# Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators

Authors: Chakkrid Klin-eam; Suthep Suantai; Wataru Takahashi
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

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.


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## Body

## 1. Introduction

LetE 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-Ty∥≤∥x-y∥ for all x,y∈C. We use F(T) to denote the set of fixed points of T; that is, F(T)={x∈C:x=Tx}. A mapping T of C into itself is called quasinonexpansive if F(T) is nonempty and ∥Tx-y∥≤∥x-y∥ for all x∈C and y∈F(T). For two mappings S and T of C into itself, Das and Debata [1] considered the following iteration scheme: x0∈C and
(1.1)xn+1=αnS(βnTxn+(1-βn)xn)+(1-αn)xn,n≥0,
where {αn} and {βn} are sequences in [0,1]. In this case of S=T, such an iteration process was considered by Ishikawa [2]; see also Mann [3]. Das and Debata [1] 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 [4].LetA 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 u∈E satisfying
(1.2)0∈Au.
We denote by A-10 the set of all points u∈C such that 0∈Au. 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: x0∈E and
(1.3)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, [5–8] and in a Banach space; see, for instance, [9–11].Next, we recall that for allx∈E 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
(1.4)Jx={x*∈E*:〈x,x*〉=∥x∥2=∥x*∥2},∀x∈E.
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→ℝ by
(1.5)ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2,∀y,x∈E.A pointu∈C is said to be an asymptotic fixed point of T [12] if C contains a sequence {xn} which converges weakly to u and limn→∞∥xn-Txn∥=0. We denote the set of all asymptotic fixed points of T by F̂(T). A mapping T:C→C is said to be relatively nonexpansive [13–15] if F̂(T)=F(T)≠∅ and ϕ(u,Tx)≤ϕ(u,x) for all u∈F(T) and x∈C. The asymptotic behavior of a relatively nonexpansive mapping was studied in [13–15].In 2004, Matsushita and Takahashi [15] 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 [16] have adapted Matsushita and Takahashi's idea [15] 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. [17] 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. [17]).
LetE be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let A⊂E×E* be a maximal monotone operator satisfying D(A)⊂C and let Jr=(J+rA)-1J for all r>0. Let S:C→C be a relatively nonexpansive mapping such that F(S)∩A-10≠∅. Let {xn} be a sequence generated by x0=x∈C and
(1.6)un=J-1(βnJxn+(1-βn)JSJrnxn),Cn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0
for all n∈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. [17] and Das and Debata [1] 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. Letℕ and ℝ 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 x∈E, the weak convergence of {xn} to x and the strong convergence of {xn} to x are denoted by xn⇀x and xn→x, respectively.LetS(E) be the unit sphere centered at the origin of E. Then the space E is said to be smooth if the limit
(2.1)limt→0∥x+ty∥-∥x∥t
exists for all x,y∈S(E). It is also said to be uniformly smooth if the limit exists uniformly in x,y∈S(E). A Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 whenever x,y∈S(E) and x≠y. 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,y∈S(E) and ∥x-y∥≥ϵ. We know the following [18]:(i)
ifE is smooth, then J is single-valued;(ii)
ifE is reflexive, then J is onto;(iii)
ifE is strictly convex, then J is one to one;(iv)
ifE is strictly convex, then J is strictly monotone;(v)
ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.A Banach spaceE is said to have the Kadec-Klee property if for a sequence {xn} of E satisfying that xn⇀x and ∥xn∥→∥x∥, xn→x. 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→ℝ by
(2.2)ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2,∀y,x∈E.
Observe that, in a Hilbert space H, (2.2) reduces to ϕ(x,y)=∥x-y∥2, for all x,y∈H. It is obvious from the definition of the function ϕ that, for all x,y∈E,(1)
(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,(2)
ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉,(3)
ϕ(x,y)=〈x,Jx-Jy〉+〈y-x,Jy〉≤∥x∥∥Jx-Jy∥+∥y-x∥∥y∥.Following Alber [20], the generalized projection ΠC from E onto C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(y,x); that is, ΠCx=x̅, where x̅ is the solution to the minimization problem
(2.3)ϕ(x̅,x)=miny∈Cϕ(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 [6]).
LetE 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 limn→∞ϕ(xn,yn)=0, then limn→∞∥xn-yn∥=0.Lemma 2.2 (Matsushita and Takahashi [15]).
LetC 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 [20] and Kamimura and Takahashi [6]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space, x∈E and let z∈C. Then, z=ΠCx if and only if 〈y-z,Jx-Jz〉≤0 for all y∈C.Lemma 2.4 (Alber [20] and Kamimura and Takahashi [6]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space. Then
(2.4)ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y),∀x∈C,y∈E.LetE 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)={z∈E:Az≠∅}, and range R(A)=∪{Az:z∈D(A)}. We denote a set-valued operator A from E to E* by A⊂E×E*. A is said to be monotone if 〈x-y,x*-y*〉≥0, for all (x,x*),(y,y*)∈A. A monotone operator A⊂E×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={z∈D(A):0∈Az} is closed and convex. The following theorem is well known.Lemma 2.5 (Rockafellar [21]).
LetE be a smooth, strictly convex, and reflexive Banach space and let A⊂E×E* be a monotone operator. Then A is maximal if and only if R(J+rA)=E* for all r>0.LetE be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E and let A⊂E×E* be a monotone operator satisfying
(2.5)D(A)⊂C⊂J-1(⋂r>0R(J+rA)).
Then we can define the resolvent Jr:C→D(A) of A by
(2.6)Jrx={z∈D(A):Jx∈Jz+rAz},∀x∈C.
We know that Jrx consists of one point. For r>0, the Yosida approximation Ar:C→E* is defined by Arx=(Jx-JJrx)/r for all x∈C.Lemma 2.6 (Kohsaka and Takahashi [22]).
LetE be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E and let A⊂E×E* be a monotone operator satisfying
(2.7)D(A)⊂C⊂J-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:(i)
ϕ(u,Jrx)+ϕ(Jrx,x)≤ϕ(u,x), for all x∈C, u∈A-10;(ii)
(Jrx,Arx)∈A, for all x∈C;(iii)
F(Jr)=A-10.Lemma 2.7 (Zălinescu [23] and Xu [24]).
LetE 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
(2.8)∥tx+(1-t)y∥2≤t∥x∥2+(1-t)∥y∥2-t(1-t)g(∥x-y∥)
for all x,y∈Br(0) and t∈[0,1], where Br(0)={z∈E:∥z∥≤r}.


## 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.
LetE be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let A⊂E×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 x0∈C and
(3.1)un=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JSJrnxn),Cn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0
for all n∈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 infn→∞βn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.Proof.
We first show thatCn and Qn are closed and convex for each n≥0. From the definitions of Cn and Qn, it is obvious that Cn is closed and Qn is closed and convex for each n≥0. Next, we prove that Cn is convex. Since ϕ(z,un)≤ϕ(z,xn) is equivalent to
(3.2)0≤∥xn∥2-∥un∥2-2〈z,Jxn-Jun〉,
which is affine in z, and hence Cn is convex. So, Cn∩Qn is a closed and convex subset of E for all n≥0. Next, we show that Ω⊂Cn for all n≥0. Indeed, let u∈Ω and yn=Jrnxn for all n≥0. Since Jrn are relatively nonexpansive mappings, we have
(3.3)ϕ(u,zn)=ϕ(u,J-1(βnJxn+(1-βn)JSyn))=∥u∥2-2〈u,βnJxn+(1-βn)JSyn〉+∥βnJxn+(1-βn)JSyn∥2≤∥u∥2-2βn〈u,Jxn〉-2(1-βn)〈u,JSyn〉+βn∥xn∥2+(1-βn)∥Syn∥2=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
(3.4)ϕ(u,un)=ϕ(u,J-1(αnJxn+(1-αn)JTzn))=∥u∥2-2〈u,αnJxn+(1-αn)JTzn〉+∥αnJxn+(1-αn)JTzn∥2≤∥u∥2-2αn〈u,Jxn〉-2(1-αn)〈u,JTzn〉+αn∥xn∥2+(1-αn)∥Tzn∥2=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, u∈Cn for all n≥0, which implies that Ω⊂Cn. Next, we show that Ω⊂Qn for all n≥0. We prove by induction. For n=0, we have Ω⊂C=Q0. Assume that Ω⊂Qn. Since xn+1 is the projection of x0 onto Cn∩Qn, by Lemma 2.3 we have
(3.5)〈xn+1-z,Jx0-Jxn+1〉≥0,∀z∈Cn∩Qn.
As Ω⊂Cn∩Qn by the induction assumptions, we have
(3.6)〈xn+1-z,Jx0-Jxn+1〉≥0,∀z∈Ω.
This together with definition of Qn+1 implies that Ω⊂Qn+1 and hence Ω⊂Qn for all n≥0. So, we have that Ω⊂Cn∩Qn for all n≥0. This implies that {xn} is well defined. From definition of Qn that xn=ΠQnx0 and xn+1=ΠCn∩Qnx0∈Cn∩Qn⊂Qn, we have
(3.7)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
Therefore, {ϕ(xn,x0)} is nondecreasing. It follows from Lemma 2.4 and xn=ΠQnx0 that
(3.8)ϕ(xn,x0)=ϕ(ΠQnx0,x0)≤ϕ(u,x0)-ϕ(u,ΠQnx0)≤ϕ(u,x0)
for all u∈Ω⊂Qn. 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
(3.9)ϕ(xn+1,xn)=ϕ(xn+1,ΠQnx0)≤ϕ(xn+1,x0)-ϕ(ΠQnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0)
for all n≥0. This implies that limn→∞ϕ(xn+1,xn)=0. From xn+1=ΠCn∩Qnx0∈Cn, we have
(3.10)ϕ(xn+1,un)≤ϕ(xn+1,xn).
Therefore, we have limn→∞ϕ(xn+1,un)=0.
Sincelimn→∞ϕ(xn+1,xn)=limn→∞ϕ(xn+1,un)=0 and E is uniformly convex and smooth, we have from Lemma 2.1 that
(3.11)limn→∞∥xn+1-xn∥=limn→∞∥xn+1-un∥=0.
So, we have limn→∞∥xn-un∥=0. Since J is uniformly norm-to-norm continuous on bounded sets, we have
(3.12)limn→∞∥Jxn+1-Jxn∥=limn→∞∥Jxn+1-Jun∥=limn→∞∥Jxn-Jun∥=0.
On the other hand, we have
(3.13)∥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∥-αn∥Jxn-Jxn+1∥.
This follows that
(3.14)∥Jxn+1-JTzn∥≤11-αn(∥Jxn+1-Jun∥+αn∥Jxn-Jxn+1∥).
From (3.12) and lim infn→∞(1-αn)>0, we obtain that limn→∞∥Jxn+1-JTzn∥=0.
SinceJ-1 is uniformly norm-to-norm continuous on bounded sets, we have
(3.15)limn→∞∥xn+1-Tzn∥=0.
From
(3.16)∥xn-Tzn∥≤∥xn-xn+1∥+∥xn+1-Tzn∥,
we have
(3.17)limn→∞∥xn-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
(3.18)ϕ(u,zn)=ϕ(u,J-1(βnJxn+(1-βn)JSyn))=∥u∥2-2〈u,βnJxn+(1-βn)JSyn〉+∥βnJxn+(1-βn)JSyn∥2≤∥u∥2-2βn〈u,Jxn〉-2(1-βn)〈u,JSyn〉+βn∥xn∥2+(1-βn)∥Syn∥2-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
(3.19)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
(3.20)limj→∞ϕ(u,xnj′)=lim supk→∞ϕ(u,xnk)=a,
where u∈Ω. By (2) and (3), we have
(3.21)ϕ(u,xnj′)=ϕ(u,Tznj′)+ϕ(Tznj′,xnj′)+2〈u-Tznj′,JTznj′-Jxnj′〉≤ϕ(u,znj′)+∥Tznj′∥∥JTznj′-Jxnj′∥+∥Tznj′-xnj′∥∥xnj′∥+2∥u-Tznj′∥∥JTznj′-Jxnj′∥.
Since limn→∞∥xn-Tzn∥=0 and hence limn→∞∥Jxn-JTzn∥=0, it follows that
(3.22)a=lim infj→∞ϕ(u,xnj′)≤lim infj→∞ϕ(u,znj′).
We also have from (3.3) that
(3.23)lim supj→∞ϕ(u,znj′)≤lim supj→∞ϕ(u,xnj′)=a,
and hence
(3.24)limj→∞ϕ(u,xnj′)=limj→∞ϕ(u,znj′)=a.
Since lim infn→∞βn(1-βn)>0, it follows from (3.19) that limj→∞g(∥Jxnj′-JSynj′∥)=0. By properties of the function g, we have limj→∞∥Jxnj′-JSynj′∥=0. Since J-1 is also uniformly norm-to-norm continuous on bounded sets, we obtain limj→∞∥xnj′-Synj′∥=0 and then
(3.25)limn→∞∥xn-Syn∥=0.
So, we have limn→∞∥Jxn-JSyn∥=0. Since
(3.26)∥Jzn-Jxn∥=∥βnJxn+(1-βn)JSyn-Jxn∥=(1-βn)∥JSyn-Jxn∥≤∥JSyn-Jxn∥,
it follows that limn→∞∥Jzn-Jxn∥=0, and hence
(3.27)limn→∞∥xn-zn∥=0.
From (3.3), we have
(3.28)11-βn(ϕ(u,zn)-βnϕ(u,xn))≤ϕ(u,yn).
Using yn=Jrnxn and Lemma 2.6, we have
(3.29)ϕ(yn,xn)=ϕ(Jrnxn,xn)≤ϕ(u,xn)-ϕ(u,Jrnxn)=ϕ(u,xn)-ϕ(u,yn).
It follows that
(3.30)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(∥xn∥2-∥zn∥2-2〈u,Jxn-Jzn〉)≤11-βn(|∥xn∥2-∥zn∥2|+2|〈u,Jxn-Jzn〉|)≤11-βn(|∥xn∥-∥zn∥|(∥xn∥+∥zn∥)+2∥u∥∥Jxn-Jzn∥)≤11-βn(∥xn-zn∥(∥xn∥+∥zn∥)+2∥u∥∥Jxn-Jzn∥).
Since lim infn→∞βn(1-βn)>0, we have that lim infn→∞(1-βn)>0. So, we have limn→∞ϕ(yn,xn)=0. Since E is uniformly convex and smooth, we have from Lemma 2.1 that
(3.31)limn→∞∥yn-xn∥=0.
Since
(3.32)∥zn-Tzn∥≤∥zn-xn∥+∥xn-Tzn∥,∥yn-Syn∥≤∥yn-xn∥+∥xn-Syn∥,
from (3.17), (3.25), (3.27), and (3.31), we obtain that
(3.33)limn→∞∥zn-Tzn∥=limn→∞∥yn-Syn∥=0.
Since {xn} is bounded, there exists a subsequence {xnk} of {xn} such that xnk⇀v. From limn→∞∥xn-yn∥=0 and limn→∞∥xn-zn∥=0, we have ynk⇀v and znk⇀v. Since S and T are relatively nonexpansive, we have that v∈F̂(S)∩F̂(T)=F(S)∩F(T). Next, we show v∈A-10. Since J is uniformly norm-to-norm continuous on bounded sets, from (3.31) we have
(3.34)limn→∞∥Jxn-Jyn∥=0.
From rn≥a, we have
(3.35)limn→∞1rn∥Jxn-Jyn∥=0.
Therefore, we have
(3.36)limn→∞∥Arnxn∥=limn→∞1rn∥Jxn-Jyn∥=0.
For (p,p*)∈A, from the monotonicity of A, we have 〈p-yn,p*-Arnxn〉≥0 for all n≥0. Replacing n by nk and letting k→∞, we get 〈p-v,p*〉≥0. From the maximallity of A, we have v∈A-10, that is, v∈Ω.
Finally, we show thatxn→ΠΩx0. Let w=ΠΩx0. From xn+1=ΠCn∩Qnx0 and w∈Ω⊂Cn∩Qn, we obtain that
(3.37)ϕ(xn+1,x0)≤ϕ(w,x0).
Since the norm is weakly lower semicontinuous, we have
(3.38)ϕ(v,x0)=∥v∥2-2〈v,Jx0〉+∥x0∥2≤lim infk→∞(∥xnk∥2-2〈xnk,Jx0〉+∥x0∥2)=lim infk→∞ϕ(xnk,x0)≤lim supk→∞ϕ(xnk,x0)≤ϕ(w,x0).
From the definition of ΠΩ, we obtain v=w. This implies that
(3.39)limk→∞ϕ(xnk,x0)=ϕ(w,x0).
Therefore we have
(3.40)0=limk→∞(ϕ(xnk,x0)-ϕ(w,x0))=limk→∞(∥xnk∥2-∥w∥2-2〈xnk-w,Jx0〉)=limk→∞(∥xnk∥2-∥w∥2).
Since E has the Kadec-Klee property, we obtain that xnk→w=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 Theorem3.1, we can obtain the following corollaries.Corollary 3.2.
LetE be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let A⊂E×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 x0∈C and
(3.41)un=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JTJrnxn),Cn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0
for all n∈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 infn→∞βn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.Proof.
PuttingS=T in Theorem 3.1, we obtain Corollary 3.2.Corollary 3.3.
LetE be a uniformly convex and uniformly smooth Banach space and let C be a nonempty closed convex subset of E. Let A⊂E×E* be a maximal monotone operator satisfying D(A)⊂C and let Jr=(J+rA)-1J for all r>0. Let S:C→C be a relatively nonexpansive mapping such that Ω=F(S)∩A-10≠∅. Let {xn} be a sequence generated by x0∈C and
(3.42)un=J-1(βnJxn+(1-βn)JSJrnxn),Cn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0
for all n∈ℕ∪{0}, where J is the duality mapping on E, {βn}⊂[0,1], and {rn}⊂[a,∞) for some a>0. If lim infn→∞βn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.Proof.
PuttingT=I and αn=0 in Theorem 3.1, we obtain Corollary 3.3.LetE be a Banach space and let f:E→(-∞,∞] be a proper lower semicontinuous convex function. Define the subdifferential of f as follows:
(3.43)∂f(x)={x*∈E:f(y)≥〈y-x,x*〉+f(x),∀y∈E}
for each x∈E. Then, we know that ∂f is a maximal monotone operator; see [18] for more details.Corollary 3.4.
LetE 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 x0∈C and
(3.44)un=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JSxn),Cn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0
for all n∈ℕ∪{0}, where J is the duality mapping on E and {αn},{βn}⊂[0,1]. If lim infn→∞(1-αn)>0 and lim infn→∞βn(1-βn)>0, then {xn} converges strongly to ΠΩx0, where ΠΩ is the generalized projection of E onto Ω.Proof.
SetA=∂iC in Theorem 3.1, where iC is the indicator function; that is,
(3.45)iC(x){0,x∈C,∞,otherwise.
Then, we have that A is a maximal monotone operator and Jr=ΠC for r>0. In fact, for any x∈E and r>0, we have from Lemma 2.3 that
(3.46)z=Jrx⇔Jz+r∂iC(z)∋Jx⇔Jx-Jz∈r∂iC(z)⇔iC(y)≥〈y-z,Jx-Jzr〉+iC(z),∀y∈E⇔0≥〈y-z,Jx-Jz〉,∀y∈C⇔z=argminy∈Cϕ(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 Theorem3.1, we obtain the following results.Theorem 4.1.
LetC be a nonempty closed convex subset of a Hilbert space H. Let A⊂H×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 x0∈C and
(4.1)un=αnxn+(1-αn)Tzn,zn=βnxn+(1-βn)SJrnxn,Cn={z∈C:∥z-un∥≤∥z-xn∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0
for all n∈ℕ∪{0}, where {αn},{βn}⊂[0,1] and {rn}⊂[a,∞) for some a>0. If lim infn→∞(1-αn)>0 and lim infn→∞βn(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 thatϕ(x,y)=∥x-y∥2 for all x,y∈H. 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 Theorem4.1.Corollary 4.2.
LetC be a nonempty closed convex subset of a Hilbert space H. Let A⊂H×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 x0∈C and
(4.2)un=αnxn+(1-αn)Tzn,zn=βnxn+(1-βn)TJrnxn,Cn={z∈C:∥z-un∥≤∥z-xn∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0
for all n∈ℕ∪{0}, where {αn},{βn}⊂[0,1] and {rn}⊂[a,∞) for some a>0. If lim infn→∞(1-αn)>0 and lim infn→∞βn(1-βn)>0, then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.Proof.
PuttingS=T in Theorem 4.1, we obtain Corollary 4.2.Corollary 4.3.
LetC be a nonempty closed convex subset of a Hilbert space H. Let A⊂H×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 x0∈C and
(4.3)un=βnxn+(1-βn)SJrnxn,Cn={z∈C:∥z-un∥≤∥z-xn∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0
for all n∈ℕ∪{0}, where {βn}⊂[0,1] and {rn}⊂[a,∞) for some a>0. If lim infn→∞βn(1-βn)>0 then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.Proof.
PuttingT=I and αn=0 in Theorem 4.1, we obtain Corollary 4.3.Corollary 4.4.
LetC 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=x∈C and
(4.4)un=αnxn+(1-αn)Tzn,zn=βnxn+(1-βn)Sxn,Cn={z∈C:∥z-un∥≤∥z-xn∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0
for all n∈ℕ∪{0}, where {αn},{βn}⊂[0,1]. If lim infn→∞(1-αn)>0 and lim infn→∞βn(1-βn)>0, then {xn} converges strongly to PΩx0, where PΩ is the metric projection of H onto Ω.Proof.
SetA=∂iC in Theorem 4.1, where iC is the indicator function; that is,
(4.5)iC(x){0,x∈C,∞,otherwise.
Then, we have that A is a maximal monotone operator and Jr=PC for r>0. In fact, for any x∈E and r>0, we have that
(4.6)z=Jrx⇔z+r∂iC(z)∋x⇔x-z∈r∂iC(z)⇔iC(y)≥〈y-z,x-zr〉+iC(z),∀y∈E⇔0≥〈y-z,x-z〉,∀y∈C⇔z=PCx.
So, from Theorem 4.1, we obtain Corollary 4.4.



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*Source: 649831-2009-11-11.xml*