Volume 2007, Article ID 48174,9pages doi:10.1155/2007/48174
Research Article
Strong Convergence of Modified Implicit Iteration Processes for
Common Fixed Points of Nonexpansive Mappings
Fang Zhang and Yongfu Su
Received 21 December 2006; Accepted 19 March 2007
Recommended by William Art Kirk
Strong convergence theorems are obtained by hybrid method for modified composite im-plicit iteration process of nonexpansive mappings in Hilbert spaces. The results presented in this paper generalize and improve the corresponding results of Nakajo and Takahashi (2003) and others.
Copyright © 2007 F. Zhang and Y. Su. 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 preliminaries
Throughout this paper, letH be a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty closed convex subset ofH, we denote byPC(·) the metric projection from H ontoC. It is known thatz=PC(x) is equivalent toz−y,x−z ≥0 for every y∈C. Recall thatT:C→Cis nonexpansive ifTx−T y ≤ x−yfor all
x,y∈C. A pointx∈Cis a fixed point ofTprovided thatTx=x. Denote byF(T) the set of fixed points ofT, that is,F(T)= {x∈C:Tx=x}. It is known thatF(T) is closed and convex.
Construction of fixed points of nonexpansive mappings (and asymptotically nonex-pansive mappings) is an important subject in the theory of nonexnonex-pansive mappings and finds application in a number of applied areas, in particular, in image recovery and signal processing (see, e.g., [1–5]). However, the sequence{Tnx}∞n=0of iterates of the mapping Tat a pointx∈Cmay not converge even in the weak topology. Thus averaged iterations prevail. Indeed, Mann’s iterations do have weak convergence. More precisely, Mann’s it-eration procedure is a sequence{xn}which is generated in the following recursive way:
xn+1=αnxn+
where the initial valuex0∈Cis chosen arbitrarily. For example, Reich [6] proved that if
Xis a uniformly convex Banach space with a Fr´echet differentiable norm and if{αn}is chosen such that∞n=1αn(1−αn)= ∞, then the sequence{xn}defined by (1.1) converges
weakly to a fixed point of T. However we note that Mann’s iterations have only weak convergence even in a Hilbert space [7].
Attempts to modify the Mann iteration method (1.1) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [8] proposed the following modification of Mann iteration method (1.1) for a single nonexpansive mappingT in a Hilbert spaceH:
x0∈Cchosen arbitrarily,
yn=αnxn+1−αnTxn,
Cn=z∈C:yn−z≤xn−z,
Qn=z∈C:xn−z,x0−xn ≥0,
xn+1=PCn∩Qn
x0.
(1.2)
They proved that if the sequence{αn}is bounded above from one, then the sequence {xn}generated by (1.2) converges strongly toPF(T)(x0).
In recent years, the implicit iteration scheme for approximating fixed points of non-linear mappings has been introduced and studied by several authors.
In 2001, Xu and Ori [9] introduced the following implicit iteration scheme for com-mon fixed points of a finite family of nonexpansive mappings{Ti}Ni=1in Hilbert spaces:
xn=αnxn−1+1−αnTnxn, n≥1, (1.3)
whereTn=TnmodN, and they proved weak convergence theorem.
In 2004, Osilike [10] extended results of Xu and Ori from nonexpansive mappings to strictly pseudocontractive mappings. By this implicit iteration scheme (1.3) he proved some convergence theorems in Hilbert spaces and Banach spaces.
We note that it is the same as Mann’s iterations that have only weak convergence the-orems with implicit iteration scheme (1.3). In this paper, we introduce the following two general composite implicit iteration schemes and modify them by hybrid method, so strong convergence theorems are obtained:
xn=αnxn−1+1−αnTnyn,
yn=βnxn+1−βnTnxn,
(1.4)
xn=αnxn−1+1−αnTnyn,
yn=βnxn−1+
1−βnTnxn,
(1.5)
Observe that ifK is a nonempty closed convex subset of a real Banach spaceEand
T:K→K is a nonexpansive mapping, then for everyu∈K,α,β∈[0, 1], and positive integern, the operatorS=S(α,β):K→Kdefined by
Sx=αu+ (1−α)Tβx+ (1−β)Tx (1.6)
satisfies
Sx−Sy =(1−α)Tβx+ (1−β)Tx−Tβy+ (1−β)T y
≤(1−α)βx+ (1−β)Tx−βy+ (1−β)T y
≤(1−α)βx−y+ (1−β)Tx−T y
≤(1−α)βx−y+ (1−β)x−y≤(1−α)x−y,
(1.7)
for allx,y∈K. Thus, ifα >0, thenSis a contraction and so has a unique fixed point
x∗∈K. Thus there exists a uniquex∗∈Ksuch that
x∗=αu+ (1−α)Tβx∗+ (1−β)Tx∗. (1.8)
This implies that ifαn>0, the general composite implicit iteration scheme (1.4) can be employed for the approximation of common fixed points of a finite family of nonexpan-sive mappings.
For the same reason, the operatorS=S(α,β):K→Kdefined by
Sx=αu+ (1−α)Tβu+ (1−β)Tx (1.9)
satisfies
Sx−Sy =(1−α)Tβu+ (1−β)Tx−Tβu+ (1−β)T y
≤(1−α)βu+ (1−β)Tx−βu+ (1−β)T y ≤(1−α)(1−β)Tx−T y ≤(1−α)(1−β)x−y,
(1.10)
for allx,y∈K. Thus, if (1−α)(1−β)<1, theSis a contractive mapping, thenShas a unique fixed pointx∗∈K. Thus there exists a uniquex∗∈Ksuch that
x∗=αu+ (1−α)Tβu+ (1−β)Tx∗. (1.11)
It is the purpose of this paper to modify iteration processes (1.4) and (1.5) by hybrid method as follows:
x0∈Cchosen arbitrarily,
yn=αnxn+1−αnTnzn,
zn=βnyn+1−βnTnyn,
Cn=z∈C:yn−z≤xn−z,
Qn=z∈C:xn−z,x0−xn ≥0,
xn+1=PCn∩Qn
x0.
(1.12)
x0∈Cchosen arbitrarily,
yn=αnxn+1−αnTnzn,
zn=βnxn+1−βnTnyn,
Cn=z∈C:yn−z≤xn−z,
Qn=z∈C:xn−z,x0−xn ≥0,
xn+1=PCn∩Qn
x0,
(1.13)
whereTn=TnmodN, for common fixed points of a finite family of nonexpansive mappings
{Ti}Ni=1in Hilbert spaces and to prove strong convergence theorems.
We will use the notation (1)for weak convergence and→for strong convergence. (2)ww(xn)= {x:∃xnjx}denotes the weakw-limit set of{xn}. We need some facts
and tools in a real Hilbert spaceHwhich are listed as lemmas below.
Lemma1.1 (see Martinez-Yanes and Xu [11]). Let H be a real Hilbert space, C a closed
convex subset ofH. Given pointsx,y∈H, the set
D=v∈C:y−vx−v (1.14)
is closed and convex.
Lemma1.2 (see Goebel and Kirk [12]). LetCbe a closed convex subset of a real Hilbert
spaceHand letT:C→Cbe a nonexpansive mapping such thatFix(T)= ∅. If a sequence
{xn}inCis such thatxnzandxn−Txn→0, thenz=Tz.
Lemma1.3 (see Martinez-Yanes and Xu[11]). LetK be a closed convex subset ofH. Let
{xn}be a sequence in H andu∈H. Letq=Pku. If {xn} is such thatww(xn)⊂K and satisfies the condition
xn−uu−q, ∀n, (1.15)
2. Main results
LetCbe a nonempty closed convex subset ofH, let{Ti}Ni=1:C→CbeNnonexpansive
mappings with nonempty common fixed points setF. Assume that{αn}and{βn}are sequences in [0, 1]. We consider the sequence{xn}generated by (1.12). We assume that
αn>0 (for alln∈N) in Lemmas2.1,2.2, and2.3.
Lemma2.1. {xn}is well defined andF⊂Cn∩Qnfor everyn∈N∪ {0}.
Proof. First observe thatCnis convex byLemma 1.1. Next, we show thatF⊂Cnfor alln. Indeed, we have, for allp∈F,
yn−p=αnxn+
1−αnTnzn−pαnxn−p+1−αnTnzn−p
αnxn−p+1−αnzn−p
αnxn−p+1−αnβnyn+1−βnTnyn−p
αnxn−p+1−αnβnyn−p+1−βnTnyn−p
αnxn−p+1−αnyn−p.
(2.1)
It follows that
yn−p≤xn−p. (2.2)
Sop∈Cnfor everyn≥0, thereforeF⊂Cnfor everyn≥0.
Next, we show thatF⊂Cn∩Qnfor alln≥0. It suffices to show thatF⊂Qn, for all
n≥0. We prove this by mathematical induction. Forn=0, we haveF⊂C=Q0. Assume thatF⊂Qn. Sincexn+1is the projection ofx0ontoCn∩Qn, we have
xn+1−z,x0−xn+1 ≥0, ∀z∈Qn∩Cn, (2.3)
asF⊂Cn∩Qn, the last inequality holds, in particular, for allz∈F. This together with the definition ofQn+1, implies thatF⊂Qn+1. Hence theF⊂Cn∩Qnholds for alln≥0.
This completes the proof.
Lemma2.2. {xn}is bounded.
Proof. SinceF is a nonempty closed convex subset ofC, there exists a unique element
z0∈Fsuch thatz0=PF(x0). Fromxn+1=PCnQn(x0), we have
xn+1−x0≤z−x0, (2.4)
for everyz∈Cn∩Qn. Asz0∈F⊂Cn∩Qn, we get
xn+1−x0≤z0−x0, (2.5)
Lemma2.3. xn+1−xn →0.
Proof. Indeed, by the definition ofQn, we have thatxn=PQn(x0) which together with the
fact thatxn+1∈Cn∩Qnimplies that
x0−xn≤x0−xn+1. (2.6)
This shows that the sequence{xn−x0}is increasing, fromLemma 2.2, we know that limn→∞xn−x0exists. Noticing again thatxn=PQn(x0) andxn+1∈Qnwhich implies thatxn+1−xn,xn−x0 ≥0, and noticing the identity
u−v2= u2− v2−2u−v,v, ∀u,v∈H, (2.7)
we have
xn+1−xn2
=xn+1−x0
−xn−x02
≤xn+1−x02−xn−x02−2xn+1−xn,xn−x0
≤xn+1−x02−xn−x02−→0,n−→ ∞.
(2.8)
Theorem2.4. If{αn} ⊂(0,a]for somea∈(0, 1)and{βn} ⊂[b, 1]for someb∈(0, 1], thenxn→z0, wherez0=PF(x0).
Proof. We first prove thatTnzn−xn →0, indeed,
Tnzn−xn= 1 1−αn
yn−xn≤ 1 1−αn
yn−xn+1+xn+1−xn. (2.9)
Sincexn+1∈Cn, then
yn−xn+1≤xn−xn+1, (2.10)
byLemma 2.3xn+1−xn →0, so thatyn−xn+1 →0, which leads to
Tnzn−xn−→0. (2.11)
On the other hand, we have
Tnxn−xn≤Tnxn−Tnzn+Tnzn−xn≤zn−xn+Tnzn−xn
≤βnyn−xn+1−βnTnyn−xn+Tnzn−xn
≤βnyn−xn+1−βnTnyn−Tnxn+Tnxn−xn+Tnzn−xn
≤βnyn−xn+1−βnyn−xn+Tnxn−xn+Tnzn−xn
≤yn−xn+1−βnTnxn−xn+Tnzn−xn,
which implies that
Tnxn−xn≤ 1
βn
yn−xn+ 1
βn
Tnzn−xn. (2.13)
By the condition 0< b≤βnand (2.11), we obtain that
Tnxn−xn−→0, asn−→ ∞, (2.14)
fromLemma 2.3, we know thatxn+1−xn →0, so that for allj=1, 2,...,N,
xn−xn+j−→0, asn−→ ∞. (2.15)
So, for anyi=1, 2,...,N, we also have
xn−Tn+ixn≤xn−xn+i+xn+i−Tn+ixn+i+Tn+ixn+i−Tn+ixn
≤xn−xn+i+xn+i−Tn+ixn+i+xn+i−xn
≤2xn−xn+i+xn+i−Tn+ixn+i.
(2.16)
Thus, it follows from (2.15) and (2.14) that
lim
n→+∞Tn+ixn−xn=0, i=1, 2, 3,...,N. (2.17)
BecauseTn=TnmodN, it is easy to see, for anyl=1, 2, 3,...,N, that
lim
n→+∞Tlxn−xn=0. (2.18)
ByLemma 1.2and (2.18), we obtain thatww(xn)⊂F(Tl). So,ww(xn)⊂F=Nl=1F(Tl), this, together withxn−x0PF(x0)−x0(for alln∈N) andLemma 1.3, guarantees
the strong convergence of{xn}toPF(x0).
Remark 2.5. If we setβn=1 for alln, thenzn=ynandyn=αnxn+ (1−αn)Tnyn, the itera-tion scheme (1.12) becomes modified implicit iteration scheme, so we, fromTheorem 2.4, obtain the convergence theorem of composite modified implicit iteration scheme.
Theorem2.6. LetCbe a nonempty closed convex subset ofH, let{Ti}Ni=1:C→CbeN nonexpansive mappings with nonempty common fixed points setF. Assume that{αn}and {βn}are sequences in[0, 1]and{αn} ⊂[0,a]for somea∈[0, 1)and{βn} ⊂[b, 1]for some
have, for allp∈F,
yn−p=αnxn+
1−αnTnzn−p
αnxn−p+1−αnTnzn−p
αnxn−p+1−αnzn−p
αnxn−p+1−αnβnxn+1−βnTnyn−p
αnxn−p+1−αnβnxn−p+1−βnTnyn−p
αn+βn−αnβnxn−p+1−αn1−βnyn−p.
(2.19)
It follows that
yn−p≤xn−p. (2.20)
Sop∈Cnfor everyn≥0, thereforeF⊂Cnfor everyn≥0.
Next, we show thatF⊂Cn∩Qnfor alln≥0. It suffices to show thatF⊂Qn, for all
n≥0. We prove this by mathematical induction. Forn=0, we haveF⊂C=Q0. Assume thatF⊂Qn. Sincexn+1is the projection ofx0ontoCn∩Qn, we have
xn+1−z,x0−xn+1 ≥0, ∀z∈Qn∩Cn, (2.21)
asF⊂Cn∩Qn, the last inequality holds, in particular, for allz∈F. This together with the definition ofQn+1implies thatF⊂Qn+1. HenceF⊂Cn∩Qnholds for alln≥0. This
completes the proof.
ByLemma 2.2 {xn} is bounded and by Lemma 2.3xn+1−xn →0, so that yn− xn →0, which leads to
Tnzn−xn= 1 1−αn
yn−xn−→0. (2.22)
On the other hand, we have
Tnxn−xn≤Tnxn−Tnzn+Tnzn−xn
≤zn−xn+Tnzn−xn
≤1−βnTnyn−xn+Tnzn−xn
≤1−βnTnyn−Tnxn+Tnxn−xn+Tnzn−xn
≤1−βnyn−xn+Tnxn−xn+Tnzn−xn,
(2.23)
which implies that
Tnxn−xn≤1−βn
βn
yn−xn+ 1
βn
By the condition 0< b≤βnand (2.22), we obtain that
Tnxn−xn−→0, asn−→ ∞. (2.25)
As in the proof ofTheorem 2.4we have for anyl=1, 2, 3,...,Nthat
lim
n→+∞Tlxn−xn=0. (2.26)
ByLemma 1.2and (2.26), we obtain thatww(xn)⊂F(Tl). So,ww(xn)⊂F=Nl=1F(Tl),
this together withxn−x0PF(x0)−x0(for alln∈N) andLemma 1.3guarantees the strong convergence of{xn}toPF(x0).
Remark 2.7. If we setβn=1 for alln, thenzn=xn and yn=αnxn+ (1−αn)Tnxn, so the iteration scheme (1.13) becomes modified Mann iteration, and if there is only one nonexpansive mapping, we can obtain the theorem of Nakajo and Takahashi [8].
References
[1] C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image re-construction,”Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.
[2] C. I. Podilchuk and R. J. Mammone, “Image recovery by convex projections using a least-squares constraint,”Journal of the Optical Society of America A, vol. 7, no. 3, pp. 517–521, 1990. [3] M. I. Sezan and H. Stark, “Applications of convex projection theory to image recovery in
tomog-raphy and related areas,” inImage Recovery: Theory and Application, H. Stark, Ed., pp. 415–462, Academic Press, Orlando, Fla, USA, 1987.
[4] D. Youla, “On deterministic convergence of iteration of relaxed projection operators,”Journal of Visual Communication and Image Representation, vol. 1, no. 1, pp. 12–20, 1990.
[5] D. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery: Theory and Application, H. Stark, Ed., pp. 29–77, Academic Press, Orlando, Fla, USA, 1987.
[6] S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979.
[7] A. Genel and J. Lindenstrauss, “An example concerning fixed points,”Israel Journal of Mathe-matics, vol. 22, no. 1, pp. 81–86, 1975.
[8] K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.
[9] 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.
[10] M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,”Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 73–81, 2004.
[11] C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point itera-tion processes,”Nonlinear Analysis, vol. 64, no. 11, pp. 2400–2411, 2006.
[12] 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.
Fang Zhang: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address:[email protected]
