A regularization method for treating zero points of the sum of two monotone operators

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R E S E A R C H

Open Access

A regularization method for treating zero

points of the sum of two monotone operators

Xiaolong Qin

1

_{, Sun Young Cho}

2

_{and Lin Wang}

3* *_{Correspondence:}

[email protected] 3_{College of Statistics and} Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, China

Full list of author information is available at the end of the article

Abstract

In this paper, a regularization method for treating zero points of the sum of two monotone operators is investigated. Strong convergence theorems are established in the framework of Hilbert spaces.

Keywords: maximal monotone operator; ﬁxed point; nonexpansive mapping; proximal point algorithm; zero point

1 Introduction

In the real world, many important problems have reformulations which require finding zero points of some nonlinear operator, for instance, evolution equations, complementar-ity problems, mini-max problems, variational inequalities and optimization problems; see [–] and the references therein. It is well known that minimizing a convex functionfcan be reduced to finding zero points of the subdifferential mappingA=∂f. Splitting meth-ods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decom-posed as the sum of two nonlinear operators. The central problem is to iteratively find a zero point of the sum of two monotone operators; that is, ∈(A+B)(x). Many problems can be formulated as a problem of the above form. For instance, a stationary solution to the initial value problem of the evolution equation ∈Fu+∂_∂u_t,u=u(), can be recast

as the above inclusion problem when the governing maximal monotoneFis of the form F=A+B; for more details; see [] and the references therein.

In this paper, we study a regularization method for treating zero points of the sum of an inverse-strongly monotone and a maximal monotone operator. The main contribution of the paper is establish a strong convergence theorem for viscosity zero points under mild re-strictions imposed on the control sequences. The main results include the corresponding results in Xu [] as a special case. The organization of this paper is as follows. In Section , we provide some necessary preliminaries. In Section , a regularization method is inves-tigated. A strong convergence theorem for zero points of the sum operator is established. In Section , applications of the main results are discussed.

2 Preliminaries

In what follows, we always assume thatHis a real Hilbert space with inner product·,· and norm · . LetCbe a nonempty, closed and convex subset ofH. LetS:C→Cbe a

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mapping.F(S) stands for the ﬁxed point set ofS; that is,F(S) :={x∈C:x=Sx}. Recall that Sis said to becontractiveiﬀ there exists a constantκ∈(, ) such that

Sx–Sy ≤κx–y, ∀x,y∈C.

It is well known that every contractive mapping has a unique ﬁxed point in metric spaces. Sis said to benonexpansiveiﬀ

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 [] and the references therein.

LetA:C→Hbe a mapping. Recall thatAis said to bemonotoneiﬀ

Ax–Ay,x–y ≥, ∀x,y∈C.

Recall thatAis said to beinverse-strongly monotoneiﬀ 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. It is not hard to see that every inverse-strongly monotone mapping is monotone and continuous.

Recall that a set-valued mappingB:H⇒His said to bemonotoneiff, for allx,y∈H, f ∈Bxandg∈Byimplyx–y,f–g> . In this paper, we useB–() to stand for the zero point ofB. A monotone mappingB:H⇒Hismaximaliff the graphGraph(B) ofBis not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingBis maximal if and only if, for any (x,f)∈H×H,x–y,f –g ≥, for all (y,g)∈Graph(B) impliesf ∈Bx. For a maximal monotone operatorBonH, and r> , we may define the single-valued resolventJr:H→Dom(B), whereDom(B) denote

the domain ofB. It is well known thatJris ﬁrmly nonexpansive, andB–() =F(Jr).

Recently, many authors studied zero points of monotone operators based on diﬀerent regularization methods; see [–] and the references therein. The main motivation is from Xu []. We propose a regularization method for treating zero points of the sum of two monotone operators. Strong convergence theorems are established in the framework of Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma .[] Let A:C→H be a mapping,and B:H⇒H a maximal monotone op-erator.Then F(Jr(I–rA)) = (A+B)–().

Lemma .[] Let{an}be a sequence of nonnegative numbers satisfying the condition

an+≤( –tn)an+tnbn+cn,∀n≥,where{tn}is a number sequence in(, )such that lim_n_→∞tn= and

∞

n=tn=∞,{bn}is a number sequence such thatlim supn→∞bn≤,

and{cn}is a positive number sequence such that∞n=cn<∞.Thenlimn→∞an= .

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3 Main results

Theorem . Let A:C→H be anα-inverse-strongly monotone mapping and let B be a maximal monotone operator on H.Assume thatDom(B)⊂C and(A+B)–_()_{is not empty.}

Let f :C→C be a ﬁxedκ-contraction and let Jrn= (I+rnB)–.Let{xn}be a sequence in C

in the following process:x∈C and

⎧ ⎨ ⎩

yn=αnf(xn) + ( –αn)xn,

xn+=Jrn(yn–rnAyn+en), ∀n≥,

where{αn}is a real number sequence in(, ),{en}is sequence in H and{rn}is a positive

real number sequence in(, α).If the control sequences satisfy the following restrictions:

(a) limn→∞αn= ,

∞

n=αn=∞and

∞

n=|αn–αn–|<∞; (b)  <a≤rn≤b< αand

∞

n=|rn–rn–|<∞; (c) ∞_n_=en<∞,

then{xn}converges strongly to a pointx¯∈(A+B)–(),wherex¯=Proj(A+B)–_()f(x).¯

Proof First, we show that{xn}is bounded. Notice thatI–rnAis nonexpansive. Indeed, we

have

(I–rnA)x– (I–rnA)y 

=x–y– rnx–y,Ax–Ay+rnAx–Ay

≤ x–y–rn(α–rn)Ax–Ay.

In view of the restriction (b), we ﬁnd thatI–rnAis nonexpansive. Fixingp∈(A+B)–(),

we ﬁnd that

yn–p ≤αnf(xn) –p+ ( –αn)xn–p ≤

 –αn( –κ)

xn–p+αnf(p) –p.

It follows that

xn+–p ≤(yn–rnAyn+en) – (I–rnA)p ≤(I–rnA)yn– (I–rnA)p+en ≤ –αn( –κ)

xn–p+αnf(p) –p+en

≤max xn–p,

f(p) –p  –κ

+en

≤max xn––p,

f(p) –p  –κ

+en–+en

.. .

≤max x–p,

f(p) –p  –κ

+

n

i= ei

≤max x–p,

f(p) –p  –κ

+

∞

i=

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This proves that the sequence{xn}is bounded, and so is{yn}. Notice that

yn–yn– ≤

 –αn( –κ)

xn–xn–+|αn–αn–|f(xn–) –xn–.

Puttingzn=yn–rnAyn+en, we ﬁnd that

zn–zn– ≤ yn–yn–+rn–rn–Ayn–+en+en– ≤ –αn( –κ)

xn–xn–+|αn–αn–|f(xn–) –xn–

+|rn–rn–|Ayn–+en+en–.

It follows from Lemma . that

xn+–xn=Jrnzn–Jrn–zn–

=Jrn–

rn–

rn

zn+

 –rn– rn

Jrnzn

–Jrn–zn–

≤rn–

rn

(zn–zn–) +

 –rn– rn

(Jrnzn–zn–)

≤(zn–zn–) +

 –rn– rn

(Jrnzn–zn)

≤ zn–zn–+|

rn–rn–|

a Jrnzn–zn

≤ –αn( –κ)

xn–xn–+fn,

where

fn=|αn–αn–|f(xn–) –xn–+|rn–rn–|

Ayn–+

Jrnzn–zn a

+en+en–.

It follows from the restrictions (a), (b), and (c) that∞_n_=fn<∞. In view of Lemma ., we

ﬁnd thatlim_n_→∞xn+–xn= . In view ofyn–xn ≤αnf(xn) –xn, we ﬁnd from the

above that

lim

n→∞yn–xn+=nlim→∞yn–xn= . (.)

Next, we show that

lim sup n→∞

f(x) –¯ x,¯ yn–x¯

≤, (.)

wherex¯is the unique ﬁxed point of the mappingProj₍_A₊_B₎–_()f. To show this inequality,

we choose a subsequence{yni}of{yn}such that

lim sup n→∞

f(x) –¯ x,¯ yn–x¯

=lim i→∞

f(x) –¯ x,¯ yni–x¯

≤.

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Now, we show thatxˆ∈(A+B)–_{(). Notice that}_y

n–rnAyn+en∈xn++rnBxn+; that is,

yn–rnAyn+en–xn+

rn ∈

Bxn+.

Letμ∈Bν. SinceBis monotone, we ﬁnd that

yn+en–xn+

rn

–Ayn–μ,xn+–ν

≥.

In view of the restriction (b), we see from (.) that–Axˆ–μ,xˆ–ν ≥. This implies that –Axˆ∈Bx, that is,ˆ xˆ∈(A+B)–_{(). This proves that (.) holds. Notice that}

yn–x¯≤αnκxn–x¯yn–x¯+αn

f(x) –¯ x,¯ yn–x¯

+ ( –αn)xn–x¯yn–x¯.

It follows thatyn–x¯≤( –αn( –κ))xn–x¯+ αnf(x) –¯ x,¯ yn–x¯. On the other hand,

we have

xn+–x¯≤Jrn(yn–rnAyn+en) –x¯ 

≤(yn–rnAyn) – (I–rnA)x¯ 

+en

en+ (yn–rnAyn) – (I–rnA)x¯ ≤ yn–x¯+en

en+ (yn–rnAyn) – (I–rnA)x¯ ≤ –αn( –κ)

xn–x¯+ αn

f(x) –¯ x,¯ yn–x¯

+en

en+ (yn–rnAyn) – (I–rnA)x¯.

An application of Lemma . to the above inequality yieldslimn→∞xn–x¯= . This

completes the proof.

4 Applications

First, we consider the problem of ﬁnding a minimizer of a proper convex lower semicon-tinuous function.

For a proper lower semicontinuous convex functiong:H→(–∞,∞], the subdiﬀeren-tial mapping∂gofgis deﬁned by

∂g(x) =x∗∈H:g(x) +y–x,x∗≤g(y),∀y∈H, ∀x∈H.

Rockafellar [] proved that∂gis a maximal monotone operator. It is easy to verify that ∈∂g(v) if and only ifg(v) =min_x_∈_Hg(x).

Theorem . Let g:H→(–∞, +∞]be a proper convex lower semicontinuous function such that(∂g)–_()_{is not empty.}_{Let f}_:_H_→_{H be a}_{κ-contraction and let}_{_x

n}be a sequence

in H in the following process:x∈H and

⎧ ⎨ ⎩

xn+=arg minz∈H{g(z) +z–yn–en 

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real number sequence.If the control sequences satisfy the following restrictions:

(a) limn→∞αn= ,

_∞

n=αn=∞and

_∞

n=|αn–αn–|<∞; (b)  <a≤rn;

(c) ∞_n_=en<∞,

then{xn}converges strongly to a pointx¯∈(∂g)–(),wherex¯=Proj(∂g)–_()f(x).¯

Proof Sinceg:H→(–∞,∞] is a proper convex and lower semicontinuous function, we see that subdiﬀerential∂gofgis maximal monotone. Noting that

xn+=arg min z∈H g(z) +

z–yn–en

rn

is equivalent to

∈∂g(xn+) +

 rn

(xn+–yn–en).

It follows that

yn+en∈xn++rn∂g(xn+).

PuttingA= , we immediately derive from Theorem . the desired conclusion.

Next, we consider the problem of ﬁnding a solution of a classical variational inequality. LetCbe a nonempty closed and convex subset of a Hilbert spaceH. LetiCbe the

indi-cator function ofC, that is,

iC(x) =

⎧ ⎨ ⎩

, x∈C,

∞, x∈/C.

SinceiCis a proper lower and semicontinuous convex function onH, the subdiﬀerential

∂iC ofiC is maximal monotone. So, we can deﬁne the resolventJr of∂iC forr> ,i.e.,

Jr:= (I+r∂iC)–. Lettingx=Jry, we ﬁnd that

y∈x+r∂iCx ⇐⇒ y∈x+rNCx

⇐⇒ y–x,v–x ≤, ∀v∈C

⇐⇒ x=Proj_Cy,

whereProj_Cis the metric projection fromHontoCandNCx:={e∈H:e,v–x,∀v∈C}.

Theorem . Let A:C→H be anα-inverse-strongly monotone mapping.Assume that VI(C,A)is not empty.Let f :C→C be a ﬁxedκ-contraction.Let{xn}be a sequence in C

in the following process:x∈C and

⎧ ⎨ ⎩

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real number sequence in(, α).If the control sequences satisfy the following restrictions:

(a) lim_n_→∞αn= ,

∞

n=αn=∞and

∞

n=|αn–αn–|<∞; (b)  <a≤rn≤b< αand

∞

n=|rn–rn–|<∞; (c) ∞_n_=en<∞,

then{xn}converges strongly to a pointx¯∈VI(C,A),wherex¯=ProjVI(C,A)f(x).¯

Proof PuttingB=∂iCin Theorem ., we ﬁnd thatJrn=ProjC. We can draw the desired

conclusion from Theorem ..

Next, we consider the problem of ﬁnding a solution of a Ky Fan inequality, which is known as an equilibrium problem in the terminology of Blum and Oettli; see [] and [] and the references therein.

LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem:

Findx∈Csuch thatF(x,y)≥, ∀y∈C. (.)

To study the equilibrium problem (.), we may assume thatF satisﬁes the following restrictions:

(A) F(x,x) = for allx∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,lim sup_t_↓_F(tz+ ( –t)x,y)≤F(x,y);

(A) for eachx∈C,y→F(x,y)is convex and lower semicontinuous.

The following lemma can be found in [].

Lemma . Let F:C×C→Rbe a bifunction satisfying(A)-(A).Then,for any r>  and x∈H,there exists z∈C such that F(z,y) +_ry–z,z–x ≥,∀y∈C.Further,deﬁne

Trx= z∈C:F(z,y) +



ry–z,z–x ≥,∀y∈C

(.)

for all r> and x∈H.Then()Tris single-valued and ﬁrmly nonexpansive; ()F(Tr) =

EP(F)is closed and convex.

Lemma .[] Let F be a bifunction from C×C toRwhich satisﬁes(A)-(A),and let AF be a multivalued mapping of H into itself deﬁned by

AFx=

⎧ ⎨ ⎩

{z∈H:F(x,y)≥ y–x,z,∀y∈C}, x∈C,

∅, x∈/C. (.)

Then AF is a maximal monotone operator with the domain D(AF)⊂C,EP(F) =A–F (),

where FP(F)stands for the solution set of(.),and Trx= (I+rAF)–x,∀x∈H,r> ,where

Tris deﬁned as in(.).

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a sequence in C in the following process:x∈C and

xn+=Trn

αnf(xn) + ( –αn)xn+en

, ∀n≥,

real number sequence.If the control sequences satisfy the following restrictions:

(a) lim_n_→∞αn= ,

∞

n=αn=∞and

∞

n=|αn–αn–|<∞; (b)  <a≤rn≤b<∞and

∞

n=|rn–rn–|<∞; (c) ∞_n_=en<∞,

then{xn}converges strongly to a pointx¯∈EP(F),wherex¯=ProjEP(F)f(x).¯

Proof PuttingA=  in Theorem ., we ﬁnd thatJrn=Trn. From Theorem ., we can draw the desired conclusion immediately.

Recall that a mappingT:C→Tis said to beα-strictly pseudocontractive if there exists a constantα∈[, ) such that

Tx–Ty≤ x–y+α(I–T)x– (I–T)y, ∀x,y∈C.

The class of strictly pseudocontractive mappings was ﬁrst introduced by Browder and Petryshyn []. It is well known that ifT isα-strictly-pseudocontractive, thenI–T is

–α

 -inverse-strongly monotone.

Finally, we consider ﬁxed point problem ofα-strictly pseudocontractive mappings.

Theorem . Let T:C→C be anα-strictly pseudocontractive mapping with a nonempty ﬁxed point set and let f :C→C be a ﬁxedκ-contraction.Let{xn}be a sequence generated

in the following manner:x∈C and

⎧ ⎨ ⎩

xn+= ( –rn)yn+rnTyn, ∀n≥,

where{αn}is a real number sequence in(, )and{rn}is a positive real number sequence

in(,  –α).If the control sequences satisfy the following restrictions:

(a) limn→∞αn= ,∞n=αn=∞and∞n=|αn–αn–|<∞; (b)  <a≤rn≤b<  –αand

∞

n=|rn–rn–|<∞;

then{xn}converges strongly to a pointx¯∈F(T),wherex¯=ProjF(T)f(x).¯

Proof PuttingA=I–T, we ﬁndAis–_α-inverse-strongly monotone. We also haveF(T) = VI(C,A) andProj_C(yn–rnAyn) = ( –rn)yn+rnTyn. In view of Theorem ., we obtain the

desired result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China.}2_{Department of Mathematics,}

Gyeongsang National University, Jinju, 6660-701, Korea.3_{College of Statistics and Mathematics, Yunnan University of} Finance and Economics, Kunming, 650221, China.

Acknowledgements

The authors are grateful to the three anonymous referees for useful suggestions, which improved the contents of the article.

Received: 5 December 2013 Accepted: 11 March 2014 Published:25 Mar 2014 References

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