ITERATIVE ALGORITHM FOR A CONVEX FEASIBILITY PROBLEM
Yu Li
Abstract
The purpose of this paper is to study convex feasibility problems in the setting of a real Hilbert space. The approximation of common elements of solution set of variational inequality problems and fixed point set of nonexpansive mappings is considered. Strong convergence theorems are established in the framework of Hilbert spaces.
1 Introduction and Preliminaries
Recently, many authors studied the following convex feasibility problem (CFP):
finding ap∈
r
\
i=1
Ci, (1.1)
wherer≥1 is an integer and eachCi is a nonempty closed and convex subset of a real Hilbert spaceH. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [9,12], computer tomography [19] and radiation therapy treatment planning [10].
In this paper, we always assume thatHis a real Hilbert space, whose inner product and norm are denoted byh·,·iandk · k. LetCbe a nonempty closed
Key Words: convex feasibility problem; inverse-strongly monotone mapping; nonexpan- sive mapping; variational inequality.
Mathematics Subject Classification: 47J05, 47H09, 47J25 Received: August, 2009
Accepted: January, 2010
205
and convex subset of H. Recall that a mapping A is said to be α-inverse- strongly monotone if there exists a real numberα >0 such that
hAx−Ay, x−yi ≥αkAx−Ayk2, ∀x, y ∈C.
Recall that the classical variational inequality problem, denoted byV I(C, A), is to findu∈C such that
hAu, v−ui ≥0, ∀v∈C. (1.2) Givenz∈H, u∈C, the following inequality holds
hu−z, v−ui ≥0, ∀v∈C,
if and only ifu=PCz.It is known that the projectionPC is firmly nonexpan- sive. That is,
kPCx−PCyk2≤ hx−y, PCx−PCyi, ∀x, y ∈H.
One can see that the variational inequality problem (1.2) is equivalent to a fixed point problem. It is easy to see that an elementu∈ C is a solution of the variational inequality (1.2) if and only ifu∈C is a fixed point of the mappingPC(I−λA),whereλ >0 is a constant andIis the identity mapping, that is,
u∈V I(C, A)⇐⇒u=PC(I−λA)u.
In [11], Iiduka and Takahashi showed that ifAisα-inverse-strongly monotone and λ ≤ 2α, then the mapping I−λA is nonexpansive. This implies that PC(I−λA) is also nonexpansive. In [1], Browder showed that ifCis a bounded closed and convex subset ofH, then nonexpansive mapping onChas a unique fixed point. Moreover, the fixe point set if closed and convex, see also [2] and [13].
In this paper, we shall consider the case that Ci is the set of solutions of the variational inequality problem (1.2). That is, Ci = V I(C, Ai) for each 1≤i≤r. LetS:C→Cbe a mapping. In this paper, we use F(S) to stand for the set of fixed points of the mappingS. Recall that the mappingS is said to be nonexpansive if
kSx−Syk ≤ kx−yk, ∀x, y∈C.
Recently, iterative algorithms for the classical variational inequality (1.2) and fixed point problem of nonexpansive mappings have received rapid devel- opment, see, for example, [5-8,11-17,20,21,23] and the references therein. Re- cently, Iiduka and Takahashi [11] constructed an iterative algorithm to study
the problem of finding a common element of the set of solution of a varia- tional inequality for an inverse-strongly monotone mapping and of the set of fixed points of a nonexpansive mapping. To be more precise, they proved the following theorem:
Theorem IT. Let C be a closed convex subset of a real Hilbert space H. Let Abe anα-inverse-strongly monotone mapping ofC intoH and let S be a nonexpansive mapping ofCinto itself such thatF(S)∩V I(C, A)6=∅. Suppose x1=x∈C and{xn}is given by
xn+1 =αnx+ (1−αn)SPC(xn−λnAxn), (1.3) for everyn= 1,2, . . . ,where{αn}is a sequence in[0,1)and{λn}is a sequence in [a, b]. If {αn} and{λn} are chosen so that{λn} ∈[a, b]for some a, bwith 0< a < b <2α,
n→∞lim αn= 0,
∞
X
n=1
αn=∞,
∞
X
n=1
|αn+1−αn|<∞ and
∞
X
n=1
|λn+1−λn|<∞, then {xn}converges strongly to PF(S)∩V I(C,A)x.
Recently, Y. Yao and J.C. Yao [23] further studied the approximation com- mon elements of solution set of the variational inequality (1.1) and of the fixed point set of a nonexpansive mapping by considering the following iterative al- gorithm:
x1=u∈C,
yn=PC(xn−λnAxn),
xn+1=αnu+βnxn+γnSPC(yn−λnAyn), n≥1,
(1.4)
where{αn},{βn}and{γn}are sequence in (0,1) such thatαn+βn+γn= 1 for eachn≥1, Ais anα-inverse-strongly monotone mapping ofCintoHandSis a nonexpansive mapping ofCinto itself such thatF(S)∩V I(C, A)6=∅. They proved that the sequence {xn} generated by the algorithm (1.4) converges strongly tox∗=PF(S)∩V I(C,A)u.
Quite recently, Ceng, Wang and Yao [5] considered the problem for a pair of inverse-strongly monotone mappings by the following iterative algorithm:
x1=u∈C,
yn =PC(xn−µBxn),
xn+1=αnu+βnxn+γnSPC(yn−λAyn), n≥1,
(1.5)
where{αn},{βn}and{γn} are sequence in (0,1) such thatαn+βn+γn= 1 for eachn≥1, AandB are two inverse-strongly monotone mappings and S
is a nonexpansive mapping. They also obtained a strong convergence theorem of the iterative algorithm (1.5).
In this paper, motivated by Ceng et al. [5], Iiduka and Takahashi [11], Y. Yao and J.C. Yao [23], we study the convex feasibility problem (1.1) by considering a family of inverse-strongly monotone mappings and a single non- expansive mapping. The results presented in this paper improve and extend the corresponding results announced by many others.
In order to prove our main results, we need the following lemmas.
Lemma 1.1 (Suzuki [18]). Let {xn} and {yn} be bounded sequences in a Banach spaceE and let{βn} be a sequence in [0,1]with
0<lim inf
n→∞ βn≤lim sup
n→∞ βn<1.
Supposexn+1= (1−βn)yn+βnxn for all integers n≥0and lim sup
n→∞
(kyn+1−ynk − kxn+1−xnk)≤0.
Thenlimn→∞kyn−xnk= 0.
Lemma 1.2(Bruck [4]). Let C be a closed convex subset of a strictly convex Banach spaceE. Let{Ti: 1≤i≤r}be a sequence of nonexpansive mappings on C. Suppose ∩ri=1F(Ti) is nonempty. Let {µi} be a sequence of positive numbers withPr
i=1µi= 1. Then a mappingS onC defined by Sx=
r
X
i=1
µiTix
forx∈C is well defined, nonexpansive andF(S) =∩∞i=1F(Ti)holds.
Recall that a mapping S :C →C is closed at zero if {xn} is a sequence in C converging strongly tox∈C and Sxn converges strongly to zero, then Sx= 0.
Recall that a mappingS:C→Cis demiclosed at zero if{xn}is a sequence in C converging weakly to x∈ C and Sxn converges strongly to zero, then Sx= 0.
Lemma 1.3(Browder [3]). Let H be a real Hilbert space, C be a nonempty closed convex subset of H andS :C →C be a nonexpansive mapping. Then I−S is demiclosed at zero.
Lemma 1.4(Xu [22]). Assume that{αn} is a sequence of nonnegative real numbers such that
αn+1≤(1−γn)αn+δn,
where{γn}is a sequence in(0,1) and{δn} is a sequence such that
(i) limn→∞γn= 0and P∞
n=1γn=∞;
(ii) lim supn→∞δn/γn≤0 orP∞
n=1|δn|<∞.
Thenlimn→∞αn= 0.
2 Main results
Theorem 2.1. Let C be a nonempty closed convex subset of a real Hilbert spaceH. LetAi:C→H be aµi-inverse-strongly monotone mapping for each 1≤i≤r, whereris some positive integer. Let S:C→C be a nonexpansive mapping with a fixed point. Assume thatF :=∩ri=1V I(C, Ai)∩F(S)6=∅. Let {xn} be a sequence defined by the following manner:
x1∈C, xn+1=αnu+βnxn+γnS
r
X
i=1
ηiPC(xn−λiAixn), n≥1, (2.1) whereu∈Cis a fixed point,λ1, λ2, . . .andλrare real numbers such thatλi∈ (0,2µi)for each 1≤i≤r, and{αn},{βn} and{γn} are sequences in(0,1).
Assume that the above control sequences satisfies the following conditions:
(i) αn+βn+γn=Pr
i=1ηi= 1,∀n≥1;
(ii) limn→∞αn = 0,P∞
n=1αn=∞;
(iii) 0<lim infn→∞βn ≤lim supn→∞βn<1.
Then the sequence {xn} generated in the iterative algorithm (2.1) converges strongly top=PFu.
Proof. The proof is split into five steps.
Step 1. Show that the sequence{xn}is bounded.
Note that the mappingI−λiAiis nonexpansive for eachi.Indeed, for any x, y ∈C,we see that
k(I−λiAi)x−(I−λiAi)yk2
=kx−yk2−2λihAix−Aiy, x−yi+λ2ikAix−Aiyk2
≤ kx−yk2−λi(2µi−λi)kAix−Aiyk2.
Since, for each 1≤i≤r,λi∈(0,2µi), we see thatI−λiAi is nonexpansive.
Putyn=Pr
i=1ηiPC(xn−λiAixn) for each n≥1. For anyx∗∈ F, we have kxn+1−x∗k=kαnu+βnxn+γnSyn−x∗k
≤αnku−x∗k+βnkxn−x∗k+γnkSyn−x∗k
≤αnku−x∗k+βnkxn−x∗k+γn r
X
i=1
ηikPC(xn−λiAixn)−x∗k
≤αnku−x∗k+βnkxn−x∗k+γnkxn−x∗k
=αnku−x∗k+ (1−αn)kxn−x∗k.
By mathematical inductions, we can obtain that
kxn−x∗k ≤ {kx1−x∗k,ku−x∗k}, ∀n≥1.
This shows that the sequence{xn}is bounded. sinceI−λiAi is nonexpansive for each i, we obtain that
kyn−x∗k=k
r
X
i=1
ηiPC(xn−λiAixn)−
r
X
i=1
ηix∗k
≤
r
X
i=1
ηikPC(xn−λiAixn)−x∗k
≤ kxn−x∗k.
This shows that{yn} is also bounded.
Step 2. Show thatxn+1−xn →0 asn→ ∞.
Note that kyn+1−ynk=k
r
X
i=1
ηiPC(xn+1−λiAixn+1)−
r
X
i=1
ηiPC(xn−λiAixn)k
≤ kxn+1−xnk.
(2.2) Putln= xn+11−β−βnxn
n , for alln≥1. That is,
xn+1= (1−βn)ln+βnxn, ∀n≥1. (2.3) Note that
ln+1−ln
= αn+1u+γn+1Syn+1
1−βn+1
−αnu+γnSyn
1−βn
= αn+1
1−βn+1
u+1−βn+1−αn+1
1−βn+1
Syn+1− αn
1−βn
u−1−βn−αn
1−βn
Syn
= αn+1
1−βn+1
u−Syn+1 + αn
1−βn
Syn−u
+Syn+1−Syn.
It follows that
kln+1−lnk ≤ αn+1
1−βn+1
ku−Syn+1k+ αn
1−βn
kSyn−uk+kSyn+1−Synk
≤ αn+1
1−βn+1
ku−Syn+1k+ αn
1−βn
kSyn−uk+kyn+1−ynk.
By virtue of (2.2), we arrive at
kln+1−lnk − kxn+1−xnk ≤ αn+1
1−βn+1
ku−Syn+1k+ αn
1−βn
kSyn−uk.
It follows from the conditions (ii) and (iii) that lim sup
n→∞
(kln+1−lnk − kxn+1−xn+1k)≤0.
Thanks to Lemma 1.1, we obtain that
n→∞lim kln−xnk= 0.
In view of (2.3), we have
xn+1−xn = (1−βn)(ln−xn).
This implies that
n→∞lim kxn+1−xnk= 0. (2.4) Step 3. Show thatSxn−xn →0 asn→ ∞.
Note that
kxn+1−x∗k2=kαnu+βnxn+γnSyn−x∗k2
≤αnku−x∗k2+βnkxn−x∗k2+γnkS
r
X
i=1
ηiPC(xn−λiAixn)−x∗k2
≤αnku−x∗k2+βnkxn−x∗k2+γnk
r
X
i=1
ηiPC(xn−λiAixn)−x∗k2
≤αnku−x∗k2+βnkxn−x∗k2+γn r
X
i=1
ηikPC(xn−λiAixn)−x∗k2. (2.5)
This implies that kxn+1−x∗k2
≤αnku−x∗k2+βnkxn−x∗k2 +γn
r
X
i=1
ηikxn−x∗−λi(Aixn−Aix∗)k2
≤αnku−x∗k2+βnkxn−x∗k2+γn r
X
i=1
ηi(kxn−x∗k2
−2λihAixn−Aix∗, xn−x∗i+λ2ikAixn−Aix∗k2)
≤αnku−x∗k2+kxn−x∗k2−γn r
X
i=1
ηiλi(2µi−λi)kAixn−Aix∗k2.
It follows that γn
r
X
i=1
ηiλi(2µi−λi)kAixn−Aix∗k2
≤αnku−x∗k2+kxn−x∗k2− kxn+1−x∗k2
≤αnku−x∗k2+ (kxn−x∗k+kxn+1−x∗k)kxn−xn+1k.
Thanks to conditions (ii) and (iii), one obtains that
n→∞lim kAixn−Aix∗k= 0, ∀1≤i≤r. (2.6) On the other hand, one has
kPC(I−λiAi)xn−x∗k2
=kPC(I−λiAi)xn−PC(I−λiAi)x∗k2
≤ h(I−λiAi)xn−(I−λiAi)x∗, PC(I−λiAi)xn−x∗i
=1
2 k(I−λiAi)xn−(I−λiAi)x∗k2+kPC(I−λiAi)xn−x∗k2
− k(I−λiAi)xn−(I−λiAi)x∗−(PC(I−λiAi)xn−x∗)k2
≤1
2 kxn−x∗k2+kPC(I−λiAi)xn−x∗k2
− kxn−PC(I−λiAi)xn−λi(Aixn−Aix∗)k2
=1
2 kxn−x∗k2+kPC(I−λiAi)xn−x∗k2− kxn−PC(I−λiAi)xnk2 + 2λihAixn−Aix∗, xn−PC(I−λiAi)xni −λ2ikAixn−Aix∗k2
.
It follows that
kPC(I−λiAi)xn−x∗k2≤ kxn−x∗k2−kxn−PC(I−λiAi)xnk2+MikAixn−Aix∗k, (2.7) where Mi is given by
Mi= sup{2λikxn−PC(I−λiAi)xnk:∀n≥1}.
On the other hand, we have kyn−xnk2=k
r
X
i=1
ηiPC(I−λiAi)xn−xnk2≤
r
X
i=1
ηikPC(I−λiAi)xn−xnk2, which combines with (2.7) yields that
r
X
i=1
ηikPC(I−λiAi)xn−x∗k2≤ kxn−x∗k2−kyn−xnk2+
r
X
i=1
ηiMikAixn−Aix∗k.
From (2.5), we see that
kxn+1−x∗k2≤αnku−x∗k2+kxn−x∗k2+γn r
X
i=1
ηiMikAixn−Aix∗k−γnkyn−xnk2, from which it follows that
γnkyn−xnk2≤αnku−x∗k2+kxn−x∗k2− kxn+1−x∗k2+ +γn
r
X
i=1
ηiMikAixn−Aix∗k
≤αnku−x∗k2+ (kxn−x∗k+kxn+1−x∗k)kxn−xn+1k +γn
r
X
i=1
ηiMikAixn−Aix∗k.
It follows from (2.4), (2.6) and the conditions (ii) and (iii) that
n→∞lim kyn−xnk= 0. (2.8) Note that
Syn−xn =(xn+1−xn)−αn(u−xn) γn
. Combining this with the condition (ii) and (iii) gives that
n→∞lim kSyn−xnk= 0. (2.9)
Observe that
kSxn−xnk ≤ kxn−Synk+kSyn−Sxnk
≤ kxn−Synk+kyn−xnk.
It follows from (2.8) and (2.9) that
n→∞lim kSxn−xnk= 0. (2.10) Step 4. Show that
lim sup
n→∞
hu−p, xn−pi ≤0, wherep=PFu.
To show it, we can choose a sequence {xni}of{xn} such that lim sup
n→∞
hu−p, xn−pi= lim
i→∞hu−p, xni−pi. (2.11) Since {xni} is bounded, there exists a subsequence {xnij} of {xni} which converges weakly tof. Without loss of generality, we can assume thatxni ⇀ f. Define a mappingW :C→C by
W x=
r
X
i=1
ηiPC(I−λiAi)x, ∀x∈C.
From Lemma 1.2, we see thatW is nonexpansive such that F(W) =∩ri=1F(PC(I−λiAi)) =∩ri=1V I(C, Ai).
From (2.8), we see that
n→∞lim kxn−W xnk= 0. (2.12) From Lemma 1.3, we can obtain thatf ∈F(W). In view of (2.10) and Lemma 1.3, we see thatf ∈F(S). This proves that
f ∈F(W)∩F(S) =∩ri=1V I(C, Ai)∩F(S).
It follows from (2.11) that lim sup
n→∞
hu−p, xn−pi ≤0.
Step 5. Show thatxn→pasn→ ∞.
Note that kxn+1−pk2
=hαnu+βnxn+γnSyn−p, xn+1−pi
=αnhu−p, xn+1−pi+βnhxn−p, xn+1−pi +γnhSyn−p, xn+1−pi
≤αnhu−p, xn+1−pi+βnkxn−pkkxn+1−pk+γnkSyn−pkkxn+1−pk
≤αnhu−p, xn+1−pi+βnkxn−pkkxn+1−pk+γnkyn−pkkxn+1−pk
≤αnhu−p, xn+1−pi+ (1−αn)kxn−pkkxn+1−p|
≤αnhu−p, xn+1−pi+1−αn
2 kxn−pk2+1
2kxn+1−pk2,
(2.13) which implies that
kxn+1−pk2≤(1−αn)kxn−pk2+ 2αnhu−p, xn+1−pi.
Applying Lemma 1.4 to (2.13), we obtain that
n→∞lim kxn−pk= 0.
This completes the proof.
PuttingS=I, the identity mapping, we have the following result.
Corollary 2.2. Let C be a nonempty closed convex subset of a real Hilbert spaceH. LetAi:C→H be aµi-inverse-strongly monotone mapping for each 1≤i≤r, whereris some positive integer. Assume thatF :=∩ri=1V I(C, Ai)6=
∅. Let{xn} be a sequence defined by the following manner:
x1∈C, xn+1=αnu+βnxn+γn r
X
i=1
ηiPC(xn−λiAixn), n≥1, whereu∈Cis a fixed point,λ1, λ2, . . .andλrare real numbers such thatλi∈ (0,2µi)for each 1≤i≤r, and{αn},{βn} and{γn} are sequences in(0,1).
Assume that the above control sequences satisfies the following conditions:
(i) αn+βn+γn=Pr
i=1ηi= 1,∀n≥1;
(ii) limn→∞αn = 0,P∞
n=1αn=∞;
(iii) 0<lim infn→∞βn ≤lim supn→∞βn<1.
Then the sequence {xn} converges strongly top=PFu.
Next, we give a special case of Theorem 2.1 on a pair of inverse-strongly monotone mappings.
Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H be a µ1-inverse-strongly monotone mapping and B : C → H a µ2-inverse-strongly monotone mapping, respectively. Let S : C → C be a nonexpansive mapping with a fixed point. Assume that F :=
V I(C, A)∩V I(C, B)∩F(S) 6= ∅. Let {xn} be a sequence defined by the following manner:
x1∈C, chosen arbitrarily
yn=ηPC(xn−λAxn) + (1−η)PC(xn−ρBxn), xn+1=αnu+βnxn+γnSyn, n≥1,
where u ∈ C is a fixed point, η is a real number in (0,1), λ and ρ are real numbers such thatλ∈(0,2µ1)andρ∈(0,2µ2), respectively, and {αn},{βn} and {γn} are sequences in (0,1). Assume that the above control sequences satisfy the following conditions:
(i) αn+βn+γn= 1,∀n≥1;
(ii) limn→∞αn= 0,P∞
n=1αn =∞;
(iii) 0<lim infn→∞βn≤lim supn→∞βn<1.
Then the sequence{xn}converges strongly to p=PFu.
Acknowledgments
The author is extremely grateful to the referee for useful suggestions that improved the contents of the paper.
This work was supported by the project of development of science and tech- nology (2009) foundation grant funded by the Department of Henan Science and Technology (092102210134).
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Henan University
Management Science and Engineering Research Institute Kaifeng 475004, China
Email: [email protected]
