R E S E A R C H
Open Access
Viscosity approximation process for a
sequence of quasinonexpansive mappings
Koji Aoyama
1and Fumiaki Kohsaka
2* *Correspondence:2Department of Computer Science
and Intelligent Systems, Oita University, Dannoharu, Oita-shi, Oita, 870-1192, Japan Full list of author information is available at the end of the article
Abstract
We study the viscosity approximation method due to Moudafi for a fixed point problem of quasinonexpansive mappings in a Hilbert space. First, we establish a strong convergence theorem for a sequence of quasinonexpansive mappings. Then we employ our result to approximate a solution of the variational inequality problem over the common fixed point set of the sequence of quasinonexpansive mappings.
MSC: 47H09; 47H10; 41A65
Keywords: viscosity approximation method; quasinonexpansive mapping; fixed point; hybrid steepest descent method
1 Introduction
LetCbe a nonempty closed convex subset of a Hilbert space. This paper is devoted to the study of strong convergence of a sequence{xn}inCdefined by an arbitrary pointx∈C
and
xn+=αnfn(xn) + ( –αn)Tnxn (.)
forn∈N, whereαnis a real number in [, ],fnis a contraction-like mapping onC, and Tnis a quasinonexpansive mapping onC. This iterative method (.) is called the viscosity approximation method []. In Section , we establish that, under some appropriate as-sumptions, the sequence{xn}converges strongly to a certain common fixed point of{Tn} by using the technique developed in []. Then, in Section , we apply our result to ap-proximate a solution of a variational inequality problem over the common fixed point set of{Tn}.
The viscosity approximation method (.) is based on the study of Moudafi [], who considered a fixed point problem of a single nonexpansive mapping and proved strong convergence of sequences generated by the method. After that, Xu [] extended Moudafi’s results [] in the framework of Hilbert spaces and Banach spaces; Suzuki [] gave simple proofs of Xu’s results []; Aoyama and Kimura [] investigated a relationship between viscosity approximation methods and Halpern [] type iterative methods for a sequence of nonexpansive mappings.
On the other hand, Maingé [] adopted the viscosity approximation method for a fixed point problem of a single quasinonexpansive mapping; Wongchan and Saejung [] ex-tended Maingé’s result []. Our main result (Theorem .) is a generalization of Wongchan and Saejung’s result [] and is closely related to the study in []. Moreover, it is also
cable to an approximation method, which is called the hybrid steepest descent method [, ], for a variational inequality problem over the common fixed point set of a sequence of quasinonexpansive mappings.
2 Preliminaries
Throughout the present paper,Hdenotes a real Hilbert space,·,·the inner product of H, · the norm ofH,Ca nonempty closed convex subset ofH,Ithe identity mapping onH,Rthe set of real numbers, andNthe set of positive integers. Strong convergence of a sequence{xn}inHtox∈His denoted byxn→xand weak convergence byxnx.
LetT:C→Hbe a mapping. The set of fixed points ofTis denoted byFix(T). A map-pingTis said to bequasinonexpansiveifFix(T)=∅andTx–p ≤ x–pfor allx∈Cand p∈Fix(T);T is said to benonexpansiveifTx–Ty ≤ x–yfor allx,y∈C;T is said to bestrongly quasinonexpansiveifTis quasinonexpansive andTxn–xn→ whenever{xn} is a bounded sequence inCandxn–p–Txn–p → for some pointp∈Fix(T);Tis demiclosed at ifTp= whenever{xn}is a sequence inCsuch thatxnpandTxn→. We know that ifT: C→His quasinonexpansive, thenFix(T) is closed and convex; see [, Theorem ].
It is known that, for eachx∈H, there exists a unique pointx∈Csuch that
x–x=min
x–y:y∈C.
Such a pointxis denoted byPC(x) andPCis called the metric projection ofHontoC. It is known that the metric projectionPCis nonexpansive; see [].
Letf :C→Cbe a mapping,F a nonempty subset ofC, andθ a real number in [, ). A mappingf is said to be aθ-contraction with respect to Fiff(x) –f(z) ≤θx–zfor allx∈Candz∈F;f is said to be aθ-contractioniff is aθ-contraction with respect toC. By definition, it is easy to check the following results.
Lemma . Let F be a nonempty subset of C and f :C→C aθ-contraction with respect to F,where≤θ< .If F is closed and convex,then PF◦f is aθ-contraction on F,where PFis the metric projection of H onto F.
Lemma . Let f :C→C be aθ-contraction,where≤θ< and T: C→C a quasi-nonexpansive mapping.Then f ◦T is aθ-contraction with respect toFix(T).
LetDbe a nonempty subset ofC. A sequence{fn}of mappings ofCintoHis said to be stable on D[] if{fn(z) :n∈N}is a singleton for everyz∈D. It is clear that if{fn}is stable onD, thenfn(z) =f(z) for alln∈Nandz∈D.
A function τ: N→Nis said to be eventually increasing[] iflimn→∞τ(n) =∞and
τ(n)≤τ(n+ ) for alln∈N. By definition, we easily obtain the following.
Lemma . Letτ :N→Nbe an eventually increasing function and{ξn}a sequence of real numbers such thatξn→ξ.Thenξτ(n)→ξ.
The following is a direct consequence of [, Lemma .].
Under the assumptions of Lemma ., we cannot choose a strictly increasing functionτ; see [, Example .].
Let{Tn}be a sequence of mappings ofCintoHsuch thatF=
∞
n=Fix(Tn) is nonempty. Then
• {Tn}is said to bestrongly quasinonexpansive typeif eachTnis quasinonexpansive and Tnxn–xn→whenever{xn}is a bounded sequence inCand
xn–p–Tnxn–p →
for some pointp∈F;
• {Tn}is said to satisfy thecondition(Z) [, –] if every weak cluster point of{xn}
belongs toFwhenever{xn}is a bounded sequence inCsuch thatTnxn–xn→.
Remark . Sinceβn–αn→ if and only ifβn–αn→ for all bounded sequences{αn} and{βn}in [,∞),{Tn}is strongly quasinonexpansive type if and only if it is a strongly relatively nonexpansive sequence in the sense of [, ]. See also [, ].
We know several examples of strongly quasinonexpansive type sequences satisfying the condition (Z); see [] and Example . in Section .
The following lemma follows from [, Lemma .] and Remark ..
Lemma . Let{Tn}be a sequence of mappings of C into H such that F=
∞
n=Fix(Tn)is nonempty,τ :N→Nan eventually increasing function,and{zn}a bounded sequence in C such thatzn–p–Tτ(n)zn–p →for some p∈F.If{Tn}is strongly quasinonexpansive type,then Tτ(n)zn–zn→.
In order to prove our main result in Section , we need the following lemmas.
Lemma .([, Lemma .]) Let{Tn}be a sequence of mappings of C into H such that
F=∞n=Fix(Tn)is nonempty,τ : N→Nan eventually increasing function,and{zn}a bounded sequence in C such that Tτ(n)zn–zn→.Suppose that{Tn}satisfies the condi-tion(Z).Then every weak cluster point of{zn}belongs to F.
Lemma .([, Lemma .]) Let{Tn}be a sequence of mappings of C into H,F a nonempty closed convex subset of H,{zn}a bounded sequence in C such that Tnzn–zn→,and u∈H. Suppose that every weak cluster point of{zn}belongs to F.Then
lim sup
n→∞ Tnzn–w,u–w ≤,
where w=PF(u).
The following lemma is well known; see [, ].
Lemma . Let{ξn}be a sequence of nonnegative real numbers,{δn}a sequence of real numbers,and{βn}a sequence in[, ].Suppose thatξn+≤( –βn)ξn+βnδnfor every n∈N, lim supn→∞δn≤,and
∞
3 Strong convergence of a viscosity approximation process
In this section, we prove the following strong convergence theorem.
Theorem . Let H be a real Hilbert space,C a nonempty closed convex subset of H,{Sn} a sequence of mappings of C into C such that F=∞n=Fix(Sn)is nonempty,{αn}a sequence in(, ]such thatαn→and
∞
n=αn=∞,and{fn}a sequence of mappings of C into C such that each fnis aθ-contraction with respect to F and{fn}is stable on F,where≤θ< . Let{xn}be a sequence defined by x∈C and
xn+=αnfn(xn) + ( –αn)Snxn (.)
for n∈N.Suppose that {Sn}is strongly quasinonexpansive type and satisfies the condi-tion(Z).Then{xn}converges strongly to w∈F,where w is the unique fixed point of a con-traction PF◦f.
Note that Lemma . implies thatPF◦fis aθ-contraction onFand hence it has a unique
fixed point onF.
First, we show some lemmas; then we prove Theorem .. In the rest of this section, we set
βn=αn
+ ( – θ)( –αn)
and
γn=αnfn(xn) –w
+ αn( –αn)Snxn–w,f(w) –w
forn∈N.
Lemma . {xn},{Snxn},and{fn(xn)}are bounded,and moreover,
xn+–w ≤αnfn(xn) –w+Snxn–w (.)
and
xn+–w≤( –βn)xn–w+γn (.)
hold for every n∈N.
Proof Sincefn is aθ-contraction with respect to F, Sn is quasinonexpansive,w∈F⊂ Fix(Sn), and{fn}is stable onF, it follows that
xn+–w
≤αnfn(xn) –w+ ( –αn)Snxn–w
≤αnfn(xn) –fn(w)+fn(w) –w+ ( –αn)Snxn–w
≤ –αn( –θ)
xn–w+αn( –θ)
f(w) –w
for everyn∈N. Thus, by induction onn, we have Snxn–w ≤ xn–w ≤max
x–w,f(w) –w/( –θ)
.
Therefore, it turns out that{xn}and{Snxn}are bounded, and moreover,{fn(xn)}is also bounded.
Equation (.) follows from (.).
Next, we show (.). By assumption, it follows that
Snxn–w,fn(xn) –w
≤ Snxn–wfn(xn) –fn(w)+ Snxn–w,fn(w) –w
≤θxn–w+ Snxn–w,f(w) –w
,
and thus
xn+–w =αnfn(xn) –w+ ( –αn)Snxn–w
+ αn( –αn)Snxn–w,fn(xn) –w
≤αnfn(xn) –w
+( –αn)+ αn( –αn)θ
xn–w
+ αn( –αn)Snxn–w,f(w) –w
= ( –βn)xn–w+γn (.)
for everyn∈N. Therefore, (.) holds.
Lemma . The following hold:
• <βn≤for everyn∈N;
• αn( –αn)/βn→/( –θ);
• α
nfn(xn) –w/βn→;
• ∞n=βn=∞.
Proof Since <αn≤ and – < – θ≤, we know that
<αn=αn
+ (–)( –αn)
≤βn≤αn
+ ( –αn)
=αn( –αn)≤.
It follows fromαn→ that αn( –αn)/βn→/( –θ). Since{fn(xn)}is bounded by Lemma . and
αn βn
= αn
+ ( – θ)( –αn) →,
it follows thatα
nfn(xn) –w/βn→.
Finally, we prove∞n=βn=∞. Suppose that – θ≥. Then it is clear thatβn≥αn for everyn∈N. Thus,∞n=βn≥∞n=αn=∞. Next, we suppose that – θ< . Then it is clear thatβn> ( –θ)αnfor everyn∈N. Thus,
∞
n=βn≥( –θ)
∞
n=αn=∞. This
completes the proof.
Proof We assume, in order to obtain a contraction, that{xn–w}is not convergent. Then Lemma . implies that there existN∈Nand an eventually increasing functionτ:N→N such that
xτ(n)–w ≤ xτ(n)+–w (.)
for everyn∈Nand
xn–w ≤ xτ(n)+–w (.)
for everyn≥N.
We show that Sτ(n)xτ(n) –xτ(n) → . SinceSτ(n) is quasinonexpansive and w∈ F⊂
Fix(Sτ(n)), it follows from (.), (.), and Lemmas . and . that
≤ xτ(n)–w–Sτ(n)xτ(n)–w
≤ xτ(n)+–w–Sτ(n)xτ(n)–w
≤ατ(n)fτ(n)(xτ(n)) –w→
asn→ ∞. Since{xτ(n)}is bounded and {Sn}is strongly quasinonexpansive type, Lem-ma . implies thatSτ(n)xτ(n)–xτ(n)→.
Since{Sn}satisfies the condition (Z), it follows from Lemma . that every weak cluster point of{xτ(n)}belongs toF. Thus Lemma . shows that
lim sup n→∞
Sτ(n)xτ(n)–w,f(w) –w
≤.
Moreover, Lemmas . and . imply thatα
τ(n)fτ(n)(xτ(n)) –w/βτ(n)→ and ατ(n)( – ατ(n))/βτ(n)→/( –θ). Therefore, we obtain
lim sup n→∞
γτ(n) βτ(n) ≤
. (.)
On the other hand, from (.) and (.), we know that
xτ(n)+–w≤( –βτ(n))xτ(n)–w+γτ(n)
≤( –βτ(n))xτ(n)+–w+γτ(n)
for everyn∈N. Thus, byβτ(n)> , this shows that
xτ(n)+–w≤ γτ(n) βτ(n)
(.)
for everyn∈N.
Finally, we obtain a contradiction thatxn–w →. Using (.), (.), and (.), we conclude that
lim sup n→∞ xn
–w≤lim sup n→∞ x
τ(n)+–w≤lim sup
n→∞
γτ(n) βτ(n) ≤
,
Proof of Theorem. We first show thatSnxn–xn→. SinceSnis quasinonexpansive, it follows from (.) that
≤ xn–w–Snxn–w ≤ xn–w–xn+–w+αnfn(xn) –w
for everyn∈N, so thatxn–w–Snxn–w → by Lemma .,αn→, and Lemma .. Since{Sn}is strongly quasinonexpansive type and{xn}is bounded, we conclude thatSnxn– xn→.
Since{Sn}satisfies the condition (Z), Lemma . implies that
lim sup
n→∞ Snxn–w,f(w) –w
≤.
This shows thatlim supn→∞γn/βn≤ by using Lemmas . and .. On the other hand, it follows from (.) that
xn+–w≤( –βn)xn–w+βn
γn
βn
for everyn∈N. Therefore, noting that∞n=βn=∞and using Lemma ., we conclude
thatxn–w→.
A direct consequence of Theorem . is the following corollary, which is a slight gener-alization of [, Theorem .].
Corollary . Let H be a real Hilbert space,C a nonempty closed convex subset of H, S:C→C a strongly quasinonexpansive mapping,{αn}a sequence in(, ]such thatαn→ and∞n=αn=∞,and f :C→C aθ-contraction with respect to F=Fix(S),where≤
θ< .Let{xn}be a sequence defined by x∈C and
xn+=αnf(xn) + ( –αn)Sxn (.)
for n∈N.Suppose that I–S is demiclosed at.Then{xn}converges strongly to w∈F,where w is the unique fixed point of a contraction PF◦f.
Proof SetSn=Sandfn=f forn∈N. Then it is clear that
∞
n=Fix(Sn) =Fix(S),{Sn}is strongly quasinonexpansive type,{Sn}satisfies the condition (Z), and{fn}is stable onC.
Thus Theorem . implies the conclusion.
4 Application to a variational inequality problem
In this section, applying Theorem ., we study an approximation method for the following variational inequality problem.
Problem . Letκandηbe positive real numbers such thatη< κ. LetFbe a nonempty closed convex subset ofHandA:H→Haκ-strongly monotone andη-Lipschitz contin-uous mapping, that is, we assume thatx–y,Ax–Ay ≥κx–yandAx–Ay ≤ηx–y
for allx,y∈H. Then findz∈Fsuch that
The solution set of Problem . is denoted byVI(F,A). Under the assumptions of Prob-lem ., it is known that the following hold; see, for example, [].
• κ≤η,≤ – κ+η< andI–Ais aθ-contraction, whereθ= – κ+η;
• Problem . has a unique solution andVI(F,A) =Fix(PF(I–A)).
Remark . The assumption thatη< κin Problem . is not restrictive. Indeed, letF be a nonempty closed convex subset ofHandA˜ aκ˜-strongly monotone andη˜-Lipschitz continuous mapping, whereκ˜> andη˜> . SetA=μA,˜ κ=μκ˜, andη=μη˜, whereμ
is a positive constant such thatμη˜< κ˜. Then it is easy to verify that Aisκ-strongly monotone andη-Lipschitz continuous,η< κ, and moreover,VI(F,A) =VI(F,A).˜
Using Theorem ., we obtain the following convergence theorem for Problem ..
Theorem . Let H,κ,η,and A be the same as in Problem..Let{Sn}be a sequence of mappings of H into H such that F=∞n=Fix(Sn)is nonempty,and{αn}the same as in Theorem..Let{xn}be a sequence defined by x∈H and
xn+=Snxn–αnASnxn (.)
for n∈N.Suppose that{Sn}is strongly quasinonexpansive type and{Sn}satisfies the con-dition(Z).Then{xn}converges strongly to the unique solution of Problem..
Proof Setfn= (I–A)Snforn∈Nandθ=
– κ+η. SinceI–Ais aθ-contraction and
Snis quasinonexpansive, Lemma . implies that eachfnis aθ-contraction with respect toF. It is obvious that{fn}is stable onF. Moreover, it follows from (.) that
xn+=αnfn(xn) + ( –αn)Snxn
forn∈N. Thus Theorem . implies that{xn} converges strongly tow= (PF ◦f)(w) =
PF(I–A)w, which is the unique solution of Problem ..
Remark . The iteration (.) is called the hybrid steepest descent method; see [, ] for more details.
We finally construct an example of{Sn}in Theorem . by using the notion of a subgra-dient projection.
Letg:H→Rbe a continuous and convex function such that C=x∈H:g(x)≤
is nonempty andh: H→H a mapping such thath(x)∈∂g(x) for allx∈H, where∂g denotes the subdifferential mapping ofgdefined by
∂g(x) =z∈H:g(x) +y–x,z ≤g(y) (∀y∈H)
the setL(x) defined by
L(x) =y∈H:g(x) + y–x,h(x)≤
for allx∈H. Note thatCis a subset ofL(x) for allx∈Hand thatL(x) is a closed half space for allx∈H\C. According to [, Section ], [, Proposition .], and [, Propo-sition ..], we know the following:
(S) Fix(Pg,h) =C;
(S) z–Pg,hx,x–Pg,hx ≤for allz∈Candx∈H;
(S) ifg(V)is bounded for each bounded subsetVofH, thenI–Pg,his demiclosed at. It is known that the metric projectionPD ofHonto a nonempty closed convex subset DofHcoincides with the subgradient projectionPg,hwith respect togandhdefined by g(x) =infy∈Dx–yfor allx∈Hand
h(x) =
⎧ ⎨ ⎩
(x∈D);
(x–PDx)/x–PDx (x∈H\D).
The subgradient projection is not necessarily nonexpansive. In fact, ifg: R→Rand h:R→Rare defined byg(x) =max{x, x– }for allx∈Randh(x) = ifx< ;h(x) = if x≥, thenPg,his given by
Pg,h(x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x (x≤);
( <x< );
/ (x≥)
and is not nonexpansive.
Using (S), (S), and (S), we show the following.
Example . Letg:H→Rbe a continuous and convex function such thatC={x∈H: g(x)≤}is nonempty andg(V) is bounded for each bounded subsetVofH,h:H→H a mapping such thath(x)∈∂g(x) for allx∈H, and{Sn}a sequence of mappings ofHinto Hdefined by
Sn=βnI+ ( –βn)Pg,h
for all n∈N, where {βn} is a sequence of real numbers such that – <infnβn and supnβn< . Then the following hold:
(i) Fix(Sn) =Cfor alln∈N;
(ii) {Sn}is strongly quasinonexpansive type;
(iii) {Sn}satisfies the condition (Z).
Proof Sinceβn= for alln∈N, the part (i) obviously follows from (S).
We first show (ii). By (i), we know that∞n=Fix(Sn) =Cis nonempty. Letn∈N,p∈C, andx∈Hbe given. Then we have
Snx–p+x–Snx–x–p= Snx–x,Snx–p
It follows from (S) that
p–Snx,x–Pg,hx ≤ Pg,hx–Snx,x–Pg,hx. (.)
On the other hand, we also know that
Pg,hx–Snx,x–Pg,hx
= –Pg,hx–x+x–Snx,x–Pg,hx
≤–
Pg,hx–x–
x–Snx
+
x–Snx
≤
x–Snx
. (.)
By (.), (.), and (.), eachSnsatisfies
Snx–p+
( +βn)x–Snx
≤ x–p (.)
for allp∈Candx∈H. Since ( +βn)/ > , we know that eachSnis quasinonexpansive. Let{xn}be a bounded sequence inHsuch thatxn–p–Snxn–p → for somep∈C. Since{Snxn}is bounded, it follows from (.) that
( +βn)xn–Snxn
≤ x
n–p–Snxn–p→
and henceSnxn–xn→ byinfn( +βn) > . Thus{Sn}is strongly quasinonexpansive type. We finally show (iii). Let{yn}be a bounded sequence inHsuch thatSnyn–yn→. By the definition ofSn, we have
Pg,hyn–yn= –βn
Snyn–yn
for alln∈N. Sinceinfn( –βn) > , we obtainPg,hyn–yn→. Consequently, by (S) and (S), we know that{Sn}satisfies the condition (Z).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba, 263-8522, Japan.2Department of
Computer Science and Intelligent Systems, Oita University, Dannoharu, Oita-shi, Oita, 870-1192, Japan.
Acknowledgements
This paper is dedicated to Professor Wataru Takahashi on the occasion of his 70th birthday.
Received: 27 September 2013 Accepted: 18 December 2013 Published:22 Jan 2014 References
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10.1186/1687-1812-2014-17
