Iterative Schemes for Zero Points of Maximal Monotone Operators and Fixed Points of

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Volume 2008, Article ID 168468,12pages doi:10.1155/2008/168468

Research Article

Iterative Schemes for Zero Points of Maximal Monotone Operators and Fixed Points of

Nonexpansive Mappings and Their Applications

Li Wei¹and Yeol Je Cho²

1School of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang 050061, China

2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea

Correspondence should be addressed to Yeol Je Cho,[email protected] Received 16 August 2007; Accepted 25 November 2007

Recommended by Massimo Furi

Two iterative schemes for finding a common element of the set of zero points of maximal monotone operators and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space are obtained. Two strong convergence theorems are obtained which extend some previous work. Moreover, the applications of the iterative schemes are demonstrated.

Copyrightq2008 L. Wei and Y. J. Cho. 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.

1. Introduction and preliminaries

In this paper, we will present two iterative schemes with errors which are proved to be strongly convergent to a common element of the set of zero points of maximal monotone operators and the set of fixed points of nonexpansive mappings with respect to the Lyapunov functional in real uniformly smooth and uniformly convex Banach spaces. Moreover, it is shown that some results proposed by Martinez-Yanes and Xu in1and Solodov and Svaiter in2are special cases of ours. Finally, we will demonstrate the applications of our iterative schemes on both finding the minimizer of a proper convex and lower semicontinuous function and solving the variational inequalities.

LetEbe a real Banach space with norm·and letE^∗be its dual space. The normalized duality mappingJ:E→2^E^∗is defined as follows:

Jx

x^∗∈E^∗:x, x^∗x²x^∗²

∀x∈E, 1.1

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wherex, x^∗denotes the value ofx^∗∈E^∗atx∈E. We use symbols “→^s ” and “→^w ” to represent strong and weak convergence inEor inE^∗, respectively.

A multivalued operatorT :E → 2^E^∗with domainDT {x∈E :Tx /∅}and range

RT

{Tx:x∈DT}is said to be monotone ifx1−x2, y1−y2 ≥0 for allxi∈DTand yi∈Txi,i1,2. A monotone operatorTis said to be a maximal monotone ifRJrT E^∗for allr >0. For a monotone operatorT, we denote byT⁻¹0{x∈DT: 0∈Tx}the set of zero points ofT. For a single-valued mappingS:E→E, we denote by FixS {x∈E:Sxx}

the set of fixed points ofS.

Lemma 1.1see3,4. The duality mappingJhas the following properties.

1IfEis a real reflexive and smooth Banach space, thenJ:E→E^∗is single-valued.

2For allx∈Eandλ∈R,Jλx λJx.

3IfEis a real uniformly convex and uniformly smooth Banach space, thenJ⁻¹:E^∗→Eis also a duality mapping. Moreover,J : E →E^∗ andJ⁻¹ : E^∗ → Eare uniformly continuous on each bounded subset ofEorE^∗, respectively.

Lemma 1.2see4. LetEbe a real smooth and uniformly convex Banach space and letT:E→2^E^∗ be a maximal monotone operator. ThenT⁻¹0 is a closed and convex subset ofEand the graph ofT,GT, is demiclosed in the following sense: for all{xn} ⊂DTwithxn→w xinEand for allyn∈Txnwith yn s

→ yinE^∗,x∈DTandy∈Tx.

Definition 1.3. LetEbe a real smooth and uniformly convex Banach space and letT :E→2^E^∗ be a maximal monotone operator. For allr > 0, define the operatorQ_r^T : E → Eby Q^T_rx JrT⁻¹Jxfor allx∈E.

Definition 1.4see 5. LetEbe a real smooth Banach space. Then the Lyapunov functional ϕ:E×E→Ris defined as follows:

ϕx, y x²−2 x, jy

y² ∀x, y∈E, jy∈Jy. 1.2 Lemma 1.5see5. LetEbe a real reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed and convex subset ofE, and letx∈E. Then there exists a unique elementx0∈Csuch thatϕx0, x min{ϕz, x:z∈C}.

Define the mappingQC ofEontoCbyQCxx0for allx∈E.QCis called the generalized projection operator fromEontoC. It is easy to see thatQCis coincident with the metric projection P_Cin a Hilbert space.

Lemma 1.6see5. LetEbe a real reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed and convex subset ofE, and letx∈E. Then, for ally∈C,

ϕ

y, QCx ϕ

QCx, x

≤ϕy, x. 1.3

Lemma 1.7see6. LetEbe a real smooth and uniformly convex Banach space and let{xn}and {yn}be two sequences ofE. If either{xn}or{yn}is bounded andϕxn, yn → 0 asn → ∞, then xn−yn s

→ 0 asn→ ∞.

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Lemma 1.8see7. LetEbe a real reflexive, strictly convex and smooth Banach space and letT : E→2^E^∗be a maximal monotone operator withT⁻¹0/∅. Then for allx∈E, y∈T⁻¹0 andr >0, one hasϕ(y, Q_r^Tx)ϕ(Q^T_rx, x)≤ϕ(y, x).

Lemma 1.9see5. LetEbe a real smooth Banach space, letCbe a convex subset ofE, letx∈E, and letx0 ∈C. Thenϕx0, x inf{ϕz, x :z ∈C}if and only ifz−x0, Jx0−Jx ≥0 for all z∈C.

Definition 1.10. LetEbe a real Banach space. ThenS:E→Eis said to be nonexpansive with respect to the Lyapunov functional ifϕSx, Sy≤ϕx, yfor allx, y∈E.

Remark 1.11. If Eis a real Hilbert space H, then Sis a nonexpansive mapping in the usual sense:Sx−Sy ≤ x−yfor allx, y∈H.

Lemma 1.12. LetEbe a real smooth and uniformly convex Banach space. IfS:E→Eis a mapping which is nonexpansive with respect to the Lyapunov functional, then FixSis a convex and closed subset ofE.

Proof. In fact, we only need to prove the case that FixS/∅. For allx, y∈FixSandt∈0,1, letztx 1−ty. Then we have

ϕz, Sz t

x²−2x, JSzSz²

1−t

y²−2y, JSzSz²

−tx²−1−ty²z²

tϕx, Sz 1−tϕy, Sz−tx²−1−ty²z²

≤tϕx, z 1−tϕy, z−tx²−1−ty²z² ϕz, z 0.

1.4

By usingLemma 1.7, we know thatzSz, which implies that FixSis a convex subset ofE.

For allxn ∈ FixSsuch thatxn s

→ x, it follows thatϕSxn, Sx ≤ ϕxn, x → 0.Lemma 1.7 implies thatSxn→s Sxasn→ ∞. Sox∈FixS.

2. Strong convergence theorems

Throughout this section, we assume that E is a real uniformly smooth and uniformly convex Banach space, S : E → E is nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous andT : E → 2^E^∗ is a maximal monotone operator with T⁻¹0 FixS/∅.

Theorem 2.1. The sequence{xn}generated by the following scheme:

x0∈E, r0>0, ynQ^T_r_n

xnen , JznαnJxn

1−αn Jyn, unSzn,

Hn

v∈E:ϕ v, un

≤ϕ v, zn

≤αnϕ v, xn

1−αn

ϕ

v, xnen , Wn

z∈E:z−xn, Jx0−Jxn ≤0 , xn1QHn∩Wnx0 ∀n≥0,

2.1

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converges strongly toQT⁻¹0 FixSx0provided

i{αn} ⊂0,1is a sequence such thatαn≤1−δ, for someδ∈0,1;

ii{rn} ⊂0,∞is a sequence such that infn≥0rn>0;

iii{en} ⊂Eis a sequence such thaten →0 asn→ ∞.

Proof. We split the proof into five steps.

Step 1. BothHnandWnare closed and convex subsets ofE.

Noting the facts that ϕ

v, zn

≤αnϕ v, xn

1−αn

ϕ

v, xnen

⇐⇒zn²−αnxn²−

1−αnxnen²≤2

v, Jzn−αnJxn− 1−αn

J

xnen , ϕ

v, un

≤ϕ v, zn

⇐⇒zn²−un²≥2v, Jzn−Jun,

2.2

we can easily know thatHnis a closed and convex subset ofE. It is obvious thatWnis also a closed and convex subset ofE.

Step 2. T⁻¹0 FixS⊂Hn∩Wnfor each nonnegative integern.

To observe this, takep ∈ T⁻¹0 FixS. From the definition of the maximal monotone operator, we know that there existsy0∈Esuch thaty0Q^T_r₀x0e0. It follows fromLemma 1.8 thatϕp, y0≤ϕp, x0e0. Then

ϕ p, u0

≤ϕ p, z0

≤α0ϕ p, x0

1−α0

ϕ p, y0

≤α0ϕ p, x0

1−α0

ϕ

p, x0e0

, 2.3 which implies thatp∈H0.

On the other hand, it is clear thatp ∈ W0 E. Thenp ∈ H0∩W0 and thereforex1 QH0∩W0x0are well defined.

Suppose thatp ∈Hn−1∩Wn−1andxnis well defined for somen≥1. Then there exists yn∈Esuch thatynQ^Tr_nxnen.Lemma 1.8implies thatϕp, yn≤ϕp, xnen. Thus

ϕ p, un

≤ϕ p, zn

≤αnϕ p, xn

1−αn

ϕ p, yn

≤ϕ p, zn

≤αnϕ p, xn

1−αn

ϕ

p, yn 2.4

which implies thatp∈Hn. It follows fromLemma 1.9that

p−xn, Jx0−Jxnp−QHn−1∩Wn−1x0, Jx0−JQHn−1∩Wn−1x0 ≤0, 2.5 which implies thatp ∈ Wn. Hencexn1 QH_n∩W_nx0 is well defined. Then, by induction, the sequence generated by2.1is well defined andT⁻¹0 FixS⊂Hn∩Wnfor eachn≥0.

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Step 3. {xn}is a bounded sequence ofE.

In fact, for allp∈T⁻¹0 FixS⊂Hn∩Wn, it follows fromLemma 1.6that ϕ

p, QW_nx0 ϕ

QW_nx0, x0

≤ϕ p, x0

. 2.6

From the definition ofWnand Lemmas1.5and1.9, we know thatxn QW_nx0, which implies thatϕp, xn ϕxn, x0≤ϕp, x0. Therefore,{xn}is bounded.

Step 4. ωxn ⊂ T⁻¹0 FixS, where ωxndenotes the set consisting all of the weak limit points of{xn}.

From the factsxnQW_nx0,xn1∈WnandLemma 1.6, we have ϕ

xn1, xn ϕ

xn, x0

≤ϕ xn1, x0

· 2.7

Therefore, limn→∞ϕxn, x0exists. Thenϕxn1, xn →0, which implies fromLemma 1.7that xn1−xn s

→ 0 asn→ ∞. Sincexn1∈Hn, we have ϕ

xn1, un

≤ϕ

xn1, zn

, 2.8

ϕ

xn1, zn

≤αnϕ

xn1, xn

1−αn ϕ

xn1, xnen

· 2.9

Notice that ϕ

xn1, xnen

−ϕ

xn1, xn

xnen²−xn²2

xn1, Jxn−J

xnen

. 2.10

SinceJ :E→E^∗is uniformly continuous on each bounded subset ofEanden →0, we know from2.10thatϕxn1, xnen→ 0, which implies thatϕxn1, zn → 0 by2.9. Moreover, 2.8implies thatϕxn1, un→0 asn→ ∞. UsingLemma 1.7, we know thatxn1−zn s

→ 0, xn1−un→s 0 asn→ ∞. Since bothJ :E→E^∗andJ⁻¹:E^∗→Eare uniformly continuous on bounded subsets, we havexn−yn s

→ 0 asn→ ∞. From Step3, we know thatωxn/∅. Then, for allq ∈ωxn, there exists a subsequence of{xn}, for simplicity, we still denote it by{xn} such thatxn w

→ qasn→ ∞. Therefore,un w

→ q,zn w

→ qandyn w

→ qasn→ ∞. SinceS:E→E is weakly continuous andunSzn, thenq∈FixS. From the iterative scheme2.1, we know that there existsvn∈Tynsuch thatrnvnJxnen−Jyn. Thenvn s

→ 0 asn→ ∞.Lemma 1.2 implies thatq∈T⁻¹0.

Step 5. xn→s q^∗QT⁻¹0 FixSx0asn→ ∞.

Let{xni}be any subsequence of{xn}which is weakly convergent toq∈T⁻¹0 FixS.

Sincexn1QHn∩Wnx0andq^∗∈T⁻¹0 FixS⊂Hn∩Wn, we haveϕxn1, x0≤ϕq^∗, x0. Then it follows that

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

, 2.11

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which yields

lim sup

n→∞ ϕ xni, q^∗

≤ϕ q^∗, x0

ϕ x0, q^∗

−2q−x0, Jq^∗−Jx0

2q^∗−q, Jq^∗−Jx0 ≤0. 2.12

Henceϕxni, q^∗→0 asi→ ∞. It follows fromLemma 1.7thatxni

→s q^∗asi→ ∞. This means that the whole sequence{xn}converges weakly toq^∗and each weakly convergent subsequence of{xn}converges strongly toq^∗. Therefore,xn s

→ q^∗QT⁻¹0 FixSx0asn→ ∞.

Remark 2.2. If Eis reduced to a real Hilbert space H and S ≡ I, thenQ^T_r_n equals to J_r^T_n IrnT⁻¹. So the iterative scheme2.1is reduced to the following one introduced by Yanes and Xu in1:

x0∈Hchosen arbitrarily, ynαnxn 1−αnJ_r^T_nxnen, Hn

v∈H :yn−v²≤xn−v²21−αnxn−v, enen² , Wn

z∈H :z−xn, x0−xn ≤0 , xn1PHn∩Wnx0, ∀n≥0.

2.13

They proved that, ifT⁻¹0/∅, then the sequence{xn}generated by2.13converges strongly toPT⁻¹0x0provided

i{αn} ⊂0,1is a sequence such thatαn≤1−δfor someδ∈0,1;

ii{rn} ⊂0,∞is a sequence such that infnrn>0;

iii{en} ⊂Eis a sequence such thaten →0.

Remark 2.3. IfE is reduced to a real Hilbert spaceH,αn ≡ 0,en ≡ 0 andS ≡ I, then 2.1 includes the following iterative scheme introduced by Solodov and Svaiter in2:

x0∈H, 0vn 1

rn

yn−xn

, vn∈Tyn, Hn

z∈H:z−yn, vn ≤0 , Wn

z∈H:z−xn, x0−xn ≤0 , xn1PHn∩Wnx0, ∀n≥0.

2.14

They proved that, ifT⁻¹0/∅ and lim infn→∞rn > 0, then the sequence generated by2.14 converges strongly toPT⁻¹0x0.

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Corollary 2.4. Suppose thatEandS are the same as those inTheorem 2.1. Fori 1,2, . . . , m, let Ti : E → 2^E^∗ be maximal monotone operators. DenoteD: ^m_i1T_i⁻¹0 Fix(S) and suppose that D /∅. Then the sequence{xn}generated by

x0∈E, r0, i>0, i1,2, . . . , m, yn,iQ^Trn,iⁱ

xnen

, i1,2, . . . , m, Jzn,iαn,iJxn

1−αn,i

Jyn,i, i1,2, . . . , m, un,iSzn,i, i1,2, . . . , m,

Hn,i

v∈E:ϕ v, un,i

≤ϕ v, zn,i

≤αn,iϕ v, xn

1−αn,i ϕ

v, xnen

, i1,2, . . . , m, Hn:^m

i1

Hn,i, Wn

z∈E:z−xn, Jx0−Jxn ≤0 , xn1QH_n∩W_nx0 ∀n≥0,

2.15 converges strongly toQDx0provided

i{αn,i} ⊂ 0,1is a sequence such thatαn,i ≤1−δ, for someδ ∈0,1,i 1,2, . . . , mand n≥0; 1,2, . . .,

ii{rn,i} ⊂0,∞is a sequence such that infn≥0rn,i>0 fori1,2, . . . , m;

iii{e} ⊂Eis a sequence such thaten →0 asn→ ∞.

Similar to the proof ofTheorem 2.1, we have the following result.

Theorem 2.5. The sequence{xn}generated by

x0∈E, r0>0, ynQ^T_r_n

xnen , JznαnJx0

1−αn Jyn, unSzn,

Hn

v∈E:ϕ v, un

≤ϕ v, zn

≤αnϕ v, x0

1−αn

ϕ

v, xnen , Wn

z∈E:z−xn, Jx0−Jxn ≤0 , xn1QHn∩Wnx0 ∀n≥0,

2.16

converges strongly toQT⁻¹0 FixSx0provided

i{αn} ⊂0,1is a sequence such thatαn→0 asn→ ∞;

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Remark 2.6. IfEis reduced to a real Hilbert spaceHandS≡I, then the iterative scheme2.16 is reduced to the following one, which is similar to that in1:

x0∈Hchosen arbitrarily, ynαnx0 1−αnJ_r^T_nxnen, Hn

v∈H:yn−v²≤xn−v²αnx0²2

xn−x0, v 2

1−αn

xn−v, en

1−αnen²−αnxn² , Wn

z∈H:

z−xn, x0−xn

≤0 , xn1PHn∩Wnx0 ∀n≥0.

2.17

Corollary 2.7. Suppose thatE,S,Ti, andDare the same as those inCorollary 2.4. IfD /∅, then the sequence{xn}generated by

x0∈E, r0,i>0, yn,iQ^Trn,iⁱ

xnen , Jzn,iαn,iJx0

1−αn,i Jyn,i, un,iSzn,i,

Hn,i

v∈E:ϕ v, un,i

≤ϕ v, zn,i

≤αn,iϕ v, x0

1−αn,i ϕ

v, xnen , Hn :^m

i1

Hn,i, i1,2, . . . , m, Wn

z∈E:z−xn, Jx0−Jxn ≤0 , xn1QH_n∩W_nx0 ∀n≥0,

2.18

converges strongly toQDx0provided

i{αn,i} ⊂0,1is a sequence such thatαn,i→0 asn→ ∞fori1,2, . . . , m;

ii{rn,i} ⊂0,∞is a sequence such that infn≥0rn,i>0 fori1,2, . . . , m;

3. Applications to minimization problem

Definition 3.1. Letf:E → −∞,∞be a proper convex and lower semicontinuous function.

Then the subdiﬀerential∂foffis defined by

∂f z

v∈E^∗:f y

≥f z

y−z, v

, ∀y∈E

∀z∈E· 3.1

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Theorem 3.2. Let E,S, {αn},{rn}, and {en}be the same as those in Theorem 2.1. Let f : E →

−∞,∞be a proper convex and lower semicontinuous function. Let{xn}be the sequence generated by

x0∈E, r0>0, ynarg min

z∈E

fz 1

2rnzn²− 1 rn

z,J

xnen , JznαnJxn

1−αn Jyn, unSzn,

Hn

v∈E:ϕ v, un

≤ϕ v, zn

≤αnϕ v, xn

1−αn

ϕ

v, xnen , Wn

z∈E:z−xn, Jx0−Jxn ≤0 , xn1QHn∩Wnx0 ∀n≥0.

3.2

If∂f⁻¹0 Fix(S)/∅, then{xn}converges strongly toQ_∂f⁻¹₀ _FixSx0.

Proof. Sincef:E→−∞,∞is a proper convex and lower semicontinuous function, the sub- diﬀerential∂foffis a maximal monotone operator fromEintoE^∗. We also know that

ynarg min

z∈E

fz 1

2rnzn²− 1 rn

z, J

xnen

3.3

is equivalent to

0∈∂f yn

1

rnJyn− 1 rnJ

xnen

· 3.4

Thus we haveyn Q^∂frnxnenand soTheorem 2.1implies that{xn}converges strongly to Q_∂f⁻¹₀ _FixSx0asn→ ∞.

Similarly, we have the following theorem.

Theorem 3.3. Let E, S, {αn}, {rn}, and {en} be the same as those in Theorem 2.5. Let f:E →

−∞,∞be a proper convex and lower semicontinuous function. Let{xn}be the sequence generated by

x0∈E, r0>0, ynarg min

z∈E

fz 1

2rn

zn²− 1 rn

z, J

xnen , JznαnJx0

1−αn Jyn, unSzn,

Hn

v∈E:ϕ v, un

≤ϕ v, zn

≤αnϕ v, x0

1−αn

ϕ

v, xnen , Wn

z∈E:z−xn, Jx0−Jxn ≤0 , xn1QHn∩Wnx0 ∀n≥0.

3.5

If∂f⁻¹0 Fix(S)/∅, then{xn}converges strongly toQ_∂f⁻¹₀ _FixSx0.

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4. Applications on solving the variational inequalities

Definition 4.1see4. LetEbe a real Banach space. A single-valued operatorA: E→E^∗is said to be hemicontinuous if it is continuous along each line segment inEwith respect to the weak^∗topology ofE^∗.

Definition 4.2. LetEbe a real Banach space and letCbe a nonempty closed and convex subset ofE. LetA:C→E^∗be a single-valued monotone operator which is hemicontinuous. Then a pointu∈Cis said to be a solution of the variational inequality forAif

y−u, Au

≥0, ∀y∈C. 4.1

We denote by VIC, Athe set of all solutions of the variational inequality forA.

Definition 4.3. LetEbe a real Banach space and letCbe a nonempty closed and convex subset ofE. We denote byNCxthe normal cone forCat a pointx∈C, that is,

NCx

x^∗∈E^∗:

y−x, x^∗

≤0, y∈C

. 4.2

In8, it is proven that the operatorT :E→2^E^∗defined by Tx

⎧⎨

⎩

AxNCx, x∈C,

∅, x /∈C, 4.3

is a maximal monotone operator. It is easy to verify thatT⁻¹0 VIC, A.

Theorem 4.4. LetE,Sbe the same as those inTheorem 2.1and letCbe a nonempty closed and convex subset ofE. LetA:C→E^∗be a single-valued monotone operator which is hemicontinuous. Let{xn} be a sequence generated by

x0∈E, r0>0, ynVI

C, A 1 rn

J−J

xnen , JznαnJxn

1−αn Jyn, unSzn,

Hn

v∈E:ϕ v, un

≤ϕ v, zn

≤αnϕ v, xn

1−αn

ϕ

v, xnen , Wn

z∈E:

z−xn, Jx0−Jxn

≤0 , xn1QHn∩Wnx0 ∀n≥0,

4.4

which converges strongly toQVIC,A FixSx0provided

i{αn} ⊂0,1is a sequence such thatαn≤1−δ, for someδ∈0,1;

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Proof. Note that

ynVI

C, A 1 rn

J−J

xnen

⇐⇒

y−yn,Ayn 1 rn

Jyn−J

xnen

≥0 ∀y∈C

⇐⇒J

xnen

∈rnTynJyn

⇐⇒yn

JrnT₋₁ J

xnen Q^Tr_n

xnen ,

4.5

whereTis the same as that inDefinition 4.3. Then the result follows fromTheorem 2.1.

Similarly, we have the following result.

Theorem 4.5. LetE,C,SandAbe the same as those inTheorem 4.4. Let{xn}be a sequence generated by

x0∈E, r0>0, ynVI

C, A 1 rn

J−J

xnen , JznαnJx0

1−αn Jyn, unSzn,

Hn

v∈E:ϕ v, un

≤ϕ v, zn

≤αnϕ v, x0

1−αn

ϕ

v, xnen , Wn

z∈E:

z−xn, Jx0−Jxn

≤0 , xn1QH_n W_nx0 ∀n≥0,

4.6

which converges strongly toQVIC,A FixSx0provided

i{αn} ⊂0,1is a sequence such thatαn→0 asn→ ∞;

Remark 4.6. It will be interesting to consider similar problems when a single mapping ”S” is replaced by an amenable semigroup S of mappings that are nonexpansive with respect to the Lyapunov functional and to combine the iterative scheme for the fixed point set deter- mined by a left regular sequence of means as demonstrated in the recent work9with that of Theorem 2.1.

Acknowledgments

This paper is supported by the National Nature Science Foundation of China Grant no.

10771050. The authors thank the reviewers for their good suggestions.

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