Tomus 45 (2009), 147–158
STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS
AND FIXED POINT PROBLEMS
Xiaolong Qin1, Shin Min Kang1, Yongfu Su2, and Meijuan Shang3
Abstract. In this paper, we introduce a general iterative scheme to investigate the problem of finding a common element of the fixed point set of a strict pseudocontraction and the solution set of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Strong convergence theorems are established in a real Hilbert space.
1. Introduction and preliminaries
Variational inequalities introduced by Stampacchia [16] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, see [4]–[22] and references therein. In this paper, we consider the problem of approximation of solutions of the classical variational inequality problem by iterative methods.
LetH be a real Hilbert space, whose inner product and norm are denoted by h·,·iandk · k,Ca nonempty closed convex subset ofH andA:C→H a nonlinear mapping.
Recall the following definitions.
(a) Ais said to be monotone if
hAx−Ay, x−yi ≥0, ∀ x, y∈C .
(b) Ais said to beν-strongly monotone if there exists a constantν >0 such that
hAx−Ay, x−yi ≥νkx−yk2, ∀x, y∈C .
(c) Ais said to be µ-cocoercive if there exists a constantµ >0 such that hAx−Ay, x−yi ≥µkAx−Ayk2, ∀ x, y∈C .
Clearly, every µ-cocoercive mapping is 1/µ-Lipschitz continuous.
2000Mathematics Subject Classification: primary 47H09; secondary 47J25.
Key words and phrases: nonexpansive mapping, strict pseudocontraction, fixed point, variatio- nal inequality, relaxed cocoercive mapping.
Received May 30, 2007, revised May 2009. Editor O. Došlý.
(d) Ais said to be relaxed µ-cocoercive if there exists a constantµ >0 such that
hAx−Ay, x−yi ≥(−µ)kAx−Ayk2, ∀x, y∈C .
(e) Ais said to be relaxed (µ, ν)-cocoercive if there exist two constantsµ, ν >0 such that
hAx−Ay, x−yi ≥(−µ)kAx−Ayk2+νkx−yk2, ∀x, y∈C . The classical variational inequality is to findu∈C such that
(1.1) hAu, v−ui ≥0, ∀ v∈C .
In this paper, we useV I(C, A) to denote the solution set of the problem (1.1).
It is easy to see that an elementu∈Cis a solution to the problem (1.1) if and only ifu∈C is a fixed point of the mappingPC(I−λA), wherePC denotes the metric projection from H ontoC, λis a positive constant and I is the identity mapping.
LetT:C→C be a mapping. In this paper, we useF(T) to denote the set of fixed points of the mapping T. Recall the following definitions.
(1) T is said to be a contraction if there exists a constantα∈(0,1) such that kT x−T yk ≤αkx−yk, ∀x, y∈C .
(2) T is said to be nonexpansive if
kT x−T yk ≤ kx−yk, ∀x, y∈C .
(3) T is said to be strictly pseudo-contractive with the coefficient k∈(0,1) if kT x−T yk2≤ kx−yk2+kk(I−T)x−(I−T)yk2, ∀x, y∈C .
For such a case,T is also said to be ak-strict pseudo-contraction.
(4) T is said to be pseudo-contractive if
hT x−T y, x−yi ≤ kx−yk2, ∀ x, y∈C .
Clearly, the class of strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions. We remark also that the class of strongly pseudo-contractive mappings is independent of the class of strict pseudo-contractions; see, for example [1, 24].
The class of strict pseudo-contractions which was introduced by Browder and Petryshyn [2] is one of the most important classes of mappings among nonli- near mappings. Within the past several decades, many authors have been devo- ting to the studies on the existence and convergence of fixed points for strict pseudo-contractions. Recently, Zhou [23] considered a convex combination method to study strict pseudo-contractions. More precisely, take t ∈(0,1) and define a mapping Stby
Stx=tx+ (1−t)T x , ∀ x∈C ,
where T is a strict pseudo-contraction. Under appropriate restrictions on t, it is proved that the mapping St is nonexpansive. Therefore, the techniques of
studying nonexpansive mappings can be applied to study more general strict pseudo-contractions.
For finding a common element of the set of fixed points of nonexpansive map- pings and the set of solution of variational inequalities forα-cocoercive mapping, Takahashi and Toyoda [19] introduced the following iterative process:
x0∈C , xn+1=αnxn+ (1−αn)SPC(xn−λnAxn), ∀n≥0, whereAisα-cocoercive mapping andS is a nonexpansive mapping with a fixed point. They showed that ifF(S)∩V I(C, A) is nonempty then the sequence{xn} converges weakly to somez∈F(S)∩V I(C, A) under some restrictions imposed on the sequence {αn}and{λn}; see [18] for more details.
On the other hand, for solving the variational inequality problem in the finite- -dimensional Euclidean spaceRn under the assumption that a setC⊂Rn is closed and convex, a mappingA ofC intoRn is monotone andk-Lipschitz-continuous and V I(C, A) is nonempty, Korpelevich [9] introduced the following so-called extragradient method:
x0=x∈C ,
yn=PC(xn−λAxn),
xn+1=PC(xn−λAyn), ∀n≥0,
whereλ∈(0,1/k). He proved that the sequences{xn}and{yn} generated by this iterative process converge to the same pointz∈V I(C, A).
To obtain strong convergence theorems, Iiduka and Takahashi [8] proposed the following iterative scheme:
x0∈C , xn+1=αnx+ (1−αn)SPC(xn−λnAxn), ∀n≥0,
whereAisα-cocoercive mapping andS is a nonexpansive mapping with a fixed point. They showed that ifF(S)∩V I(C, A) is nonempty then the sequence{xn} converges strongly to somez∈F(S)∩V I(C, A) under some restrictions imposed on the sequence {αn}and{λn}; see [8] for more details.
Recently, Yao and Yao [22], further improved Iiduka and Takahashi [9]’ results by considering the following iterative process
x0∈C ,
yn =PC(xn−λnAxn),
xn+1=αnu+βnxn+γnSPC(I−λnA)yn, ∀ n≥0,
whereAisα-cocoercive mapping andS is a nonexpansive mapping with a fixed point. A strong convergence theorem was also established in the framework of Hilbert space under some restrictions imposed on the sequence{αn}and{λn}; see [22] for more details.
In this paper, motivated by the research working going on in this direction, we continue to study the variational inequality problem and the fixed point problem by the viscosity approximation method which was first considered by Moudafi [10]. To be more precise, we introduce a general iterative process to find a common element of the set of fixed points of a strict pseudocontraction and the set of solutions of
the variational inequality problem (1.1) for a relaxed cocoercive mapping in a real Hilbert space. Strong convergence of the purposed iterative process is obtained.
In order to prove our main results, we need the following lemmas.
Lemma 1.1 ([23]). Let C be a nonempty closed convex subset of a real Hilbert space H and T:C →C a k-strict pseudo-contraction with a fixed point. Define S:C→C by Sx=kx+ (1−k)T x for eachx∈C. ThenS is nonexpansive with F(S) =F(T).
The following lemma is a corollary of Bruck’s result in [3].
Lemma 1.2. LetC be a nonempty closed convex subset of a real Hilbert space H. LetT1 andT2 be two nonexpansive mappings fromC intoC with a common fixed point. Define a mappingS:C→C by
Sx=λT1x+ (1−λ)T2x , ∀x∈C ,
where λis a constant in (0,1). ThenS is nonexpansive andF(S) =F(T1)∩F(T2) Lemma 1.3([2]). LetH be a Hilbert space,Cbe a nonempty closed convex subset ofH andS:C→C be a nonexpansive mapping. ThenI−S is demi-closed at zero Lemma 1.4 ([17]). Let{xn} and {yn} be bounded sequences in a Banach space X 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 integersn≥0 and lim sup
n→∞
(kyn+1−ynk − kxn+1−xnk)≤0, . Thenlimn→∞kyn−xnk= 0.
Lemma 1.5 ([21]). 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 (a) P∞
n=1γn=∞;
(b) lim supn→∞δn/γn≤0or P∞
n=1|δn|<∞.
Thenlimn→∞αn= 0.
2. Main results
Theorem 2.1. LetH be a real Hilbert space,Ca nonempty closed convex subset of H and A:C→H a relaxed(µ, ν)-cocoercive andL-Lipschitz continuous mapping.
Let f: C → C be a contraction with the coefficient α ∈ (0,1) and T:C → C a strict pseudocontraction with a fixed point. Define a mapping S: C → C by
Sx=kx+ (1−k)T xfor each x∈C. Assume thatF =F(T)∩V I(C, A)6=∅. Let {xn} be a sequence generated by the following algorithm:x1∈C and
zn=ωnxn+ (1−ωn)PC(xn−tnAxn), yn=δnSxn+ (1−δn)zn,
xn+1=αnf(xn) +βnxn+γnyn, ∀ n≥1,
where {αn}, {βn}, {γn}, {δn} and {ωn} are sequences in (0,1) and {tn} is a positive sequence. Assume that the above control sequences satisfy the following restrictions:
(a) αn+βn+γn= 1, for each n≥1;
(b) 0<lim infn→∞βn≤lim supn→∞βn<1;
(c) limn→∞αn= 0,P∞
n=1αn=∞;
(d) 0< t≤tn ≤2(ν−LL22µ), wheret is some constant, for eachn≥1;
(e) limn→∞|tn−tn+1|= 1
(f) limn→∞δn =δ∈(0,1),limn→∞ωn=ω∈(0,1).
Then the sequence {xn} converges strongly to x¯∈ F, where ¯x=PFf(¯x), which solves the following variational inequality
hf(¯x)−x,¯ x¯−xi ≤0, ∀ x∈ F.
Proof. First, we show the mappingI−tnAis nonexpansive for eachn≥1. Indeed, from the relaxed (µ, ν)-cocoercive andL-Lipschitz definition onA, we have
k(I−tnA)x−(I−tnA)yk2=k(x−y)−tn(Ax−Ay)k2
=kx−yk2−2tnhx−y, Ax−Ayi+t2nkAx−Ayk2
≤ kx−yk2−2tn[−µkAx−Ayk2+νkx−yk2] +t2nkAx−Ayk2
≤ kx−yk2+ 2tnµL2kx−yk2−2tnνkx−yk2+L2t2nkx−yk2
= (1 + 2tnL2µ−2tnν+L2t2n)kx−yk2
≤ kx−yk2,
which implies the mappingI−tnAis nonexpansive for eachn≥1.
Next, we show that the sequence{xn} is bounded. Fixp∈F(T)∩V I(C, A).
From Lemma 1.1, we see that F(T) =F(S). It follows that
kzn−pk=kωn(xn−p) + (1−ωn)(PC(I−tnA)xn−p)k
≤ωnkxn−pk+ (1−ωn)kxn−pk
=kxn−pk. It follows that
kyn−pk=kδn(Sxn−p) + (1−δn)zn−p)k
≤δnkSxn−pk+ (1−δn)kzn−pk
≤ kxn−pk.
This implies that
kxn+1−pk=kαn(f(xn)−p) +βn(xn−p) +γnyn−p)k
≤αnkf(xn)−pk+βnkxn−pk+γnkyn−pk
≤αnkf(xn)−f(p)k+αnkp−f(p)k+βnkxn−pk+γnkxn−pk
≤ααnkxn−pk+αnkp−f(p)k+ (1−αn)kxn−pk
= [1−αn(1−α)]kxn−pk+αnkp−f(p)k
≤maxn
kxn−pk,kp−f(p)k 1−α
o .
By simple inductions, we have kxn−pk ≤maxn
kx1−pk,kp−f(p)k 1−α
o
, ∀ n≥1,
which gives that the sequence{xn} is bounded, so are{yn}and{zn}. Putρn= PC(I−tnA)yn. It follows that
(2.1)
kρn−ρn+1k=kPC(I−tnA)xn−PC(I−tn+1A)xn+1k
≤ k(I−tnA)xn−(I−tn+1A)xn+1k
=k(I−tnA)xn−(I−tnA)xn+1+ (tn+1−tn)Axn+1k
≤ kxn−xn+1k+|tn+1−tn| kAxn+1k. Note that
(zn =ωnxn+ (1−ωn)ρn,
zn+1=ωn+1xn+1+ (1−ωn+1)ρn+1. It follows that
zn−zn+1=ωn(xn−xn+1) + (1−ωn)(ρn−ρn+1) + (ρn+1−xn+1)(ωn+1−ωn), which yields that
kzn−zn+1k ≤ωnkxn−xn+1k+ (1−ωn)kρn−ρn+1k +kρn+1−xn+1k |ωn+1−ωn|.
(2.2)
Substituting (2.1) into (2.2), we see that
(2.3) kzn−zn+1k ≤ kxn−xn+1k+ (|tn+1−tn|+|ωn+1−ωn|)M1, whereM1 is an appropriate constant such that
M1= max sup
n≥1
{kAxnk},sup
n≥1
{kρn−xnk} . On the other hand, we have
yn−yn+1=δn(Sxn−Sxn+1) + (1−δn)(zn−zn+1) + (zn+1−Sxn+1)(δn+1−δn), which yields that
kyn−yn+1k ≤δnkxn−xn+1k+ (1−δn)kzn−zn+1k +kzn+1−Sxn+1k|δn+1−δn|. (2.4)
Substituting (2.3) into (2.4), we see that
(2.5) kyn−yn+1k ≤ kxn−xn+1k+ (|tn+1−tn|+|ωn+1−ωn|+|δn+1−δn|)M2, whereM2 is an appropriate constant such that
M2= max{M1,sup
n≥1
{kzn−Sxnk}}.
Putln = xn+11−β−βnxn
n for eachn≥1. That is,xn+1 = (1−βn)ln+βnxn for each n≥1. Now, we compute kln+1−lnk.Observing that
ln+1−ln=αn+1f(xn+1) +γn+1yn+1
1−βn+1 −αnf(xn) +γnyn
1−βn
=αn+1f(xn+1) + (1−αn+1−βn+1)yn+1
1−βn+1
−αnf(xn) + (1−αn−βn)yn
1−βn
= αn+1
1−βn+1
(f(xn+1)−yn+1)− αn
1−βn
(f(xn)−yn) +yn+1−yn
we obtain that kln+1−lnk ≤ αn+1
1−βn+1
kf(xn+1)−yn+1k − αn 1−βn
kf(xn)−ynk+kyn+1−ynk, which combines with (2.5) yields that
kln+1−lnk − kxn−xn+1k ≤ αn+1
1−βn+1
kf(xn+1)−yn+1k+ αn
1−βn
kf(xn)−ynk +kyn+1−ynk+ (|tn+1−tn|+|ωn+1−ωn|+|δn+1−δn|)M2. It follows from the conditions (a), (b), (c), (e) and (f) that
lim sup
n→∞
(kln+1−lnk − kxn+1−xnk)≤0.
In view of Lemma 1.4, we obtain that limn→∞kln−xnk= 0. It follows that
n→∞lim kxn+1−xnk= lim
n→∞(1−βn)kln−xnk= 0. Observing that
xn+1−xn=αn f(xn)−xn
+γn(yn−xn), we have
(2.6) lim
n→∞kyn−xnk= 0.
Note thatPFf is a contraction. Indeed, for allx, y∈C, we have kPFf(x)−PFf(y)k ≤ kf(x)−f(y)k ≤αkx−yk.
Banach’s contraction mapping principle guarantees thatPFf has a unique fixed point, say ¯x∈C. That is, ¯x=PFf(¯x).
Next, we show that
lim sup
n→∞
hf(¯x)−x, x¯ n−xi ≤¯ 0. To show it, we choose a subsequence{xni} of{xn}such that
lim sup
n→∞
hf(¯x)−x, x¯ n−xi¯ = lim
i→∞hf(¯x)−x, x¯ ni−xi¯ .
Since {xni} is bounded, we can choose a subsequence{xnij}of {xni} converging weakly tox. We may without loss of generality assume thatb xni *x, whereb * denotes the weak convergence. From the condition (d), we see that there exits a subsequence{tni} of{tn}such that tni →s∈
t,2(ν−LL22µ)
. Next, we prove that xb∈ F. Indeed, define a mappingR1:C→C by
R1x=ωx+ (1−ω)PC(I−sA)x , ∀x∈C . From Lemma 1.2, we see that R1 is nonexpansive with
F(R1) =F(I)∩F(PC(I−sA)) =V I(C, A). Now, we define another mappingR2:C→C by
R2x=δSx+ (1−δ)R1x , ∀x∈C . From Lemma 1.2, we also obtain thatR2 is nonexpansive with
F(R2) =F(S)∩F(R1) =F(T)∩V I(C, A) =F. Note that
kR1xni−znik=kωxni+ (1−ω)PC(I−sA)xni−znik
=kωxni+ (1−ω)PC(I−sA)xni−[ωnixni+ (1−ωni)PC(I−tniA)xni]k
≤ |ω−ωni|(kxnik+kPC(I−tniA)xnik) + (1−ω)kPC(I−sA)xni−PC(I−tniA)xnik
≤ |ω−ωni|(kxnik+kPC(I−tniA)xnik) + (1−ω)|s−tni|kAxnik. (2.7)
In view of the condition (f), we obtain that limi→∞kR1xni−znik = 0. On the other hand, we have
kR2xni−ynik=kδSxni+ (1−δ)R1xni−ynik
=kδSxni+ (1−δ)R1xni−[δniSxni+ (1−δni)zni]k
≤ |δ−δni|(kSxnik+kznik) + (1−δ)kR1xni−znik. From the condition (f) and limi→∞kR1xni−znik= 0, we obtain that
(2.8) lim
i→∞kR2xni−ynik= 0. Note that
kR2xni−xnik ≤ kR2xni−ynik+kyni−xnik. Combining (2.6) and (2.8), we see that
i→∞lim kR2xni−xnik= 0.
Note thatxni*x. From Lemma 1.3, we obtain thatb bx∈ F. It follows that lim sup
n→∞
hf(¯x)−x, x¯ n−¯xi= lim
i→∞hf(¯x)−x, x¯ ni−xi¯ = lim
i→∞hf(¯x)−x,¯ bx−xi ≤¯ 0. Finally, we show thatxn→x¯ asn→ ∞. Note that
kxn+1−xk¯ 2=hαnf(xn) +βnxn+γnyn−x, x¯ n+1−xi¯
=αnhf(xn)−x, x¯ n+1−xi¯ +βnhxn−x, x¯ n+1−xi¯ +γnhyn−x, x¯ n+1−xi¯
≤αnhf(xn)−f(¯x), xn+1−xi¯ +αnhf(¯x)−x, x¯ n+1−xi¯ +βnkxn−xk kx¯ n+1−xk¯ +γnkyn−xk kx¯ n+1−xk¯
≤αnαkxn−xk kx¯ n+1−xk¯ +αnhf(¯x)−x, x¯ n+1−xi¯ +βnkxn−xk kx¯ n+1−xk¯ +γnkxn−xk kx¯ n+1−xk¯
≤ 1−αn(1−α)
2 (kxn−xk¯ 2+kxn+1−xk¯ 2) +αnhf(¯x)−¯x, xn+1−xi¯
≤ 1−αn(1−α)
2 kxn−¯xk2+1
2kxn+1−xk¯ 2+αnhf(¯x)−x, x¯ n+1−xi¯ . This implies that
kxn+1−xk¯ 2≤[1−αn(1−α)]kxn−xk¯ 2+ 2αnhf(¯x)−x, x¯ n+1−¯xi. In view of Lemma 1.5, we can conclude the desired conclusion easily.
As corollaries of Theorem 2.1, we have the following results immediately.
Corollary 2.1. LetH be a real Hilbert space,Ca nonempty closed convex subset of H and A:C→H a relaxed(µ, ν)-cocoercive andL-Lipschitz continuous mapping.
Let f: C →C be a contraction with the coefficient α∈(0,1) and T: C →C a nonexpansive mapping with a fixed point. Assume that F=F(T)∩V I(C, A)6=∅.
Let {xn} be a sequence generated by the following algorithm:x1∈C and
zn=ωnxn+ (1−ωn)PC(xn−tnAxn), yn =δnT xn+ (1−δn)zn,
xn+1=αnf(xn) +βnxn+γnyn, ∀n≥1,
where {αn}, {βn}, {γn}, {δn} and {ωn} are sequences in (0,1) and {tn} is a positive sequence. Assume that the above control sequences satisfy the following restrictions:
(a) αn+βn+γn = 1, for eachn≥1;
(b) 0<lim infn→∞βn ≤lim supn→∞βn <1;
(c) limn→∞αn= 0,P∞
n=1αn =∞;
(d) 0< t≤tn≤ 2(ν−LL22µ), wheret is some constant, for eachn≥1;
(e) limn→∞|tn−tn+1|= 1;
(f) limn→∞δn =δ∈(0,1),limn→∞ωn=ω∈(0,1).
Then the sequence {xn} converges strongly to x¯∈ F, where ¯x=PFf(¯x), which solves the following variational inequality
hf(¯x)−x,¯ x¯−xi ≤0, ∀x∈ F.
Corollary 2.2. LetH be a real Hilbert space, C a nonempty closed convex subset of H and A : C → H a relaxed (µ, ν)-cocoercive and L-Lipschitz continuous mapping. Let f :C→C be a contraction with the coefficient α∈(0,1). Assume thatV I(C, A)6=∅. Let {xn} be a sequence generated by the following algorithm:
x1∈C and
(yn= [δ+ (1−δ)ω]xn+ (1−δ)(1−ω)PC(xn−tnAxn), xn+1=αnf(xn) +βnxn+γnyn, ∀n≥1,
where{αn},{βn} and {γn} are sequences in (0,1),δand ω are two constant in (0,1) and {tn} is a positive sequence. Assume that the above control sequences satisfy the following restrictions:
(a) αn+βn+γn = 1, for eachn≥1;
(b) 0<lim infn→∞βn ≤lim supn→∞βn <1;
(c) limn→∞αn= 0,P∞
n=1αn =∞;
(d) 0< t≤tn≤ 2(ν−LL22µ), wheret is some constant, for eachn≥1;
(e) limn→∞|tn−tn+1|= 1;
Then the sequence{xn}converges strongly tox¯∈V I(C, A), wherex¯=PV I(C,A)f(¯x), which solves the following variational inequality
hf(¯x)−x,¯ x¯−xi ≤0, ∀x∈V I(C, A).
Corollary 2.3. LetH be a real Hilbert space,Ca nonempty closed convex subset of H and A:C→H a relaxed(µ, ν)-cocoercive andL-Lipschitz continuous mapping.
Let f: C → C be a contraction with the coefficient α ∈ (0,1) and T:C → C a strict pseudocontraction with a fixed point. Define a mapping S: C → C by Sx = kx+ (1−k)T x for each x ∈ C. Assume that F(T) 6= ∅. Let {xn} be a sequence generated by the following algorithm: x1∈C and
(yn =δnSxn+ (1−δn)xn,
xn+1=αnf(xn) +βnxn+γnyn, ∀ n≥1,
where{αn},{βn},{γn}and {δn} are sequences in(0,1). Assume that the above control sequences satisfy the following restrictions:
(a) αn+βn+γn = 1, for eachn≥1;
(b) 0<lim infn→∞βn ≤lim supn→∞βn <1;
(c) limn→∞αn= 0,P∞
n=1αn =∞;
(d) limn→∞δn=δ∈(0,1).
Then the sequence {xn} converges strongly to x¯ ∈ F(T), where x¯ = PF(T)f(¯x), which solves the following variational inequality
hf(¯x)−x,¯ x¯−xi ≤0, ∀ x∈F(T).
Acknowledgement. The authors are grateful to the referees for useful suggestions that improved the contents of the paper.
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1Department of Mathematics and the RINS Gyeongsang National University
Jinju 660-701, Korea
E-mail:[email protected](S.M. Kang)
2 Department of Mathematics Tianjin Polytechnic University Tianjin 300160, China
3 Department of Mathematics Shijiazhuang University Shijiazhuang 050035, China
