R E S E A R C H
Open Access
On solutions of inclusion problems and fixed
point problems
Yuan Hecai
**Correspondence:
[email protected] School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China
Abstract
An inclusion problem and a fixed point problem are investigated based on a hybrid projection method. The strong convergence of the hybrid projection method is obtained in the framework of Hilbert spaces. Variational inequalities and fixed point problems of quasi-nonexpansive mappings are also considered as applications of the main results.
MSC: 47H05; 47H09; 47J25
Keywords: nonexpansive mapping; inverse-strongly monotone mapping; maximal monotone operator; fixed point
1 Introduction and preliminaries
Splitting methods have recently received much attention due to the fact that many nonlin-ear problems arising in applied areas such as image recovery, signal processing, and ma-chine learning are mathematically modeled as a nonlinear operator equation and this op-erator is decomposed as the sum of two nonlinear opop-erators. Splitting methods for linear equations were introduced by Peaceman and Rachford [] and Douglas and Rachford []. Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg [] and Lions and Mercier []. The central problem is to iteratively find a zero of the sum of two monotone operatorsAandBin a Hilbert spaceH. In this paper, we consider the prob-lem of finding a solution to the following probprob-lem: find anxin the fixed point set of the mappingSsuch that
x∈(A+B)–(),
whereAandBare two monotone operators. The problem has been addressed by many au-thors in view of the applications in image recovery and signal processing; see, for example, [–] and the references therein.
Throughout this paper, we always assume thatHis a real Hilbert space with the inner product·,·and norm · , respectively. LetCbe a nonempty closed convex subset of
HandPCbe the metric projection fromHontoC. LetS:C→Cbe a mapping. In this paper, we useF(S) to denote the fixed point set ofS; that is,F(S) :={x∈C:x=Sx}.
Recall thatSis said to be nonexpansive iff Sx–Sy ≤ x–y, ∀x,y∈C.
If Cis a bounded, closed, and convex subset ofH, thenF(S) is not empty, closed, and convex; see [].
Sis said to be quasi-nonexpansive iffF(S)=∅and Sx–y ≤ x–y, ∀x∈C,y∈F(S).
It is easy to see that nonexpansive mappings are Lipschitz continuous; however, the nonexpansive mapping is discontinuous on its domain generally. Indeed, the quasi-nonexpansive mapping is only continuous in its fixed point set.
LetA:C→Hbe a mapping. Recall thatAis said to be monotone iff Ax–Ay,x–y ≥, ∀x,y∈C.
Ais said to be strongly monotone iff there exists a constantα> such that Ax–Ay,x–y ≥αx–y, ∀x,y∈C.
For such a case,Ais also said to beα-strongly monotone.Ais said to be inverse-strongly monotone iff there exists a constantα> such that
Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.
For such a case,Ais also said to beα-inverse-strongly monotone. Notice that
αAx–Ay≤ Ax–Ay,x–y ≤ Ax–Ayx–y
clearly shows thatAis
α-Lipschitz continuous.
Recall that the classical variational inequality is to find anx∈Csuch that
Ax,y–x ≥, ∀y∈C. (.)
In this paper, we useVI(C,A) to denote the solution set of (.). It is known thatx*∈Cis a solution to (.) iffx*is a fixed point of the mappingP
C(I–λA), whereλ> is a constant,
Istands for the identity mapping, andPCstands for the metric projection fromHontoC. A multivalued operatorT :H→H with the domainD(T) ={x∈H:Tx=∅}and the rangeR(T) ={Tx:x∈D(T)}is said to be monotone if forx∈D(T),x∈D(T),y∈Tx,
andy∈Tx, we havex–x,y–y ≥. A monotone operatorTis said to be maximal
if its graph G(T) ={(x,y) :y∈Tx}is not properly contained in the graph of any other monotone operator. LetIdenote the identity operator onHandT:H→Hbe a maximal monotone operator. Then we can define, for eachλ> , a nonexpansive single-valued mappingJλ:H→HbyJλ= (I+λT)–. It is called theresolventofT. We know thatT– =
F(Jλ) for allλ> andJλis firmly nonexpansive.
In [], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator by considering the following iterative algorithm:
x∈H, xn+=αnxn+ ( –αn)Jλnxn, n= , , , . . . , (.)
where{αn}is a sequence in (, ),{λn}is a positive sequence,T :H→H is a maximal monotone, andJλn= (I+λnT)–. They showed that the sequence{xn}generated in (.) converges weakly to somez∈T–() provided that the control sequence satisfies some restrictions. Further, using this result, they also investigated the case thatT=∂f, wheref :
H→(–∞,∞] is a proper lower semicontinuous convex function. Convergence theorems
are established in the framework of real Hilbert spaces.
In [], Takahashi an Toyoda investigated the problem of finding a common solution of the variational inequality problem (.) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
x∈C, xn+=αnxn+ ( –αn)SPC(xn–λnAxn), ∀n≥, (.) where{αn}is a sequence in (, ),{λn}is a positive sequence,S:C→Cis a nonexpansive mapping, andA:C→His an inverse-strongly monotone mapping. They showed that the sequence{xn}generated in (.) converges weakly to somez∈VI(C,A)∩F(S) provided that the control sequence satisfies some restrictions.
The above convergence theorems are weak. In this paper, motivated by the above results, we consider the problem of finding a common solution to the zero point problems and fixed point problems based on hybrid iterative methods with errors. Strong convergence theorems are established in the framework of Hilbert spaces.
To obtain our main results in this paper, we need the following lemmas and definitions. LetCbe a nonempty, closed, and convex subset ofH. LetS:C→Cbe a mapping. Then the mappingI–Sis demiclosed at zero, that is, if{xn}is a sequence inCsuch thatxnx¯ andxn–Sxn→, thenx¯∈F(S).
Lemma[] Let C be a nonempty,closed,and convex subset of H,A:C→H be a mapping,
and B:H⇒H be a maximal monotone operator.Then F(Jr(I–λA)) = (A+B)–().
2 Main results
Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H,A:
C→H be anα-inverse-strongly monotone mapping,S:C→C be a quasi-nonexpansive mapping such that I–S is demiclosed at zero,and B be a maximal monotone operator on H such that the domain of B is included in C.Assume thatF=F(S)∩(A+B)–() =∅.Let
{λn}be a positive real number sequence.Let{αn}be a real number sequence in[, ].Let {xn}be a sequence in C generated in the following iterative process:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C,
C=C,
yn=αnxn+ ( –αn)SJλn(xn–λnAxn),
Cn+={z∈Cn:yn–z ≤ xn–z},
where Jλn= (I+λnB)–.Suppose that the sequences{αn}and{λn}satisfy the following
re-strictions:
(a) ≤αn≤a< ;
(b) <b≤λn≤c< α.
Then the sequence{xn}converges strongly to PFx.
Proof First, we show thatCnis closed and convex. Notice thatC=Cis closed and convex.
Suppose thatCiis closed and convex for somei≥. We show thatCi+is closed and convex
for the samei. Indeed, for anyv∈Ci, we see that yi–z ≤ xi–z
is equivalent to
yi–xi– z,yi–xi ≥.
ThusCi+is closed and convex. This shows thatCnis closed and convex. Next, we prove thatI–λnAis a nonexpansive mapping. Indeed, we have
(I–λnA)x– (I–λnA)y
=(x–y) –λn(Ax–Ay)
=x–y– λnx–y,Ax–Ay+λnAx–Ay ≤ x–y–λn(α–λn)Ax–Ay.
In view of the restriction (b), we obtain thatI–λnAis nonexpansive. Next, we show that F⊂Cnfor eachn≥. From the assumption, we see thatF⊂C=C. Assume thatF⊂Ci for somei≥. For anyz∈F⊂Ci, we find from Lemma that
z=Sz=Jλi(z–λiAz).
Putzn=Jλn(xn–λnAxn). SinceJλnandI–λnAare nonexpansive, we have
zn–p ≤(xn–λnAxn) – (p–λnAp)
≤ xn–p. (.)
It follows from (.) that
yi–z=αixi+ ( –αi)Szi–z ≤αixi–z+ ( –αi)zi–z ≤ xi–z.
This shows thatz∈Ci+. This proves thatF⊂Cn. Notice thatxn=PCnx. For everyz∈
F⊂Cn, we have
In particular, we have
x–xn ≤ x–PFx.
This implies that{xn}is bounded. Sincexn=PCnx andxn+ =PCn+x∈Cn+⊂Cn, we
arrive at
≤ x–xn,xn–xn+
≤–x–xn+x–xnx–xn+.
It follows that
xn–x ≤ xn+–x.
This implies thatlimn→∞xn–xexists. On the other hand, we have
xn–xn+
=xn–x+ xn–x,x–xn++x–xn+
=xn–x– xn–x+ xn–x,xn–xn++x–xn+
≤ x–xn+–xn–x.
It follows that
lim
n→∞xn–xn+= . (.) Notice thatxn+=PCn+x∈Cn+. It follows that
yn–xn+ ≤ xn–xn+.
This in turn implies that
yn–xn ≤ yn–xn++xn–xn+ ≤xn–xn+.
In view of (.), we obtain that
lim
n→∞xn–yn= . (.) On the other hand, we have
xn–yn= ( –αn)xn–Szn.
It follows from (.) that
lim
For anyp∈F, we see that
zn–p=Jλn(xn–λnAxn) –Jλn(p–λnAp)
≤ xn–p– xn–p,Axn–Ap+λnAxn–Ap
≤ xn–p–λn(α–λn)Axn–Ap. (.)
Notice that
yn–p≤αnxn–p+ ( –αn)Szn–p
≤αnxn–p+ ( –αn)zn–p. (.)
Substituting (.) into (.), we see that
yn–p≤ xn–p– ( –αn)λn(α–λn)Axn–Ap.
It follows that
( –αn)λn(α–λn)Axn–Ap≤ xn–p–yn–p ≤xn–p+yn–p
xn–yn.
This implies from (.) that
lim
n→∞Axn–Ap= . (.) On the other hand, we have
zn–p=Jλn(xn–λnAxn) –Jλn(p–λnAp)
≤ (xn–λnAxn) – (p–λnAp),zn–p
=
(xn–λnAxn) – (p–λnAp)
+zn–p –(xn–λnAxn) – (p–λnAp) – (zn–p)
≤
xn–p+zn–p–xn–zn–λn(Axn–Ap)
≤
xn–p+zn–p–xn–zn–λnAxn–Ap + λnxn–znAxn–Ap
≤
xn–p+zn–p–xn–zn+ λnxn–znAxn–Ap
.
It follows that
Substituting (.) into (.), we see that
yn–p≤ xn–p– ( –αn)xn–zn+ ( –αn)λnxn–znAxn–Ap.
It follows that
( –αn)xn–zn
≤ xn–p–yn–p+ ( –αn)λnxn–znAxn–Ap ≤xn–p+yn–p
xn–yn+ ( –αn)λnxn–znAxn–Ap.
In view of the restriction (a), we obtain from (.) that
lim
n→∞xn–zn= . (.) Since{xn}is bounded, we may assume that there is a subsequence{xni}of{xn}converging
weakly to some pointx*. It follows from (.) thatz
niconverges weakly tox*. Notice that
Szn–zn ≤ Szn–xn+xn–zn.
It follows from (.) and (.) that
lim
n→∞Szn–zn= .
In view of the assumption thatSis demiclosed at zero, we see thatx*∈F(S).
Next, we show thatx*∈(A+B)–(). Notice thatz
n=Jλn(xn–λnAxn). This implies that
xn–λnAxn∈(I+λnB)zn. That is,
xn–zn
λn
–Axn∈Bzn.
SinceBis monotone, we get for any (u,v)∈B, that
zn–u,
xn–zn
λn
–Axn–v
≥. (.)
Replacingnbyniand lettingi→ ∞, we obtain from (.) that ω–u, –Aω–v ≤.
This means –Aω∈Bω, that is, ∈(A+B)(ω). Hence, we getω∈(A+B)–(). This com-pletes the proof thatx*∈F.
Notice thatPFx⊂Cn+andxn+=PCn+x, we have
On the other hand, we have
x–PFx ≤x–x*
≤lim inf
i→∞ x–xni
≤lim sup
i→∞ x–xni ≤ x–PFx.
We, therefore, obtain that
x–x*= lim
i→∞x–xni=x–PFx.
This impliesxni→x*=PFx. Since{xni}is an arbitrary subsequence of{xn}, we obtain
thatxn→PFxasn→ ∞. This completes the proof.
From Theorem ., we have the following results immediately.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H,A:C→ H be anα-inverse-strongly monotone mapping,and B be a maximal monotone operator on H such that the domain of B is included in C.Assume that(A+B)–()=∅.Let{λ
n}be
a positive real number sequence.Let{αn}be a real number sequence in[, ].Let{xn}be a
sequence in C generated in the following iterative process:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C,
C=C,
yn=αnxn+ ( –αn)Jλn(xn–λnAxn),
Cn+={z∈Cn:yn–z ≤ xn–z},
xn+=PCn+x, n≥,
where Jλn= (I+λnB)–.Suppose that the sequences{αn}and{λn}satisfy the following
re-strictions:
(a) ≤αn≤a< ;
(b) <b≤λn≤c< α.
Then the sequence{xn}converges strongly to P(A+B)–()x.
Letf :H→(–∞,∞] be a proper lower semicontinuous convex function. Define the
subdifferential
∂f(x) =z∈H:f(x) +y–x,z ≤f(y),∀y∈H
for allx∈H. Then∂f is a maximal monotone operator ofHinto itself; see [] for more details. LetCbe a nonempty closed convex subset ofHandiCbe the indicator function ofC, that is,
iCx=
⎧ ⎨ ⎩
Furthermore, we define the normal coneNC(v) ofCatvas follows:
NCv=
z∈H:z,y–v ≤,∀y∈H
for anyv∈C. TheniC:H→(–∞,∞] is a proper lower semicontinuous convex function onH and∂iCis a maximal monotone operator. LetJλx= (I+λ∂iC)–xfor anyλ> and
x∈H. From∂iCx=NCxandx∈C, we get
v=Jλx ⇔ x∈v+λNCv
⇔ x–v,y–v ≤, ∀y∈C, ⇔ v=PCx,
where PC is the metric projection from H into C. Similarly, we can get thatx∈(A+
∂iC)–()⇔x∈VI(A,C). PuttingB=∂iCin Theorem ., we can seeJλn=PC. The
fol-lowing is not hard to derive.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H,A:C→ H be anα-inverse-strongly monotone mapping,and S:C→C be a quasi-nonexpansive mapping such that I–S is demiclosed at zero.Assume thatF=F(S)∩VI(C,A) =∅.Let
{λn}be a positive real number sequence.Let{αn}be a real number sequence in[, ].Let {xn}be a sequence in C generated in the following iterative process:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C,
C=C,
yn=αnxn+ ( –αn)SPC(xn–λnAxn),
Cn+={z∈Cn:yn–z ≤ xn–z},
xn+=PCn+x, n≥.
Suppose that the sequences{αn}and{λn}satisfy the following restrictions:
(a) ≤αn≤a< ;
(b) <b≤λn≤c< α.
Then the sequence{xn}converges strongly to PFx.
In view of Corollary ., we have the following corollary on variational inequalities.
Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H and A:C→H be anα-inverse-strongly monotone mapping.Assume thatF=VI(C,A)=∅.
Let{λn}be a positive real number sequence.Let{αn}be a real number sequence in[, ].
Let{xn}be a sequence in C generated in the following iterative process:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x∈C,
C=C,
yn=αnxn+ ( –αn)PC(xn–λnAxn),
Cn+={z∈Cn:yn–z ≤ xn–z},
Suppose that the sequences{αn}and{λn}satisfy the following restrictions:
(a) ≤αn≤a< ;
(b) <b≤λn≤c< α.
Then the sequence{xn}converges strongly to PVI(C,A)x.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author is grateful to the reviewers’ suggestions which improved the contents of the article.
Received: 20 November 2012 Accepted: 27 December 2012 Published: 14 January 2013 References
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doi:10.1186/1687-1812-2013-11
