Volume 2009, Article ID 208692,13pages doi:10.1155/2009/208692
Research Article
On a Two-Step Algorithm for Hierarchical Fixed
Point Problems and Variational Inequalities
Filomena Cianciaruso,
1Giuseppe Marino,
1Luigi Muglia,
1and Yonghong Yao
21Dipartimento di Matematica, Universit´a della Calabria, 87036 Arcavacata di Rende (CS), Italy
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Correspondence should be addressed to Giuseppe Marino,[email protected]
Received 5 May 2009; Accepted 12 September 2009
Recommended by Ram U. Verma
A common method in solving ill-posed problems is to substitute the original problem by a family of well-posedi.e., with a unique solutionregularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters.
Copyrightq2009 Filomena Cianciaruso et al. 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.
1. Introduction and Preliminar Results
A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed i.e., with a unique solution regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters. We will see that choosing appropriate hypotheses on the parameters, we will obtain convergence to the solution of well-posed problems. Changing these assumptions, we will obtain convergence to one of the solutions of a ill-posed problem. The results are situaded on the lines of research of Byrne1, Yang and Zhao2, Moudafi3, and Yao and Liou4.
In this paper, we consider variational inequalities of the form
x∗∈FixTsuch thatI−Sx∗, x−x∗ ≥0, ∀x∈FixT 1.1
whereT, S:C → Care nonexpansive mappings such that the fixed points set ofTFixTis nonempty andCis a nonempty closed convex subset of a Hilbert spaceH. If we denote with
Variational inequalities of1.1cover several topics recently investigated in literature as monotone inclusion5and the references therein, convex optimization6, quadratic minimization over fixed point setsee, e.g.,5,7–10and the references therein.
It is well known that the solutions of1.1are the fixed points of the nonexpansive mappingPFixTS.
There are in literature many papers in which iterative methods are defined in order to solve1.1.
Recently, in3Moudafi defined the following explicit iterative algorithm
xn1 1−αnxnαnσnSxn 1−σnTxn, 1.2
whereαnn∈Nandσnn∈Nare two sequences in0,1,and he proved a weak-convergence’s result. In order to obtain a strong-convergence result, Maing´e and Moudafi in11introduced and studied the following iterative algorithm
wn1 αnfwn 1−αnzn, n≥1,
zn βnSwn1−βnTwn,
1.3
whereαnn∈Nandβnn∈Nare two sequences in0,1.
Letf :C → Cbe a contraction with coefficientρ∈0,1.In this paper, under different conditions on involved parameters, we study the algorithm
xn1 αnfxn 1−αnTyn, n≥1,
yn βnSxn1−βnxn,
1.4
and give some conditions which assure that the method converges to a solution which solves some variational inequality.
We will confront the two methods1.3and1.4later.
We recall some general results of the Hilbert spaces theory and of the monotone operators theory.
Lemma 1.1. For allx, y∈H, there holds the inequality xy2
≤ x22y, xy. 1.5
IfKis closed convex subset of a real Hilbert spaceH, the metric projectionPK:H →
K is the mapping defined as follows: for eachx ∈ H,PKxis the only point inK with the
property
x−PKx inf
y∈Kx−y. 1.6
Lemma 1.2. LetKbe a nonempty closed convex subset of a real Hilbert spaceHand letPKbe the metric projection fromHontoK. Givenx∈Handz∈K,z PKxif and only if
Lemma 1.3see7. Letf :C → Cbe a contraction with coefficientρ∈0,1andW :C → C be a nonexpansive mapping. Then, for allx, y∈C:
athe mappingI−fis strongly monotone with coefficient1−ρ, that is,
x−y,I−fx−I−fy≥1−ρx−y2, 1.8
bthe mappingI−Wis monotone, that is,
x−y,I−Wx−I−Wy≥0. 1.9
Finally, we conclude this section with a lemma due to Xu on real sequences which has a fundamental role in the sequel.
Lemma 1.4see9. Assumeann∈Nis a sequence of nonnegative numbers such that
an1≤
1−γnanδn, n≥0, 1.10
whereγnnis a sequence in0,1,andδnnis a sequence inRsuch that,
1∞n 1γn ∞;
2lim supn→ ∞δn/γn≤0or∞n 1|δn|<∞.
Thenlimn→ ∞an 0.
2. Convergence of the Two-Step Iterative Algorithm
Let us consider the scheme
xn1 αnfxn 1−αnTyn, n≥1,
yn βnSxn1−βnxn.
2.1
As we will see the convergence of the scheme depends on the choice of the parameters
αnn∈N⊂0,1andβnn∈N⊂0,1. We list some possible hypotheses on them:
H1there existsγ >0 such thatβn≤γαn; H2 limn→ ∞βn/αn τ ∈0,∞;
H3αn → 0 asn → ∞andn∈Nαn ∞; H4n∈N|αn−αn−1|<∞;
H5n∈N|βn−βn−1|<∞;
H6limn→ ∞|αn−αn−1|/αn 0; H7limn→ ∞|βn−βn−1|/βn 0;
H8limn→ ∞|βn−βn−1||αn−αn−1|/αnβn 0;
Proposition 2.1. Assume that (H1) holds. Thenxnn∈Nandynn∈Nare bounded.
Proof. Letz∈FixT. Then,
xn1−z ≤αnfxn−fzαnfz−z 1−αnTyn−z
≤αnρxn−zαnfz−z 1−αnβnSxn−z 1−αn1−βnxn−z
≤αnρxn−zαnfz−z 1−αnβnSz−z 1−αnxn−z
1−1−ραnxn−zαnfz−zγSz−z .
2.2
So, by induction, one can see that
xn−z ≤max
x0−z,
1
1−ρfz−zγSz−z . 2.3
Of courseynn∈Nis bounded too.
Proposition 2.2. Suppose that (H1), (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then
1 xnn∈Nis asymptotically regular, that is,
lim
n→ ∞xn1−xn 0, 2.4
2the weak cluster points setωwxn⊂FixT.
Proof. Observing that
xn1−xn αnfxn 1−αnTyn−αn−1fxn−1−1−αn−1Tyn−1
αnfxn−fxn−1
fxn−1−Tyn−1
αn−αn−1 1−αn
Tyn−Tyn−1
,
2.5
then, passing to the norm we have
xn1−xn ≤αnfxn−fxn−1fxn−1−Tyn−1|αn−αn−1| 1−αnTyn−Tyn−1
≤αnρxn−xn−1fxn−1−Tyn−1|αn−αn−1| 1−αnyn−yn−1.
By definition ofynone obtain that
yn−yn−1 βnSxn−Sxn−1 Sxn−1−xn−1
βn−βn−1
1−βnxn−xn−1
≤βnxn−xn−1Sxn−1−xn−1βn−βn−1
1−βnxn−xn−1
xn−xn−1Sxn−1−xn−1βn−βn−1,
2.7
so, substituting2.7in2.6we obtain
xn1−xn ≤αnρxn−xn−1|αn−αn−1|fxn−1−Tyn−1
1−αnxn−xn−1Sxn−1−xn−1βn−βn−1 .
2.8
ByProposition 2.1, we callM: max{supn∈Nfxn−1−Tyn−1,supn∈NSxn−1−xn−1}so we
have
xn1−xn ≤1−αn1−ρ xn−xn−1M
|αn−αn−1|βn−βn−1 . 2.9
So, ifH4andH5hold, we obtain the asymptotic regularity byLemma 1.4. If, instead,H6andH7hold, fromH1we can write
xn1−xn ≤1−αn1−ρ xn−xn−1Mαn
|αn−αn−1|
αn
βn−βn−1
αn
≤1−αn1−ρ xn−xn−1Mαn
|αn−αn−1|
αn γ
βn−βn−1
βn
,
2.10
so, the asymptotic regularity follows byLemma 1.4also. In order to prove2, we can observe that
xn−Txn ≤ xn−xn1xn1−Txn
≤ xn−xn1αnfxn−Txn 1−αnTyn−Txn ≤ xn−xn1αnfxn−Txn 1−αnyn−xn
xn−xn1αnfxn−Txn 1−αnβnSxn−xn.
2.11
ByH1, andH3it follows thatβn → 0, asn → ∞, so thatxn−Txn → 0 sincexnn∈Nis asymptotically regular. By demiclosedness principle we obtain the thesis.
Corollary 2.3. Suppose that the hypotheses ofProposition 2.2hold. Then
Proof. To provei,we can observe that
xn−Tyn≤ xn−xn1xn1−Tyn xn−xn1αnγfxn−Tyn. 2.12
The asymptotical regularity ofxnn∈Ngives the claim. Moreover, noting that
yn−xn βnSxn−xn, 2.13
sinceβn → 0 asn → ∞we obtainii. In the endiiifollows easily byiandii.
Theorem 2.4. Suppose (H2) withτ 0and (H3). Moreover Suppose that either (H4) and (H5) hold, or (H6) and (H7) hold. If one denote byz∈Cthe unique element inFixTsuch thatz PFixTfz,
then
1
lim sup
n→ ∞ fz−z, xn−z ≤0, 2.14
2xn → zasn → ∞.
Proof. First of all, PFixTf is a contraction, so there exists a unique z ∈ FixT such that
PFixTfz z. Moreover, fromLemma 1.2,zis characterized by the fact that
fz−z, y−z≤0, ∀y∈FixT. 2.15
SinceH2impliesH1, thusxnn∈Nis bounded. Letxnkk∈Nbe a subsequence ofxnn∈N such that
lim sup
n→ ∞
fz−z, xn−z lim
k→ ∞
fz−z, xnk−z, 2.16
and xnk x. Thanks to eitherH4and H5orH6 andH7, byProposition 2.2it follows thatx∈FixT. Then
lim
k→ ∞
Now we observe that, byLemma 1.1
xn1−z2 αnfxn−fz αnfz−z 1−αnTyn−z2
≤αnfxn−fz 1−αnTyn−z22αnfz−z, xn1−z
≤αnρxn−z2 1−αnyn−z22αnfz−z, xn1−z
≤αnρxn−z22αnfz−z, xn1−z
1−αnβnSxn−Sz βnSz−z 1−βnxn−z2
≤αnρxn−z22αnfz−z, xn1−z
1−αnxn−z221−αnβnSz−z, yn−z
1−1−ραnxn−z221−αnβnSz−z, yn−z2αnfz−z, xn1−z
.
2.18
Sinceτ 0,then
lim
n→ ∞
βn
αn
Sz−z, yn−z 0. 2.19
Thus, byLemma 1.4,xn → zasn → ∞.
Theorem 2.5. Suppose that (H2) with0< τ <∞, (H3), (H8), (H9) hold. Thenxn → x, asn → ∞,
wherex∈FixTis the unique solution of the variational inequality
1
τ
I−fx I−Sx, x−y
≤0, ∀y∈FixT. 2.20
Proof. First of all, we show that2.20cannot have more than one solution. Indeed, letxand
xbe two solutions. Then, sincexis solution, fory xone has
I−fx, x−x≤τI−Sx, x −x. 2.21
Analogously
I−fx, x−x≤τI−Sx,x−x. 2.22
Adding2.21and2.22, we obtain
1−ρx−x2≤I−fx−I−fx,x−x≤ −τI−Sx−I−Sx,x−x ≤0 2.23
Similarly, byProposition 2.2, the weak cluster points set ofxn,ωwxn, is a subset of FixT.
Now we have
xn1−xn αnfxn−xn 1−αnTyn−xn
αnfxn−xn 1−αnTyn−yn 1−αnyn−xn
αnf−Ixn 1−αnT−Iyn 1−αnβnS−Ixn,
2.24
so that
xn−xn1
1−αnβn I−Sxn
1
βnI−Tyn
αn 1−αnβn
I−fxn, 2.25
and denoting byvn: xn−xn1/1−αnβn,we have
vn I−Sxnβ1
nI−Tyn
αn 1−αnβn
I−fxn. 2.26
Dividing byβnin2.9, one observe that
xn1−xn
βn ≤
1−αn1−ρ xnβ−xn−1 n−1
1−αn1−ρ xn−xn−1
β1n −β1 n−1
M
|αn−αn−1|
βn
βn−βn−1
βn
≤1−αn1−ρ xnβ−xn−1
n−1 xn1−xn
β1n − β1 n−1
M
|αn−αn−1|
βn
βn−βn−1
βn
,
byH9≤1−αn1−ρ xnβ−xn−1
n−1 αnKxn1−xn
M
|αn−αn−1|
βn
βn−βn−1
βn
.
2.27
ByLemma 1.4, we have
lim
n→ ∞
xn1−xn
so, alsovn : xn−xn1/1−αnβnis a null sequence asn → ∞. Fixingz∈FixT, by2.26
it results
vn, xn−z I−Sxn, xn−zβ1 n
I−Tyn, xn−z
αn
1−αnβn
I−fxn, xn−z
I−Sxn−I−Sz, xn−zI−Sz, xn−z
β1
n
I−Tyn−I−Tz, xn−z
αn
1−αnβn
I−fxn−I−fz, xn−z
αn
1−αnβn
I−fz, xn−z.
2.29
ByLemma 1.3, we obtain that
vn, xn−z ≥ I−Sz, xn−zβ1 n
I−Tyn−I−Tz, xn−yn
β1 n
I−Tyn−I−Tz, yn−z
αn
1−αnβn
I−fz, xn−z
1−ραn
1−αnβnxn−z
2
≥ I−Sz, xn−zI−Tyn−I−Tz, xn−Sxn
1−ραn
1−αnβnxn−z
2 αn
1−αnβn
I−fz, xn−z.
2.30
Now, we observe that
xn−z2≤ 1−αnβn
1−ραn
vn, xn−z − I−Sz, xn−z − I−Tyn, xn−Sxn
− 1 1−ρ
I−fz, xn−z,
2.31
By Proposition 2.2, xnn∈N is bounded, thus there exists a subsequence xnkk∈N converging tox. For allz∈FixTby2.26
I−fxnk, xnk −z 1−ααnkβnk
nk vnk, xnk−z −
1−αnkβnk
αnk I−Sxnk, xnk−z
−1−αnk
αnk I−Tynk, xnk−z
by Lemma 1.3≤ 1−αnkβnk
αnk vnk, xnk−z −
1−αnkβnk
αnk I−Sz, xnk−z
−1−ααnk
nk I−Tynk−I−Tz, xnk−ynk
≤ 1−αnkβnk
αnk vnk, xnk−z −
1−αnkβnk
αnk I−Sz, xnk−z
−1−αnk
αnk βnkI−Tynk, xnk−Sxnk.
2.32
Passing tok → ∞,we obtain
I−fx, x−z≤ −τI−Sz, x−z, ∀z∈FixT, 2.33
which 2.20. Thus, since the 2.20 cannot have more than one solution, it follows that
ωwxn ωsxn {x}and this, of course, ensures thatxn → x, asn → ∞.
Proposition 2.6. Suppose that (H2) holds withτ ∞. Suppose that (H3), (H8) and (H9) hold. Moreover letxnn∈Nbe bounded andβnn∈Nbe a null sequence. Then everyv∈ωwxnis solution of the variational inequality
I−Sv, v−x ≤0, ∀x∈FixT, 2.34
that is,v∈Ω.
Proof. Since H8 implies H6 and H7, by boundedness of xnn∈N, we can obtain its asymptotical regularity as in proof ofProposition 2.2. Moreover, sinceβn → 0, as in proof ofProposition 2.2,ωwxn⊂FixT. With the same notation in proof ofTheorem 2.5we have that
vn, xn−z ≥ I−Sz, xn−zI−Tyn, xn−Sxn
1−ραn
1−αnβnxn−z
2 αn
1−αnβn
I−fz, xn−z
≥ I−Sz, xn−zI−Tyn, xn−Sxn1−ααnnβ n
holds for allz∈FixT. So, ifv∈ωwxnandxnk v, byH2, the boundedness ofxnn∈N,
vn → 0 andiiiofCorollary 2.3we have
I−Sz, w−z lim
k I−Sz, xnk−z
≤lim
k
vnk, xnk−zI−Tynk, Sxnk −xnk
αnk
1−αnkβnk
I−fz, z−xnk
≤0, ∀z∈FixT.
2.36
If we changezwithvμz−v, μ∈0,1, we have
I−Svμz−v, v−z ≤0. 2.37
Lettingμ → 0 finally
I−Sv, v−z ≤0, ∀z∈FixT. 2.38
Remark 2.7. If we chooseαn n−ξ andβn n−γ withξ, γ > 0, since|αn−αn−1| ≈ n−ξ and
|αn−αn−1| ≈ n−γ it is not difficult to prove thatH8is satisfied for 0 < ξ, γ < 1 andH9is
satisfied ifξγ≤1.
Remark 2.8. It is clear that our algorithm1.4is different from1.3. At the same time, our algorithm1.4includes some algorithms in the literature as special cases. For instance, if we takeS I in1.4, then we getxn1 αnfxn 1−αnTxn which is well-known as the
viscosity method studied by Moudafi8and Xu10.
Remark 2.9. We do not know the rate of convergence of our method. Nevertheless, the rates of convergence of our method1.4that generates the sequencexnand the Mainge-Moudafi
method1.3, seem not comparable. To see this, we consider three examples. In such examples we takeH R, C −1,1, αn n1−1/4, βn n1−1/6, f 1/2I,x0 1.
In all three examples all the assumptionsthat are the same of the Mainge-Moudafi methodare satisfied and the point at which both the sequencesxnandwnconverge isx PΩfx 0.
Example 2.10. TakeS IandT −I. Then
xn1 −
1− 3
2n11/4
xn 2.39
while
wn1
−
1− 3
2n11/4
2
n11/6
1− 1
n11/4
Nowx1 w1 0.5, while, for 1< n≤ 63, it results|xn|<|wn|. For instance, we report here
some value
x2 0.130672, w2 0.272417
x10 −2.52698e−11, w10 0.00086297
x20 −1.40916e−17, w20 6.2157e−08
x30 −2.19772e−22, w30 5.95813e−13
x40 −1.55804e−26, w40 1.0547e−18
x50 −2.79005e−30, w50 4.10857e−25
x60 −9.53183e−34, w60 3.8945e−32
x63 9.61011e−35, w63 2.33246e−34.
2.41
However from the 64th iteration onward,wnbecomes quickly very exiguous with respect to
xn. For instance,w259 −1.4822e−323 whilexn 7.18026e−83.
Example 2.11. TakeS T −I. Then
xn1
−
1− 3
2n11/4
2
n11/6
1− 1
n11/4
xn, 2.42
while
wn1 −
1− 3
2n11/4
wn, 2.43
that is the sequencesxn andwn are interchanged with respect to the previous example. So
this time|xn|>|wn|for 1< n <64 and|xn|<|wn|forn≥64.
Example 2.12. TakeS −I,T P−1/2,1/2. Then
xn1 1
2n11/4xn
1− 1
n11/4
P−1/2,1/2
1− 2
n11/6
xn
,
wn1 1
2n11/4wn
1− 1
n11/4
− 1
n11/6wn
1− 1
n11/6
P−1/2,1/2wn
2.44
so this timexn wn.
Acknowledgment
The authors are extremely grateful to the anonymous referees for their useful comments and suggestions. This work was supported in part by Ministero dell’Universit´a e della Ricerca of Italy.
References
1 C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,”Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
2 Q. Yang and J. Zhao, “Generalized KM theorems and their applications,”Inverse Problems, vol. 22, no. 3, pp. 833–844, 2006.
3 A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,”Inverse Problems, vol. 23, no. 4, pp. 1635–1640, 2007.
4 Y. Yao and Y.-C. Liou, “Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems,”Inverse Problems, vol. 24, no. 1, Article ID 015015, 8 pages, 2008.
5 I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
6 M. Solodov, “An explicit descent method for bilevel convex optimization,”Journal of Convex Analysis, vol. 14, no. 2, pp. 227–237, 2007.
7 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
8 A. Moudafi, “Viscosity approximation methods for fixed-points problems,”Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
9 H.-K. Xu, “Iterative algorithms for nonlinear operators,”Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
10 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
