AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation64983110.1155/2009/649831649831Research ArticleStrong Convergence of Generalized Projection Algorithms for Nonlinear OperatorsKlin-eamChakkrid1SuantaiSuthep1TakahashiWataru2ReichSimeon1Department of MathematicsFaculty of ScienceChiang Mai UniversityChiang Mai 50200Thailandcmu.ac.th2Department of Mathematical and Computing SciencesTokyo Institute of Technology Tokyo 152-8552Japantitech.ac.jp200920102009200922062009151020092009Copyright © 2009This 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.

We establish strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

1. Introduction

Let E be a real Banach space with · and let C be a nonempty closed convex subset of E. A mapping T of C into itself is called nonexpansive if Tx-Tyx-y for all x,yC. We use F(T) to denote the set of fixed points of T; that is, F(T)={xC:x=Tx}. A mapping T of C into itself is called quasinonexpansive if F(T) is nonempty and Tx-yx-y for all xC and yF(T). For two mappings S and T of C into itself, Das and Debata  considered the following iteration scheme: x0C and xn+1=αnS(βnTxn+(1-βn)xn)+(1-αn)xn,n0, where {αn} and {βn} are sequences in [0,1]. In this case of S=T, such an iteration process was considered by Ishikawa ; see also Mann . Das and Debata  proved the strong convergence of the iterates {xn} defined by (1.1) in the case when E is strictly convex and S, T are quasinonexpansive mappings. Fixed point iteration processes for nonexpansive mappings in a Hilbert space and a Banach space including Das and Debata's iteration and Ishikawa's iteration have been studied by many researchers to approximating a common fixed point of two mappings; see, for instance, Takahashi and Tamura .

Let A be a maximal monotone operator from E to E*, where E* is the dual space of E. It is well known that many problems in nonlinear analysis and optimization can be formulated as follows. Find a point uE satisfying 0Au. We denote by A-10 the set of all points uC such that 0Au. Such a problem contains numerous problems in economics, optimization, and physics. A well-known method to solve this problem is called the proximal point algorithm: x0E and xn+1=Jrnxn,n=0,1,2,3,, where {rn}(0,) and Jrn are the resovents of A. Many researchers have studied this algorithm in a Hilbert space; see, for instance,  and in a Banach space; see, for instance, .

Next, we recall that for all xE and x*E*, we denote the value of x* at x by x,x*. Then, the normalized duality mapping J on E is defined by Jx={x*E*:x,x*=x2=x*2},xE. We know that if E is smooth, then the duality mapping J is single valued. Next, we assume that E is a smooth Banach space and define the function ϕ:E×E by ϕ(y,x)=y2-2y,Jx+x2,y,xE.

A point uC is said to be an asymptotic fixed point of T  if C contains a sequence {xn} which converges weakly to u and limnxn-Txn=0. We denote the set of all asymptotic fixed points of T by F̂(T). A mapping T:CC is said to be relatively nonexpansive  if F̂(T)=F(T) and ϕ(u,Tx)ϕ(u,x) for all uF(T) and xC. The asymptotic behavior of a relatively nonexpansive mapping was studied in .

In 2004, Matsushita and Takahashi  proposed the following modification of Mann's iteration for a relatively nonexpansive mapping by using the hybrid method in a Banach space. Four years later, Qin and Su  have adapted Matsushita and Takahashi's idea  to modify Halpern's iteration and Ishikawa's iteration for a relatively nonexpansive mapping in a Banach space. In particular, in a Hilbert space Mann's iteration, Halpern's iteration, and Ishikawa's iteration were considered by many researchers.

Very recently, Inoue et al.  proved the following strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method.

Theorem 1.1 (Inoue et al. [<xref ref-type="bibr" rid="B7">17</xref>]).

Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let AE×E* be a maximal monotone operator satisfying D(A)C and let Jr=(J+rA)-1J for all r>0. Let S:CC be a relatively nonexpansive mapping such that F(S)A-10. Let {xn} be a sequence generated by x0=xC and un=J-1(βnJxn+(1-βn)JSJrnxn),Cn={zC:ϕ(z,un)ϕ(z,xn)},Qn={zC:xn-z,Jx0-Jxn0},xn+1=ΠCnQnx0 for all n{0}, where J is the duality mapping on E, {βn}[0,1], and {rn}[a,) for some a>0. If lim infn(1-βn)>0, then {xn} converges strongly to ΠF(S)A-10x0, where ΠF(S)A-10 is the generalized projection of E onto F(S)A-10.

The purpose of this paper is to employ the idea of Inoue et al.  and Das and Debata  to introduce a new hybrid method for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings. We prove a strong convergence theorem of the new hybrid method. Moreover we apply our main results to obtain strong convergence for a maximal monotone operator and two nonexpansive mappings in a Hilbert space.

2. Preliminaries

Throughout this paper, all linear spaces are real. Let and be the sets of all positive integers and real numbers, respectively. Let E be a Banach space and let E* be the dual space of E. For a sequence {xn} of E and a point xE, the weak convergence of {xn} to x and the strong convergence of {xn} to x are denoted by xnx and xnx, respectively.

Let S(E) be the unit sphere centered at the origin of E. Then the space E is said to be smooth if the limit limt0x+ty-xt exists for all x,yS(E). It is also said to be uniformly smooth if the limit exists uniformly in x,yS(E). A Banach space E is said to be strictly convex if (x+y)/2<1 whenever x,yS(E) and xy. It is said to be uniformly convex if for each ϵ(0,2], there exists δ>0 such that (x+y)/2<1-δ whenever x,yS(E) and x-yϵ. We know the following :

if E is smooth, then J is single-valued;

if E is reflexive, then J is onto;

if E is strictly convex, then J is one to one;

if E is strictly convex, then J is strictly monotone;

if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

A Banach space E is said to have the Kadec-Klee property if for a sequence {xn} of E satisfying that xnx and xnx, xnx. It is known that if E is uniformly convex, then E has the Kadec-Klee property; see [18, 19] for more details. Let E be a smooth, strictly convex, and reflexive Banach space and let C be a closed convex subset of E. Throughout this paper, define the function ϕ:E×E by ϕ(y,x)=y2-2y,Jx+x2,y,xE. Observe that, in a Hilbert space H, (2.2) reduces to ϕ(x,y)=x-y2, for all x,yH. It is obvious from the definition of the function ϕ that, for all x,yE,

(x-y)2ϕ(x,y)(x+y)2,

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2x-z,Jz-Jy,

ϕ(x,y)=x,Jx-Jy+y-x,JyxJx-Jy+y-xy.

Following Alber , the generalized projection ΠC from E onto C is a map that assigns to an arbitrary point xE the minimum point of the functional ϕ(y,x); that is, ΠCx=x̅, where x̅ is the solution to the minimization problem ϕ(x̅,x)=minyCϕ(y,x). Existence and uniqueness of the operator ΠC follows from the properties of the functional ϕ(y,x) and strict monotonicity of the mapping J. In a Hilbert space, ΠC is the metric projection of H onto C. We need the following lemmas for the proof of our main results.

Lemma 2.1 (Kamimura and Takahashi [<xref ref-type="bibr" rid="B10">6</xref>]).

Let E be a uniformly convex and smooth Banach space and let {xn} and {yn} be two sequences in E such that either {xn} or {yn} is bounded. If limnϕ(xn,yn)=0, then limnxn-yn=0.

Lemma 2.2 (Matsushita and Takahashi [<xref ref-type="bibr" rid="B17">15</xref>]).

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E and let T be a relatively nonexpansive mapping from C into itself. Then F(T) is closed and convex.

Lemma 2.3 (Alber [<xref ref-type="bibr" rid="B1">20</xref>] and Kamimura and Takahashi [<xref ref-type="bibr" rid="B10">6</xref>]).

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, xE and let zC. Then, z=ΠCx if and only if y-z,Jx-Jz0 for all yC.

Lemma 2.4 (Alber [<xref ref-type="bibr" rid="B1">20</xref>] and Kamimura and Takahashi [<xref ref-type="bibr" rid="B10">6</xref>]).

Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then ϕ(x,ΠCy)+ϕ(ΠCy,y)ϕ(x,y),xC,yE.

Let E be a smooth, strictly convex, and reflexive Banach space, and let A be a set-valued mapping from E to E* with graph G(A)={(x,x*):x*Ax}, domain D(A)={zE:Az}, and range R(A)={Az:zD(A)}. We denote a set-valued operator A from E to E* by AE×E*. A is said to be monotone if x-y,x*-y*0, for all (x,x*),(y,y*)A. A monotone operator AE×E* is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then A-10={zD(A):0Az} is closed and convex. The following theorem is well known.

Lemma 2.5 (Rockafellar [<xref ref-type="bibr" rid="B21">21</xref>]).

Let E be a smooth, strictly convex, and reflexive Banach space and let AE×E* be a monotone operator. Then A is maximal if and only if R(J+rA)=E* for all r>0.

Let E be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E and let AE×E* be a monotone operator satisfying D(A)CJ-1(r>0R(J+rA)). Then we can define the resolvent Jr:CD(A) of A by Jrx={zD(A):JxJz+rAz},xC. We know that Jrx consists of one point. For r>0, the Yosida approximation Ar:CE* is defined by Arx=(Jx-JJrx)/r for all xC.

Lemma 2.6 (Kohsaka and Takahashi [<xref ref-type="bibr" rid="B13">22</xref>]).

Let E be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E and let AE×E* be a monotone operator satisfying D(A)CJ-1(r>0R(J+rA)). Let r>0 and let Jr and Ar be the resolvent and the Yosida approximation of A, respectively. Then, the following hold:

ϕ(u,Jrx)+ϕ(Jrx,x)ϕ(u,x), for all xC, uA-10;

(Jrx,Arx)A, for all xC;

F(Jr)=A-10.

Lemma 2.7 (Zălinescu [<xref ref-type="bibr" rid="B29">23</xref>] and Xu [<xref ref-type="bibr" rid="B28">24</xref>]).

Let E be a uniformly convex Banach space and let r>0. Then there exists a strictly increasing, continuous, and convex function g:[0,)[0,) such that g(0)=0 and tx+(1-t)y2tx2+(1-t)y2-t(1-t)g(x-y) for all x,yBr(0) and t[0,1], where Br(0)={zE:zr}.

3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using the hybrid method.

Theorem 3.1.

Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let AE×E* be a maximal monotone operator satisfying D(A)C and let Jr=(J+rA)-1J for all r>0. Let S and T be relatively nonexpansive mappings from C into itself such that Ω=F(S)F(T)A-10. Let {xn} be a sequence generated by x0C and un=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JSJrnxn),Cn={zC:ϕ(z,un)ϕ(z,xn)},Qn={zC:xn-z,Jx0-Jxn0},xn+1=ΠCnQnx0 for all n{0}, where J is the duality mapping on E, {αn},{βn}[0,1] and {rn}[a,) for some a>0 . If lim infn(1-αn)>0 and lim infnβn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.

Proof.

We first show that Cn and Qn are closed and convex for each n0. From the definitions of Cn and Qn, it is obvious that Cn is closed and Qn is closed and convex for each n0. Next, we prove that Cn is convex. Since ϕ(z,un)ϕ(z,xn) is equivalent to 0xn2-un2-2z,Jxn-Jun, which is affine in z, and hence Cn is convex. So, CnQn is a closed and convex subset of E for all n0. Next, we show that ΩCn for all n0. Indeed, let uΩ and yn=Jrnxn for all n0. Since Jrn are relatively nonexpansive mappings, we have ϕ(u,zn)=ϕ(u,J-1(βnJxn+(1-βn)JSyn))=u2-2u,βnJxn+(1-βn)JSyn+βnJxn+(1-βn)JSyn2u2-2βnu,Jxn-2(1-βn)u,JSyn+βnxn2+(1-βn)Syn2=βnϕ(u,xn)+(1-βn)ϕ(u,Syn)βnϕ(u,xn)+(1-βn)ϕ(u,yn)=βnϕ(u,xn)+(1-βn)ϕ(u,Jrnxn)βnϕ(u,xn)+(1-βn)ϕ(u,xn)=ϕ(u,xn). It follows that ϕ(u,un)=ϕ(u,J-1(αnJxn+(1-αn)JTzn))=u2-2u,αnJxn+(1-αn)JTzn+αnJxn+(1-αn)JTzn2u2-2αnu,Jxn-2(1-αn)u,JTzn+αnxn2+(1-αn)Tzn2=αnϕ(u,xn)+(1-αn)ϕ(u,Tzn)αnϕ(u,xn)+(1-αn)ϕ(u,zn)αnϕ(u,xn)+(1-αn)ϕ(u,xn)=ϕ(u,xn). So, uCn for all n0, which implies that ΩCn. Next, we show that ΩQn for all n0. We prove by induction. For n=0, we have ΩC=Q0. Assume that ΩQn. Since xn+1 is the projection of x0 onto CnQn, by Lemma 2.3 we have xn+1-z,Jx0-Jxn+10,zCnQn. As ΩCnQn by the induction assumptions, we have xn+1-z,Jx0-Jxn+10,zΩ. This together with definition of Qn+1 implies that ΩQn+1 and hence ΩQn for all n0. So, we have that ΩCnQn for all n0. This implies that {xn} is well defined. From definition of Qn that xn=ΠQnx0 and xn+1=ΠCnQnx0CnQnQn, we have ϕ(xn,x0)ϕ(xn+1,x0),n0. Therefore, {ϕ(xn,x0)} is nondecreasing. It follows from Lemma 2.4 and xn=ΠQnx0 that ϕ(xn,x0)=ϕ(ΠQnx0,x0)ϕ(u,x0)-ϕ(u,ΠQnx0)ϕ(u,x0) for all uΩQn. Therefore, {ϕ(xn,x0)} is bounded. Moreover, by definition of ϕ, we know that {xn} is bounded. So, we have {yn} and {zn} are bounded. So, the limit of {ϕ(xn,x0)} exists. From xn=ΠQnx0 and Lemma 2.4, we have ϕ(xn+1,xn)=ϕ(xn+1,ΠQnx0)ϕ(xn+1,x0)-ϕ(ΠQnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0) for all n0. This implies that limnϕ(xn+1,xn)=0. From xn+1=ΠCnQnx0Cn, we have ϕ(xn+1,un)ϕ(xn+1,xn). Therefore, we have limnϕ(xn+1,un)=0.

Since limnϕ(xn+1,xn)=limnϕ(xn+1,un)=0 and E is uniformly convex and smooth, we have from Lemma 2.1 that limnxn+1-xn=limnxn+1-un=0. So, we have limnxn-un=0. Since J is uniformly norm-to-norm continuous on bounded sets, we have limnJxn+1-Jxn=limnJxn+1-Jun=limnJxn-Jun=0. On the other hand, we have Jxn+1-Jun=Jxn+1-αnJxn-(1-αn)JTzn=αn(Jxn+1-Jxn)+(1-αn)(Jxn+1-JTzn)=(1-αn)(Jxn+1-JTzn)-αn(Jxn-Jxn+1)(1-αn)Jxn+1-JTzn-αnJxn-Jxn+1. This follows that Jxn+1-JTzn11-αn(Jxn+1-Jun+αnJxn-Jxn+1). From (3.12) and lim infn(1-αn)>0, we obtain that limnJxn+1-JTzn=0.

Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have limnxn+1-Tzn=0. From xn-Tznxn-xn+1+xn+1-Tzn, we have limnxn-Tzn=0. Since {xn} and {yn} are bounded, we also obtain that {Jxn} and {JSyn} are bounded. So, there exists r>0 such that {Jxn},{JSyn}Br(0). Therefore Lemma 2.7 is applicable and we observe that ϕ(u,zn)=ϕ(u,J-1(βnJxn+(1-βn)JSyn))=u2-2u,βnJxn+(1-βn)JSyn+βnJxn+(1-βn)JSyn2u2-2βnu,Jxn-2(1-βn)u,JSyn+βnxn2+(1-βn)Syn2-βn(1-βn)g(Jxn-JSyn)=βnϕ(u,xn)+(1-βn)ϕ(u,Syn)-βn(1-βn)g(Jxn-JSyn)=βnϕ(u,xn)+(1-βn)ϕ(u,SJrnxn)-βn(1-βn)g(Jxn-JSyn)βnϕ(u,xn)+(1-βn)ϕ(u,xn)-βn(1-βn)g(Jxn-JSyn)=ϕ(u,xn)-βn(1-βn)g(Jxn-JSyn), where g:[0,)[0,) is a continuous, strictly increasing, and convex function with g(0)=0. That is βn(1-βn)g(Jxn-JSyn)ϕ(u,xn)-ϕ(u,zn).

Let {xnk-Synk} be any subsequence of {xn-Syn}. Since {xnk} is bounded, there exists a subsequence {xnj} of {xnk} such that limjϕ(u,xnj)=lim supkϕ(u,xnk)=a, where uΩ. By (2) and (3), we have ϕ(u,xnj)=ϕ(u,Tznj)+ϕ(Tznj,xnj)+2u-Tznj,JTznj-Jxnjϕ(u,znj)+TznjJTznj-Jxnj+Tznj-xnjxnj+2u-TznjJTznj-Jxnj. Since limnxn-Tzn=0 and hence limnJxn-JTzn=0, it follows that a=lim infjϕ(u,xnj)lim infjϕ(u,znj). We also have from (3.3) that lim supjϕ(u,znj)lim supjϕ(u,xnj)=a, and hence limjϕ(u,xnj)=limjϕ(u,znj)=a. Since lim infnβn(1-βn)>0, it follows from (3.19) that limjg(Jxnj-JSynj)=0. By properties of the function g, we have limjJxnj-JSynj=0. Since J-1 is also uniformly norm-to-norm continuous on bounded sets, we obtain limjxnj-Synj=0 and then limnxn-Syn=0. So, we have limnJxn-JSyn=0. Since Jzn-Jxn=βnJxn+(1-βn)JSyn-Jxn=(1-βn)JSyn-JxnJSyn-Jxn, it follows that limnJzn-Jxn=0, and hence limnxn-zn=0. From (3.3), we have 11-βn(ϕ(u,zn)-βnϕ(u,xn))ϕ(u,yn). Using yn=Jrnxn and Lemma 2.6, we have ϕ(yn,xn)=ϕ(Jrnxn,xn)ϕ(u,xn)-ϕ(u,Jrnxn)=ϕ(u,xn)-ϕ(u,yn). It follows that ϕ(yn,xn)ϕ(u,xn)-ϕ(u,yn)ϕ(u,xn)-11-βn(ϕ(u,zn)-βnϕ(u,xn))=11-βn(ϕ(u,xn)-ϕ(u,zn))=11-βn(xn2-zn2-2u,Jxn-Jzn)11-βn(|xn2-zn2|+2|u,Jxn-Jzn|)11-βn(|xn-zn|(xn+zn)+2uJxn-Jzn)11-βn(xn-zn(xn+zn)+2uJxn-Jzn). Since lim infnβn(1-βn)>0, we have that lim infn(1-βn)>0. So, we have limnϕ(yn,xn)=0. Since E is uniformly convex and smooth, we have from Lemma 2.1 that limnyn-xn=0. Since zn-Tznzn-xn+xn-Tzn,yn-Synyn-xn+xn-Syn, from (3.17), (3.25), (3.27), and (3.31), we obtain that limnzn-Tzn=limnyn-Syn=0. Since {xn} is bounded, there exists a subsequence {xnk} of {xn} such that xnkv. From limnxn-yn=0 and limnxn-zn=0, we have ynkv and znkv. Since S and T are relatively nonexpansive, we have that vF̂(S)F̂(T)=F(S)F(T). Next, we show vA-10. Since J is uniformly norm-to-norm continuous on bounded sets, from (3.31) we have limnJxn-Jyn=0. From rna, we have limn1rnJxn-Jyn=0. Therefore, we have limnArnxn=limn1rnJxn-Jyn=0. For (p,p*)A, from the monotonicity of A, we have p-yn,p*-Arnxn0 for all n0. Replacing n by nk and letting k, we get p-v,p*0. From the maximallity of A, we have vA-10, that is, vΩ.

Finally, we show that xnΠΩx0. Let w=ΠΩx0. From xn+1=ΠCnQnx0 and wΩCnQn, we obtain that ϕ(xn+1,x0)ϕ(w,x0). Since the norm is weakly lower semicontinuous, we have ϕ(v,x0)=v2-2v,Jx0+x02lim infk(xnk2-2xnk,Jx0+x02)=lim infkϕ(xnk,x0)lim supkϕ(xnk,x0)ϕ(w,x0). From the definition of ΠΩ, we obtain v=w. This implies that limkϕ(xnk,x0)=ϕ(w,x0). Therefore we have 0=limk(ϕ(xnk,x0)-ϕ(w,x0))=limk(xnk2-w2-2xnk-w,Jx0)=limk(xnk2-w2). Since E has the Kadec-Klee property, we obtain that xnkw=ΠΩx0. Since {xnk} is an arbitrary weakly convergent subsequence of {xn}, we can conclude that {xn} converges strongly to ΠΩx0. This completes the proof.

As direct consequences of Theorem 3.1, we can obtain the following corollaries.

Corollary 3.2.

Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let AE×E* be a maximal monotone operator satisfying D(A)C and let Jr=(J+rA)-1J for all r>0. Let T be a relatively nonexpansive mapping from C into itself such that Ω=F(T)A-10. Let {xn} be a sequence generated by x0C and un=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JTJrnxn),Cn={zC:ϕ(z,un)ϕ(z,xn)},Qn={zC:xn-z,Jx0-Jxn0},xn+1=ΠCnQnx0 for all n{0}, where J is the duality mapping on E, {αn},{βn}[0,1], and {rn}[a,) for some a>0 . If lim infn(1-αn)>0 and lim infnβn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.

Proof.

Putting S=T in Theorem 3.1, we obtain Corollary 3.2.

Corollary 3.3.

Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let AE×E* be a maximal monotone operator satisfying D(A)C and let Jr=(J+rA)-1J for all r>0. Let S:CC be a relatively nonexpansive mapping such that Ω=F(S)A-10. Let {xn} be a sequence generated by x0C and un=J-1(βnJxn+(1-βn)JSJrnxn),Cn={zC:ϕ(z,un)ϕ(z,xn)},Qn={zC:xn-z,Jx0-Jxn0},xn+1=ΠCnQnx0 for all n{0}, where J is the duality mapping on E, {βn}[0,1], and {rn}[a,) for some a>0. If lim infnβn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.

Proof.

Putting T=I and αn=0 in Theorem 3.1, we obtain Corollary 3.3.

Let E be a Banach space and let f:E(-,] be a proper lower semicontinuous convex function. Define the subdifferential of f as follows: f(x)={x*E:f(y)y-x,x*+f(x),yE} for each xE. Then, we know that f is a maximal monotone operator; see  for more details.

Corollary 3.4.

Let E be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let S and T be relatively nonexpansive mappings from C into itself such that Ω=F(S)F(T). Let {xn} be a sequence generated by x0C and un=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JSxn),Cn={zC:ϕ(z,un)ϕ(z,xn)},Qn={zC:xn-z,Jx0-Jxn0},xn+1=ΠCnQnx0 for all n{0}, where J is the duality mapping on E and {αn},{βn}[0,1]. If lim infn(1-αn)>0 and lim infnβn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.

Proof.

Set A=iC in Theorem 3.1, where iC is the indicator function; that is, iC(x){0,xC,,otherwise. Then, we have that A is a maximal monotone operator and Jr=ΠC for r>0. In fact, for any xE and r>0, we have from Lemma 2.3 that z=JrxJz+riC(z)JxJx-JzriC(z)iC(y)y-z,Jx-Jzr+iC(z),yE0y-z,Jx-Jz,yCz=argminyCϕ(y,x)z=ΠCx. So, from Theorem 3.1, we obtain Corollary 3.4.

4. Applications

In this section, we discuss the problem of strong convergence concerning a maximal monotone operator and two nonexpansive mappings in a Hilbert space. Using Theorem 3.1, we obtain the following results.

Theorem 4.1.

Let C be a nonempty closed convex subset of a Hilbert space H. Let AH×H be a monotone operator satisfying D(A)C and let Jr=(I+rA)-1 for all r>0. Let S and T be nonexpansive mappings from C into itself such that Ω=F(S)F(T)A-10. Let {xn} be a sequence generated by x0C and un=αnxn+(1-αn)Tzn,zn=βnxn+(1-βn)SJrnxn,Cn={zC:z-unz-xn},Qn={zC:xn-z,x0-xn0},xn+1=PCnQnx0 for all n{0}, where {αn},{βn}[0,1] and {rn}[a,) for some a>0. If lim infn(1-αn)>0 and lim infnβn(1-βn)>0, then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.

Proof.

We know that every nonexpansive mapping with a fixed point is a relatively nonexpansive one. We also know that ϕ(x,y)=x-y2 for all x,yH. Using Theorem 3.1, we are easily able to obtain the desired conclusion by putting J=I. This completes the proof.

The following corollary follows from Theorem 4.1.

Corollary 4.2.

Let C be a nonempty closed convex subset of a Hilbert space H. Let AH×H be a monotone operator satisfying D(A)C and let Jr=(I+rA)-1 for all r>0. Let T be a nonexpansive mapping from C into itself such that Ω=F(T)A-10. Let {xn} be a sequence generated by x0C and un=αnxn+(1-αn)Tzn,zn=βnxn+(1-βn)TJrnxn,Cn={zC:z-unz-xn},Qn={zC:xn-z,x0-xn0},xn+1=PCnQnx0 for all n{0}, where {αn},{βn}[0,1] and {rn}[a,) for some a>0. If lim infn(1-αn)>0 and lim infnβn(1-βn)>0, then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.

Proof.

Putting S=T in Theorem 4.1, we obtain Corollary 4.2.

Corollary 4.3.

Let C be a nonempty closed convex subset of a Hilbert space H. Let AH×H be a maximal monotone operator satisfying D(A)C and let Jr=(I+rA)-1 for all r>0. Let S be a nonexpansive mapping from C into itself such that Ω=F(S)A-10. Let {xn} be a sequence generated by x0C and un=βnxn+(1-βn)SJrnxn,Cn={zC:z-unz-xn},Qn={zC:xn-z,x0-xn0},xn+1=PCnQnx0 for all n{0}, where {βn}[0,1] and {rn}[a,) for some a>0. If lim infnβn(1-βn)>0 then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.

Proof.

Putting T=I and αn=0 in Theorem 4.1, we obtain Corollary 4.3.

Corollary 4.4.

Let C be a nonempty closed convex subset of a Hilbert space H. Let S and T be nonexpansive mappings from C into itself such that Ω=F(S)F(T). Let {xn} be a sequence generated by x0=xC and un=αnxn+(1-αn)Tzn,zn=βnxn+(1-βn)Sxn,Cn={zC:z-unz-xn},Qn={zC:xn-z,x0-xn0},xn+1=PCnQnx0 for all n{0}, where {αn},{βn}[0,1]. If lim infn(1-αn)>0 and lim infnβn(1-βn)>0, then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.

Proof.

Set A=iC in Theorem 4.1, where iC is the indicator function; that is, iC(x){0,xC,,otherwise. Then, we have that A is a maximal monotone operator and Jr=PC for r>0. In fact, for any xE and r>0, we have that z=Jrxz+riC(z)xx-zriC(z)iC(y)y-z,x-zr+iC(z),yE0y-z,x-z,yCz=PCx. So, from Theorem 4.1, we obtain Corollary 4.4.

Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve this manuscript and the Thailand Research Fund (RGJ Project) and Commission on Higher Education for their financial support during the preparation of this paper. The first author was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and the Graduate School, Chiang Mai University, Thailand. The third author is supported by Grant-in-Aid for Scientific Research no. 19540167 from Japan Society for the Promotion of Science.

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