Nonlinear Analysis 71 (2009) 341–346

Contents lists available at ScienceDirect

Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Strong convergence of projection scheme for zeros of maximalmonotone operatorsI

Li Wei a,∗, Haiyun Zhou ba School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, Hebei Province, 050061, PR Chinab Institute of Applied Mathematics and Mechanics, Ordnance Engineering College, Shijiazhuang, Hebei Province, 050003, PR China

a r t i c l e i n f o

Article history:Received 5 August 2006Accepted 29 October 2008

MSC:47H0947H05

Keywords:Maximal monotone operatorStrong convergenceZeros

a b s t r a c t

A projection scheme with errors for zeros of maximal monotone operators is introducedand some strong convergence theorems are obtained in a real uniformly smooth anduniformly convexBanach space E. It is also shownhow to compute commonzeros of finitelymany maximal monotone operators in E.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Constructing iterative algorithms to approximate zeros of maximal monotone operators is a very active topic in appliedmathematics. The proximal point method (see Rockafellar [1]) is among the main tools for finding zeros of maximalmonotone operators in Hilbert spaces. However, it was shown in [1] that the iterative sequence converges weakly butnot strongly. To get the result of strong convergence, Solodov and Svaiter [2] modified the proximal point algorithm andintroduced the projection scheme as follows:

x0 ∈ H,

0 = vn +1rn(yn − xn) , vn ∈ Tyn,

Hn = z ∈ H : 〈z − yn, vn〉 ≤ 0 ,Wn = z ∈ H : 〈z − xn, x0 − xn〉 ≤ 0 ,xn+1 = PHn∩Wnx0, n = 0, 1, 2, . . . .

(1.1)

They proved that if T−10 6= ∅ and lim infn→∞ rn > 0, then the sequence generated by (1.1) converges strongly to PT−10x0 ina Hilbert space H , where PT−10 is the metric projection of H onto T

−10.

I This paper is supported by National Natural Science Foundation of China (No. 10771050).∗ Corresponding author.E-mail address: diandianba@yahoo.com (L. Wei).

0362-546X/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.10.081

342 L. Wei, H. Zhou / Nonlinear Analysis 71 (2009) 341–346

Later, projection scheme (1.1)was extended by Kamimura and Takahashi in [3] fromHilbert spaceH to uniformly smoothand uniformly convex Banach space E:

x0 ∈ E,

0 = vn +1rn(Jyn − Jxn) , vn ∈ Tyn,

Hn = z ∈ E : 〈z − yn, vn〉 ≤ 0 ,Wn = z ∈ E : 〈z − xn, Jx0 − Jxn〉 ≤ 0 ,xn+1 = QHn∩Wnx0, n = 0, 1, 2, . . . .

(1.2)

They proved that if T−10 6= ∅ and lim infn→∞ rn > 0, then the sequence generated by (1.2) converges strongly to QT−10x0 inE, where QT−10 is the so-called generalized projection of E onto T

−10.Actually,many important problems related to practical problems are generally defined in Banach spaces. For example, the

maximal monotone operator related to elliptic boundary value problem has Sobolev spaceWm,p (Ω) as its natural domainof definition [4]. Therefore, the work done by Kamimura and Takahashi is meaningful. Following their ideas, we began ourresearch and have obtained some strong and weak convergence results of revised proximal point algorithms for zeros ofmaximal monotone operators in Banach spaces [5–7]. This paper is a continuation of our previous work.This paper is motivated by the ideas of Yanes and Xu [8] in 2006, where a projection algorithm with errors in Hilbert

space is introduced:

Lemma 1.1 ([8], Theorem 4.1). The sequence xn generated by the algorithm:x0 ∈ H chosen arbitrarily,yn = αnx0 + (1− αn)Jcn (xn + en) ,Hn =

v ∈ H : ‖yn − v‖2 ≤ ‖xn − v‖2 + αn

(‖x0‖2 + 2 〈xn − x0, v〉

)+ 2 (1− αn) 〈xn − v, en〉 + ‖en‖2

,

Wn = z ∈ C : 〈z − xn, x0 − xn〉 ≤ 0 ,xn+1 = PHn∩Wnx0, n = 0, 1, 2, . . .

(1.3)

converges strongly to PT−10x0 provided(i) αn → 0,(ii) infn cn > 0, and(iii) ‖en‖ → 0.

Lemma 1.2 ([8], Theorem 4.2). The sequence xn generated by the algorithm:x0 ∈ H chosen arbitrarily,yn = αnxn + (1− αn)Jcn (xn + en) ,Hn =

v ∈ H : ‖yn − v‖2 ≤ ‖xn − v‖2 + 2 (1− αn) 〈xn − v, en〉 + ‖en‖2

, n ≥ 0

Wn = z ∈ C : 〈z − xn, x0 − xn〉 ≤ 0 , n ≥ 0xn+1 = PHn∩Wnx0, n ≥ 0

(1.4)

converges strongly to PT−10x0 provided(i) αn ≤ 1− δ for some δ ∈ (0, 1),(ii) infn cn > 0, and(iii) ‖en‖ → 0.

The purpose of this paper is to employ Kamimura and Takahashi’s [3] idea to prove some strong convergence theoremsin uniformly smooth and uniformly convex Banach spaces which can be regarded as extensions of those in Yanes and Xu [8].The paper is organized as follows: in Section 2, we will list some definitions and lemmas which will be used in our paper.In Section 3, we shall extend (1.3) and (1.4) to new projection algorithms with errors and show the strong convergencetheorems. In Section 4, we will extend the results introduced in Section 3 to the cases of approximating common zeros offinitely many maximal monotone operators.

2. Preliminaries

Let E be a real Banach spacewith norm ‖·‖, and let E∗ be the dual space of E. The normalized dualitymapping J : E → 2E∗

is defined as follows:

Jx =x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖2 =

∥∥x∗∥∥2 , for x ∈ E,

where 〈x, x∗〉denotes the value of x∗ ∈ E∗ at x ∈ E.Weuse symbols ‘‘s−→’’ and ‘‘

w−→’’ to represent strong andweak convergence

in E or in E∗, respectively.A multi-valued operator T : E → 2E

∗

with domain D(T ) = x ∈ E : Tx 6= ∅ and range R(T ) =⋃Tx : x ∈ D(T ) is said

to be monotone if 〈x1 − x2, y1 − y2〉 ≥ 0 for ∀xi ∈ D(T ) and yi ∈ Txi, i = 1, 2. A monotone operator T is said to be maximalmonotone if R(J + rT ) = E∗, for ∀r > 0. For a monotone operator T , we denote by T−10 = x ∈ D(T ) : 0 ∈ Tx the set ofzeros of T .

L. Wei, H. Zhou / Nonlinear Analysis 71 (2009) 341–346 343

Lemma 2.1 ([9,10]). The duality mapping J has the following properties:(1) If E is a real reflexive and smooth Banach space, then J : E → E∗ is single-valued and JE = E∗;(2) If E is a real uniformly smooth Banach space, then J : E → E∗ is uniformly continuous on each bounded subset of E;(3) If E is a real smooth and uniformly convex Banach space, then J−1 : E∗ → E is also a duality mapping and is uniformly

continuous on each bounded subset of E∗.

Lemma 2.2 ([5–7,10]). Let E be a real smooth and uniformly convex Banach space, T : E → 2E∗

be amaximalmonotone operator,then T−10 is a closed and convex subset of E and the graph of T , G(T ), is demi-closed in the following sense: ∀ xn ⊂ D(T ),xn

w−→ x(n→∞), ∀yn ∈ Txn, yn

s−→ y(n→∞)⇒ x ∈ D(T ) and y ∈ Tx.

Definition 2.1. Let E be a real smooth and uniformly convex Banach space and T : E → 2E∗

be a maximal monotoneoperator. For ∀r > 0, define the operator Q Tr : E → E by Q Tr x = (J + rT )

−1Jx.

Definition 2.2. Let E be a real smooth Banach space, then the Lyapunov functional ϕ : E × E → R+ is defined as follows:

ϕ(x, y) = ‖x‖2 − 2〈x, Jy〉 + ‖y‖2 , ∀x, y ∈ E.

Lemma 2.3 ([3,5–7]). Let E be a real reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed and convexsubset of E, and let x ∈ E. Then there exists a unique element x0 ∈ C such that ϕ(x0, x) = min ϕ(z, x) : z ∈ C.

In this case, we can define the mapping QC of E onto C by QCx = x0, for ∀x ∈ E. It is easy to see that, in a Hilbert space,the mapping QC is coincident with the metric projection PC .

Lemma 2.4 ([3,5–7]). Let E be a real reflexive, strictly convex and smooth Banach space, let C be a nonempty closed and convexsubset of E, and let x ∈ E. Then

ϕ(y,QCx)+ ϕ(QCx, x) ≤ ϕ(y, x)

for ∀y ∈ C.

Lemma 2.5 ([3,5–7]). Let E be a real smooth and uniformly convex Banach space, and let xn and yn be two sequences of E. Ifeither xn or yn is bounded and ϕ(xn, yn)→ 0 as n→∞, then xn − yn

s−→ 0, as n→∞.

Lemma 2.6 ([5]). Let E be a real reflexive, strictly convex and smooth Banach space, T : E → 2E∗

be a maximal monotoneoperator with T−10 6= ∅, then for ∀x ∈ E, y ∈ T−10 and r > 0, we have

ϕ(y,Q Tr x)+ ϕ(QTr x, x) ≤ ϕ(y, x).

Lemma 2.7 ([3,5–7]). Let E be a real smooth Banach space, let C be a convex subset of E, let x ∈ E and let x0 ∈ C. Then

ϕ(x0, x) = infϕ(z, x) : z ∈ C

if and only if

〈z − x0, Jx0 − Jx〉 ≥ 0, ∀z ∈ C .

3. Strong convergence to zeros of maximal monotone operator

Throughout this section, unless otherwise stated, we assume that E is a real uniformly smooth and uniformly convexBanach space, and T : E → 2E

∗

is a maximal monotone operator with T−10 6= ∅.

Theorem 3.1. The sequence xn generated by the algorithm:

x0 ∈ E, r0 > 0yn = Q Trn (xn + en) ,Jzn = αnJxn + (1− αn)Jyn,Hn = v ∈ E : ϕ (v, zn) ≤ αnϕ (v, xn)+ (1− αn) ϕ (v, xn + en) ,Wn = z ∈ E : 〈z − xn, Jx0 − Jxn〉 ≤ 0 ,xn+1 = QHn∩Wnx0, n = 0, 1, 2, . . .

(3.1)

converges strongly to QT−10x0 provided(i) αn ⊂ [0, 1] with αn ≤ 1− δ for some δ ∈ (0, 1);(ii) rn ⊂ (0,+∞) with infn≥0 rn > 0, and(iii) the error sequence en ⊂ E such that ‖en‖ → 0, as n→∞.

344 L. Wei, H. Zhou / Nonlinear Analysis 71 (2009) 341–346

Proof. We split the proof into five steps.Step 1. Hn is a closed and convex subset of E.Notice the fact that

ϕ (v, zn) ≤ αnϕ (v, xn)+ (1− αn) ϕ (v, xn + en)⇔‖zn‖2 − αn ‖xn‖2 − (1− αn) ‖xn + en‖2 ≤ 2 〈v, Jzn − αnJxn − (1− αn) J (xn + en)〉 .

We can easily know that Hn is a closed and convex subset of E.It is obvious thatWn is also a closed and convex subset of E. Therefore, both Hn andWn are closed convex sets of E.Step 2. T−10 ⊂ Hn ∩Wn for each nonnegative integer n.To observe this, take p ∈ T−10.From Definition 2.1, we know that there exists y0 ∈ E such that y0 = Q Tr0 (x0 + e0). It follows from Lemma 2.6 that

ϕ (p, y0) ≤ ϕ (p, x0 + e0). Then

ϕ (p, z0) ≤ α0ϕ (p, x0)+ (1− α0) ϕ (p, y0) ≤ α0ϕ (p, x0)+ (1− α0) ϕ (p, x0 + e0) ,

which implies that p ∈ H0. On the other hand, it is clear that p ∈ W0 = E. Then p ∈ H0 ∩W0, and therefore x1 = QH0∩W0x0is well defined.Suppose that p ∈ Hn−1 ∩Wn−1 and xn is well defined for some n ≥ 1. Again by Definition 2.1, we obtain yn ∈ E such that

yn = Q Trn (xn + en). Then Lemma 2.6 implies that ϕ (p, yn) ≤ ϕ (p, xn + en). Thus

ϕ (p, zn) ≤ αnϕ (p, xn)+ (1− αn) ϕ (p, yn) ≤ αnϕ (p, xn)+ (1− αn) ϕ (p, xn + en) ,

which implies that p ∈ Hn. It follows from Lemma 2.7 that

〈p− xn, Jx0 − Jxn〉 =⟨p− QHn−1∩Wn−1x0, Jx0 − JQHn−1∩Wn−1x0

⟩≤ 0

which implies that p ∈ Wn. Therefore p ∈ Hn ∩ Wn, and hence xn+1 = QHn∩Wnx0 is well defined. Then by induction, thesequence generated by (3.1) is well defined, for each n ≥ 0. Moreover, T−10 ⊂ Hn ∩Wn for each nonnegative integer n.Step 3. xn is a bounded sequence of E.In fact, for ∀p ∈ T−10 ⊂ Hn ∩Wn, it follows from Lemma 2.4 that

ϕ(p,QWnx0)+ ϕ(QWnx0, x0) ≤ ϕ(p, x0).

From the definition ofWn and Lemmas 2.3 and 2.7, we know that xn = QWnx0, which implies that

ϕ(p, xn)+ ϕ(xn, x0) ≤ ϕ(p, x0).

Therefore xn is bounded.Step 4. ω (xn) ⊂ T−10, where ω (xn) denotes the set consisting all of the weak limit points of xn.From the facts that xn = QWnx0, xn+1 ∈ Wn and Lemma 2.4, we have

ϕ(xn+1, xn)+ ϕ(xn, x0) ≤ ϕ(xn+1, x0).

Therefore limn→∞ ϕ (xn, x0) exists. Thenϕ (xn+1, xn)→ 0,which implies from Lemma2.5 that xn+1−xns−→ 0, as n→∞.

Since xn+1 ∈ Hn, then

ϕ (xn+1, zn) ≤ αnϕ (xn+1, xn)+ (1− αn) ϕ (xn+1, xn + en) . (3.2)

Notice that

ϕ (xn+1, xn + en)− ϕ (xn+1, xn) = ‖xn + en‖2 − ‖xn‖2 + 2 〈xn+1, Jxn − J (xn + en)〉 . (3.3)

Since J : E → E∗ is uniformly continuous on each bounded subset of E and ‖en‖ → 0, we know from (3.3) thatϕ (xn+1, xn + en) → 0, which implies that ϕ (xn+1, zn) → 0 by (3.2). Using Lemma 2.5, we know that xn+1 − zn

s−→ 0,

as n→∞. Since both J : E → E∗ and J−1 : E∗ → E are uniformly continuous on bounded subsets, then we have from (3.1)that xn − yn

s−→ 0, as n→∞.

From step 3, we know thatω (xn) 6= ∅. Then ∀q ∈ ω (xn), there exists a subsequence of xn, for simplicity we still denoteit by xn, such that xn

w−→ q, as n→∞. Therefore yn

w−→ q, as n→∞.

From iterative scheme (3.1), we know that there exists vn ∈ Tyn such that rnvn = J(xn + en) − Jyn. Then vns−→ 0, as

n→∞. Lemma 2.2 implies that q ∈ T−10.Step 5. xn

s−→ q∗ = QT−10x0, as n→∞.

Letxnibe any subsequence of xn which is weakly convergent to q ∈ T−10. Since xn+1 = QHn∩Wnx0 and q

∗∈ T−10 ⊂

Hn ∩Wn, we have ϕ(xn+1, x0) ≤ ϕ(q∗, x0). Then

ϕ(xn, q∗

)= ϕ (xn, x0)+ ϕ

(x0, q∗

)− 2〈xn − x0, Jq∗ − Jx0〉

≤ ϕ(q∗, x0

)+ ϕ

(x0, q∗

)− 2〈xn − x0, Jq∗ − Jx0〉,

L. Wei, H. Zhou / Nonlinear Analysis 71 (2009) 341–346 345

which yields

lim supi→∞

ϕ(xni , q

∗)≤ ϕ

(q∗, x0

)+ ϕ

(x0, q∗

)− 2〈q− x0, Jq∗ − Jx0〉 = 2〈q∗ − q, Jq∗ − Jx0〉 ≤ 0.

Hence ϕ(xni , q∗) → 0, as i → ∞. It follows from Lemma 2.5 that xni

s−→ q∗, as i → ∞. This means that the whole

sequence xn converges weakly to q∗ and that each weakly convergent subsequence of xn converges strongly to q∗.Therefore, xn

s−→ q∗ = QT−10x0, as n→∞. This completes the proof.

Theorem 3.2. The sequence xn generated by the algorithm:

x0 ∈ E, r0 > 0yn = Q Trn (xn + en) ,Jzn = αnJx0 + (1− αn)Jyn,Hn = v ∈ E : ϕ (v, zn) ≤ αnϕ (v, x0)+ (1− αn) ϕ (v, xn + en) ,Wn = z ∈ E : 〈z − xn, Jx0 − Jxn〉 ≤ 0 ,xn+1 = QHn∩Wnx0, n = 0, 1, 2, . . .

(3.4)

converges strongly to QT−10x0 provided(i) αn ⊂ [0, 1] with αn → 0;(ii) rn ⊂ (0,+∞) with infn≥0 rn > 0, and(iii) the error sequence en ⊂ E such that ‖en‖ → 0, as n→∞.Proof. By slightly modifying the corresponding proof in Theorem 3.1, we can easily obtain the result.

Remark 3.1. If E is reduced to a Hilbert space, then QT−10 is reduced to the metric projection PT−10 and Q Trn is reduced toJTrnx = (I + rnT )

−1x in Lemmas 1.1 and 1.2. Therefore Theorems 3.1 and 3.2 can be considered as extensions of Lemmas 1.1and 1.2 to a real uniformly smooth and uniformly convex Banach space.

Remark 3.2. Compared with algorithm (1.2) introduced by Kamimura and Takahashi in [3], algorithms (3.1) and (3.2) havethe error sequence en.

4. Strong convergence to common zeros of finitely many maximal monotone operators

Throughout this section, unless otherwise stated, we assume that E is a real uniformly smooth and uniformly convexBanach space, and Ti : E → 2E

∗

are maximal monotone operators, for i = 1, 2, . . . ,m. Denote by D :=⋂mi=1 T

−1i 0 and

suppose that D 6= ∅.

Theorem 4.1. The sequence xn generated by the algorithm:

x0 ∈ E, r0,i > 0, i = 1, 2, . . . ,myn,i = Q Tirn,i (xn + en) , i = 1, 2, . . . ,mJzn,i = αn,iJxn + (1− αn,i)Jyn,i, i = 1, 2, . . . ,mHn,i =

v ∈ E : ϕ

(v, zn,i

)≤ αn,iϕ (v, xn)+

(1− αn,i

)ϕ (v, xn + en)

, i = 1, 2, . . . ,m

Hn :=m⋂i=1

Hn,i,

Wn = z ∈ E : 〈z − xn, Jx0 − Jxn〉 ≤ 0 ,xn+1 = QHn∩Wnx0, n = 0, 1, 2, . . .

(4.1)

converges strongly to QDx0 provided(i)αn,i⊂ [0, 1] with αn,i ≤ 1− δ, for some δ ∈ (0, 1), i = 1, 2, . . . ,m and n = 0, 1, 2, . . .;

(ii)rn,i⊂ (0,+∞) with infn≥0 rn,i > 0, i = 1, 2, . . . ,m; and

(iii) ‖en‖ → 0, as n→∞.

Theorem 4.2. The sequence xn generated by the algorithm

x0 ∈ E, r0,i > 0, i = 1, 2, . . . ,myn,i = Q Tirn,i (xn + en) , i = 1, 2, . . . ,mJzn,i = αn,iJx0 + (1− αn,i)Jyn,i, i = 1, 2, . . . ,mHn,i =

v ∈ E : ϕ

(v, zn,i

)≤ αn,iϕ (v, x0)+

(1− αn,i

)ϕ (v, xn + en)

, i = 1, 2, . . . ,m

Hn :=m⋂i=1

Hn,i,

Wn = z ∈ E : 〈z − xn, Jx0 − Jxn〉 ≤ 0 ,xn+1 = QHn∩Wnx0, n = 0, 1, 2, . . .

(4.2)

346 L. Wei, H. Zhou / Nonlinear Analysis 71 (2009) 341–346

converges strongly to QDx0 provided

(i) αn,i → 0, n→∞, i = 1, 2, . . . ,m;(ii) infn≥0 rn,i > 0, i = 1, 2, . . . ,m, and(iii) ‖en‖ → 0, as n→∞.

Proof. Similar to the proof of Theorem 3.1, we can prove that

(1) Both Hn,i andWn are closed convex sets of E, for i = 1, 2, . . . ,m and n ≥ 0.(2) D ⊂ Hn ∩Wn for each nonnegative integer n.(3) xn is a bounded sequence of E.(4) There exists q ∈ E such that xn

s−→ q, as n→∞.

(5) q = QDx0.

So both of the results of Theorems 4.1 and 4.2 are available.

References

[1] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898.[2] M.V. Solodov, B.F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. 87 (2000) 189–202.[3] S. Kamimura, W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (3) (2003) 938–945.[4] U. Mosco, Perturbation of variational inequalities, in: Nonlinear Functional Analysis (Proc. Sympos. Pure Math. vol. XVIII, Part1, Chicago, IL,1968),American Mathematical Society, Providence, RI, 1970, pp. 182–194.

[5] L. Wei, H.Y. Zhou, The new iterative scheme with errors of zero point for maximal monotone operator in Banach space, Math. Appl. 19 (1) (2006)101–105 (in Chinese).

[6] L. Wei, H.Y. Zhou, The iterative convergence theorem of zero point for maximal monotone operator in Banach space and its application, Math. PracticeTheory 36 (5) (2006) 235–241 (in Chinese).

[7] L. Wei, A new iterative algorithm with errors for maximal monotone operators and its applications. in: Proc. ICMLC2005 Conference, Guangzhou,August 2005, pp. 969–975.

[8] C.M. Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (11) (2006) 2400–2411.[9] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.[10] D. Pascali, S. Sburlan, Nonlinear Mappings of Monotone type, Sijthoff & Noordhoff, Romania, 1978.