Iterative Approaches to Find Zeros of Maximal Monotone Operators by Hybrid Approximate Proximal Point Methods

(1)

Volume 2011, Article ID 282171,18pages doi:10.1155/2011/282171

Research Article

Iterative Approaches to Find Zeros of Maximal

Monotone Operators by Hybrid Approximate

Proximal Point Methods

Lu Chuan Ceng,

1

_{Yeong Cheng Liou,}

2

_{and Eskandar Naraghirad}

3

1_{Department of Mathematics, Shanghai Normal University, Shanghai 200234, China} 2_{Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan} 3_{Department of Mathematics, Yasouj University, Yasouj 75914, Iran}

Correspondence should be addressed to Eskandar Naraghirad,[email protected]

Received 18 August 2010; Accepted 23 September 2010

Academic Editor: Jen Chih Yao

Copyrightq2011 Lu Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is to introduce and investigate two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.

1. Introduction

LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. In this paper, we always assume thatT :C → 2His a maximal monotone operator. A classical method to solve the following set-valued equation:

0∈Tx 1.1

is the proximal point method. To be more precise, start with any pointx0 ∈H, and update xn1iteratively conforming to the following recursion:

xn∈xn1λnTxn1, ∀n≥0, 1.2

where{λn} ⊂λ,∞ λ >0is a sequence of real numbers. However, as pointed out in1, the

(2)

exactly is either impossible or has the same diﬃculty as the original problem1.1. Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an eﬃcient iterative algorithm to compute approximate zeros ofT.

In 1976, Rockafellar2gave an inexact variant of the method

x0∈H, xnen1∈xn1λnTxn1, ∀n≥0, 1.3

where{en}is regarded as an error sequence. This is an inexact proximal point method. It was

shown that, if

∞

n0

en<∞, 1.4

the sequence{xn}defined by1.3converges weakly to a zero ofT provided thatT−10/∅.

In3, G ¨uler obtained an example to show that Rockafellar’s inexact proximal point method

1.3does not converge strongly, in general.

Recently, many authors studied the problems of modifying Rockafellar’s inexact proximal point method1.3in order to strong convergence to be guaranteed. In 2008, Ceng et al.4gave new accuracy criteria to modified approximate proximal point algorithms in Hilbert spaces; that is, they established strong and weak convergence theorems for modified approximate proximal point algorithms for finding zeros of maximal monotone operators in Hilbert spaces. In the meantime, Cho et al.5proved the following strong convergence result.

Theorem CKZ 1. Let H be a real Hilbert space, Ωa nonempty closed convex subset of H, and T :Ω → 2Ha maximal monotone operator withT−10/∅. LetPΩbe the metric projection ofHonto

Ω. Suppose that, for any givenxn ∈H,λn >0, anden ∈H, there existsxn ∈Ωconforming to the

following set-valued mapping equation:

xnen∈xnλnTxn, 1.5

where{λn} ⊂0,∞withλn → ∞asn → ∞and

∞

n1

en2<∞. 1.6

Let{αn}be a real sequence in0,1such that

iαn → 0asn → ∞,

ii∞_n₀αn∞.

For any fixedu∈Ω, define the sequence{xn}iteratively as follows:

xn1αnu 1−αnPΩxn−en, ∀n≥0. 1.7

(3)

They also derived the following weak convergence theorem.

Theorem CKZ 2. Let H be a real Hilbert space, Ωa nonempty closed convex subset of H, and T :Ω → 2Ha maximal monotone operator withT−10/∅. LetPΩbe the metric projection ofHonto

Ω. Suppose that, for any givenxn ∈H,λn >0, anden ∈H, there existsxn ∈Ωconforming to the

following set-valued mapping equation:

xnen∈xnλnTxn, 1.8

wherelim infn→ ∞λn>0and

∞

n0

en2<∞. 1.9

Let{αn}be a real sequence in0,1withlim sup_n_{→ ∞}αn <1, and define a sequence{xn}iteratively

as follows:

x0∈Ω, xn1αnxnβnPΩxn−en, ∀n≥0, 1.10

whereαnβn 1for alln≥0. Then the sequence{xn}converges weakly to a zerox∗ofT.

Very recently, Qin et al.6extended1.7and1.10to the iterative scheme

x0∈H, xn1αnuβnPCxn−en γnPCfn, ∀n≥0, 1.11

and the iterative one

x0∈C, xn1 αnxnβnPCxn−en γnPCfn, ∀n≥0, 1.12

respectively, whereαnβnγn1, supn≥0fn<∞, anden ≤ηnxn−xnwith supn≥0ηnη <

1. Under appropriate conditions, they derived one strong convergence theorem for1.11and another weak convergence theorem for1.12. In addition, for other recent research works on approximate proximal point methods and their variants for finding zeros of monotone maximal operators, see, for example,7–10and the references therein.

In this paper, motivated by the research work going on in this direction, we continue to consider the problem of finding a zero of the maximal monotone operatorT. The iterative algorithms1.7and1.10are extended to develop the following new iterative ones:

x0∈H, xn1αnuβnPC1−γn−δnxnγnxn−en δnfn, ∀n≥0, 1.13

x0∈C, xn1αnxnβnPC1−γn−δnxnγnxn−en δnfn, ∀n≥0, 1.14

respectively, whereuis any fixed point inC,αnβn 1,γnδn ≤ 1, sup_n_≥0fn < ∞, and

en ≤ηnxn−xnwith sup_n_≥0ηn η < 1. Under mild conditions, we establish one strong

(4)

presented in this paper improve the corresponding results announced by many others. It is easy to see that in the case whenγn1 andδn0 for alln≥0, the iterative algorithms1.13

and1.14reduce to1.7and1.10, respectively. Moreover, the iterative algorithms1.13

and1.14are very diﬀerent from1.11and1.12, respectively. Indeed, it is clear that the iterative algorithm1.13is equivalent to the following:

x0 ∈H,

yn 1−γn−δnxnγnxn−en δnfn,

xn1αnuβnPCyn, ∀n≥0.

1.15

Here, the first iteration step yn 1−γn −δnxn γnxn −en δnfn, is to compute the

prediction value of approximate zeros ofT; the second iteration step,xn1 αnuβnPCyn,

is to compute the correction value of approximate zeros ofT. Similarly, it is obvious that the iterative algorithm1.14is equivalent to the following:

x0∈C,

yn 1−γn−δnxnγnxn−en δnfn,

xn1αnxnβnPCyn, ∀n≥0.

1.16

Here, the first iteration step, yn 1−γn−δnxn γnxn −en δnfn, is to compute the

prediction value of approximate zeros ofT; the second iteration step,xn1αnxnβnPCyn, is

to compute the correction value of approximate zeros ofT. Therefore, there is no doubt that the iterative algorithms1.13and1.14are very interesting and quite reasonable.

In this paper, we consider the problem of finding zeros of maximal monotone operators by hybrid proximal point method. To be more precise, we introduce two kinds of iterative schemes, that is,1.13and 1.14. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we also consider a problem of finding a minimizer of a convex function.

2. Preliminaries

In this section, we give some preliminaries which will be used in the rest of this paper. LetH be a real Hilbert space with inner product·,·and norm · . LetTbe a set-valued mapping. The setDTdefined by

DT {u∈H:Tu/∅} 2.1

is called the eﬀective domain ofT. The setRTdefined by

RT

u∈H

(5)

is called the range ofT. The setGTdefined by

GT {x, u∈H×H:x∈DT, u∈Tx} 2.3

is called the graph ofT. A mappingTis said to be monotone if

x−y, u−v ≥0, ∀x, u,y, v∈GT. 2.4

T is said to be maximal monotone if its graph is not properly contained in the one of any other monotone operator.

The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoted to the study of the existence and iterative algorithms of zeros for maximal monotone mappings; see1–5,7,11–30. In order to prove our main results, we need the following lemmas. The first lemma can be obtained from Eckstein1, Lemma 2immediately.

Lemma 2.1. LetCbe a nonempty, closed, and convex subset of a Hilbert spaceH. For any given xn ∈ H,λn > 0, anden ∈ H, there existsxn ∈ Cconforming to the following set-valued mapping

equation (SVME ):

xnen∈xnλnTxn. 2.5

Furthermore, for anyp∈T−10, we have

xn−xn, xn−xnen ≤ xn−p, xn−xnen,

xn−en−p2≤ xn−p2− xn−xn2en2.

2.6

Lemma 2.2see30, Lemma 2.5, page 243. Let{sn}be a sequence of nonnegative real numbers

satisfying the inequality

sn1≤1−αnsnαnβnγn, ∀n≥0, 2.7

where{αn},{βn}, and{γn}satisfy the conditions

i{αn} ⊂0,1,∞n0αn∞, or equivalently

_∞

n01−αn 0,

iilim sup_n_{→ ∞}βn≤0,

iii{γn} ⊂0,∞,∞n0γn<∞. Thenlimn→ ∞sn0.

Lemma 2.3see28, Lemma 1, page 303. Let{an}and{bn}be sequences of nonnegative real

numbers satisfying the inequality

an1≤anbn, ∀n≥0. 2.8

(6)

Lemma 2.4see11. LetEbe a uniformly convex Banach space, letCbe a nonempty closed convex subset ofE, and letS:C → Cbe a nonexpansive mapping. ThenI−Sis demiclosed at zero.

Lemma 2.5see31. LetEbe a uniformly convex Banach space, and andBr0be a closed ball

ofE. Then there exists a continuous strictly increasing convex functiong : 0,∞ → 0,∞with g0 0such that

λxμyνz2 _≤_λ_x2_μ_y2_ν_z2₋_λμg_x₋_y _2.9

for allx, y, z∈Br0andλ, μ, ν∈0,1withλμν1.

It is clear that the following lemma is valid.

Lemma 2.6. LetHbe a real Hilbert space. Then there holds

xy2_≤_x2₂_{y, x}_y_, _∀_{x, y}_∈_H. _2.10

3. Main Results

LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. We always assume thatT : C → 2H is a maximal monotone operator. Then, for eacht > 0, the resolventJt

ItT−1is a single-valued nonexpansive mapping whose domain is allH. Recall also that the Yosida approximation ofTis defined by

Tt 1_tI−Jt. 3.1

Assume thatT−10/∅, whereT−10is the set of zeros ofT. ThenT−10 FixJtfor allt >0,

where FixJtis the set of fixed points of the resolventJt.

Theorem 3.1. LetH be a real Hilbert space,C a nonempty, closed, and convex subset ofH, and T :C → 2Ha maximal monotone operator withT−10/∅. LetPC be a metric projection fromH

ontoC. For any givenxn ∈H,λn >0, anden∈H, findxn∈Cconforming to SVME2.5, where

{λn} ⊂0,∞withλn → ∞asn → ∞anden ≤ηnxn−xnwithsup_n_≥0ηnη <1. Let{αn},

{βn},{γn}, and{δn}be real sequences in0,1satisfying the following control conditions:

iαnβn 1andγnδn≤1,

iilimn→ ∞αn0and∞n0αn ∞,

iiilimn→ ∞γn1and∞n0δn<∞.

Let{xn}be a sequence generated by the following manner:

x0 ∈H, xn1αnuβnPC1−γn−δnxnγnxn−en δnfn, ∀n≥0, 3.2

whereu∈Cis a fixed point and{fn}is a bounded sequence inH. Then the sequence{xn}generated

(7)

Proof. First, let us show the necessity. Assume thatxn → zasn → ∞, wherez∈T−10. It

follows from2.5that

xn−zJλnxnen−Jλnz

≤ xn−zen

≤ xn−zηnxn−xn

≤1ηnxn−zηnxn−z,

3.3

and hence

xn−z ≤ 1ηn

1−ηnxn−z ≤

1η

1−ηxn−z. 3.4

This implies thatxn → zasn → ∞. Note that

en ≤ηnxn−xn ≤ηnxn−zz−xn. 3.5

This shows thaten → 0 asn → ∞.

Next, let us show the suﬃciency. The proof is divided into several steps.

Step 1{xn}is bounded. Indeed, from the assumptionsen ≤ ηnxn−xnand supn≥0ηn

η <1, it follows that

en ≤ xn−xn. 3.6

Take an arbitraryp∈T−10. Then it follows fromLemma 2.1that

xn−en−p2≤ xn−p2− xn−xn2en2≤ xn−p2, 3.7

and hence

PC1−γn−δnxnγnxn−en δnfn−p2

≤ 1−γn−δnxnγnxn−en δnfn−p2

1−γn−δnxn−pγnxn−en−pδnfn−p2

≤1−γn−δnxn−p2γnxn−en−p2δnfn−p2

≤1−γn−δnxn−p2γnxn−p2δnfn−p2

1−δnxn−p2δnfn−p2.

(8)

This implies that

xn1−p2αnuβnPC1−γn−δnxnγnxn−en δnfn−p2

≤αnu−p2βnPC1−γn−δnxnγnxn−en δnfn−p2

≤αnu−p2βn

1−δnxn−p2δnfn−p2

αnu−p2βn1−δnxn−p2βnδnfn−p2

≤αnu−p2βn1−δnxn−p2βnδnsup

n≥0fn−p 2_.

3.9

Putting

Mmax

x0−p2,u−p2,sup

n≥0fn−p 2

, 3.10

we show thatxn−p2 ≤ Mfor alln ≥ 0. It is easy to see that the result holds for n 0.

Assume that the result holds for some n ≥ 0. Next, we prove that xn1 −p2 ≤ M. As a

matter of fact, from3.9, we see that

xn1−p2≤M. 3.11

This shows that the sequence{xn}is bounded.

Step 2lim sup_n_{→ ∞}u−z, xn1−z ≤0, wherezlimt→ ∞Jtu. The existence of limt→ ∞Jtuis

guaranteed by Lemma 1 of Bruck12.

SinceT is maximal monotone,Ttu∈TJtuandTλnxn ∈TJλnxn, we deduce that

u−Jtu, Jλnxn−Jtu−tTtu, Jtu−Jλnxn

−tTtu−Tλnxn, Jtu−Jλnxn −tTλnxn, Jtu−Jλnxn

≤ −_λt

nxn−Jλnxn, Jtu−Jλnxn.

3.12

Sinceλn → ∞asn → ∞, for eacht >0, we have

lim sup

n→ ∞ u−Jtu, Jλnxn−Jtu ≤0. 3.13

On the other hand, by the nonexpansivity ofJλn, we obtain that

(9)

From the assumptionen → 0 asn → ∞and3.13, we get

lim sup

n→ ∞ u−Jtu, Jλnxnen−Jtu ≤0. 3.15

From2.5, we see that

PC1−γn−δnxnγnxn−en δnfn−Jλnxnen

≤ 1−γn−δnxnγnxn−en δnfn−Jλnxnen

≤1−γn−δnxn−Jλnxnenγnxn−en−Jλnxnenδnfn−Jλnxnen

1−γn−δnxn−Jλnxnenγnenδnfn−Jλnxnen.

3.16

Since limn→ ∞γn 1 and∞n0δn <∞, we conclude fromen → 0 and the boundedness of

{fn}that

lim

n→ ∞PC

1−γn−δnxnγnxn−en δnfn−Jλnxnen0. 3.17

Combining3.15with3.17, we have

lim sup

n→ ∞ u−Jtu, PC

1−γn−δnxnγnxn−en δnfn−Jtu≤0. _3.18

In the meantime, from algorithm3.2and assumptionαnβn1, it follows that

xn1−PC

1−γn−δnxnγnxn−en δnfn

αnu−PC1−γn−δnxnγnxn−en δnfn.

3.19

Thus, from the condition limn→ ∞αn0, we have

xn1−PC1−γn−δnxnγnxn−en δnfn−→0 as n−→ ∞. 3.20

This together with3.18implies that

lim sup

n→ ∞ u−Jtu, xn1−Jtu ≤0, ∀t >0. 3.21

Fromzlimt→ ∞Jtuand3.21, we can obtain that

lim sup

(10)

Step 3xn → zasn → ∞. Indeed, utilizing3.8, we deduce from algorithm3.2that

xn1−z21−αn

PC1−γn−δnxnγnxn−en δnfn−zαnu−z2

≤1−αn2PC1−γn−δnxnγnxn−en δnfn−z2

2αnu−z, xn1−z

≤1−αn

1−δnxn−z2δnfn−z2

2αnu−z, xn1−z

≤1−αnxn−z2αn·2u−z, xn1−zδnfn−z2.

3.23

Note that∞_n₀δn < ∞and{fn}is bounded. Hence it is known that∞n0δnfn−z2 < ∞.

Since∞_n₀αn ∞, lim supn→ ∞2u−z, xn1−z ≤ 0, and∞n0δnfn−z2 <∞, in terms of

Lemma 2.2, we conclude that

xn−z −→0 asn−→ ∞. 3.24

This completes the proof.

Remark 3.2. The maximal monotonicity of T is only used to guarantee the existence of solutions of SVME 2.4, for any givenxn ∈ H,λn > 0, and en ∈ H. If we assume that

T :C → 2His monotonenot necessarily maximaland satisfies the range condition

DT C⊂

r>0

RIrT, _3.25

we can see thatTheorem 3.1still holds.

Corollary 3.3. LetH be a real Hilbert space,Ca nonempty, closed, and convex subset of H, and S:C → Ca demicontinuous pseudocontraction with a fixed point inC. LetPCbe a metric projection

fromHontoC. For anyxn∈C,λn>0, anden∈H, findxn∈Csuch that

xnen 1λnxn−λnSxn, 3.26

where{λn} ⊂0,∞withλn → ∞asn → ∞anden ≤ηnxn−xnwithsupn≥0ηnη <1. Let

{αn},{βn},{γn}, and{δn}be real sequences in0,1satisfying the following control conditions:

(11)

whereu∈Cis a fixed point and{fn}is a bounded sequence inH. If the sequence{en}satisfies the

conditionen → 0asn → ∞, then the sequence{xn}converges strongly to a fixed pointzofS, where

zlimt→ ∞ItI−S−1u.

Proof. LetT I−S. ThenT :C → His demicontinuous, monotone, and satisfies the range condition:

DT C⊂

r>0

RIrT. _3.28

For anyy∈C, define an operatorG:C → Cby

Gx t

1tSx 1

1ty. 3.29

ThenGis demicontinuous and strongly pseudocontractive. By the study of Lan and Wu21, Theorem 2.2, we see thatGhas a unique fixed pointx∈C; that is,

yxtI−Sx. 3.30

This implies thaty ∈RItTfor allt > 0. In particular, for any givenxn ∈ C,λn > 0, and

en∈H, there existsxn∈Csuch that

xnenxnλnTxn, ∀n≥0, 3.31

that is,

xnen 1λnxn−λnSxn. 3.32

Finally, from the proof ofTheorem 3.1, we can derive the desired conclusion immediately.

FromTheorem 3.1, we also have the following result immediately.

Corollary 3.4. LetH be a real Hilbert space,Ca nonempty, closed, and convex subset of H, and T :C → 2Ha maximal monotone operator withT−10/∅. LetPC be a metric projection fromH

ontoC. For anyxn ∈ H,λn > 0 and en ∈ H, findxn ∈ Cconforming to SVME 2.5, where

{λn} ⊂0,∞withλn → ∞asn → ∞anden ≤ηnxn−xnwithsupn≥0ηnη <1. Let{αn},

{βn}, and{γn}be real sequences in0,1satisfying the following control conditions:

iαnβn 1,

(12)

x0∈H, xn1αnuβnPC1−γnxnγnxn−en, ∀n≥0, 3.33

whereu ∈ C is a fixed point. Then the sequence{xn} converges strongly to a zero zofT, where

zlimt→ ∞Jtu, if and only ifen → 0asn → ∞.

Proof. InTheorem 3.1, putδn0 for alln≥0. Then, fromTheorem 3.1, we obtain the desired

result immediately.

Next, we give a hybrid Mann-type iterative algorithm and study the weak convergence of the algorithm.

Theorem 3.5. LetH be a real Hilbert space,C a nonempty, closed, and convex subset ofH, and T :C → 2Ha maximal monotone operator withT−10/∅. LetPC be a metric projection fromH

ontoC. For any givenxn ∈C,λn >0, anden ∈H, findxn∈Cconforming to SVME2.5, where

lim infn→ ∞λn>0anden ≤ηnxn−xnwithsup_n_≥0ηnη <1. Let{αn},{βn},{γn}, and{δn}

be real sequences in0,1satisfying the following control conditions:

iilim infn→ ∞βn>0,

iiilim infn→ ∞γn>0and∞n0δn<∞.

x0∈C, xn1 αnxnβnPC1−γn−δnxnγnxn−en δnfn, ∀n≥0, 3.34

where{fn}is a bounded sequence inH. Then the sequence{xn}generated by3.34converges weakly

to a zerox∗ofT.

Proof. Take an arbitraryp∈T−10. UtilizingLemma 2.1, from the assumptionen ≤ηnxn−

xnwith sup_n_≥0ηnη <1, we conclude that

xn−en−p2≤ xn−p2− xn−xn2en2

≤ xn−p2− xn−xn2η2nxn−xn2

≤ xn−p2−

1−η2xn−xn2.

(13)

It follows fromLemma 2.5that

xn1−p2αnxnβnPC

1−γn−δnxnγnxn−en δnfn−p2

≤αnxn−p2βnPC1−γn−δnxnγnxn−en δnfn−p2

≤αnxn−p2βn1−γn−δnxnγnxn−en δnfn−p2

≤αnxn−p2βn1−γn−δnxn−p2γnxn−en−p2δnfn−p2

≤αnxn−p2βn1−γn−δnxn−p2γn

xn−p2−

1−η2

xn−xn2

δnfn−p2

αnxn−p2βn1−δnxn−p2−βnγn

1−η2xn−xn2βnδnfn−p2

≤ xn−p2−βnγn

1−η2xn−xn2δnfn−p2

≤ xn−p2δnfn−p2.

3.36

Utilizing Lemma 2.3, we know that limn→ ∞xn −p exists. We, therefore, obtain that the

sequence{xn}is bounded. It follows from3.36that

βnγn1−η2xn−xn2≤ xn−p2− xn1−p2δnfn−p2. 3.37

From the conditions lim infn→ ∞βn>0, lim infn→ ∞γn>0, and∞n0δn <∞, we conclude that

lim

n→ ∞xn−xn0. 3.38

Note that

xn−Jλnxnxn−xnxn−Jλnxn

≤ xn−xnxn−Jλnxn

≤1ηnxn−xn.

3.39

In view of3.38, we obtain that

lim

(14)

Also, note that

Jλnxn−J1JλnxnT1Jλnxn

≤inf{w:w∈TJλnxn}

≤ Tλnxn

xn−_λJλnxn

n .

3.41

In view of the assumption lim infn→ ∞λn>0 and3.40, we see that

lim

n→ ∞Jλnxn−J1Jλnxn0. 3.42

Let x∗ ∈ C be a weakly subsequential limit of {xn} such that{xni} converges weakly to

x∗ _as _i _{→ ∞}_{. From} _3.40_{, we see that} _J_λ

nixni also converges weakly to x∗. Since J1 is

nonexpansive, we can obtain thatx∗∈FixJ1 T−10byLemma 2.4. Opial’s conditionsee

23guarantees that the sequence{xn}converges weakly tox∗. This completes the proof.

By the careful analysis of the proof ofCorollary 3.3andTheorem 3.5, it is not hard to derive the following result.

Corollary 3.6. LetH be a real Hilbert space,Ca nonempty, closed, and convex subset of H, and S:C → Ca demicontinuous pseudocontraction with a fixed point inC. LetPCbe a metric projection

fromHontoC. For anyxn∈C,λn>0, anden∈H, findxn∈Csuch that

xnen 1λnxn−λnSxn, ∀n≥0, 3.43

wherelim infn→ ∞λn>0anden ≤ηnxn−xnwithsup_n_≥0ηnη <1. Let{αn},{βn},{γn}, and

{δn}be real sequences in0,1satisfying the following control conditions:

x0∈C, xn1αnxnβnPC

1−γn−δnxnγnxn−en δnfn, ∀n≥0, 3.44

where{fn}is a bounded sequence inH. Then the sequence{xn}converges weakly to a fixed pointx∗

ofS.

UtilizingTheorem 3.5, we also obtain the following result immediately.

(15)

ontoC. For anyxn ∈ C,λn > 0, anden ∈ H, findxn ∈ Cconforming to SVME 2.5, where

lim infn→ ∞λn>0anden ≤ηnxn−xnwithsupn≥0ηnη <1. Let{αn},{βn}, and{γn}be real

sequences in0,1satisfying the following control conditions:

iαnβn 1,

iilim sup_n_{→ ∞}αn<1,

iiilim infn→ ∞γn>0.

x0∈C, xn1αnxnβnPC1−γnxnγnxn−en, ∀n≥0. 3.45

Then the sequence{xn}converges weakly to a zerox∗ofT.

4. Applications

In this section, as applications of the main Theorems3.1and3.5, we consider the problem of finding a minimizer of a convex functionf.

LetHbe a real Hilbert space, and letf :H → −∞,∞be a proper convex lower semi-continuous function. Then the subdiﬀerential∂foffis defined as follows:

∂fx y∈H:fz≥fx z−x, y, z∈H, ∀x∈H. 4.1

Theorem 4.1. LetHbe a real Hilbert space andf :H → −∞,∞a proper convex lower semi-continuous function such that∂f−10/∅. Let{λn}be a sequence in0,∞withλn → ∞as

n → ∞and{en}a sequence inHsuch thaten ≤ηnxn−xnwithsup_n_≥0ηnη <1. Letxnbe

the solution of SVME2.5withTreplaced by∂f; that is, for any givenxn∈H,

xnen∈xnλn∂fxn, ∀n≥0. 4.2

Let{αn},{βn},{γn}, and{δn}be real sequences in0,1satisfying the following control conditions:

x0 ∈H,

xnarg minx∈H

fx 1

2λnx−xn−en 2

,

xn1αnuβnPC1−γn−δnxnγnxn−en δnfn, ∀n≥0,

4.3

whereu∈His a fixed point and{fn}is a bounded sequence inH. If the sequence{en}satisfies the

conditionen → 0asn → ∞, then the sequence{xn}converges strongly to a minimizer offnearest

(16)

Proof. Sincef :H → −∞,∞is a proper convex lower semi-continuous function, we have that the subdiﬀerential∂f offis maximal monotone by the study of Rockafellar2. Notice that

xnarg minx∈H

fx 1

2λnx−xn−en 2

4.4

is equivalent to the following:

0∈∂fxn _λ1

nxn−xn−en. 4.5

It follows that

By usingTheorem 3.1, we can obtain the desired result immediately.

Theorem 4.2. LetHbe a real Hilbert space andf :H → −∞,∞a proper convex lower semi-continuous function such that∂f−10/∅. Let{λn}be a sequence in0,∞withlim infn→ ∞λn>

0and{en}a sequence inHsuch thaten ≤ ηnxn−xnwith supn≥0ηn η <1. Letxn be the

solution of SVME2.5withTreplaced by∂f; that is, for any givenxn∈H,

Let{αn},{βn},{γn}, and{δn}be real sequences in0,1satisfying the following control conditions:

x0 ∈H,

xnarg min_x_∈_H

fx 1

2λnx−xn−en 2

,

xn1αnxnβnPC

1−γn−δnxnγnxn−en δnfn, ∀n≥0,

4.8

where{fn}is a bounded sequence inH. Then the sequence{xn}converges weakly to a minimizer of

f.

(17)

Acknowledgment

This research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai.

References

1 J. Eckstein, “Approximate iterations in Bregman-function-based proximal algorithms,”Mathematical Programming, vol. 83, no. 1, Ser. A, pp. 113–123, 1998.

2 R. T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.

3 O. G ¨uler, “On the convergence of the proximal point algorithm for convex minimization,” SIAM Journal on Control and Optimization, vol. 29, no. 2, pp. 403–419, 1991.

4 L.-C. Ceng, S.-Y. Wu, and J.-C. Yao, “New accuracy criteria for modified approximate proximal point algorithms in Hilbert spaces,”Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1691–1705, 2008. 5 Y. J. Cho, S. M. Kang, and H. Zhou, “Approximate proximal point algorithms for finding zeroes

of maximal monotone operators in Hilbert spaces,”Journal of Inequalities and Applications, vol. 2008, Article ID 598191, 10 pages, 2008.

6 X. Qin, S. M. Kang, and Y. J. Cho, “Approximating zeros of monotone operators by proximal point algorithms,”Journal of Global Optimization, vol. 46, no. 1, pp. 75–87, 2010.

7 L.-C. Ceng and J.-C. Yao, “On the convergence analysis of inexact hybrid extragradient proximal point algorithms for maximal monotone operators,”Journal of Computational and Applied Mathematics, vol. 217, no. 2, pp. 326–338, 2008.

8 L.-C. Ceng and J.-C. Yao, “Generalized implicit hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space,”Taiwanese Journal of Mathematics, vol. 12, no. 3, pp. 753–766, 2008.

9 L. C. Ceng, A. Petrus¸el, and S. Y. Wu, “On hybrid proximal-type algorithms in Banach spaces,”

Taiwanese Journal of Mathematics, vol. 12, no. 8, pp. 2009–2029, 2008.

10 L.-C. Ceng, S. Huang, and Y.-C. Liou, “Hybrid proximal point algorithms for solving constrained minimization problems in Banach spaces,”Taiwanese Journal of Mathematics, vol. 13, no. 2, pp. 805– 820, 2009.

11 F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,”

Proceedings of Symposia in Pure Mathematics, vol. 18, pp. 78–81, 1976.

12 R. E. Bruck, Jr., “A strongly convergent iterative solution of 0∈Uxfor a maximal monotone operator

Uin Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 48, pp. 114–126, 1974. 13 R. S. Burachik, A. N. Iusem, and B. F. Svaiter, “Enlargement of monotone operators with applications

to variational inequalities,”Set-Valued Analysis, vol. 5, no. 2, pp. 159–180, 1997.

14 Y. Censor and S. A. Zenios, “Proximal minimization algorithm with D-functions,” Journal of Optimization Theory and Applications, vol. 73, no. 3, pp. 451–464, 1992.

15 G. Cohen, “Auxiliary problem principle extended to variational inequalities,”Journal of Optimization Theory and Applications, vol. 59, no. 2, pp. 325–333, 1988.

16 K. Deimling, “Zeros of accretive operators,”Manuscripta Mathematica, vol. 13, pp. 365–374, 1974. 17 R. S. Dembo, S. C. Eisenstat, and T. Steihaug, “Inexact Newton methods,”SIAM Journal on Numerical

Analysis, vol. 19, no. 2, pp. 400–408, 1982.

18 D. Han and B. He, “A new accuracy criterion for approximate proximal point algorithms,”Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 343–354, 2001.

19 S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,”Journal of Approximation Theory, vol. 106, no. 2, pp. 226–240, 2000.

20 S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,”SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.

21 K. Q. Lan and J. H. Wu, “Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces,”Nonlinear Analysis, vol. 49, no. 6, pp. 737–746, 2002.

(18)

mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.

24 A. Pazy, “Remarks on nonlinear ergodic theory in Hilbert space,”Nonlinear Analysis, vol. 3, no. 6, pp. 863–871, 1979.

25 X. Qin and Y. Su, “Approximation of a zero point of accretive operator in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 415–424, 2007.

26 M. V. Solodov and B. F. Svaiter, “An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions,”Mathematics of Operations Research, vol. 25, no. 2, pp. 214–230, 2000.

27 M. Teboulle, “Convergence of proximal-like algorithms,”SIAM Journal on Optimization, vol. 7, no. 4, pp. 1069–1083, 1997.

28 K.-K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,”Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993. 29 R. U. Verma, “Rockafellar’s celebrated theorem based on A-maximal monotonicity design,”Applied

Mathematics Letters, vol. 21, no. 4, pp. 355–360, 2008.

30 H.-K. Xu, “Iterative algorithms for nonlinear operators,”Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.