PROJECTIONS AND MAXIMAL MONOTONE OPERATORS IN BANACH SPACES

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PROJECTIONS AND MAXIMAL MONOTONE OPERATORS IN BANACH SPACES

TAKANORI IBARAKI, YASUNORI KIMURA, AND WATARU TAKAHASHI

Received 20 January 2002

We study a sequence of generalized projections in a reflexive, smooth, and strictly convex Banach space. Our result shows that Mosco convergence of their ranges implies their pointwise convergence to the generalized projection onto the limit set. Moreover, using this result, we obtain strong and weak convergence of resolvents for a sequence of maximal monotone operators.

1. Introduction

LetCbe a nonempty closed convex subset of a Hilbert spaceH. For an arbitrary pointxofH, consider the set{z∈C:x−z =min_y_∈_Cx−y}. It is known that this set is always a singleton. LetPCbe a mapping fromHontoCsatisfying

x−PCx=min

y∈Cx−y. (1.1)

Such a mappingPCis called themetric projection. The metric projection has the following important property:x0=PCxif and only ifx−x0,x0−y ≥0, for all y∈C.

IfCis a nonempty closed convex subset of a Banach spaceEwhose norm is Gˆateaux diﬀerentiable, then the metric projectionPChas the following property:

x0=PCxif and only if

Jx−x0

,x0−y≥0 ∀y∈C, (1.2)

whereJis a normalized duality mapping fromEtoE^∗. Likewise, ifQCis a surjec- tive sunny nonexpansive retraction on a smooth Banach spaceE, thenx0=QCx if and only if

x−x0,Jx0−y≥0 ∀y∈C. (1.3)

Copyright©2003 Hindawi Publishing Corporation Abstract and Applied Analysis 2003:10 (2003) 621–629 2000 Mathematics Subject Classification: 41A65, 47H05, 46B20 URL:http://dx.doi.org/10.1155/S1085337503207065

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Notice thatQCis identical with the metric projection ifEis a Hilbert space.

Let{Cn}be a sequence of nonempty closed convex subsets ofEand suppose that{Cn}converges toC0in a sense of Mosco [4]. In [7], Tsukada proved that {PCn}converges weakly toPC0ifEis reflexive and strictly convex. Moreover, ifE has the Kadec-Klee property, the convergence is in the strong topology. On the other hand, Kimura and Takahashi [3] proved the following. Suppose that each Cnis a sunny nonexpansive retract,Eis a reflexive Banach space with a uniformly Gˆateaux diﬀerentiable norm, and every weakly compact convex subset ofEhas the fixed-point property for nonexpansive mappings. If the duality mappingJis weakly sequentially continuous, thenQCnconverges strongly toQC0.

One of the purposes of this paper is to obtain an analogous result for a generalized projectionΠCwhich was defined by Alber [1]. A weak convergence theorem is inSection 3and a strong convergence theorem appears inSection 4.

InSection 5, we discuss sequences of maximal monotone operators. For a single operatorAwithA⁻¹0= ∅, it is known that, for everyx^∗∈E^∗, (J+λA)⁻¹x^∗ converges strongly toπ_A^∗⁻¹0x^∗asλ→ ∞whenEis smooth andE^∗has a Fr´echet diﬀerentiable norm [5]. The mappingπ_A^∗⁻¹0is defined byπ_A^∗⁻¹0=ΠA⁻¹0◦J⁻¹. Us- ing convergence theorems shown in Sections3and4, we obtain a result which replaces a single operatorAwith a sequence of operators{An}.

2. Preliminaries

LetEbe a real Banach space with its dualE^∗. We denote byJ the normalized duality mapping fromEtoE^∗. IfEis smooth, reflexive, and strictly convex,Jis a bijection. LetCbe a nonempty closed convex subset ofE. DefineV:E×E→R by

V(x, y)= x²−2J(x), y+y². (2.1) Suppose that Eis smooth, reflexive, and strictly convex. Then, for arbitrarily fixedx∈E, there exists a unique pointyx∈Csuch that

Vx, yx

=min

y∈CV(x, y). (2.2)

Following the notation of [1], we letΠC(x)=yx and callΠC ageneralized pro- jectionontoC. Notice that ifEis a Hilbert space, thenΠCis identical with the metric projection ontoC.

The following is a well-known result. See, for example, [1,5].

Proposition2.1. LetCbe a nonempty closed convex subset of a smooth Banach spaceEandx∈E. Then,x0=ΠCxif and only if

J(x)−Jx0

,x0−y≥0 ∀y∈C. (2.3)

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Using a generalized projectionΠC, we define a mappingπC^∗fromE^∗toEby

π_C^∗=ΠC◦J⁻¹. (2.4)

FromProposition 2.1, we obtain that, forx^∗∈E^∗,x0=π_C^∗x^∗if and only if x^∗−Jx0

,x0−y≥0 ∀y∈C. (2.5)

LetEbe a Banach space and letC1,C2,C3,...be a sequence of weakly closed subsets ofE. We denote by s-LinCnthe set of limit points of{Cn}, that is,x∈ s-Li_nCnif and only if there exists{xn} ⊂Esuch that{xn}converges strongly tox and thatxn∈Cnfor alln∈N. Similarly, we denote by w-Li_nCnthe set of cluster points of{Cn}; y∈ w-Li_nCn if and only if there exists{yni} such that {yni} converges weakly toyand thatyni∈Cnifor alli∈N. Using these definitions, we define the Mosco convergence [4] of{Cn}. IfC0satisfies

s-LinCn=C0=w-Ls

n Cn, (2.6)

we say that{Cn}is a Mosco convergent sequence toC0and write C0= M-lim

n→∞Cn. (2.7)

Notice that the inclusion s-LinCn⊂w-LsnCnis always true. Therefore, to show the existence of M-lim_n_→∞Cn, it is suﬃcient to prove w-Ls_nCn⊂s-Li_nCn. For more details, see [2].

3. Weak convergence of a sequence of generalized projections

In this section, we prove a pointwise weak convergence theorem for a sequence of generalized projections. The sequence of ranges of these projections is assumed to converge in the sense of Mosco.

Theorem3.1. LetEbe a smooth, reflexive, and strictly convex Banach space andC a nonempty closed convex subset ofE. LetC1,C2,C3,...be nonempty closed convex subsets ofC. IfC0=M-lim_n_→∞Cnexists and nonempty, thenC0is a closed convex subset ofCand, for eachx∈C,ΠCn(x)converges weakly toΠC0(x).

Proof. It is easy to prove thatC0 is closed and convex ifCnis a closed convex subset ofCfor eachn∈N. Fixx∈C. For the sake of simplicity, we writexn

instead ofΠCn(x) forn∈N. SinceC0=M-lim_n_→∞Cn, we have, for eachy∈C0

there exists{yn} ⊂Esuch thatyn→yasn→ ∞and thatyn∈Cnfor eachn∈N. FromProposition 2.1, we have

J(x)−Jxn

,xn−yn

≥0. (3.1)

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Hence, we obtain 0≤

J(x)−Jxn

,xn−x+J(x)−Jxn ,x−yn

≤ −

x −xn²+x+xnx−yn, (3.2) thus

x −xn²≤

x+xnx−yn. (3.3) Assume that{xn}is unbounded. Then there exists a subsequence{xni}of{xn} such that lim_i_→∞xni = ∞. From yn→y and (3.3), we get a contradiction.

Hence{xn}is bounded.

Since {xn} is bounded, there exists a subsequence, again denoted by{xn}, such that it converges weakly tox0∈C. From the definition ofC0, we getx0∈C0. Now, we prove thatΠC0(x)=x0. From lower semicontinuity of the norm, we have

lim inf

n→∞ Vx,xn

=lim inf

n→∞

x²−2J(x),xn

+xn²

≥ x²−2J(x),x0 +x0²

=Vx,x0 .

(3.4)

On the other hand, we get lim inf

n→∞ Vx,xn

≤lim inf

n→∞ Vx, yn

=V(x, y), (3.5)

that is,

Vx,x0

=min

y∈C0V(x, y). (3.6) Hence we getΠC0(x)=x0.

According to our consideration above, each sequence{xn}has, in turn, a subsequence which converges weakly to the unique pointΠC0(x). Therefore, the se-

quence{xn}converges weakly toΠC0(x).

4. Strong convergence of a sequence of generalized projections

A Banach spaceEis said to have theKadec-Klee propertyif a sequence{xn}of Esatisfying that w-lim_n_→∞xn=x0and lim_n_→∞xn = x0converges strongly tox0. It is known thatE^∗has a Fr´echet diﬀerentiable norm if and only ifEis reflexive, strictly convex, and has the Kadec-Klee property; see, for example, [6].

Theorem4.1. LetEbe a smooth Banach space such thatE^∗has a Fr´echet diﬀer- entiable norm. LetCbe a nonempty closed convex subset ofE. LetC1,C2,C3,...be nonempty closed convex subsets ofC. IfC0=M-limn→∞Cnexists and nonempty, then for eachx∈C,ΠCn(x)converges strongly toΠC0(x).

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Proof. Fixx∈Carbitrarily. We writexn=ΠCn(x) andx0=ΠC0(x). ByTheorem 3.1, we obtain w-lim_n_→∞xn=x0. SinceE^∗has a Fréchet differentiable norm,E has the Kadec-Klee property. Therefore, it is sufficient to prove thatxn → x0 asn→ ∞. Sincex0∈C0, there exists a sequence{yn} ⊂Csuch thatyn→x0as n→ ∞andyn∈Cnfor eachn∈N. It follows that

Vx,x0

≤lim inf

n→∞ Vx,xn

≤lim sup

n→∞ Vx,xn

≤lim

n→∞Vx, yn

≤Vx,x0

. (4.1)

Hence we obtainV(x,x0)=lim_n_→∞V(x,xn). SinceJ(x),xnconverges toJ(x), x0, we get

nlim→∞xn=x0. (4.2)

Using the Kadec-Klee property ofE, we obtain that{xn}converges strongly to

x0.

On the other hand, the following theorem shows that the pointwise strong convergence of{ΠCn(x)}implies the Mosco convergence of{Cn}under certain conditions.

Theorem4.2. LetEbe a reflexive and strictly convex Banach space with a Fr´echet diﬀerentiable norm, andCa nonempty closed convex subset ofE. LetC0,C1,C2,...

be nonempty closed convex subsets ofC. Suppose that

nlim→∞ΠCn(x)=ΠC0(x) ∀x∈C. (4.3) Then

C0= M-lim

n→∞Cn. (4.4)

Proof. For the sake of simplicity, we writeΠninstead ofΠCnforn∈N∪ {0}. For an arbitraryx∈C0, we have

x=Π0(x)=lim

n→∞Πn(x) (4.5)

andΠn(x)∈Cnfor alln∈N. This means thatx∈s-Li_nCnand hence we have C0⊂s-LinCn. Next, we show that w-LsnCn⊂C0. For anyz∈w-LsnCn, there exists{zi}such that {zi}converges weakly tozasi→ ∞ and thatzi∈Cni for eachi∈N. UsingProposition 2.1, we have

J(z)−JΠi(z),Πi(z)−zi

≥0. (4.6)

SinceEhas a Fr´echet diﬀerentiable norm, the duality mappingJis strongly continuous. Thus we get

J(z)−JΠ0(z),Π0(z)−z≥0. (4.7)

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By the strict convexity ofE,Jis strictly monotone. Hencez=Π0(z)∈C0. This means that w-Ls_nCn⊂C0, and consequently, we obtainC0=M-lim_n_→∞Cn. 5. Convergence of resolvents for a sequence of maximal

monotone operators

In this section, we consider a set-valued mapping called monotone operator. A set-valued mappingTfromXintoY is denoted byT: X⇒Y.

LetEbe a real Banach space. A set-valued mapping A:E⇒E^∗is called a monotone operatorif, for anyx, y∈Eandx^∗, y^∗∈E^∗withx^∗∈Axand y^∗∈ Ay,

x^∗−y^∗,x−y≥0. (5.1)

If a monotone operatorAhas no monotone extension, thenAis said to bemax- imal monotone.

For a maximal monotone operatorAand a real numberλwith 0< λ <∞, we define a set-valued mappingJλ:E^∗⇒Eby

Jλ:E^∗x^∗−→(J+λA)⁻¹x^∗⊂E. (5.2)

It is known thatJλ is a single-valued mapping if E is reflexive, smooth, and strictly convex.

First we show the following lemma.

Lemma5.1. LetEbe a reflexive Banach space andCa nonempty closed convex subset ofE. Let{xn}be a sequence ofEconverging weakly tox0∈C. For a sequence {Cn}of nonempty closed convex subsets ofEsuch thatM-lim_n_→∞Cn=C, it follows that

C= M-lim

n→∞coxn ∪Cn

. (5.3)

Proof. We writeDn=co({xn} ∪Cn) for alln∈N. Fixy∈w-Ls_nDn. Then there exist{yi∈Dni},{zi∈Cni}, and{αi} ⊂[0,1] such that

yi=αixni+1−αi

zi; w-lim

i→∞yi=y;

w-lim

i→∞zi=z0∈C; lim

i→∞αi=α0∈[0,1]. (5.4)

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Hence, we have y=α0x0+ (1−α0)z0∈Cand therefore w-Ls_nDn⊂C. On the other hand, it is obvious that

C⊂s-Li

nCn⊂s-Li

nDn. (5.5)

Thus we haveC=M-limn→∞Dn=M-limn→∞co({xn} ∪Cn).

Theorem5.2. LetEbe a reflexive, smooth, and strictly convex Banach space and let{A0,A1,A2,...}be a sequence of maximal monotone operators fromEintoE^∗. Suppose thatM-lim_n_→∞A⁻_n¹0=A⁻0¹0= ∅and that

w-LsnA⁻_n¹y^∗_n ⊂A⁻0¹0 (5.6) for any{y_n^∗} ⊂E^∗, converging strongly to0. Forx^∗∈E^∗and{λn} ∈]0,∞[with λn→ ∞, define a single-valued mappingJλn(x^∗)=(J+λnAn)⁻¹x^∗. Then Jλnx^∗ converges weakly toπ_A^∗⁻₀¹₀x^∗.

Proof. For the sake of simplicity, we writexn=Jλnx^∗for eachn∈N. SinceJ(xn) + λnAnxnx^∗, there existsw_n^∗∈Anxnsuch that

Jxn

+λnw_n^∗=x^∗ ∀n∈N. (5.7) From the assumption, there exists a bounded sequence{un}such thatun∈A⁻_n¹0 for eachn∈N. SinceAnis monotone, we have

Jxn

−Jun

,xn−un

=

x^∗−λnw_n^∗−Jun

,xn−un

=

x^∗−Jun

,xn−un

−λn

w^∗_n,xn−un

≤

x^∗−Jun

,xn−un .

(5.8)

Thus we get

xn²−2xnun+un²≤x^∗−Junxn+un. (5.9) Suppose that{xn}is not bounded. Then there exists a subsequence{xni}of{xn} such thatxni → ∞. It follows that

xni−2uni+uni²

xni ≤x^∗−J(u)

1 +uni xni

(5.10) for a suﬃciently large numberi∈N. Asi→ ∞, we obtain +∞ ≤ x^∗−J(u)<

+∞. This is a contradiction. Hence we have that{xn}is bounded.

Fix an arbitrary subsequence {xni} of {xn}converging weakly to x0. Since J(xni) +λniAnixnix^∗, we have

xni∈A⁻_n_i¹

x^∗−Jxni

λni

. (5.11)

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Using (5.6), we get

x0=w-lim

i→∞xni∈ M-lim

n→∞A⁻_n¹0. (5.12) Let Ci =co({xni} ∪A⁻ni¹0) for each i ∈N. Then Lemma 5.1 implies that A⁻0¹0=M-lim_i_→∞A⁻ni¹0=M-lim_i_→∞Ci. Now we fixi∈N. For anyv∈Ci, there existα∈[0,1] andu∈A⁻_n_i¹0 such thatv=αxni+ (1−α)u. SinceAni is monotone, we obtain

x^∗−Jxni

λni

−0,xni−u

≥0. (5.13)

This implies thatx^∗−J(xni),xni−v ≥0. Hence, we have xni=π_C^∗_ix^∗. Using Theorem 3.1, we obtain w-lim_i_→∞xni =π_A^∗⁻₀¹₀x^∗. Since {xni} is an arbitrary weakly convergent subsequence of a bounded sequence{xn}, it follows that

w-lim

n→∞xn=π_A^∗⁻₀¹₀x^∗. (5.14)

This completes the proof.

Assuming thatEhas the Kadec-Klee property, we obtain a strong convergence theorem. The proof is almost the same as the previous one.

Theorem5.3. LetEbe a smooth Banach space and suppose thatE^∗has a Fr´echet diﬀerentiable norm. Let{A0,A1,A2,...},x^∗,{λn},{Jλn}be the same asTheorem 5.2 and suppose that (5.6) holds. ThenJλnx^∗converges strongly toπ_A^∗⁻₀¹₀x^∗.

We can apply Theorems5.2and5.3to a single maximal monotone operator AwithA⁻¹0= ∅. Namely, for an arbitrary sequence{y_n^∗}ofE^∗converging to 0, it holds that

w-LsnA⁻¹y^∗_n ⊂A⁻¹0. (5.15) Indeed, forx∈w-Ls_nA⁻¹y^∗_n, there exists a sequence{xi}such thatxi∈A⁻¹y^∗_n_i for eachi∈Nand thatxi converges weakly tox. For anyv∈E andv^∗∈E^∗ satisfyingv^∗∈Av, we have

y_n^∗_i−v^∗,xni−v≥0. (5.16) Asi→ ∞, it follows that

0−v^∗,x−v≥0 (5.17)

and hencex∈A⁻¹0.

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References

[1] Y. I. Alber,Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, vol. 178, Dekker, New York, 1996, pp. 15–50.

[2] G. Beer,Topologies on Closed and Closed Convex Sets, Mathematics and Its Applica- tions, vol. 268, Kluwer Academic Publishers Group, Dordrecht, 1993.

[3] Y. Kimura and W. Takahashi,Strong convergence of sunny nonexpansive retractions in Banach spaces, Panamer. Math. J.9(1999), no. 4, 1–6.

[4] U. Mosco,Convergence of convex sets and of solutions of variational inequalities, Ad- vances in Math.3(1969), 510–585.

[5] S. Reich,Constructive techniques for accretive and monotone operators, Applied Non- linear Analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex, 1978), Academic Press, New York, 1979, pp. 335–345.

[6] W. Takahashi,Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000.

[7] M. Tsukada,Convergence of best approximations in a smooth Banach space, J. Approx.

Theory40(1984), no. 4, 301–309.

Takanori Ibaraki: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Current address: Institute of Economic Research, Hitotsubashi University, Tokyo 186-8603, Japan

E-mail address:[email protected], [email protected]

Yasunori Kimura: Institute of Economic Research, Hitotsubashi University, Tokyo 186-8603, Japan

Current address: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

E-mail address:[email protected], [email protected]

Wataru Takahashi: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

E-mail address:[email protected]

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