# Iterative Schemes for Fixed Points of Relatively Nonexpansive Mappings and Their Applications

Authors: Somyot Plubtieng; Wanna Sriprad
Journal: Abstract and Applied Analysis
 (2010)

---

## Abstract

We present two iterative schemes with errors which are proved to be strongly convergent
to a common element of the set of fixed points of a countable family of relatively nonexpansive mappings
and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly
smooth and uniformly convex Banach space. Using the result we consider strong convergence theorems for
variational inequalities and equilibrium problems in a real Hilbert space and strong convergence theorems
for maximal monotone operators in a real uniformly smooth and uniformly convex Banach space.


---

## Body

## 1. Introduction

LetE be a real Banach space, and E* the dual space of E. The function ϕ:E→E* is denoted by (1.1)ϕ(y,x)=‖y‖2-2〈y,Jx〉+‖x‖2
for all x,y∈E, where J is the normalized duality mapping from E to E. Let C be a closed convex subset of E, and let T be a mapping from C into itself. We denote by F(T) the set of fixed points of T. A point p in C is said to be an asymptotic fixed point of T [1] if C contains a sequence {xn} which converges weakly to p such that the strong limn→∞(xn-Txn) equals 0. The set of asymptotic fixed points of T will be denoted by F̂(T). A mapping T from C into itself is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C and nonexpansive with respect to the Lyapunov functional [2] if ϕ(Tx,Ty)≤ϕ(x,y) for all x,y∈C and it is called relatively nonexpansive [3–6] if F̂(T)=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of relatively nonexpansive mapping was studied in [3–6].There are many methods for approximating fixed points of a nonexpansive mapping. In 1953, Mann [7] introduced the iteration as follows: a sequence {xn} is defined by(1.2)xn+1=αnxn+(1-αn)Txn,
where the initial guess element x0∈C is arbitrary and {αn} is a real sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results was proved by Reich [1]. In an infinite-dimensional Hilbert space, Mann iteration can yield only weak convergence (see [8, 9]). Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [10] proposed the following modification of Mann iteration method (1.2) for nonexpansive mapping T in a Hilbert space: in particular, they studied the strong convergence of the sequence {xn} generated by(1.3)x0=x∈C,yn=αnxn+(1-αn)Txn,Cn={z∈C:‖z-yn‖≤‖z-xn‖},Qn={z∈C:〈xn-z,x-xn〉≥0},xn+1=PCn∩Qnx,n=0,1,2,…,
where {αn}⊂[0,1] and PCn∩Qn is the metric projection from C onto Cn∩Qn.Recently, Takahashi et al. [11] extended iteration (1.6) to obtain strong convergence to a common fixed point of a countable family of nonexpansive mappings; let C be a nonempty closed convex subset of a Hilbert space H. Let {Tn} and 𝒯 be families of nonexpansive mappings of C into itself such that ∅≠F(𝒯)=⋂n=1∞F(Tn) and let x0∈H. Suppose that {Tn} satisfies the NST-condition (I) with 𝒯; that is, for each bounded sequence {zn}⊂C, limn→∞∥zn-Tnzn∥=0 implies that limn→∞∥zn-Tnzn∥=0 for all T∈𝒯. For x1=PCx0, define a sequence {xn} of C as follows:(1.4)yn=αnxn+(1-αn)Tnxn,Cn={z∈C:‖z-yn‖≤‖z-xn‖},Qn={z∈C:〈xn-z,x0-xn〉≥0},un+1=PCn∩Qnx0,n∈N,
where 0≤αn≤α<1. Then, {un} converges strongly to z0=PF(T)x0.On the other hand, Halpern [12] introduced the following iterative scheme for approximating a fixed point of T:(1.5)xn+1=αnx+(1-αn)Txn
for all n∈ℕ, where x1=x∈C and {αn} is a sequence of [0,1]. Strong convergence of this type of iterative sequence has been widely studied: for instance, see [13, 14] and the references therein. In 2006, Martinez-Yanes and Xu [15] have adapted Nakajo and Takahashi's [10] idea to modify the process (1.5) for a nonexpansive mapping T in a Hilbert space:(1.6)x0=x∈C,yn=αnx0+(1-αn)Txn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+αn(‖x0‖2+2〈xn-x0,z〉)},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,n=0,1,2,…,
where PCn∩Qn is the metric projection from C onto Cn∩Qn. They proved that if {αn}⊂(0,1) and limn→∞αn=0, then the sequence {xn} generated by (1.6) converges strongly to PF(T)x.The ideas to generalize the process (1.3) and (1.6) from Hilbert space to Banach space have been studied by many authors. Matsushita and Takahashi [6], Qin and Su [16], and Plubtieng and Ungchittrakool [17] generalized the process (1.3) and (1.6) and proved the strong convergence theorems for relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach spacey; see, for instance, [2, 6, 16, 18–22] and the references therein.Recently, Nakajo et al. [18] introduced the following condition. Let C be a nonempty closed convex subset of a Hilbert space H, let {Tn} be a family of mappings of C into itself with F:=⋂n=1∞F(Tn)≠∅, and let ωw(zn) denote the set of all weak subsequential limits of a bounded sequence {zn} in C. {Tn} is said to satisfy the NST-condition (II) if, for every bounded sequence {zn} in C, (1.7)lim⁡n→∞‖zn-Tnzn‖=0impliesωw(zn)⊂F.
Very recently, Nakajo et al. [19] introduced the more general condition so-called the NST*-condition, {Tn} is said to satisfy the NST*-condition if, for every bounded sequence {zn} in C, (1.8)lim⁡n→∞‖zn-Tnzn‖=lim⁡n→∞‖zn-zn+1‖=0impliesωw(zn)⊂F.
It follows directly from the definitions above that if {Tn} satisfies the NST-condition (I), then {Tn} satisfies the NST*-condition.Motivated and inspired by Wei and Cho [2], in this paper, we introduce two iterative schemes (3.1) and (3.14) and use the NST*-condition for a countable family of relatively nonexpansive mappings to obtain the strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space. Using this result, we also discuss the problem of strong convergence concerning variational inequality, equilibrium, and nonexpansive mappings in Hilbert spaces. Moreover, we also apply our convergence theorems to the maximal monotone operators in Banach spaces. The results obtained in this paper improve and extend the corresponding result of Matsushita and Takahashi [6], Qin and Su [16], Wei and Cho [2], Wei and Zhou [22], and many others.


## 2. Preliminaries

LetE be a real Banach space with norm ∥·∥ and let E* be the dual of E. For all x∈E and x*∈E*, we denote the value of x* at x by 〈x,x*〉. We denote the strong convergence and the weak convergence of a sequence {xn} to x in E by xn→x and xn⇀x, respectively. We also denote the weak* convergence of a sequence {xn*} to x* in E* by xn*⇀*x*. An operator T⊂E×E* is said to be monotone if 〈x-y,x*-y*〉≥0 whenever (x,x*),(y,y*)∈T. We denote the set {x∈E:0∈Tx} by T-10. A monotone T is said to be maximal if its graph G(T)={(x,y):y∈Tx} is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the solution set T-10 is closed and convex.The normalized duality mappingJ from E to E* is defined by (2.1)Jx={x*∈E*:〈x,x*〉=‖x‖2=‖x*‖2}
for x∈E. By Hahn-Banach theorem, J(x) is nonempty; see [23] for more details. A Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. It is also said to be uniformly convex if, for each ɛ∈(0,2], there exists δ>0 such that ∥x+y∥/2≤δ for x,y∈E with ∥x∥=∥y∥=1 and ∥x-y∥≥ɛ. Let S(E)={x∈E:∥x∥=1} be the unit sphere of E. Then the Banach space E is said to be smooth provided that(2.2)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈S(E). It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈S(E). It is well known that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single valued, one-to-one, and onto.LetE be a smooth, strictly convex, and reflexive Banach space and let C be a nonempty closed convex subset of E. Throughout this paper, we denote by ϕ the function defined by (2.3)ϕ(y,x)=‖y‖2-2〈y,Jx〉+‖x‖2,∀x,y∈C.
It is obvious from the definition of the function ϕ that (∥x∥-∥y∥)2⩽ϕ(y,x)⩽(∥y∥2+∥x∥2) for all x,y∈E.Following Alber [24], the generalized projection PC from E onto C is defined by (2.4)PC(x)=argmin⁡y∈Cϕ(y,x),∀x∈E.
If E is a Hilbert space, then ϕ(y,x)=∥y-x∥2 and PC 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 [25]).
LetE be a uniformly convex and smooth Banach space and let {xn} and {yn} be 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 (Alber [24], Kamimura and Takahashi [25]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Then
(2.5)ϕ(x,PCy)+ϕ(PCy,y)≤ϕ(x,y),∀x∈C,y∈E.Lemma 2.3 (Alber [24], Kamimura and Takahashi [25]).
LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let x∈E, and let x0∈C. Then
(2.6)x0=PCx⟺〈z-x0,Jx0-Jx〉≥0,∀z∈C.Lemma 2.4 (Matsushita and Takahashi [6]).
LetC be a nonempty 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 2.5 (see [2]).
LetE be a real smooth and uniformly convex Banach space. If S:E→E is a mapping which is nonexpansive with respect to the Lyapunov functional, then F(S) is convex and closed subset of E.


## 3. Main Results

In this section, by using theNST*-condition, we proved two strong convergence theorems for finding a common element of the set of fixed points of a countable family of relatively nonexpansive mappings and the set of fixed points of nonexpansive mappings in the sense of Lyapunov functional in a real uniformly smooth and uniformly convex Banach space.Theorem 3.1.
LetE be a real uniformly smooth and uniformly convex Banach space, let C be a nonempty closed convex subset of E, and let S:C→C be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let {Tn} be a family of relatively nonexpansive mappings of C into itself such that F:=⋂n=1∞F(Tn)∩F(S)≠∅ and satisfy the NST*-condition. Then the sequence {xn} generated by
(3.1)x0∈C,yn=Tn(xn+en),zn=J-1(αnJxn+(1-αn)Jyn),un=Szn,Hn={z∈C:ϕ(z,un)⩽ϕ(z,zn)⩽αnϕ(z,xn)+(1-αn)ϕ(z,xn+en)},Wn={z∈C:〈z-xn,Jx0-Jxn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
the error sequence is{en}⊂C such that ∥en∥→0 as n→∞.Proof.
We split the proof into five steps.Step 1 (BothHn and Wn are closed and convex subset of C).
Noting the facts that(3.2)ϕ(z,zn)⩽αnϕ(z,xn)+(1-αn)ϕ(z,xn+en)⟺‖zn‖2-αn‖xn‖2-(1-αn)‖xn+en‖2≤2〈z,Jzn-αnJxn-(1-αn)J(xn+en)〉,ϕ(z,un)⩽ϕ(z,zn)⟺‖zn‖2-‖un‖2≥2〈z,Jzn-Jun〉,
we can easily know that Hn is closed and convex subset of E. It is obvious that Wn is also a closed and convex subset of E.Step 2 (F:=(⋂n=1∞F(Tn))∩F(S)⊆Hn∩Wn for all n∈ℕ∪{0}).
To observe this, takep∈F. Then it follows from the convexity of ∥·∥2 that
(3.3)ϕ(p,un)=ϕ(Sp,Szn)≤ϕ(p,zn)=‖p‖2-2〈p,αnJxn+(1-αn)Jyn〉+‖αnJxn+(1-αn)Jyn‖2≤‖p‖2-2αn〈p,Jxn〉-2(1-αn)〈p,Jyn〉+αn‖xn‖2+(1-αn)‖yn‖2=αnϕ(p,xn)+(1-αn)ϕ(p,yn)=αnϕ(p,xn)+(1-αn)ϕ(p,Tn(xn+en))≤αnϕ(p,xn)+(1-αn)ϕ(p,xn+en)
for all n∈ℕ∪{0}. Hence F⊂Hn for all n∈ℕ∪{0}. On the other hand, it is clear that p∈W0=C. Then p∈H0∩W0 and x1=PH0∩W0x0 is well defined. Suppose that p∈Wn-1 for some n≥1. Then p∈Hn-1∩Wn-1 and xn=PHn-1∩Wn-1x0 is well defined. It follows from Lemma 2.3 that
(3.4)〈p-xn,Jx0-Jxn〉=〈p-PHn-1∩Wn-1x0,Jx0-JPHn-1∩Wn-1x0〉≤0,
which implies that p∈Wn. Therefore p∈Hn∩Wn, and hence xn+1=PHn∩Wnx0 is well defined. Then by induction, the sequence {xn} generated by (3.1) is well defined, for each n≥0. Moreover F⊂Hn∩Wn for each nonnegative integer n.Step 3 ({xn} is bounded sequence of C).
In fact, for allp∈F⊂Hn∩Wn⊂Wn, it follows from Lemma 2.2 that
(3.5)ϕ(p,PWnx0)+ϕ(PWnx0,x0)≤ϕ(p,x0).
By the definition of Wn and Lemma 2.2, we note that xn=PWnx0 and hence
(3.6)ϕ(p,xn)+ϕ(xn,x0)≤ϕ(p,x0).
Therefore {xn} is bounded.Step 4 (ωw(xn)⊂F:=(⋂n=1∞F(Tn))∩F(S)).
From the factsxn=PWnx0, xn+1∈Wn, and Lemma 2.2, we have
(3.7)ϕ(xn+1,xn)+ϕ(xn,x0)≤ϕ(xn+1,x0).
Therefore, limn→∞ϕ(xn,x0) exists. Then ϕ(xn+1,xn)→0, which implies from Lemma 2.1 that ∥xn+1-xn∥→0 as n→∞. Since xn+1∈Hn, we have
(3.8)ϕ(xn+1,un)⩽ϕ(xn+1,zn)⩽αnϕ(xn+1,xn)+(1-αn)ϕ(xn+1,xn+en).
Note that
(3.9)ϕ(xn+1,xn+en)-ϕ(xn+1,xn)=‖xn+en‖2-‖xn‖2+2〈xn+1,Jxn-J(xn+en)〉.
Since J is uniform continuous on each bounded subset of E and ∥en∥→0, we have ϕ(xn+1,xn+en)→0 as n→∞. This implies that ϕ(xn+1,un)→0 and ϕ(xn+1,zn)→0 as n→∞. Using Lemma 2.1, we obtain limn→∞∥xn+1-(xn+en)∥=0, limn→∞∥xn+1-zn∥=0, and limn→∞∥xn+1-un∥=0. Since both J:E→E* and J-1:E*→E are uniformly continuous on bounded subsets, it follows that limn→∞∥xn-yn∥=0.
Putvn:=xn+en for all n∈ℕ∪{0}. Since {xn} is bounded and ∥en∥→0, we have {vn} is bounded. Note that
(3.10)‖vn+1-vn‖=‖(xn+1+en+1)-(xn+en)‖≤‖xn+1-(xn+en)‖+‖en+1‖,‖vn-Tnvn‖=‖(xn+en)-Tn(xn+en)‖=‖(xn+en)-yn‖≤‖(xn+en)-xn+1‖+‖xn+1-xn‖+‖xn-yn‖
for all n∈ℕ∪{0}. It implies that limn→∞∥vn+1-vn∥=limn→∞∥vn-Tnvn∥=0. Therefore, we have ωw(vn)⊂⋂n=1∞F(Tn) since {Tn} satisfies NST*-condition. Note that ∥vn-xn∥=∥(xn+en)-xn∥=∥en∥→0 as n→∞. Hence, we also have ωw(xn)⊂⋂n=1∞F(Tn).
On the other hand, from Step3, we know that ωw(xn)≠∅. Then, for all q∈ωw(xn), there exists a subsequence {xni} of {xn} such that xni⇀q as i→∞. Therefore, uni⇀q and zni⇀q as i→∞. Since S:C→C is weakly continuous and uni=Szni, we have q∈F(S). Hence ωw(xn)⊂F:=(⋂n=1∞F(Tn))∩F(S).Step 5 (xn→q*=PFx0, as n→∞).
Let{xnj} be any subsequence of {xn} which weakly converges to q∈F. Since xn+1=PHn∩Wnx0 and q*∈F⊂Hn∩Wn, we have ϕ(xn+1,x0)≤ϕ(q*,x0). Then,
(3.11)ϕ(xn,q*)=ϕ(xn,x0)+ϕ(x0,q*)-2〈xn-x0,Jq*-Jx0〉≤ϕ(q*,x0)+ϕ(x0,q*)-2〈xn-x0,Jq*-Jx0〉,
which yields
(3.12)limsup⁡j→∞ϕ(xnj,q*)≤ϕ(q*,x0)+ϕ(x0,q*)-2〈q-x0,Jq*-Jx0〉=-2〈q*-q,Jq*-Jx0〉≤0.
Hence ϕ(xnj,q*)→0 as j→∞. It follows from Lemma 2.1 that xnj→q* as j→∞. Therefore, xn→q*=PFx0 as n→∞. This completes the proof.Corollary 3.2 (see Matsushita and Takahashi [6]).
LetE be a real uniformly smooth and uniformly convex Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mappings of C into itself, and let {αn} be a sequence of real numbers such that 0≤αn<1 and lim supn→∞αn<1. Suppose that {xn} is given by
(3.13)x0=x∈C,yn=J-1(αnJxn+(1-αn)JTxn),Hn={z∈C:ϕ(z,yn)⩽ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wnx0,n=0,1,2,…,
where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to PF(T)x, where PF(T) is the generalized projection from C onto F(T).Proof.
Suppose thatF(T)≠∅ and put Tn≡T, S≡I, and en=0 for all n∈ℕ∪{0}. Let {zn} be a bounded sequence in C with limn→∞∥zn-Tzn∥=0 and let z∈ωw(zn). Then there exists subsequence {znk} of {zn} such that znk⇀z. It follows directly from the definition of T that z∈F̂(T)=F(T). Hence T satisfies NST-condition; by Theorem 3.1, {xn} converges strongly to PF(T)x.Theorem 3.3.
LetE be a real uniformly smooth and uniformly convex Banach space, let C be a nonempty closed convex subset of E, and let S:C→C be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let {Tn} be a family of relatively nonexpansive mappings of C into itself such that F:=⋂n=1∞F(Tn)∩F(S)≠∅ and satisfy the NST*-condition. Then the sequence {xn} generated by
(3.14)x0∈C,yn=Tn(xn+en),zn=J-1(αnJx0+(1-αn)Jyn),un=Szn,Hn={z∈C:ϕ(z,un)⩽ϕ(z,zn)⩽αnϕ(z,x0)+(1-αn)ϕ(z,xn+en)},Wn={z∈C:〈z-xn,Jx0-Jxn〉≤0},xn+1=PHn∩Wnx0
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
the error sequence is{en}⊂E such that ∥en∥→0 as n→∞.Proof.
By slightly modifying the corresponding proof in Theorem3.1, we can easily obtain the result.SettingTn≡T, S≡I, and en=0 for all n∈ℕ∪{0} in Theorem 3.3, we obtain the following corollary.Corollary 3.4 (see Qin and Su [16]).
LetE be a real uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let T be a relatively nonexpansive mappings of C into itself, and let {αn} be a sequence in (0,1) and limn→∞αn=0. Define a sequence {xn} in C by
(3.15)x0∈C,yn=J-1(αnJx0+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)⩽αnϕ(z,x0)+(1-αn)ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qnx0,n=0,1,2,…,
where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to PF(T)x, where PF(T) is the generalized projection from C onto F(T).


## 4. Applications to the Variational Inequality Problem, Equilibrium  Problem, and Fixed Points Problem of Nonexpansive Mappings in  a Real Hilbert Space

In this section, using Theorems3.1 and 3.3, we prove the strong convergence theorems for finding a common element of the set of fixed points of nonexpansive mapping, the solution set of the variational inequality problems, and the solution set of an equilibrium problems in a Hilbert space.


### 4.1. Common Solutions of a Fixed Point Problem and a Variational Inequality Problem

LetC be a nonempty closed convex subset of a real Hilbert space H and let A be a mapping of C into H. A classical variational inequality problem, denoted by VI(C,A), is to find an element u*∈C such that 〈u-u*,Au*〉≥0 for all u∈C. It is known that for λ>0, x∈C is a solution of the variational inequality of A if and only if x=PC(I-λA)x, where PC is the metric projection from H onto C. We denote by NC(v) the normal cone for C at a point v∈C, that is, NC(v)={w∈H:〈w,v-u〉≥0 for all u∈C}. Define f:H→H by(4.1)f(v)={Av+NC(v),∀v∈C,∅,otherwise.
Then f is maximal monotone and f-10=VI(C,A) (see [26]). A mapping A:C→H is called α-inverse strongly monotone if there exists an α>0 such that (4.2)〈Au-Av,u-v〉≥α‖Au-Av‖2,∀u,v∈C.
It is well known that if 0<λ<2α, then PC(I-λA) is nonexpansive of C into itself.Lemma 4.1 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let A be α-inverse strongly monotone mapping of C into H. Let {λn}⊂(0,2α) and let PC be a metric projection from H onto C. Let Tn=PC(I-λnA), for all n∈ℕ. Then {Tn} is a sequence of nonexpansive mappings and satisfies NST-condition.By using Lemma4.1 and Theorem 3.1 with en=0, for all n∈ℕ∪{0}, we obtain the following theorem.Theorem 4.2.
LetC be a nonempty closed convex subset of a real Hilbert space H and let A be α-inverse strongly monotone mapping of C into H. Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩VI(C,A)≠∅. Let PC be a metric projection from H onto C. Then the sequence {xn} generated by
(4.3)x0∈C,yn=αnxn+(1-αn)PC(xn-λnAxn),zn=Syn,Hn={z∈C:‖z-zn‖≤‖z-xn‖},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
{λn}⊂[a,b] for some a,b∈(0,2α) with a≤b.Next, by using Lemma4.1 and Theorem 3.3 with en=0, for all n∈ℕ∪{0}, we obtain the following theorem.Theorem 4.3.
LetC be a nonempty closed convex subset of a real Hilbert space H and let A be α-inverse strongly monotone mapping of C into H. Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩VI(C,A)≠∅. Let PC be a metric projection from H onto C. Then the sequence {xn} generated by
(4.4)x0∈C,yn=αnx0+(1-αn)PC(xn-λnAxn),zn=Syn,Hn={z∈C:‖z-zn‖2≤‖z-xn‖2+αn(‖x0‖2+2〈xn-x0,v〉)},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
{λn}⊂[a,b] for some a,b∈(0,2α) with a≤b.


### 4.2. Common Solutions of a Fixed Point Problem and an Equilibrium Problem

LetC be a nonempty closed convex subset of H and let F be a bifunction of C×C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for F:C×C→ℝ is to find x∈C such that(4.5)F(x,y)≥0,∀y∈C.
The set of solutions of (4.5) is denoted by EP(F). Numerous problems in physics, optimization, and economics reduce to find a solution of (4.5).  In 1997, Combettes and Hirstoaga [15] introduced an iterative  scheme of finding the best approximation to the initial data when EP(F) is nonempty and proved a strong convergence theorem.For solving the equilibrium problem for a bifunctionF:C×C→ℝ, let us assume that F satisfies the  following conditions:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 forall x,y∈C;(A3)
for eachx,y,z∈C,
(4.6)lim⁡t→0F((tz+(1-t)x),y)≤F(x,y);(A4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.The following lemma appears implicitly in [15, 28].Lemma 4.4 (see [15, 28]).
LetC be a nonempty closed convex subset of H and let F be a bifunction of C×C in to ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(4.7)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Define a mapping Tr:H→C as follows:
(4.8)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all z∈H. Then, the following holds:(1)
Tr is single-valued;(2)
Tr is firmly nonexpansive, that is, for any x,y∈H,
(4.9)‖Trx-Try‖2≤〈Trx-Try,x-y〉;(3)
F(Tr)=EP(F);(4)
EP(F) is closed and convex.Lemma 4.5 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f be a bifunction from C×C into ℝ satisfying (A1)–(A4) and EP(f)≠∅. If {rn} is a sequence in (0,∞) satisfying lim infn→∞rn>0, then {Trn} is a family of firmly nonexpansive mappings of H into C and satisfies NST-condition.The following theorem follows directly from Lemma4.5 and Theorem 3.1.Theorem 4.6.
LetC be a nonempty closed convex subset of a real Hilbert space H and let f be a bifunction from C×C into ℝ satisfying (A1)–(A4). Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩EP(f)≠∅. Then the sequence {xn} generated by
(4.10)x0∈C,un∈Csuchthatf(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnxn+(1-αn)un,zn=Syn,Hn={z∈C:‖z-zn‖≤‖z-xn‖},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
{rn}⊂(0,∞) such that lim infn→∞rn>0,Proof.
PutTn≡Trn and en=0 for all n∈ℕ∪{0}. Then, it follows from Lemma 4.5 and Theorem 3.1 that {xn} converges strongly to PFx0.Next, by using Lemma4.5 and Theorem 3.3 with en=0, for all n∈ℕ∪{0}, we obtain the following result.Theorem 4.7.
LetC be a nonempty closed convex subset of a real Hilbert space H and let f be a bifunction from C×C into ℝ satisfying (A1)–(A4). Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩EP(f)≠∅. Then the sequence {xn} generated by
(4.11)x0∈C,un∈Csuchthatf(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnx0+(1-αn)un,zn=Syn,Hn={z∈C:‖z-zn‖2≤‖z-xn‖2+αn(‖x0‖2+2〈xn-x0,v〉)},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
{rn}⊂(0,∞) such that lim infn→∞rn>0,


## 4.1. Common Solutions of a Fixed Point Problem and a Variational Inequality Problem

LetC be a nonempty closed convex subset of a real Hilbert space H and let A be a mapping of C into H. A classical variational inequality problem, denoted by VI(C,A), is to find an element u*∈C such that 〈u-u*,Au*〉≥0 for all u∈C. It is known that for λ>0, x∈C is a solution of the variational inequality of A if and only if x=PC(I-λA)x, where PC is the metric projection from H onto C. We denote by NC(v) the normal cone for C at a point v∈C, that is, NC(v)={w∈H:〈w,v-u〉≥0 for all u∈C}. Define f:H→H by(4.1)f(v)={Av+NC(v),∀v∈C,∅,otherwise.
Then f is maximal monotone and f-10=VI(C,A) (see [26]). A mapping A:C→H is called α-inverse strongly monotone if there exists an α>0 such that (4.2)〈Au-Av,u-v〉≥α‖Au-Av‖2,∀u,v∈C.
It is well known that if 0<λ<2α, then PC(I-λA) is nonexpansive of C into itself.Lemma 4.1 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let A be α-inverse strongly monotone mapping of C into H. Let {λn}⊂(0,2α) and let PC be a metric projection from H onto C. Let Tn=PC(I-λnA), for all n∈ℕ. Then {Tn} is a sequence of nonexpansive mappings and satisfies NST-condition.By using Lemma4.1 and Theorem 3.1 with en=0, for all n∈ℕ∪{0}, we obtain the following theorem.Theorem 4.2.
LetC be a nonempty closed convex subset of a real Hilbert space H and let A be α-inverse strongly monotone mapping of C into H. Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩VI(C,A)≠∅. Let PC be a metric projection from H onto C. Then the sequence {xn} generated by
(4.3)x0∈C,yn=αnxn+(1-αn)PC(xn-λnAxn),zn=Syn,Hn={z∈C:‖z-zn‖≤‖z-xn‖},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
{λn}⊂[a,b] for some a,b∈(0,2α) with a≤b.Next, by using Lemma4.1 and Theorem 3.3 with en=0, for all n∈ℕ∪{0}, we obtain the following theorem.Theorem 4.3.
LetC be a nonempty closed convex subset of a real Hilbert space H and let A be α-inverse strongly monotone mapping of C into H. Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩VI(C,A)≠∅. Let PC be a metric projection from H onto C. Then the sequence {xn} generated by
(4.4)x0∈C,yn=αnx0+(1-αn)PC(xn-λnAxn),zn=Syn,Hn={z∈C:‖z-zn‖2≤‖z-xn‖2+αn(‖x0‖2+2〈xn-x0,v〉)},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
{λn}⊂[a,b] for some a,b∈(0,2α) with a≤b.


## 4.2. Common Solutions of a Fixed Point Problem and an Equilibrium Problem

LetC be a nonempty closed convex subset of H and let F be a bifunction of C×C into ℝ, where ℝ is the set of real numbers. The equilibrium problem for F:C×C→ℝ is to find x∈C such that(4.5)F(x,y)≥0,∀y∈C.
The set of solutions of (4.5) is denoted by EP(F). Numerous problems in physics, optimization, and economics reduce to find a solution of (4.5).  In 1997, Combettes and Hirstoaga [15] introduced an iterative  scheme of finding the best approximation to the initial data when EP(F) is nonempty and proved a strong convergence theorem.For solving the equilibrium problem for a bifunctionF:C×C→ℝ, let us assume that F satisfies the  following conditions:(A1)
F(x,x)=0 for all x∈C;(A2)
F is monotone, that is, F(x,y)+F(y,x)≤0 forall x,y∈C;(A3)
for eachx,y,z∈C,
(4.6)lim⁡t→0F((tz+(1-t)x),y)≤F(x,y);(A4)
for eachx∈C, y↦F(x,y) is convex and lower semicontinuous.The following lemma appears implicitly in [15, 28].Lemma 4.4 (see [15, 28]).
LetC be a nonempty closed convex subset of H and let F be a bifunction of C×C in to ℝ satisfying (A1)–(A4). Let r>0 and x∈H. Then, there exists z∈C such that
(4.7)F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C.
Define a mapping Tr:H→C as follows:
(4.8)Tr(x)={z∈C:F(z,y)+1r〈y-z,z-x〉≥0,∀y∈C}
for all z∈H. Then, the following holds:(1)
Tr is single-valued;(2)
Tr is firmly nonexpansive, that is, for any x,y∈H,
(4.9)‖Trx-Try‖2≤〈Trx-Try,x-y〉;(3)
F(Tr)=EP(F);(4)
EP(F) is closed and convex.Lemma 4.5 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let f be a bifunction from C×C into ℝ satisfying (A1)–(A4) and EP(f)≠∅. If {rn} is a sequence in (0,∞) satisfying lim infn→∞rn>0, then {Trn} is a family of firmly nonexpansive mappings of H into C and satisfies NST-condition.The following theorem follows directly from Lemma4.5 and Theorem 3.1.Theorem 4.6.
LetC be a nonempty closed convex subset of a real Hilbert space H and let f be a bifunction from C×C into ℝ satisfying (A1)–(A4). Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩EP(f)≠∅. Then the sequence {xn} generated by
(4.10)x0∈C,un∈Csuchthatf(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnxn+(1-αn)un,zn=Syn,Hn={z∈C:‖z-zn‖≤‖z-xn‖},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
{rn}⊂(0,∞) such that lim infn→∞rn>0,Proof.
PutTn≡Trn and en=0 for all n∈ℕ∪{0}. Then, it follows from Lemma 4.5 and Theorem 3.1 that {xn} converges strongly to PFx0.Next, by using Lemma4.5 and Theorem 3.3 with en=0, for all n∈ℕ∪{0}, we obtain the following result.Theorem 4.7.
LetC be a nonempty closed convex subset of a real Hilbert space H and let f be a bifunction from C×C into ℝ satisfying (A1)–(A4). Let S be a nonexpansive mapping of C into itself such that F:=F(S)∩EP(f)≠∅. Then the sequence {xn} generated by
(4.11)x0∈C,un∈Csuchthatf(un,y)+1rn〈y-un,un-xn〉≥0,∀y∈C,yn=αnx0+(1-αn)un,zn=Syn,Hn={z∈C:‖z-zn‖2≤‖z-xn‖2+αn(‖x0‖2+2〈xn-x0,v〉)},Wn={z∈C:〈z-xn,x0-xn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
{rn}⊂(0,∞) such that lim infn→∞rn>0,


## 5. Applications to Maximal Monotone Operators in Banach Space

In this section, we discuss the problem of strong convergence concerning maximal monotone operators in a real uniformly smooth and uniformly convex Banach space.LetE be a smooth, strictly convex, and reflexive Banach space and let T⊂E×E* be a maximal monotone operator. Then for each r>0 and x∈E, there corresponds a unique element xr∈D(T) satisfying (5.1)J(x)∈J(xr)+rT(xr);
see [23]. We define the resolvent of T by Jrx=xr. In other words, Jr=(J+rT)-1J for all r>0. We know that Jr is relatively nonexpansive and T-10=F(Jr) for all r>0 (see [6, 23]), where F(Jr) denotes the set of all fixed points of Jr. We can also define, for each r>0, the Yosida approximation of T by Ar=λ-1(J-JJr). We know that (Jrx,Arx)∈T for all r>0.Lemma 5.1.
LetE be a real uniformly smooth and uniformly convex Banach space and let T:E→E* be a maximal monotone operator with F:=T-10≠∅ and Jr=(J+rT)-1J for all r>0. Let {Tn} be a sequence of relatively nonexpansive mappings of E into itself defined by Tn≡Jrn for all n∈ℕ∪{0}, where {rn} is a sequence in (0,∞) such that lim infn≥0rn>0. Then, {Tn} satisfies the NST-condition.Proof.
It easy to see that⋂n=0∞F(Tn)=⋂n=0∞F(Jrn)=T-10. Let {vn} be a bounded  sequence in E such that limn→∞∥vn-Jrnvn∥=0 and let v0∈ωw(vn). Then, there exists a subsequence {vnk} of {vn} such that vnk⇀v0. By the uniform smoothness of E, we have limn→∞∥Jvn-JJrnvn∥=0. Since lim infn→∞rn>0, we have
(5.2)lim⁡n→∞‖Arnvn‖=lim⁡n→∞1rn‖Jvn-JJrnvn‖=0.
Let (u,u*)∈T. Then it holds from the monotonicity of T that
(5.3)〈u-Jrnkvnk,u*-Arnkvnk〉≥0
for all k∈ℕ. Letting k→∞, we get 〈u-v0,u*〉≥0. Then, the maximality of T implies v0∈T-10:=⋂n=1∞F(Jrn)=⋂n=0∞F(Tn). Hence ωw(vn)⊂⋂n=0∞F(Tn). Therefore, {Tn} satisfies the NST-condition.Using Theorem3.1 and Lemma 5.1, we first obtain the result of [2].Theorem 5.2 (see [2]).
LetE be a real uniformly smooth and uniformly convex Banach space and let S:E→E be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let T:E→E* be a maximal monotone operator with F:=T-10∩F(S)≠∅ and Jr=(J+rT)-1J for all r>0. Then, sequence {xn} generated by the following scheme
(5.4)x0∈E,r0>0,yn=Jrn(xn+en),zn=J-1(αnJxn+(1-αn)Jyn),un=Szn,Hn={z∈E:ϕ(z,un)⩽ϕ(z,zn)⩽αnϕ(z,xn)+(1-αn)ϕ(z,xn+en)},Wn={z∈E:〈z-xn,Jx0-Jxn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
{rn}⊂(0,+∞) is a sequence such that lim infn≥0rn>0;(iii)
the error sequence is{en}⊂E such that ∥en∥→0 as n→∞.Proof.
PutTn≡Jrn for all n∈ℕ∪{0}. Hence by using Lemma 5.1 and Theorem 3.1, we obtain the result.PuttingS≡I in Theorem 5.2, we obtain the following corollary.Corollary 5.3 (see [22]).
LetE be a real uniformly smooth and uniformly convex Banach space and let T:E→E* be a maximal monotone operator with F:=T-10∩F(S)≠∅. and Jr=(J+rT)-1J for all r>0. Then, sequence {xn} generated by the following scheme
(5.5)x0∈E,r0>0,yn=Jrn(xn+en),zn=J-1(αnJxn+(1-αn)Jyn),Hn={z∈E:ϕ(z,zn)⩽αnϕ(z,xn)+(1-αn)ϕ(z,xn+en)},Wn={z∈E:〈z-xn,Jx0-Jxn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] with αn≤1-β for some β∈(0,1);(ii)
{rn}⊂(0,+∞) is a sequence such that lim infn≥0rn>0;(iii)
the error sequence is{en}⊂E such that ∥en∥→0 as n→∞.The following theorem follows directly from Lemma5.1 and Theorem 3.3Theorem 5.4 (see [2]).
LetE be a real uniformly smooth and uniformly convex Banach space and let S:E→E be nonexpansive with respect to the Lyapunov functional and weakly sequentially continuous. Let T:E→E* be a maximal monotone operator with F:=T-10∩F(S)≠∅. and Jr=(J+rT)-1J for all r>0. Then, sequence {xn} generated by the following scheme
(5.6)x0∈E,r0>0,yn=Jrn(xn+en),zn=J-1(αnJx0+(1-αn)Jyn),un=Szn,Hn={z∈E:ϕ(z,un)⩽ϕ(z,zn)⩽αnϕ(z,x0)+(1-αn)ϕ(z,xn+en)},Wn={z∈E:〈z-xn,Jx0-Jxn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
{rn}⊂(0,+∞) is a sequence such that lim infn≥0rn>0;(iii)
the error sequence is{en}⊂E such that ∥en∥→0 as n→∞.PuttingS≡I in Theorem 5.4, we obtain the following corollary.Corollary 5.5 (see [22]).
LetE be a real uniformly smooth and uniformly convex Banach space, let C be a nonempty closed convex subset of E, and let T:E→E* be a maximal monotone operator with F:=T-10∩F(S)≠∅ and Jr=(J+rT)-1J for all r>0. Then, sequence {xn} generated by the following scheme
(5.7)x0∈E,r0>0,yn=Jrn(xn+en),zn=J-1(αnJx0+(1-αn)Jyn),Hn={z∈E:ϕ(z,zn)⩽αnϕ(z,x0)+(1-αn)ϕ(z,xn+en)},Wn={z∈E:〈z-xn,Jx0-Jxn〉≤0},xn+1=PHn∩Wnx0,n=0,1,2,…
converges strongly to PFx0 provided that(i)
{αn}⊂[0,1] is a sequence such that αn→0 as n→∞;(ii)
{rn}⊂(0,+∞) is a sequence such that lim infn≥0rn>0;(iii)
the error sequence is{en}⊂E such that ∥en∥→0 as n→∞.



---

*Source: 301305-2010-12-08.xml*