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-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
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].

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 u∈E satisfying
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
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 all x∈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
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
ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2,∀y,x∈E.

A point u∈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. [<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 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
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. Let ℕ 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.

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
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]:

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 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
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,

(∥x∥-∥y∥)2≤ϕ(x,y)≤(∥x∥+∥y∥)2,

ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+2〈x-z,Jz-Jy〉,

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
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 [<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 limn→∞ϕ(xn,yn)=0, then limn→∞∥xn-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, 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 [<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),∀x∈C,y∈E.

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)={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.

Let E 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.

Let E 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
D(A)⊂C⊂J-1(⋂r>0R(J+rA)).
Then we can define the resolvent Jr:C→D(A) of A by
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 [<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 A⊂E×E* be a monotone operator satisfying
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:

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)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.

Let E 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
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 that Cn 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
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
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
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
〈xn+1-z,Jx0-Jxn+1〉≥0,∀z∈Cn∩Qn.
As Ω⊂Cn∩Qn by the induction assumptions, we have
〈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
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
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
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
ϕ(xn+1,un)≤ϕ(xn+1,xn).
Therefore, we have limn→∞ϕ(xn+1,un)=0.

Since limn→∞ϕ(xn+1,xn)=limn→∞ϕ(xn+1,un)=0 and E is uniformly convex and smooth, we have from Lemma 2.1 that
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
limn→∞∥Jxn+1-Jxn∥=limn→∞∥Jxn+1-Jun∥=limn→∞∥Jxn-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∥-αn∥Jxn-Jxn+1∥.
This follows that
∥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.

Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
limn→∞∥xn+1-Tzn∥=0.
From
∥xn-Tzn∥≤∥xn-xn+1∥+∥xn+1-Tzn∥,
we have
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
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
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
limj→∞ϕ(u,xnj′)=lim supk→∞ϕ(u,xnk)=a,
where u∈Ω. By (2) and (3), we have
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
a=lim infj→∞ϕ(u,xnj′)≤lim infj→∞ϕ(u,znj′).
We also have from (3.3) that
lim supj→∞ϕ(u,znj′)≤lim supj→∞ϕ(u,xnj′)=a,
and hence
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
limn→∞∥xn-Syn∥=0.
So, we have limn→∞∥Jxn-JSyn∥=0. Since
∥Jzn-Jxn∥=∥βnJxn+(1-βn)JSyn-Jxn∥=(1-βn)∥JSyn-Jxn∥≤∥JSyn-Jxn∥,
it follows that limn→∞∥Jzn-Jxn∥=0, and hence
limn→∞∥xn-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(∥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
limn→∞∥yn-xn∥=0.
Since
∥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
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
limn→∞∥Jxn-Jyn∥=0.
From rn≥a, we have
limn→∞1rn∥Jxn-Jyn∥=0.
Therefore, we have
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 that xn→ΠΩx0. Let w=ΠΩx0. From xn+1=ΠCn∩Qnx0 and w∈Ω⊂Cn∩Qn, we obtain that
ϕ(xn+1,x0)≤ϕ(w,x0).
Since the norm is weakly lower semicontinuous, we have
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
limk→∞ϕ(xnk,x0)=ϕ(w,x0).
Therefore we have
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 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 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
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.

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 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
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.

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),∀y∈E}
for each x∈E. Then, we know that ∂f is a maximal monotone operator; see [18] 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 x0∈C and
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.

Set A=∂iC in Theorem 3.1, where iC is the indicator function; that is,
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
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 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 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
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 that ϕ(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 Theorem 4.1.

Corollary 4.2.

Let C 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
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.

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 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
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.

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=x∈C and
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.

Set A=∂iC in Theorem 4.1, where iC is the indicator function; that is,
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
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.

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.

DasG.DebataJ. P.Fixed points of quasinonexpansive mappingsIshikawaS.Fixed points by a new iteration methodMannW. R.Mean value methods in iterationTakahashiW.TamuraT.Convergence theorems for a pair of nonexpansive mappingsKamimuraS.TakahashiW.Approximating solutions of maximal monotone operators in Hilbert spacesKamimuraS.TakahashiW.Strong convergence of a proximal-type algorithm in a Banach spaceRockafellarR. T.Monotone operators and the proximal point algorithmSolodovM. V.SvaiterB. F.Forcing strong convergence of proximal point iterations in a Hilbert spaceBruckR. E.ReichS.Nonexpansive projections and resolvents of accretive operators in Banach spacesKamimuraS.KohsakaF.TakahashiW.Weak and strong convergence theorems for maximal monotone operators in a Banach spaceKohsakaF.TakahashiW.Strong convergence of an iterative sequence for maximal monotone operators in a Banach spaceReichS.KatrosatosA. G.A weak convergence theorem for the alternating method with Bregman distancesButnariuD.ReichS.ZaslavskiA. J.Asymptotic behavior of relatively nonexpansive operators in Banach spacesButnariuD.ReichS.ZaslavskiA. J.Weak convergence of orbits of nonlinear operators in reflexive Banach spacesMatsushitaS.TakahashiW.A strong convergence theorem for relatively nonexpansive mappings in a Banach spaceQinX.SuY.Strong convergence theorems for relatively nonexpansive mappings in a Banach spaceInoueG.TakahashiW.ZembayashiK.Strong convergence theorems by hybrid methods for maximal monotone operator and relatively nonexpansive mappings in Banach spacesto appear in Journal of Convex AnalysisTakahashiW.TakahashiW.AlberY. I.KatrosatosA. G.Metric and generalized projection operators in Banach spaces: properties and applicationsRockafellarR. T.On the maximality of sums of nonlinear monotone operatorsKohsakaF.TakahashiW.Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spacesZălinescuC.On uniformly convex functionsXuH. K.Inequalities in Banach spaces with applications