R E S E A R C H
Open Access
An implicitly defined iterative sequence for
monotone operators in Banach spaces
Fumiaki Kohsaka
**Correspondence:
Department of Computer Science and Intelligent Systems, Oita University, Dannoharu, Oita-shi, Oita, 870-1192, Japan
Abstract
Given a monotone operator in a Banach space, we show that an iterative sequence, which is implicitly defined by a fixed point theorem for mappings of firmly
nonexpansive type, converges strongly to a minimum norm zero point of the given operator. Applications to a convex minimization problem and a variational inequality problem are also included.
MSC: 47H05; 47J25; 47N10
Keywords: Banach space; convex minimization; mapping of firmly nonexpansive type; fixed point; monotone operator; proximal point method; variational inequality
1 Introduction
Many nonlinear problems can be formulated as the problem of finding a zero point of a maximal monotone operator in a Banach space. The proximal point method, which was first introduced by Martinet [] and generally studied by Rockafellar [], is an iterative method for approximating a solution to this problem.
The proximal point method generates a sequence{xn}byx∈Xand
xn+= (J+λnA)–Jxn (.)
for alln∈N, whereA:X→X∗is a maximal monotone operator,Xis a smooth, strictly
convex, and reflexive real Banach space,J: X→X∗ is the normalized duality mapping, and{λn}is a sequence of positive real numbers.
The following result was obtained in []: If∞n=λn=∞, then{xn}is bounded if and only
ifA– is nonempty. Further, ifXis uniformly convex, the norm ofXis uniformly Gâteaux differentiable,A– is nonempty,inf
nλn> , andJis weakly sequentially continuous, then {xn}converges weakly to an element ofA–. This is a generalization of the result due to
Rockafellar [] in Hilbert spaces. See also [, ] for some related results.
The aim of the present paper is to study the asymptotic behavior of the sequence{xn}
generated by
xn=βn(J+λnA)–Jxn (.)
for alln∈N, whereA,X,J, and{λn}are the same as in (.) and{βn}is a sequence of
[, ). Under some additional assumptions, we show that{xn}is well defined and is strongly
convergent to an element ofA– of minimal norm; see Theorem ..
The schemes (.) and (.) above are similar to each other, though their properties are quite different. In fact, the former fails to converge strongly even in Hilbert spaces [], whereas the latter converges strongly in Banach spaces. Further, the former is well defined since R(J+λA) =X∗for eachλ> , whereas the latter is not necessarily well defined. To study the well definedness and the asymptotic behavior of{xn}in (.), we exploit some
techniques in [, ].
This paper is organized as follows: In Section , we give some definitions, recall some known results, and briefly study the existence of a zero point of a monotone operator. In Section , using the results in the previous section, we first obtain a convergence theorem for a monotone operator satisfying a range condition; see Theorem .. Using this result, we show a convergence theorem for a maximal monotone operator; see Theorem .. In Section , we apply Theorem . to a convex minimization problem and a variational inequality problem.
2 Preliminaries
Throughout the present paper, we denote byNthe set of all positive integers,Rthe set of all real numbers,Xa smooth, strictly convex, and reflexive real Banach space with dualX∗,
· the norms ofXandX∗,x,x∗the value ofx∗∈X∗atx∈X,xn→xthe strong
conver-gence of a sequence{xn}ofXtox∈X,xnxthe weak convergence of a sequence{xn}of Xtox∈X,Uthe norm closure ofU⊂X,coUthe convex hull ofU⊂X,coUthe closed convex hull ofU⊂X, andSXthe unit sphere ofX, respectively.
Under the assumptions onX, we know that for eachx∈X, there is a corresponding uniqueJxinX∗ such thatx,Jx=xandJx=x. The mappingJis called the
nor-malized duality mapping of X into X∗. We know the following: J: X→X∗ is a bijec-tion;J(αx) =αJxfor allα∈Randx∈X;Jis norm-to-weak continuous, that is,JxnJx
whenever{xn} is a sequence ofXsuch thatxn→x∈X; Jis strictly monotone, that is, x–y,x∗–y∗> for all distinctx,y∈X. The norm ofXis said to be uniformly Gâteaux differentiable if the limit
lim
t→
x+ty–x
t (.)
converges uniformly inx∈SXfor ally∈SX. The spaceXis said to be uniformly convex if
for eachε∈(, ], there existsδ> such that(x+y)/ ≤ –δwheneverx,y∈SXand x–y ≥ε. The spaceXis said to have the Kadec-Klee property ifxn→xwhenever{xn}
is a sequence ofXsuch thatxnx∈Xandxn → x. Every uniformly convex Banach
space is both strictly convex and reflexive and has the Kadec-Klee property; see [, ]. For a nonempty closed convex subsetCofXandx∈X, there exists a unique pointxˆin
Csuch thatˆx–x ≤ y–xfor ally∈C. The metric projectionPCofXontoCis defined
byPCx=xˆfor allx∈X. It is well known [] that
z=PC(x) ⇐⇒ sup y∈C
y–z,J(x–z)≤ (.)
for (x,z)∈X×C. The functionφ: X×X→Ris defined by
for allx,y∈X; see [, ]. IfXis a Hilbert space, thenφ(x,y) =x–yfor allx,y∈X. It
is easy to see that
x–y≤φ(x,y) (.)
and
φ(x,y) +φ(y,x) = x–y,Jx–Jy (.)
for allx,y∈X.
LetCbe a nonempty subset ofXandT: C→Xa mapping. The set of all fixed points ofTis denoted by F(T). The mappingTis said to be of firmly nonexpansive type [] if
Tx–Ty,JTx–JTy ≤ Tx–Ty,Jx–Jy (.)
for allx,y∈C; see also []. IfXis a Hilbert space, thenT:C→Xis firmly nonexpansive if and only if it is of firmly nonexpansive type.
For an operator A: X→X∗, the domain D(A), the range R(A), and the graph G(A) ofAare defined by D(A) ={x∈X:Ax=∅}, R(A) =x∈XAx, and G(A) ={(x,x∗)∈X× X∗:x∗∈Ax}, respectively. The operatorAis said to be monotone ifx–y,x∗–y∗ ≥ whenever (x,x∗), (y,y∗)∈G(A). It is also said to be maximal monotone ifAis monotone and there is no monotone operatorB: X→X∗such thatA=Band G(A)⊂G(B). LetC
be a nonempty closed convex subset ofXandA:X→X∗a monotone operator such that
D(A)⊂C⊂J–R(J+A). Then the mappingT: C→Cdefined byTx= (J+A)–Jxfor all x∈Cis of firmly nonexpansive type and F(T) =A–; see [, ]. We know the following
lemma.
Lemma .([]) Suppose that the norm of X is uniformly Gâteaux differentiable.Let C be a nonempty closed convex subset of X,A:X→X∗a monotone operator such that
D(A)⊂C⊂ λ>
J–R(J+λA), (.)
{λn}a sequence of(,∞)such thatinfnλn> ,and Qλn: C→C the mapping defined by Qλnx= (J+λnA)–Jx for all x∈C and n∈N.If{xn}is a sequence of C such that xnu and xn–Qλnxn→,then u is an element of A
–.
We know the following result for mappings of firmly nonexpansive type.
Lemma .([]) Let C be a nonempty closed convex subset of X and T:C→C a mapping of firmly nonexpansive type.Then the following hold:
(i) F(T)is nonempty if and only if{Tnx}is bounded for somex∈C;
(ii) F(T)is closed and convex.
Using Lemma ., we can show the following.
Proof SetS=βT. SinceJis monotone,T is of firmly nonexpansive type, andβ∈[, ), we have
≤ Sx–Sy,JSx–JSy=βTx–Ty,JTx–JTy
≤βTx–Ty,Jx–Jy
=βSx–Sy,Jx–Jy ≤ Sx–Sy,Jx–Jy (.)
for allx,y∈C. This implies thatSis of firmly nonexpansive type. SinceCis convex, ∈C, andβ∈[, ), we know thatSis a mapping ofCinto itself. Further, sinceS(C) =βT(C) and
T(C) is bounded, the sequence{Snx}is bounded for allx∈C. Thus Lemma . implies
that F(S) is nonempty.
We next show that F(S) consists of one point. Suppose thatp,p∈F(S). Then it follows from (.) that ( –β)p–p,Jp–Jp= . Since –β> , we obtainp–p,Jp–Jp= .
Thus the strict monotonicity ofJimplies thatp=p.
As a direct consequence of Lemmas . and ., we obtain the following.
Corollary . Let A:X→X∗ be a monotone operator such thatD(A)is bounded and
D(A)⊂C⊂J–R(J+A)for some nonempty closed convex subset C of X.Then the following hold:
(i) A–is nonempty,closed,and convex;
(ii) if∈Candβ∈[, ),then there exists a uniquep∈Csuch that
p=β(J+A)–Jp. (.)
Proof LetT:C→Cbe the mapping defined byTx= (J+A)–Jxfor allx∈C. Then we
know thatTis of firmly nonexpansive type andT(C)⊂D(A). Hence{Tnx}is bounded for
allx∈C. On the other hand, we know that F(T) =A–. Therefore, part (i) follows from
Lemma .. Part (ii) follows from Lemma ..
3 Strong convergence of an iterative sequence
In this section, we first show the following strong convergence theorem for a monotone operator satisfying a range condition.
Theorem . Let X be a smooth, strictly convex, and reflexive real Banach space,C a nonempty closed convex subset of X such that∈C,and A:X→X∗a monotone operator such thatD(A)is bounded and
D(A)⊂C⊂ λ>
J–R(J+λA). (.)
Let{λn}be a sequence of positive real numbers,{βn}a sequence of[, ),and Qλn: C→C the mapping defined by Qλnx= (J+λnA)–Jx for all x∈C and n∈N.Then the following hold:
(i) For eachn∈N,there exists a uniquexn∈Csuch thatxn=βnQλnxn;
Part (i) of Theorem . follows from Corollary ..
The proof of(i)of Theorem. Letn∈Nbe given and setB=λnA. ThenB: X→X
∗ is monotone and D(B) = D(A). Thus we know that D(B) is bounded and D(B)⊂C⊂J–R(J+
B). Therefore, part (ii) of Corollary . ensures the conclusion.
Before proving (ii) of Theorem ., we show the following lemma.
Lemma . The following hold:
(i) Qλnxn–y,JQλnxn ≤for ally∈A–andn∈N;
(ii) φ(Qλnxn,y) +φ(y,Qλnxn)≤y–Qλnxn,Jyfor ally∈A
–andn∈N.
Proof We show (i). Lety∈A– andn∈Nbe given. SinceQλn is of firmly nonexpansive type andQλny=y, we know that
Qλnxn–y,JQλnxn–Jxn ≤. (.)
On the other hand, by the definition of{xn}, we also know that
JQλnxn–Jxn=JQλnxn–J(βnQλnxn) = ( –βn)JQλnxn. (.)
By (.), (.), and –βn> , the result follows.
By (.) and (i), we have
φ(Qλnxn,y) +φ(y,Qλnxn) = Qλnxn–y,JQλnxn–Jy
≤Qλnxn–y, –Jy (.)
for ally∈A– andn∈N. Thus the result follows.
We next show (ii) of Theorem ..
The proof of(ii)of Theorem. Suppose thatXhas the Kadec-Klee property, the norm ofXis uniformly Gâteaux differentiable,infnλn> , andlimnβn= . Setyn=Qλnxnfor all n∈N. By (i) of Corollary ., the setA– is nonempty, closed, and convex. HenceP
A– is well defined. We denotePA–byP.
Sinceyn∈D(A) for alln∈Nand D(A) is bounded,{yn}is bounded. By the definition of {xn}, we havexn=βnyn ≤ ynfor alln∈N. Thus{xn}is also bounded.
Let{xni}be any subsequence of{xn}. To see thatxn→P(), it is sufficient to see that
there exists a subsequence{xnij}of{xni}which converges strongly toP(). SinceXis
re-flexive and{xn}is a bounded sequence ofC, there existu∈Cand a subsequence{xnij}of {xni}such thatxniju. Since{yn}is bounded andlimnβn= , we have
xn–yn=βnyn–yn= ( –βn)yn →. (.)
Sinceinfjλnij ≥infnλn> , Lemma . shows thatuis an element ofA–. It also follows
We next show thatxnij→u. By (.) and (ii) of Lemma ., we have
ynij–u
≤
φ(ynij,u)
≤φ(ynij,u) +φ(u,ynij)≤u–ynij,Ju →. (.)
Thus we obtainynij → u. SinceX has the Kadec-Klee property, we haveynij →u.
Consequently, it follows from (.) thatxnij→u.
We next show thatu=P(). To see this, lety∈A– be given. SinceJis norm-to-weak
continuous andynij→u, we know thatJynij Ju. On the other hand, by (i) of Lemma .,
we have
ynij–y,Jynij ≤ (.)
for allj∈N. Lettingj→ ∞in (.), we obtainu–y,Ju ≤ and hence
sup
y∈A–
y–u,J( –u)≤. (.)
Noting thatu∈A–, we have from (.) and (.) thatu=P(). Therefore, we conclude
that{xn}converges strongly toP().
As a direct consequence of Theorem ., we obtain the following corollary.
Corollary . Let X be a smooth,strictly convex,and reflexive real Banach space, C a nonempty closed convex subset of X such that∈C,T:C→C a mapping of firmly non-expansive type such that T(C)is bounded,{λn}a sequence of positive real numbers,and {βn}a sequence of[, ).Then the following hold:
(i) For eachn∈N,there exists a uniquexn∈Csuch thatxn=βnTxn;
(ii) ifXhas the Kadec-Klee property,the norm ofXis uniformly Gâteaux differentiable,
andlimnβn= ,then the sequence{xn}converges strongly toPF(T)().
Proof LetA: X→X∗ be the mapping defined byA=JT––J, whereT–: X→X is defined by
T–x= ⎧ ⎨ ⎩
{u∈C:Tu=x} (x∈T(C));
∅ (x∈/T(C)) (.)
for allx∈X. Then, by [], we know that the following hold: • Ais a monotone operator andA– = F(T);
• D(A) =T(C)⊂C=J–R(J+A); • Tx= (J+A)–Jxfor allx∈C.
Thus the result follows from Theorem ..
Using Theorem ., we can also show the following strong convergence theorem for a maximal monotone operator.
Theorem . Let X be a smooth,strictly convex,and reflexive real Banach space,A:X→
X∗ a maximal monotone operator such that∈D(A)andD(A)is bounded,{λ n}a se-quence of positive real numbers,and{βn}a sequence of[, ).Then for each n∈N,there exists a unique xn∈D(A)satisfying(.).Moreover,if X has the Kadec-Klee property,the norm of X is uniformly Gâteaux differentiable,infnλn> ,andlimnβn= ,then the se-quence{xn}converges strongly to PA–().
Proof The maximal monotonicity ofAimplies that D(A) is nonempty and hence so is D(A). It is obvious that D(A) is closed. It is well known [] that D(A) is convex. In fact, we know that
lim
μ↓
I+μJ–A–x=x (.)
for allx∈coD(A), where I denotes the identity mapping onX; see [, ]. Since (I+ μJ–A)–x∈D(A) for allμ> andx∈X, it follows from (.) thatcoD(A)⊂D(A). Thus
coD(A) = D(A) and hence D(A) is convex.
On the other hand, sinceAis maximal monotone, we know that R(J+λA) =X∗for all λ> ; see []. PuttingC= D(A), we have
D(A)⊂C⊂X=
λ>
J–R(J+λA). (.)
Noting that ∈D(A) =C, we obtain the desired result by Theorem ..
4 Results deduced from Theorem 3.5
In this final section, we study two applications of Theorem .. Throughout this section, we suppose the following:
• Xis a uniformly convex real Banach space whose norm is uniformly Gâteaux
differentiable;
• {λn}is a sequence of positive real numbers such thatinfnλn> ;
• {βn}is a sequence of[, )such thatlimnβn= .
We first study a convex minimization problem. For a functionf:X→(–∞,∞], we de-note byargminf orargminy∈Xf(y) the set of allu∈Xsuch thatf(u) =inff(X). In the case whenargminf={p}for somep∈X, we identifyargminf withp. The set of allx∈Xsuch thatf(x)∈Ris denoted by D(f). We denote by∂f the subdifferential mapping off; see [, ] for more details.
Corollary . Let f: X→(–∞,∞]be a proper lower semicontinuous convex function such that∈D(f)andD(f)is bounded.Then for each n∈N,there exists a unique xn∈D(f) such that
xn=βnargmin y∈X
f(y) + λn
φ(y,xn)
. (.)
Proof LetA:X→X∗be the operator defined byA=∂f. It is well known thatAis
max-imal monotone [] and A– =argminf. Since D(f) is bounded and D(A)⊂D(f), we
know that D(A) is bounded. By Brøndsted and Rockafellar’s theorem [], we know that D(f) = D(A). Thus we have ∈D(A). Further, the equality
(J+λA)–Jx=argmin
y∈X
f(y) + λφ(y,x)
(.)
holds for allλ> andx∈X. Thus we know thatxn=βn(J+λnA)–Jxnfor alln∈N.
Con-sequently, Theorem . implies the conclusion.
We finally study a variational inequality problem. For a nonempty closed convex subset
CofXand an operatorB:C→X∗, we denote byVI(C,B) the set of allu∈Csuch that
y–u,Bu ≥ for ally∈C. In the case whenVI(C,B) ={p}for somep∈C, we identify VI(C,B) withp. The operatorBis said to be hemicontinuous if the mappingg: [, ]→X∗
defined byg(t) =B(tx+ ( –t)y) for allt∈[, ] is continuous with respect to the weak* topology inX∗for allx,y∈C.
Corollary . Let C be a nonempty bounded closed convex subset of X such that∈C and B:C→X∗a monotone and hemicontinuous operator.Then for each n∈N,there exists a unique xn∈C such that
xn=βnVI
C,B+ λn
(J–Jxn)
. (.)
Moreover,the sequence{xn}converges strongly to PVI(C,B)().
Proof LetA:X→X∗be the operator defined by
A(x) = ⎧ ⎨ ⎩
B(x) +∂iC(x) (x∈C);
∅ (x∈X\C)
(.)
for allx∈X, whereiCdenotes the indicator function ofC. It is well known thatAis
max-imal monotone [],A– =VI(C,B), and D(A) =C. Thus we know that D(A) is bounded
and ∈D(A). Further, the equality
(J+λA)–Jx=VI
C,B+ λ(J–Jx)
(.)
holds for all λ> andx∈X. Thus we havexn=βn(J+λnA)–Jxnfor alln∈N.
Conse-quently, Theorem . implies the conclusion.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author would like to thank the anonymous referees for carefully reading the original version of the manuscript. The author is supported by Grant-in-Aid for Young Scientists No. 25800094 from the Japan Society for the Promotion of Science.
References
1. Martinet, B: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Opér.4, 154-158 (1970) (in French)
2. Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim.14, 877-898 (1976) 3. Aoyama, K, Kohsaka, F, Takahashi, W: Proximal point methods for monotone operators in Banach spaces. Taiwanese
J. Math.15, 259-281 (2011)
4. Kamimura, S: The proximal point algorithm in a Banach space. In: Nonlinear Analysis and Convex Analysis, pp. 143-148. Yokohama Publishers, Yokohama (2004)
5. Kamimura, S, Kohsaka, F, Takahashi, W: Weak and strong convergence theorems for maximal monotone operators in a Banach space. Set-Valued Anal.12, 417-429 (2004)
6. Güler, O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim.29, 403-419 (1991)
7. Kohsaka, F, Takahashi, W: Strongly convergent net given by a fixed point theorem for firmly nonexpansive type mappings. Appl. Math. Comput.202, 760-765 (2008)
8. Cioranescu, I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)
9. Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
10. Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp. 15-50. Dekker, New York (1996)
11. Kamimura, S, Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim.13, 938-945 (2002)
12. Kohsaka, F, Takahashi, W: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim.19, 824-835 (2008)
13. Aoyama, K, Kohsaka, F, Takahashi, W: Three generalizations of firmly nonexpansive mappings: their relations and continuity properties. J. Nonlinear Convex Anal.10, 131-147 (2009)
14. Kohsaka, F, Takahashi, W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math.91, 166-177 (2008)
15. Rockafellar, RT: On the virtual convexity of the domain and range of a nonlinear maximal monotone operator. Math. Ann.185, 81-90 (1970)
16. Barbu, V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei Republicii Socialiste România, Bucharest (1976)
17. Takahashi, W: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000) (in Japanese)
18. Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc.149, 75-88 (1970) 19. Z˘alinescu, C: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)
20. Rockafellar, RT: On the maximal monotonicity of subdifferential mappings. Pac. J. Math.33, 209-216 (1970) 21. Brøndsted, A, Rockafellar, RT: On the subdifferentiability of convex functions. Proc. Am. Math. Soc.16, 605-611 (1965)
10.1186/1029-242X-2014-181
Cite this article as:Kohsaka:An implicitly defined iterative sequence for monotone operators in Banach spaces.