·T1tx−TNt · · ·T1ty ..
.
≤1−λtτTit· · ·T1tx−Tit· · ·T1ty
.. .
≤1−λtτT1tx−T1ty≤1−λtτx−y ∀x, y∈H.
2.2
Next, we show that{xt} is bounded. Indeed, for a fixed pointp ∈ C, we have that Tt
ippfori1, . . . , N, and hence
xt−pTtx
t−pTtxt−TNt · · ·T1tp
I−λtμF
Tt
N· · ·T1txt−
I−λtμF
Tt
N· · ·T1tp−λtμF
p
≤1−λtτTNt · · ·T1txt−TNt · · ·T1tpλtμF
p
≤1−λtτTNt−1· · ·T1txt−TNt−1· · ·T1tpλtμF
p
.. .
≤1−λtτTit· · ·T1txt−Tit· · ·T1tpλtμF
p
.. .
≤1−λtτT1txt−T1tpλtμF
p
≤1−λtτxt−pλtμF
p.
2.3
Therefore,
xt−p≤ μ τF
p 2.4
that implies the boundedness of{xt}. So, are the nets{FyNt },{yit}, i1, . . . , N.
Put
y1t 1−β1txtβt1T1xt,
y2t 1−β2ty1tβ2tT2y1t,
.. .
yti 1−βityti−1βitTiyti−1,
.. .
yNt
1−βNt
yNt −1βtNTNyNt −1.
2.5
Then,
xt
I−λtμF
Moreover,
xt−p2
I−λtμF
yNt −p
2
ytN−p
2 −2λtμ
FyNt
, yNt −p
λ2tμ2F
yNt
2
≤ytN−1−p2−2λtμ
FytN, yNt −pλ2tμ2FytN2
.. .
≤yt1−p
2 −2λtμ
FyNt , yNt −pλ2tμ2F
yNt 2
≤xt−p2−2λtμ
FytN, yNt −pλ2tμ2FytN2.
2.7
Thus,
ηyNt −p
2
Fp, ytN−p
≤ λtμ
2
FyNt
2
. 2.8
Further, for the sake of simplicity, we puty0t xtand prove that
yit−1−Tiyti−1−→0, 2.9
ast → 0 fori1, . . . , N.
Let{tk} ⊂0,1be an arbitrary sequence converging to zero ask → ∞andxk:xtk. We have to prove that yik−1−Tiyki−1 → 0, where yik are defined by2.5with t tk and yikyitk. Let{xl}be a subsequence of{xk}such that
lim sup
k→ ∞
yki−1−Tiyik−1 lim l→ ∞
yli−1−Tiyli−1. 2.10
Let{xkj}be a subsequence of{xl}such that
lim sup
k→ ∞
xk−p lim
j→ ∞
xkj−p
From2.6andLemma 2.1, it implies that
xkj−p
2 I−λkjμF
yN kj −p
2
≤yNkj −p2−2λkjμ
FykNj, xkj−p
1−βNkjykNj−1−pβkNjTNykNj−1−TNp
2
−2λkjμ
FykNj, xkj−p
≤1−βNkjyNkj−1−p2βkNjTNyNkj−1−TNp 2
−2λkjμ
FykNj, xkj−p
≤yNkj−1−p2−2λkjμ
FyNkj, xkj−p
≤ · · · ≤yk1j−p2−2λkjμ
FyNkj, xkj−p
≤xkj−p
2−2λkjμ
FyNkj, xkj−p
.
2.12
Hence,
lim
j→ ∞
xkj−p
lim
j→ ∞
yikj−p, i1, . . . , N. 2.13
ByLemma 2.1,
yi
kj−p
21−βi kj
yi−1
kj −p 2βi
kj
Tiyki−j1−p
2
−βikj1−βikjyki−j1−Tiyik−j1
2
≤1−βikjyik−j1−p2βikjyki−j1−p2
−βikj1−βikjyki−j1−Tiyik−j1
2
yki−j1−p2−βikj1−βkijyki−j1−Tiyik−j1 2
≤ · · ·y0kj−p2−βikj1−βkijyki−j1−Tiyik−j1
2
xkj−p 2−βi
kj
1−βi
kj yi−1
kj −Tiy
i−1
kj
2, i1, . . . , N.
Without loss of generality, we can assume thatα≤βit≤βfor someα, β∈0,1. Then, we have
α1−βyki−j1−Tiyki−j1
2
≤xkj−p
2−ykij−p2. 2.15
This together with2.13implies that
lim
j→ ∞
yik−j1−Tiyki−j1
2
0, i1, . . . , N. 2.16
It means thatyi−1
t −Tiyit−1 → 0 ast → 0 fori1, . . . , N.
Next, we show thatxt−Tixt → 0 ast → 0. In fact, in the case thati 1 we have y0t xt. So,xt−T1xt → 0 ast → 0. Further, since
y1
t−T1xt
1−β1
t
xt−T1xt, 2.17
andxt−T1xt → 0, we have thaty1t −T1xt → 0. Therefore, from
xt−y1t≤ xt−T1xtT1xt−yt1, 2.18
it follows thatxt−y1t → 0 ast → 0. On the other hand, since
y2t−T2y1t
1−βt2yt1−T2yt1−→0,
yt2−xt≤
1−βt2yt1−xtβ2tT2yt1−xt
≤1−βt2yt1−xtβ2tT2yt1−yt1yt1−xt,
2.19
we obtain thaty2
t−xt → 0 ast → 0. Now, from
xt−T2xt ≤xt−yt2yt2−T2yt1T2yt1−T2xt
≤xt−yt2yt2−T2yt1yt1−xt,
2.20
andxt−y2t,y2t−T2y1t,y1t−xt → 0, it follows thatxt−T2xt → 0. Similarly, we obtain
thatxt−Tixt → 0, fori1, . . . , NandyNt −xt → 0 ast → 0.
Let {xk} be any sequence of {xt} converging weakly top ask → ∞. Then, xk − Tixk → 0, fori 1, . . . , Nand{ykN}also converges weakly top. ByLemma 2.3, we have
p∈C∩N
i1FixTiand from2.8, it follows that
Sincep,p∈C, by replacingpbytp 1−tpin the last inequality, dividing bytand taking
t → 0 in the just obtained inequality, we obtain
Fp, p−p≥0 ∀p∈C. 2.22
The uniqueness ofp∗in1.1guarantees that p p∗. Again, replacingpin2.8byp∗, we obtain the strong convergence for{xt}. This completes the proof.
3. Application
Recall that a mappingS : H → His called aγ-strictly pseudocontractive if there exists a constantγ ∈0,1such that
Sx−Sy2≤
x−y2γI−Sx−I−Sy2, ∀x, y∈H. 3.1
It is well known10that a mappingT :H → HbyTxαx1−αSxwith a fixedα∈γ,1 for allx∈His a nonexpansive mapping and FixT FixS. Using this fact, we can extend our result to the caseC∩Ni1FixSi, whereSiisγi-strictly pseudocontractive as follows.
Letαi∈γi,1be fixed numbers. Then,C∩Ni1FixTiwithTiyαiy 1−αiSiy, a
nonexpansive mapping, fori1, . . . , N, and hence
Tity1−βtiyβitTiy
1−βit1−αi
yβit1−αiSiy, i1, . . . , N.
3.2
So, we have the following result.
Theorem 3.1. Let H be a real Hilbert space and F : H → H a mapping such that for some constantsL, η >0,FisL-Lipschitz continuous andη-strongly monotone. Let{Si}Ni1beN γi-strictly
pseudocontractive self-maps ofHsuch thatC∩Ni1FixSi/∅. Letαi∈γi,1, μ∈0,2η/L2and
lett∈0,1,{λt},{βti} ⊂0,1, such that
λt−→0, ast−→0, 0<lim inf t→0β
i
t≤lim sup t→0
βti<1, i1, . . . , N. 3.3
Then, the net{xt}defined by
xtTtxt, Tt:T0tTNt · · ·T1t, t∈0,1, 3.4
whereTt
i, fori 1, . . . , N, are defined by3.2andT0tx I−λtμFx, converges strongly to the
unique elementp∗in1.1.
It is known in11that FixS CwhereS Ni1ξiSi withξi > 0 and Ni1ξi 1
forN γi-strictly pseudocontractions{Si}iN1. Moreover,Sisγ-strictly pseudocontractive with
Theorem 3.2. Let H be a real Hilbert space and F : H → H a mapping such that for some constants L, η > 0, F is L-Lipschitz continuous and η-strongly monotone. Let{Si}Ni1 beN γi
-strictly pseudocontractive self-maps of H such that C ∩N
i1FixSi/∅. Let α ∈ γ,1, where
γmax{γi : 1≤i≤N},μ∈0,2η/L2, and lett∈0,1,{λt},{βt} ⊂0,1, such that
λt−→0, ast−→0, 0<lim inf
t→0βt≤lim supt→0βt<1. 3.5
Then, the net{xt}, defined by
xtTtxt, Tt:T0t
1−βt1−α
Iβt1−α N
i1
ξiSi
, t∈0,1, 3.6
whereT0t I−λtμF,ξi>0, and Ni1ξi1, converges strongly to the unique elementp∗in1.1.
Acknowledgment
This work was supported by the Vietnamese National Foundation of Science and Technology Development.
References
1 D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications, vol. 88 ofPure and Applied Mathematics, Academic Press, New York, NY, USA, 1980.
2 R. Glowinski,Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer, New York, NY, USA, 1984.
3 E. Zeidler,Nonlinear Functional Analysis and Its Applications. III, Springer, New York, NY, USA, 1985.
4 H.-K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.
5 L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,”Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 11, pp. 2507–2515, 2006.
6 L.-C. Ceng, N.-C. Wong, and J.-C. Yao, “Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption,”
Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2312–2322, 2008.
7 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.
8 Y. Yamada, “The hybrid steepest-descent method for variational inequalities problems over the intesectionof the fixed point sets of nonexpansive mappings,” in Inhently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., pp. 473– 504, North-Holland, Amsterdam, Holland, 2001.
9 K. Goebel and W. A. Kirk,Topics in Metric Fixed Point Theory, vol. 28 ofCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
10 H. Zhou, “Convergence theorems of fixed points forκ-strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 2, pp. 456–462, 2008.
11 G. L. Acedo and H.-K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
