R E S E A R C H
Open Access
Viscosity iteration algorithm for a
-strictly
pseudononspreading mapping in a Hilbert
space
Bin-Chao Deng, Tong Chen and Zhi-Fang Li
**Correspondence: [email protected] School of Management, Tianjin University, Tianjin, 300072, China
Abstract
In this paper, we discuss the strong convergence of the viscosity approximation method in Hilbert spaces relatively to the computation of fixed points of an operator in
-strictly pseudononspreading. Under suitable conditions, some strong
convergence theorems are proved. Our work improves previous results for nonspreading mappings.
Keywords: nonspreading mapping;
-strictly pseudononspreading; demicontractive; fixed point; quasi-nonexpansive
1 Introduction
Throughout this paper, we always assume thatH is a real Hilbert space endowed with an inner product and its induced norm denoted by·,·and| · |, respectively. LetCbe a nonempty, closed and convex subset ofHand letA:C→Hbe a nonlinear mapping.
Definition . Ais said to be
(i) monotone if
Ax–Ay,x–y ≥, ∀x,y∈C;
(ii) strongly monotone if there exists a constantα> such that
Ax–Ay,x–y ≥αx–y, ∀x,y∈C.
For such a case,Ais said to beα-strongly-monotone;
(iii) inverse-strongly monotone if there exists a constantα> such that
Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.
For such a case,Ais said to beα-inverse-strongly-monotone; (iv) k-Lipschitz continuous if there exists a constantk≥such that
Ax–Ay ≤kx–y, ∀x,y∈C.
Remark . LetF=μB–γf, whereBis aθ-Lipschitz andη-strongly monotone operator onHwithθ> andf is a Lipschitz mapping onHwith coefficientL> , <γ ≤μηL. It is a simple matter to see that the operatorFis (μη–γL)-strongly monotone overH,i.e.,
Fx–Fy,x–y ≥(μη–γL)x–y, ∀(x,y)∈H×H.
The classical variational inequality, which is denoted byVI(A,C), is to findx∈Csuch that
Ax,y–x ≥, ∀y∈C. (.)
The variational inequality has been extensively studied in literature (see [–] and the references therein).
A mappingT:C→Cis said to be nonexpansive if
Tx–Ty ≤ x–y, ∀x,y∈C.
A mappingTis said to be firmly nonexpansive if
Tx–Ty≤ x–Tx,y–Ty, ∀x,y∈C;
see, for instance, [–]. It is known that a mappingT:C→Cis firmly nonexpansive if and only if
Tx–Ty+(I–T)x– (I–T)y≤ x–y, ∀x,y∈C.
Tis said to be nonspreading in [] if
Tx–Ty≤ Tx–y+Ty–x, ∀x,y∈C. (.)
It is shown in [] that (.) is equivalent to
Tx–Ty≤ x–y+ x–Tx,y–Ty, ∀x,y∈C.
These mappings are generalization of a firmly nonexpansive mapping in a Hilbert space.
T:C→Cis said to be firmly nonexpansive if
Tx–Ty≤ x–y,Tx–Ty, ∀x,y∈C.
See [–] for more information on firmly nonexpansive mappings.
Definition . T:H→His called demicontractive onHif there exists a constantα< such that
Definition .[] T:D(T)⊆H→His-strictly pseudononspreading if there exists
∈[, ) such that
Tx–Ty≤ x–y+x–Tx– (y–Ty)+ x–Tx,y–Ty, (.)
for allx,y∈D(T).
Remark . It is easy to claim that firmly nonexpansive mapping⇒nonspreading map-ping⇒-strictly pseudononspreading mapping.
Indeed, from the definition of those mappings,∀x,y∈C, we obtain
Tx–Ty≤ x–Tx,y–Ty, Tis firmly nonexpansive mapping.
⇓
Tx–Ty≤ x–y+ x–Tx,y–Ty, Tis nonspreading mapping.
⇓
Tx–Ty≤ x–y+x–Tx– (y–Ty)+ x–Tx,y–Ty,
Tis-strictly pseudononspreading mapping.
Clearly, every nonspreading mapping is-strictly pseudononspreading. The following example shows that the class of-strictly pseudononspreading mappings is more general than the class of nonspreading mappings. Let us give an example of a-strictly pseudonon-spreading mapping satisfying the condition of Definition ..
Example . LetX=lwith the norm · defined by
x= ∞
i=
x
i, ∀x= (x,x, . . . ,xn, . . .)∈X,
C={x= (x,x, . . . ,xn, . . .)|xi∈R,i= , , . . .}, and letC be an orthogonal subspace ofX
(i.e.,∀x,y∈C, we havex,y= ). Then it is obvious thatCis a nonempty closed convex subset ofX. Now, for anyx= (x,x, . . . ,xn, . . .)∈C, define a mappingT:C→Cas follows:
Tx= ⎧ ⎨ ⎩
(x,x, . . . ,xn, . . .), ∞i=xi< ,
(–x, –x, . . . , –xn, . . .),
∞
i=xi≥.
(.)
To see thatTis-strictly pseudononspreading, we break the process of proof into three cases.∀x,y∈C,
Case :∞i=xi< and
∞
i=yi< , observe that
Tx–Ty≤ x–y+
x–Tx– (y–Ty)
+ x–Tx,y–Ty,
∈[, ), sinceTx–Ty=x–yandx–Tx– (y–Ty)= x–Tx,y–Ty= .
Case :∞i=xi≤ and
∞
i=yi≥, we obtainTx–Ty=x+ y=x+ x,y+
y, x–Tx,y–Ty= and
Hence,
x–y+
x–Tx– (y–Ty)
+ x–Tx,y–Ty
=x– x,y+ y
=x+ x,y+ y– x,y
=x+ x,y+ y x,y= =x+ y=Tx–Ty.
Case :∞i=xi≥ and
∞
i=yi≥, we haveTx–Ty= x–y,x–Tx– (y–Ty)=
x–yand x–Tx,y–Ty= x,y= . Thus
Tx–Ty = x–y
=x–y+
x–Tx– (y–Ty)
≤ x–y+
x–Tx– (y–Ty)
+ x–Tx,y–Ty.
From (), () and (), we obtain thatTis-strictly pseudononspreading,i.e.,
Tx–Ty=x–y+
x–Tx– (y–Ty)
+ x–Tx,y–Ty, ∀x,y∈R. We can easily know thatFix(T) ={(x,x, . . . ,xn, . . .),
∞
i=xi< } ∪ {}, whereFix(T) is
de-fined by the set of fixed points ofT.
T is not nonspreading, since forx={, , . . . , , . . .},y={, , . . . , , . . .}, we haveTx–
Ty= ,x–y= and x–Tx,y–Ty= , we obtain
Tx–Ty= > =x–y+ x–Tx,y–Ty.
Since our class of maps contains the class of nonspreading mappings, it also contains the class of firmly nonexpansive mappings.
Remark .[] LetT be anα-demicontractive mapping onHwithFix(T)=∅andTω= ( –ω)I+ωTforω∈(,∞):
(A) T α-demicontractive is equivalent to
x–Tωx,x–q ≥
ω
x–Tx
, ∀(x,q)∈H×F
ix(T).
(A) Fix(T) =Fix(Tω)ifω= .
Remark . Observe that ifTis-strictly pseudononspreading andFix(T)=∅, then∀x∈
D(T) and∀p∈Fix(T), we obtain
Tx–p≤ x–p+x–Tx.
Thus, every -strictly pseudononspreading mapping with a nonempty fixed point set
Remark . According to Remark .(A) and the fact that the-strictly pseudonon-spreading mapping ofTis demicontractive, letI–Tω=ω(I–T). Then we obtain
x–Tωx,x–q ≥
ω( –)
x–Tx
, ∀(x,q)∈H×F
ix(T). (.)
In , Osilike and Isiogugu [] introduced the following propositions and proved a strong convergence theorem somewhat related to a Halpern-type iteration algorithm for a-strictly pseudononspreading mapping in Hilbert spaces.
Proposition .[] Let C be a nonempty closed convex subset of a real Hilbert space H and let T:C→C be a-strictly pseudononspreading mapping.If Fix(T)=∅,then it is
closed and convex.
Proposition .[] Let C be a nonempty closed convex subset of a real Hilbert space H and let T:C→C be a-strictly pseudononspreading mapping.Then(I–T)is demiclosed at.
Theorem .[] Let C be a nonempty closed convex subset of a real Hilbert space H and let T:C→C be a-strictly pseudononspreading mapping with a nonempty fixed point set Fix(T).Letα∈[, )and let{αn}∞n=be a real sequence in[, )such thatlimn→∞αn= and
∞
n=αn=∞.Let u∈C,{xn}and{zn}be sequences in C generated from an arbitrary x∈C
by
⎧ ⎨ ⎩
xn+=αnu+ (I–αn)zn, n> ,
zn=n
n–
k=Tαkxn, n≥.
(.)
Then{xn}and{zn}converge strongly to PFix(T)u,where PFix(T):H→Fix(T)is a metric
pro-jection of H onto Fix(T).
In , Tian [] introduced the following theorem for finding an element of a set of solutions to the fixed point of a nonexpansive mapping in a Hilbert space.
Theorem .[] Let f be a contraction on a real Hilbert space H and T be a nonex-pansive mapping on H.Starting with an arbitrary initial x∈H,define a sequence{xn}
generated by
xn+=αnγf(xn) + (I–μαnB)Txn, n≥, (.)
where B is aθ-Lipschitz andη-strongly monotone operator on H withθ> ,η> and <
μ< η/θ.Assume also that a sequence{α
n}is a sequence in(, )satisfying the following
conditions:
(c) limn→∞αn= and
∞
n=αn=∞,
(c) ∞n=|αn+–αn|<∞orlimn→∞αn+/αn= .
Then the sequence{xn}generated by (.)converges strongly to the unique solution x*∈
Fix(T)of the variational inequality
In this paper, we combine Theorem . and Theorem . and introduce the following general iterative algorithm for a-strictly pseudononspreading mappingT.
Algorithm . Letx∈Hbe arbitrary ⎧
⎨ ⎩
xn+=αnγf(xn) + (I–μαnB)zn, n> ,
zn=n
n
k=Tαkxn, n≥,
whereB:H→Hisη-strongly monotone and boundedly Lipschitzian,f is anL-Lipschitz mapping onHwith coefficientL> andTk
α= ( –α)I+αTk,α∈(k,).
Under suitable conditions, some strong convergence theorems are proved in the follow-ing chapter.
2 Preliminaries
Throughout this paper, we writexnx to indicate that the sequence{xn} converges
weakly tox.xn→ximplies that{xn}converges strongly tox. The following lemmas are
useful for main results.
Definition . A mappingT is said to be demiclosed if for any sequence {xn} which
weakly converges toy, and if the sequence{Txn}strongly converges toz, thenT(y) =z.
Lemma .[] Assume{αn}is a sequence of nonnegative real numbers such that
αn+≤( –γn)αn+δn, n≥,
where{γn}is a sequence in(, )and{δn}is a sequence inRsuch that
(i) ∞n=γn=∞,
(ii) lim supn→∞δn γn = or
∞
n=|δn|<∞.
Thenlimn→∞αn= .
Lemma .[] Let{Tn}be a sequence of real numbers that does not decrease at infinity,
in the sense that there exists a subsequence{Tnj}j≥of{Tn}which satisfiesTnj<Tnj+for all
j≥.Also,consider the sequence of integers{δ(n)}n≥ndefined by
δ(n) =max{k≤n|Tk<Tk+}. (.)
Then{δ(n)}n≥nis a nondecreasing sequence verifyinglimn→∞δ(n) =∞,∀n≥n.It holds
that Tδ(n)<Tδ(n)+,and we have
Tn<Tδ(n)+.
Lemma . Let K be a closed convex subset of a real Hilbert space H given x∈H and y∈K.Then y=PKx if and only if the following inequality holds:
3 Main results
LetCbe a nonempty closed convex subset of a real Hilbert spaceH and letTk:C→C
be ak-strictly pseudononspreading mapping with a common nonempty fixed point set
n
kFix(Tk). Letf be anL-Lipschitz mapping onHwith coefficientL> . Assume the set
n
kFix(Tk) is nonempty. SincenkFix(Tk) is closed and convex, the nearest point
projec-tion from ContonkFix(Tk) is well defined. RecallB:H→Hisη-strongly monotone
andθ-Lipschitzian onHwithθ> ,η> . Let <μ< η/θ, <γ <μ(η–μθ)/L=τ/L, consider the following sequence{xn}defined by
⎧ ⎨ ⎩
xn+=αnγf(xn) + (I–μαnB)zn, n> ,
zn=n
n
k=Tαkxn, n≥,
(.)
where Tαk= ( –α)I+αTk,α∈(k,),k={, , . . . ,n}, and{αn}is a sequence in (, )
satisfying the following conditions:
(c) limn→∞αn= ,
(c) ∞n=αn=∞orlimn→∞αn+/αn= .
Remark .[] LetHbe a real Hilbert space. LetBbe aθ-Lipschitzian andη-strongly monotone operator onHwithθ> ,η> . Let <μ< η/θ, letS= (I–tμB) andμ(η–
μθ
) =τ, then fort∈(,min{,
τ}),Sis a contraction with a constant –tτ.
Before stating our main result, we introduce some lemmas for algorithm (.) as follows.
Lemma . The sequence{xn}is generated by(.)with Tkbeing a-strictly
pseudonon-spreading mapping on H and{αn} ⊂(, ).Then{xn}is bounded.
Proof LetTk
αx= ( –α)x+αTkxand <k<α<. Then∀x,y∈C, we have
Tαkx–Tαky
=αx–y+ ( –α)Tkx–Tky–α( –α)x–Tkx–y–Tky
≤αx–y+ ( –α)x–y+kx–Tkx–
y–Tky
+ x–Tkx,y–Tky–α( –α)x–Tkx–y–Tky
=x–y+ ( –α)x–Tkx,y–Tky
– ( –α)(α–k)x–Tkx–
y–Tky
≤ x–y+ ( –α)x–Tkx,y–Tky
=x–y+( –α)
α
x–Tαkx,y–Tαky
. (.)
Fromp∈nkFix(Tk) and (.), we also have
Tαkxn–p≤ xn–p. (.)
According to (.), (.) and Remark ., we obtain
zn–p=
nn
k=
Tαkxn–p
≤
n
n
k=
Tαkxn–p≤
n
n
k=
Thus,
xn+–p=αnγ
f(xn) –f(p)
+αn
γf(p) –μBp+ (I–μαnB)(zn–p)
≤αnγf(xn) –f(p)+αnγf(p) –μBp+ ( –αnτ)zn–p
≤αnγf(xn) –f(p)+αnγf(p) –μBp+ ( –αnτ)xn–p, (.)
which combined withf(xn) –f(p) ≤Lxn–pamounts to
xn+–p ≤
–αn(τ–γL)
xn–p+αnγf(p) –μBp. (.)
PuttingM=max{x–p,γf(p) –μBp}, we clearly obtainxn–p ≤M. Hence{xn}∞n= and{zn}∞n=are bounded. From (.), we have that{Tαkxn}∞n=is also bounded.
Now we are in a position to claim the main result.
Theorem . Assume C is a nonempty closed convex subset of a real Hilbert space H and let Tk:C→C be a
k-strictly pseudononspreading mapping with a common nonempty
fixed point setnkFix(Tk). Let f be an L-Lipschitz mapping on H with coefficient L>
and B:H→H beη-strongly monotone andθ-Lipschitzian on H withθ > ,η> .Let
<μ< η/θ, <γ <μ(η–μθ)/L=τ/L.Consider the sequences{xn}∞n=and{zn}∞n= to
be sequences in C generated from an arbitrary x∈C by(.),where Tαk= ( –α)I+αTk,
α∈(k,), k={, , . . . ,n},{αn}∞n=∈[, )andlimn→∞αn= .Then{xn}∞n= and{zn}∞n=
converge strongly to the unique element x*innkFix(Tk)verifying
Pn
kFix(Tk)(I–μB+γf)x
*=x*, (.)
which equivalently solves the following variational inequality problem:
x*∈
n
k
Fix
Tk, (γf–μB)x*,v–x*≤, ∀v∈
n
k
Fix
Tk. (.)
Proof According to Lemma ., it is simple to know that{xn}∞n=,{zn}∞n=and{Tαkxn}∞n=are bounded. Thus, for∀y∈Cand∀k= , , , . . . ,n– and according to (.) and (.), we have
Tαk+xn–Tαy
=Tα
Tαkxn
–Tαy
≤Tαkxn–y+
( –α)
Tαkxn–Tαk+xn,y–Tαy
=Tαkxn–Tαy
+Tαy–y+
Tαkxn–Tαy,Tαy–y
+
( –α)
Tαkxn–Tαk+xn,y–Tαy
. (.)
Summing (.) fromk= tonand dividing byn, we obtain
nT
k+
α xn–Tαy≤
nxn–Tαy
+T
αy–y+ zn–Tαy,Tαy–y
+
n( –α)
xn–Tαnxn,Tαy–y
Since{zn}∞n=is bounded, then there exists a subsequence{znj}∞j=of{zn}∞n=which converges weakly toω∈C. Replacingnbynjin (.), we obtain
nj
Tαkxnj–Tαy
≤
nj
xnj–Tαy +T
αy–y+ znj–Tαy,Tαy–y
+
nj( –α)
xnj–T
nj
αxnj,Tαy–y
. (.)
Since{xn}∞n=and{Tαnxn}∞n=are bounded, lettingj→ ∞in (.) yields
≤ Tαy–y+ ω–Tαy,Tαy–y. (.)
Lety=ωin (.). We obtain thatω∈Fix(Tα) =Fix(T).
Observe that since{xn}∞n=and{zn}∞n=are bounded, andlimn→∞αn= , then
xn+–zn=αnγf(xn) –μBzn
≤αnγf(xn) –f(p)+αnγf(p) –μBp+αnμB(zn–p)
≤αnγLxn–p+αnγf(p) –μBp+αnτzn–p,
then
lim
n→∞xn+–zn= . (.)
We next show that
lim sup
n→∞
(γf –μB)z,xn+–z
≤. (.)
Indeed, take{xnj+} ∞
n=of{xn+}∞n=such that
lim sup
n→∞
(γf –μB)x*,xn+–x*
=lim
j→∞
(γf–μB)x*,xnj+–x *,
wherex*is obtained in (.). We may assume thatx
nj+zasj→ ∞. From (.), we haveznjzasj→ ∞, then to arbitrary bounded linear functionalgonH, we have
g(znj) –g(z) ≤ g(znj) –g(xnj+)+g(xnj+) –g(z)
≤ gznj–xnj++g(xnj+) –g(z)
→, asj→.
Thus, we obtainznj→zasj→, andz∈Fix(T). Hence, we have
lim
j→∞
(γf –μB)x*,xnj+–x
Moreover, from (.), (.) and (.), we have
lim sup
n→∞
(γf –μB)z,zn–z
=lim sup
n→∞
(γf–μB)z,xn+–z
+lim sup
n→∞
(γf–μB)z,zn–xn+
≤lim sup
n→∞
(γf –μB)z,xn+–z
≤. (.)
As required, finally we show thatxn→x*andzn→x*.
According to (.), (.) and (.), we obtain
xn+–x* =αn
γf(xn) –μBx*
+ (I–μαnB)zn– (I–μαnB)x*
=αnγf(xn) –μBx*
+(I–μαnB)zn– (I–μαnB)x*
+ αn
(I–μαnB)zn– (I–μαnB)x*,γf(xn) –μBx*
≤αnγf(xn) –μBx*+ ( –αnτ)zn–x*
+ αn
zn–x*,γf(xn) –μBx*
–μαn
Bzn–Bx*,γf(xn) –μBx*
≤( –αnτ)+ αnγLxn–x*
+αn
zn–x*,γf(xn) –μBx*
+αnγf(xn) –μBx*
+ μαnBzn–Bx*γf(xn) –μBx*
≤ – αn(τ–γL)xn–x*+αn
xn–x*,γf(xn) –μBx*
+αnγf(xn) –μBx*
+ μαnBzn–Bx*γf(xn) –μBx*
+αnτxn–x*
= ( –α¯n)xn–x*
+α¯nβ¯n,
whereα¯n= αn(τ–γL),
¯
βn=
(τ–γL)
xn–x*,γf(xn) –μBx*
+αnγf(xn) –μBx*
+ μαnBzn–Bx*γf(xn) –μBx*+αnτxn–x*
.
It is easily seen that limn→∞α¯n,
¯
αn=∞andlim supn→∞β¯n≤. By Lemma ., we
conclude thatxn→x*asn→ ∞, andznalso converges strongly to the unique elementx*
inFix(T). In addition, the variational inequality (.) can be written as
(I–μB+γf)x*–x*,z–x*≥, z∈
n
k
Fix
Tk.
So, by Lemma ., it is equivalent to the fixed point equation
Pn
kFix(Tk)(I–μB+γf)x
*=x*.
Corollary . Assume C is a nonempty closed convex subset of a real Hilbert space H and let Tk:C→C be a nonspreading mapping with a common nonempty fixed point set
n
kFix(Tk).Let f be an L-Lipschitz mapping on H with coefficient L> and B:H→H
beη-strongly monotone andθ-Lipschitzian on H withθ> ,η> .Let <μ< η/θ, <
γ <μ(η– μθ)/L=τ/L,consider the sequences{xn}∞n=and{zn}∞n= to be sequences in C
generated from an arbitrary x∈C by ⎧
⎨ ⎩
xn+=αnγf(xn) + (I–μαnB)zn, n> ,
zn=n
n
k=Tαkxn, n≥,
where Tk
α= ( –α)I+αTk,α∈(, ),{αn}
∞
n=∈[, )andlimn→∞αn= .Then{xn}∞n=and
{zn}∞n=converge strongly to the unique element x*in n
kFix(Tk)verifying
Pn
kFix(Tk)(I–μB+γf)x *=x*,
which equivalently solves the following variational inequality problem:
x*∈
n
k
Fix
Tk, (γf–μB)x*,v–x*≤, ∀v∈
n
k
Fix
Tk.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039) and China Postdoctoral Science Foundation (Grant No. 20100470783).
Received: 12 September 2012 Accepted: 25 January 2013 Published: 28 February 2013
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