# Strong Convergence Theorems for a Common Fixed Point of Two Countable Families of Relatively Quasi Nonexpansive Mappings and Applications

Authors: Jingling Zhang; Yongfu Su; Qingqing Cheng
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalizedf-projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.


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## Body

## 1. Introduction

LetE be a real Banach space with the dual E*. We denote by J the normalized duality mapping from E to 2E* defined by
(1.1)Jx={f∈E*:〈x,f〉=∥x∥2=∥f∥2},
where 〈·,·〉 denotes the generalized duality pairing. The duality mapping J has the following properties:(i)
ifE is smooth, then J is single-valued;(ii)
ifE is strictly convex, then J is one-to-one;(iii)
ifE is reflexive, then J is surjective;(iv)
ifE is uniformly smooth, then J is  uniformly norm-to-norm continuous on each bounded subset of E;(v)
ifE* is uniformly convex, then J is uniformly continuous on bounded subsets of E and J is single valued  and also one-to-one (see [1–4]).LetE be a smooth Banach space with the dual E*. The functional ϕ:E×E→R is defined by
(1.2)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,
for all x,y∈E.LetC 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 [5] if C contains a sequence {xn} which converges weakly to p such that the strong limn→∞(xn-Txn)=0. The set of asymptotic fixed points of T will be denoted by F^(T). A mapping T from C into itself is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C and relatively nonexpansive [1, 6–8] if F(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 [1, 6–8].Three classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced in 1953 by Mann [9] which is well known as Mann’s iteration process and is defined as follows:
(1.3)x0chosenarbitrarily,xn+1=αnxn+(1-αn)Txn,n≥0,
where the sequence {αn} is chosen in [0, 1]. Fourteen years later, Halpern [10] proposed the new innovation iteration process which resembles  Manns iteration (1.3), it is defined by
(1.4)x0chosenarbitrarily,xn+1=αnu+(1-αn)Txn,n≥0,
where the element u∈C is fixed. Seven years later, Ishikawa [2] enlarged and improved Mann's iteration (1.3) to the new iteration method, it is often cited as Ishikawa's iteration process which is defined recursively by
(1.5)x0chosenarbitrarily,yn=βnxn+(1-βn)Txn,xn+1=αnxn+(1-αn)Tyn,n≥0,
where {αn} and {βn} are sequences in the interval [0, 1].In both Hilbert space [10–12] and uniformly smooth Banach space [13–15] the iteration process (1.4) has been proved to be strongly convergent if the sequence {αn} satisfies the following conditions:(i)
αn→0;(ii)
∑n=0∞‍αn=∞;(iii)
∑n=0∞‍|αn+1-αn|<∞ or limn→∞(αn/αn+1)=1. By the restriction of condition (ii), it is widely believed that Halpern’s iteration process (1.4) has slow convergence though the rate of convergence has not been determined. Halpern [10] proved that conditions (i) and (ii) are necessary in the strong convergence of (1.4) for a nonexpansive mapping T on a closed convex subset C of a Hilbert space H. Moreover, Wittmann [12] showed that (1.4) converges strongly to PF(T)u when {αn} satisfies (i), (ii), and (iii), where PF(T)(·) is the metric projection onto F(T).Both iterations processes (1.3) and (1.5) have only weak convergence, in general Banach space (see [16] for more details). As a matter of fact, process (1.3) may fail to converge while process (1.5) can still converge for a Lipschitz pseudo contractive mapping in a Hilbert space [17]. For example, Reich [18] proved that if E is a uniformly convex Banach space with Frechet differentiable norm and if {αn} is chosen such that ∑n=0∞‍αn(1-αn)=∞, then the sequence {xn} defined by (1.3) converges weakly to a fixed point of T. However, we note that Manns iteration process (1.3) has only weak convergence even in a Hilbert space [16].Some attempts to modify the Mann’s iteration method so that strong convergence guaranteed has recently been made. Nakajo and Takahashi [19] proposed the following modification of the Mann iteration method for a single nonexpansive mapping T in a Hilbert space H:
(1.6)x0∈Cchosenarbitrarily,yn=αnxn+(1-αn)Txn,Cn={z∈C:∥yn-z∥≤∥xn-z∥},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qn(x0),
where C is a closed convex subset of H, PK denotes the metric projection from H onto a closed convex subset K of H. They proved that if the sequence {αn} is bounded above from one then the sequence {xn} generated by (1.6) converges strongly to PF(T)(x0), where F(T) denotes the fixed points set of T.The ideas to generalize the process (1.6) from Hilbert space to Banach space have recently been made. By using available properties on uniformly convex and uniformly smooth Banach space, Matsushita and Takahashi [8] presented their ideas as the following method for a single relatively nonexpansive mapping T in a Banach space E:
(1.7)x0∈Cchosenarbitrarily,yn=J-1(αnJx0+(1-αn)JTxn),Cn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠCn∩Qn(x0).They proved the following convergence theorem.Theorem MT.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let T be a relatively nonexpansive mapping from C into itself, and let {αn} be a sequence of real numbers such that 0≤αn<1 and limsupn→∞αn<1. Suppose that {xn} is given by (1.7), where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to ΠF(T)x0, where ΠF(T)(·) is the generalized projection from C onto F(T).In 2007, Plubtieng and Ungchittrakool [20] proposed the following hybrid algorithms for two relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.Theorem SK 1.
LetE be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty closed convex subset of E, and let T,S be two relatively nonexpansive mappings from C into itself with F:=F(T)∩F(S) is nonempty. Let a sequence {xn} be defined by
(1.8)x0∈Cchosenarbitrarily,yn=J-1(αnJxn+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wn(x0),
with the following restrictions:(i)
0≤αn<1,limsupn→∞αn<1;(ii)
0≤βn(1),βn(1),βn(3)≤1,limn→∞βn(1)=0,liminfn→∞βn(2)βn(3)>0.
Then the{xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Theorem SK 2.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let T,S be two relatively nonexpansive mappings from C into itself with F:=F(T)∩F(S) is nonempty. Let a sequence {xn} be defined by
(1.9)x0∈Cchosenarbitrarily,yn=J-1(αnJx0+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTxn+βn(3)JSxn),Hn={z∈C:ϕ(z,yn)≤ϕ(z,xn)+αn(∥x0∥2+2〈z,Jxn-Jx0〉)},Wn={z∈C:〈xn-z,Jx0-Jxn〉≥0},xn+1=ΠHn∩Wn(x0),
with the following restrictions:(i)
0<αn<1,limsupn→∞αn<1;(ii)
0≤βn(1),βn(1),βn(3)≤1,limn→∞βn(1)=0,liminfn→∞βn(2)βn(3)>0.
Then the{xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.In 2010, Su et al. [21] proposed the following hybrid algorithms for two countable families of weak relatively nonexpansive mappings in a Banach space and proved the following convergence theorems.Theorem SXZ 1.
LetE be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty closed convex subset of E, and let {Tn},{Sn} be two countable families of weak relatively nonexpansive mappings from C into itself such that F:=(∩n=0∞F(Tn))⋂‍(∩n=0∞F(Sn))≠∅. Define a sequence {xn} in C by the following algorithm:
(1.10)x0∈Cchosenarbitrarily,zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤ϕ(z,xn)},C0={z∈C:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn∩Qn(x0)
with the conditions:(i)
βn(1)+βn(2)+βn(3)=1;(ii)
liminfn→∞βn(1)βn(2)>0;(iii)
liminfn→∞βn(1)βn(3)>0;(iv)
0≤αn≤α<1 for some α∈(0,1).Then {xn} converges strongly to ΠFx0, where ΠF is the generalized projection from C onto F.Theorem SXZ 2.
LetE be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty closed convex subset of E, and let {Tn},{Sn} be two countable families of weak relatively nonexpansive mappings from C into itself such that F:=(∩n=0∞F(Tn))⋂‍(∩n=0∞F(Sn))≠∅. Define a sequence {xn} in C by the following algorithm:
(1.11)x0∈Cchosenarbitrarily,zn=J-1(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤(1-αnβn(1))ϕ(z,xn)+αnβn(1)ϕ(z,x0)},C0={z∈C:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn∩Qn(x0),
with the conditions:(i)
βn(1)+βn(2)+βn(3)=1;(ii)
limn→∞βn(1)=0;(iii)
limsupn→∞βn(2)βn(3)>0;(iv)
0≤αn≤α<1forsomeα∈(0,1).Then {xn} converges strongly to ΠFx0.Recently, Li et al. [22] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: x0∈C,
(1.12)yn=J-1(αnJxn+(1-αn)JTxn),Cn={w∈Cn:G(w,Jyn)≤G(w,Jxn)},xn+1=ΠCn+1f(x0),n≥1.
They proved a strong convergence theorem for finding an element in the fixed points set of T. We remark here that the results of Li et al. [22] extended and improved on the results of Matsushita and Takahashi [8].Motivated by the above-mentioned results and the ongoing research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalizedf-projection operator. Our results extend the results of Li et al. [22], Takahashi and Zembayashi [23], and many other recent known results in the literature.


## 2. Preliminaries

LetE be a smooth Banach space with the dual E*. The functional ϕ:E×E→R is defined by
(2.1)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,
for all x,y∈E. Observe that, in a Hilbert space H, (2.1) reduces to ϕ(x,y)=∥x-y∥2,x,y∈H.Recall that ifC is a nonempty, closed, and convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This is true only when H is a real Hilbert space. In this connection, Alber [24] has recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces. The generalized projection ΠC:E→C is a map that assigns to an arbitrary point x∈E, the minimum point of the functional ϕ(y,x), that is, ΠCx=x-, where x- is the solution to the minimization problem
(2.2)ϕ(x-,x)=miny∈Cϕ(y,x),
existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(y,x) and strict monotonicity of the mapping J. In Hilbert space, ΠC=PC. It is obvious from the definition of the functional ϕ that
(2.3)(∥x∥-∥y∥)2≤ϕ(y,x)≤(∥y∥2+∥x∥2),(2.4)ϕ(x,y)=ϕ(x,z)+ϕ(z,y)-2〈x-z,Jz-Jy〉
for all x,y∈E. See [25] for more details.This section collects some definitions and lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.Remark 2.1.
IfE is a reflexive strictly convex and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (2.3), we have ∥x∥=∥y∥. This implies 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definition of J, we have Jx=Jy. Since J is one-to-one, then we have x=y; see [12, 15, 26] for more details.
LetC be a closed convex subset of E, and let {Tn}n=0∞ be a countable family of mappings from C into itself. We denote by F the set of common fixed points of {Tn}n=0∞, that is, F=⋂n=0∞‍F(Tn), where F(Tn) denote the set of fixed points of Tn, for all n≥0. A point p in C is said to be an asymptotic fixed point of {Tn}n=0∞ if C contains a sequence {xn} which converges weakly to p such that limn→∞∥Tnxn-xn∥=0. The set of asymptotic fixed point of {Tn}n=0∞ will be denoted by F^({Tn}n=0∞). A point p in C is said to be a strong asymptotic fixed point of {Tn}n=0∞ if C contains a sequence {xn} which converges strongly to p such that limn→∞∥Tnxn-xn∥=0. The set of strong asymptotic fixed point of {Tn}n=0∞ will be denoted by F~({Tn}n=0∞) [21].Definition 2.2.
Countable family of mappings{Tn}n=0∞ is said to be countable family of relatively nonexpansive mappings if the following conditions are satisfied:(i)
F({Tn}n=0∞) is nonempty;(ii)
ϕ(u,Tnx)≤ϕ(u,x), for allu∈F(Tn),x∈C,n≥0;(iii)
F^({Tn}n=0∞)=⋂n=0∞‍F(Tn).Definition 2.3.
Countable family of mappings{Tn}n=0∞ is said to be countable family of weak relatively nonexpansive mappings if the following conditions are satisfied:(i)
F({Tn}n=0∞) is nonempty;(ii)
ϕ(u,Tnx)≤ϕ(u,x), for allu∈F(Tn),x∈C,n≥0;(iii)
F~({Tn}n=0∞)=⋂n=0∞‍F(Tn).Definition 2.4.
Countable family of mappings{Tn}n=0∞ is said to be countable family of relatively quasi nonexpansive mappings if the following conditions are satisfied:(i)
F({Tn}n=0∞) is nonempty;(ii)
ϕ(u,Tnx)≤ϕ(u,x), for allu∈F(Tn),x∈C,n≥0.Definition 2.5.
A mappingT is said to be relatively nonexpansive mappings if the following conditions are satisfied:(i)
F(T)is nonempty;(ii)
ϕ(u,Tx)≤ϕ(u,x),for allu∈F(T),x∈C;(iii)
F^(T)=F(T).Definition 2.6.
A mappingT is said to be weak relatively nonexpansive mappings if the following conditions are satisfied:(i)
F(T)isnonempty;(ii)
ϕ(u,Tx)≤ϕ(u,x),for allu∈F(T),x∈C;(iii)
F~(T)=F(T).Definition 2.7.
A mappingT is said to be relatively quasi nonexpansive mappings if the following conditions are satisfied:(i)
F(T) is nonempty;(ii)
ϕ(u,Tx)≤ϕ(u,x),  for allu∈F(T),x∈C.The Definition2.5 (Definitions 2.6 and 2.7) is a special form of the Definition 2.2 (Definitions 2.3 and 2.4) as Tn≡T for all n≥0. The following conclusions are obvious: (1) relatively nonexpansive mapping must be weak relatively nonexpansive mapping; (2) weak relatively nonexpansive mapping must be relatively quasi nonexpansive mapping.The hybrid algorithms for fixed point of relatively nonexpansive mappings and applications have been studied by many authors, for example, [1, 6, 7, 17, 27, 28]. In recent years, the definition of relatively quasi nonexpansive mapping has been presented and studied by many authors [7, 17, 26, 28]. Now we give an example which is a countable family of relatively quasi nonexpansive mappings but not a countable family of relatively nonexpansive mappings.Example 2.8.
LetE=l2, where
(2.5)l2={ξ=(ξ1,ξ2,ξ3,...,ξn,...):∑n=1∞‍|xn|2<∞},∥ξ∥=(∑n=1∞‍|ξn|2)1/2,∀ξ∈l2,〈ξ,η〉=∑n=1∞‍ξnηn,∀ξ=(ξ1,ξ2,ξ3,...,ξn,...),η=(η1,η2,η3,...,ηn....)∈l2.
It is well known that l2 is a Hilbert space, so that (l2)*=l2. Let {xn}⊂E be a sequence defined by
(2.6)x0=(1,0,0,0,…),x1=(1,1,0,0,…),x2=(1,0,1,0,0,…),x3=(1,0,0,1,0,0,…),⋮xn=(ξn,1,ξn,2,ξn,3,…,ξn,k,…),⋮
where
(2.7)ξn,k={1ifk=1,n+1,0ifk≠1,k≠n+1,
for all n≥1.
Define a countable family of mappingsTn:E→E as follows:
(2.8)Tn(x)={nn+1xnifx=xn,-xifx≠xn,
for all n≥0.Conclusion 2.9.
{
x
n
} converges weakly to x0.Proof.
For anyf=(ζ1,ζ2,ζ3,...,ζk,...)∈l2=(l2)*, we have
(2.9)f(xn-x0)=〈f,xn-x0〉=∑k=2∞‍ζkξn,k=ζn+1→0,
as n→∞. That is, {xn} converges weakly to x0Conclusion 2.10.
{
x
n
} is not a Cauchy sequence, so that, it does not converge strongly to any element of l2.Proof.
In fact, we have∥xn-xm∥=2 for any n≠m. Then {xn} is not a Cauchy sequence.Conclusion 2.11.
T
n has a unique fixed point 0, that is, F(Tn)={0}, for all n≥0.Proof.
The conclusion is obvious.Conclusion 2.12.
x
0 is an asymptotic fixed point of {Tn}n=0∞.Proof.
Since{xn} converges weakly to x0 and
(2.10)∥Tnxn-xn∥=∥nn+1xn-xn∥=1n+1∥xn∥→0
as n→∞, so that, x0 is an asymptotic fixed point of {Tn}n=0∞.Conclusion 2.13.
{
T
n
}
n
=
0
∞ is a countable family of relatively quasi nonexpansive mappings.Proof.
SinceE=L2 is a Hilbert space, for any n≥0 we have
(2.11)ϕ(0,Tnx)=∥0-Tnx∥2=∥Tnx∥2≤∥x∥2=∥x-0∥2=ϕ(0,x),∀x∈E.
then {Tn}n=0∞ is a countable family of relatively quasi nonexpansive mappingsConclusion 2.14.
{
T
n
}
n
=
0
∞ is not a countable family of relatively nonexpansive mappings.Proof.
From Conclusions2.11 and 2.12, we have ⋂n=0∞‍F(Tn)≠F^({Tn}n=0∞), so that, {Tn}n=0∞ is not a countable family of relatively nonexpansive mapping.Next, we recall the concept of generalizedf-projector operator, together with its properties. Let G:C×E*→R∪{+∞} be a functional defined as follows:
(2.12)G(ξ,φ)=∥ξ∥2-2〈ξ,φ〉+∥φ∥2+2ρf(ξ),
where ξ∈C,φ∈E,ρ is a positive number, and f:C→R∪{+∞} is proper, convex, and lower semicontinuous. From the definitions of G and f, it is easy to see the following properties:(i)
G(ξ,φ)  is convex and continuous with respect to φ when  ξ  is  fixed;(ii)
G(ξ,φ) is  convex  and  lower semi-continuous with  respect to  ξ when φ is  fixed.Definition 2.15 (see [29]).
LetE be a real Banach space with its dual E*. Let C be a nonempty, closed and convex subset of E. We say that ΠCf:E*→2C is a generalized f-projection operator if
(2.13)ΠCfφ={u∈C:G(u,φ)=infξ∈CG(ξ,φ)},∀φ∈E*.For the generalizedf-projection operator, Wu and Huang [42] proved the following theorem basic propertiesLemma 2.16 (see [29]).
LetE be a real reflexive Banach space with its dual E*. Let C be a nonempty, closed, and convex subset of E. Then the following statements hold:(i)
ΠCf is  a  nonempty  closed convex subset of  C for  all φ∈E*;(ii)
ifE is smooth, then for all φ∈E*, x∈ΠCf if and  only  if
(2.14)〈x-y,φ-Jx〉+ρf(y)-ρf(x)≥0,∀y∈C;(iii)
ifE is  strictly  convex and f:C→R∪{+∞} is  positive  homogeneous (i.e.,f(tx)=tf(x)forallt>0such  that tx∈C,wherex∈C), then ΠCf is  a  single valued mapping.Fan et al. [30] showed that the condition f is a positive homogeneous which appeared in Lemma 2.13 can be removed.Lemma 2.17 (see [30]).
LetE be a real reflexive Banach space with its dual E* and C a nonempty, closed, and convex subset of E. Then if E is strictly convex, then ΠCf is a single-valued mapping.Recall thatJ is a single-valued mapping when E is a smooth Banach space. There exists a unique element φ∈E. such that φ=Jx for each x∈E. This substitution in (2.15) gives
(2.15)G(ξ,Jx)=∥ξ∥2-2〈ξ,Jx〉+∥x∥2+2ρf(ξ).
Now, we consider the second generalized f-projection operator in a Banach space.Definition 2.18.
LetE be a real Banach space and C a nonempty, closed, and convex subset of E. We say that ΠCf:E→2C is a generalized f-projection operator if
(2.16)ΠCf(x)={u∈C:G(u,Jx)=infξ∈CG(ξ,Jx)},∀∈E.
Obviously, the definition of relatively quasi nonexpansive mapping T is equivalent to(R1)
F(T)≠∅;(R2)
G(p,JTx)≤G(p,Jx),for allx∈C,p∈F(T).Lemma 2.19 (see [31]).
LetE be a Banach space and let f:E→R∪{+∞} be a lower semi-continuous convex functional. Then there exists x*∈E and α∈R such that
(2.17)f(x)≥(x,x*)+α,∀x∈E.We know that the following lemmas hold for operatorΠCf.Lemma 2.20 (see [22]).
LetC be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Then the following statements hold:(i)
ΠCfx is a nonempty closed and convex subset of C for all x∈E;(ii)
for allx∈E, x^∈ΠCfx if and only  if
(2.18)〈x^-y,Jx-Jx^〉+ρf(y)-ρf(x)≥0,∀y∈C;(iii)
ifE is strictly convex, then ΠCfx is a single  valued mapping.Lemma 2.21 (see [22]).
LetC be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. let x∈E and x^∈ΠCf, then
(2.19)ϕ(y,x^)+G(x^,Jx)≤G(y,Jx),∀y∈C.The fixed points setF(T) of a relatively quasi nonexpansive mapping is closed convex as given in the following lemma.Lemma 2.22 (see [32, 33]).
LetC be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. let T be a closed relatively quasi nonexpansive mapping of C into itself. Then F(T) is closed and convex.Also, this following lemma will be used in the sequel.Lemma 2.23 (see [25]).
LetE be a uniformly convex and smooth real Banach space and let {xn},{yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.Lemma 2.24 (see [24, 25, 27]).
LetC be a nonempty closed convex subset of a smooth real Banach space E and x∈E. Then, x0=ΠCx if and only if
(2.20)〈x0-y,Jx-Jx0〉≥0,∀y∈C.Lemma 2.25 (see [28]).
LetE be a uniformly convex Banach space and let Br(0)={x∈E:∥x∥≤r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(2.21)∥λx+μy+γz∥2≤λ∥x∥2+μ∥y∥2+γ∥z∥2-λμg(∥x-y∥)
for all x,y,andz∈Br(0) and λ,μ,andγ∈[0,1] with λ+μ+γ=1.It is easy to prove the following result.Lemma 2.26.
LetE be a strictly convex and smooth real Banach space, let C be a closed convex subset of E, and let T be a relatively quasi nonexpansive mapping from C into itself. Then F(T) is closed and convex.Lemma 2.27.
LetE be a p-uniformly convex Banach space with p≥2. Then, for all x,y∈E, j(x)∈Jp(x) and j(y)∈Jp(y),
(2.22)〈x-y,j(x)-j(y)〉≥cpcp-2p∥x-y∥p,
where Jp is the generalized duality mapping from E into E* and 1/c is the p-uniformly convexity constant of E.Observe that an infinite family of operators{Tn}n=1∞ in a Banach space is said to be uniformly closed, if xn→x, xn-Tnxn→0 then Tnx=x (i.e., x∈∩n=1∞F(Tn)). Obviously, a countable family of uniformly closed of relatively quasi nonexpansive mappings is a countable family of weak relatively nonexpansive mappings.


## 3. Main Results

Theorem 3.1.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let {Tn}n=1∞,{Sn}n=1∞ be two countable families of uniformly closed of relatively quasi nonexpansive mappings of C into itself such that F:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))≠∅. Let f:E→R be a convex and lower semicontinuous mapping with C⊂int(D(f)). For any given gauss x0∈C, define a sequence {xn} in C by the following algorithm:
(3.1)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={z∈Cn-1⋂‍Qn-1:G(z,Jyn)≤G(z,Jxn)},C0={z∈C:G(z,Jy0)≤G(z,Jx0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn⋂‍Qnf(x0),
with the conditions(i)
βn(1)+βn(2)+βn(3)=1;(ii)
liminfn→∞βn(1)βn(2)>0;(iii)
liminfn→∞βn(1)βn(3)>0;(iv)
0≤αn≤α<1forsomeα∈(0,1). Then {xn} converges strongly to x*, where x*=ΠFfx0.Proof.
Step1. We show that Cn and Qn are closed and convex for each n≥0.
From the definitions ofCn and Qn, it is obvious that Qn is closed and convex and Cn is closed for each n≥0. Moreover, since G(z,Jyn)≤G(z,Jxn) is equivalent to
(3.2)ϕ(z,yn)≤ϕ(z,xn),
and is equivalent to
(3.3)2〈z,Jxn-Jyn〉≤∥xn∥2+∥yn∥2,
it follows that Cn is convex for each n≥0. So, Cn⋂‍Qn is a closed convex subset of E for all n∈N∪{0}.
Step2. We show that F⊂Cn⋂‍Qn for all n≥0. Observe that
(3.4)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn).
Hence from the definition of G(x,Jy) and the convexity of ∥·∥2we have, for all p∈F, that
(3.5)G(p,Jzn)=G(p,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn)=∥p∥2-2〈p,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn〉+∥βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn∥2+2ρf(p)≤∥p∥2-2〈p,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn〉+βn(1)∥Jxn∥2+βn(2)∥JTnxn∥2+βn(3)∥JSnxn∥2+2ρf(p)=βn(1)G(p,Jxn)+βn(2)G(p,JTnxn)+βn(3)G(p,JSnxn)≤βn(1)G(p,Jxn)+βn(2)G(p,Jxn)+βn(3)G(p,Jxn)=G(p,Jxn).
By the similar reason we have, for allp∈F, that
(3.6)yn=J-1(αnJxn+(1-αn)Jzn),(3.7)G(p,Jyn)=G(p,αnJxn+(1-αn)Jzn)=∥p∥2-2〈p,αnJxn+(1-αn)Jzn〉+∥αnJxn+(1-αn)Jzn∥2+2ρf(p)≤∥p∥2-2〈p,αnJxn+(1-αn)Jzn〉+αn∥Jxn∥2+(1-αn)∥Jzn∥2+2ρf(p)=αnG(p,Jxn)+(1-αn)G(p,Jzn)≤αnG(p,Jxn)+(1-αn)G(p,Jxn)=G(p,Jxn).
That is, p∈Cn for all n≥0.
Next, we show thatF⊂Qn for all n≥0, we prove this by induction. For n=0, we have F⊂C=Q0. Assume that F⊂Qn. Since xn+1=ΠCn⋂‍Qnf(x0), by Definition 2.15 we have
(3.8)G(xn+1,Jx0)≤G(z,Jx0),∀z∈Cn⋂‍Qn.
It is equivalent to
(3.9)ϕ(xn+1,Jx0)≤ϕ(z,Jx0),
and is equivalent to
(3.10)〈xn+1-z,Jx0-Jxn+1〉≥0,∀z∈Cn⋂‍Qn.
As F⊂Cn⋂‍Qn, by the induction assumptions, the last inequality holds, in particular, for all z∈F. This together with the definition of Qn+1 implies that F⊂Qn+1.
Step3. We show that xn→x* as n→∞, and x*∈F.
We now show thatlimn→∞G(xn,Jx0) exists. Since f:E→R is a convex and lower semi-continuous, applying Lemma 2.19, we see that there exists u*∈E* and α∈R such that
(3.11)f(y)≥〈y,u*〉+α,∀y∈E.
It follows that
(3.12)G(xn+1,Jx0)=∥xn+1∥2-2〈xn+1,Jx0〉+∥x0∥2+2ρf(xn+1)≥∥xn+1∥2-2〈xn+1,Jx0〉+∥x0∥2+2ρ〈xn+1,u*〉+2ρα=∥xn+1∥2-2〈xn+1,Jx0-ρu*〉+∥x0∥2+2ρα≥∥xn+1∥2-2∥xn+1∥∥Jx0-ρu*∥+∥x0∥2+2ρα=(∥xn+1∥-∥Jx0-ρu*∥)2-∥Jx0-ρu*∥2+∥x0∥2+2ρα.
Sincexn+1=ΠCn⋂‍Qnf(x0), for each p∈F, it follows from (3.12) that
(3.13)G(p,Jx0)≥G(xn+1,Jx0)≥(∥xn+1∥-∥Jx0-ρu*∥)2-∥Jx0-ρu*∥2+∥x0∥2+2ρα.
This implies that {xn}n=1∞ is bounded and so is {G(xn,Jx0)}n=0∞.
Sincexn+1=ΠCn⋂‍Qnf(x0)∈Cn⋂‍Qn⊂Cn-1⋂‍Qn-1, by Lemma 2.21 we have
(3.14)ϕ(xn+1,xn)+G(xn,Jx0)≤G(xn+1,Jx0).
It is obvious that
(3.15)G(xn,Jx0)≤G(xn+1,Jx0),
and so {G(xn,Jx0)}n=0∞ is nondecreasing. It follows that the limit of {G(xn,Jx0)}n=0∞ exists.
By the fact thatCn+m⋂‍Qn+m⊂Cn-1⋂‍Qn-1 and xn+1=ΠCn⋂‍Qnf(x0)∈Cn-1⋂‍Qn-1, by Lemma 2.21 we obtain
(3.16)ϕ(xn+m,xn)+G(xn,Jx0)≤G(xn+m,Jx0).
Taking the limit as m,n→∞ in (3.16), we obtain
(3.17)limn→∞ϕ(xn+m,xn)=0,
which holds uniformly for all m. By using Lemma 2.21, we get that
(3.18)limn→∞∥xn+m-xn∥=0,
which holds uniformly for all m. Then {xn} is a Cauchy sequence, therefore there exists a point x*∈C such that xn→x*. In particular, we have
(3.19)limn→∞∥xn+1-xn∥=0.
Since xn+1=ΠCn⋂‍Qnf(x0)∈Cn, from the definition of Cn, we know that
(3.20)G(xn+1,Jyn)≤G(xn+1,Jxn)
is equivalent to
(3.21)ϕ(xn+1,yn)≤ϕ(xn+1,xn).
So, ϕ(xn+1,yn)→0.
By using Lemma2.23, we get that
(3.22)∥xn+1-yn∥→0,
as n→∞.
Henceyn→x* as n→∞. Since J is uniformly norm-to-norm continuous on bounded sets, we have
(3.23)limn→∞(1-αn)∥Jzn-Jxn∥=∥Jyn-Jxn∥=0.
Since 0≤αn≤α<1, then
(3.24)limn→∞∥Jzn-Jxn∥=0.
Since J-1 is also uniformly norm-to-norm continuous on bounded sets, we have
(3.25)limn→∞∥zn-xn∥=0
so that zn→x* as n→∞.
Since{xn} is convergent, then {xn} is bounded, so are {zn}, {JTnxn}, and {JSnxn}. From the definition of ϕ(x,y) and
(3.26)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),
we have, for all p∈F that
(3.27)ϕ(p,zn)=ϕ(p,J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn))=∥p∥2-2〈p,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn〉+∥βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn∥2.
Therefore by using Lemma 2.25 (inequality (2.21)), for all p∈F, we have
(3.28)ϕ(p,zn)≤∥p∥2-2〈p,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn〉+βn(1)∥Jxn∥2+βn(2)∥JTnxn∥2+βn(3)∥JSnxn∥2-βn(1)βn(2)g(∥Jxn-JTnxn∥)=βn(1)ϕ(p,xn)+βn(2)ϕ(p,Tnxn)+βn(3)ϕ(p,Snxn)-βn(1)βn(2)g(∥Jxn-JTnxn∥)≤βn(1)ϕ(p,xn)+βn(2)ϕ(p,xn)+βn(3)ϕ(p,xn)-βn(1)βn(2)g(∥Jxn-JTnxn∥)=ϕ(p,xn)-βn(1)βn(2)g(∥Jxn-JTnxn∥),
and hence
(3.29)βn(1)βn(2)g(∥Jxn-JTnxn∥)≤ϕ(p,xn)-ϕ(p,zn)→0,
as n→∞. By using the same way, we can prove that
(3.30)βn(1)βn(3)g(∥Jxn-JSnxn∥)≤ϕ(p,xn)-ϕ(p,zn)→0,
as n→∞. From the properties of the mapping g, we have
(3.31)∥Jxn-JTnxn∥→0,
as n→∞, and
(3.32)∥Jxn-JSnxn∥→0,
as n→∞. Since J-1 is also uniformly norm-to-norm continuous on any bounded sets, we have
(3.33)∥xn-Tnxn∥→0,
as n→∞, and
(3.34)∥Jxn-JSnxn∥→0,
as n→∞. Since xn→x* and Tn, Sn are uniformly closed, x*=Tnx*, and x*=Snx*.
Step4. we show that x*=ΠFfx0.
SinceF:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn)) is a closed and convex set, from Lemma 2.19, we know that ΠFfx0 is single valued and denote w=ΠFfx0. Since xn=ΠCn-1⋂‍Qn-1f(x0) and w∈F⊂Cn-1⋂‍Qn-1, we have
(3.35)G(xn,Jx0)≤G(w,Jx0),∀n≥0.
We know that G(ξ,Jφ) is convex and lower semicontinuous with respect to ξ when φ is fixed. This implies that
(3.36)G(x*,Jx0)≤liminfn→∞G(xn,Jx0)≤limsupn→∞G(xn,Jx0)≤G(w,Jx0).
From the definition of ΠFfx0 and x*∈F, we see that x*=w. This completes the proof.Based on Theorem3.1, we have the following.Corollary 3.2.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let {Tn}n=1∞,{Sn}n=1∞ be two countable families of weak relatively nonexpansive mappings of C into itself such that F:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))≠∅. For any given gauss x0∈C, define a sequence {xn} in C by the following algorithm:
(3.37)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤ϕ(z,xn)},C0={z∈C:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn⋂‍Qn(x0),
with the conditions:(i)
βn(1)+βn(2)+βn(3)=1;(ii)
liminfn→∞βn(1)βn(2)>0;(iii)
liminfn→∞βn(1)βn(3)>0;(iv)
0≤αn≤α<1forsomeα∈(0,1). Then {xn} converges strongly to x*, where x*=ΠFx0.Proof.
Puttingf(x)≡0, we can conclude from Theorem 3.1 the desired conclusion immediately.Theorem 3.3.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let {Tn}n=1∞,{Sn}n=1∞ be two countable families of uniformly closed of relatively quasi nonexpansive mappings of C into itself such that F:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))≠∅. Let f:E→R be a convex and lower semicontinuous mapping with C⊂int(D(f)). For any given gauss x0∈C, define a sequence {xn} in C by the following algorithm:
(3.38)zn=J-1(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={z∈Cn-1⋂‍Qn-1:G(z,Jyn)≤(βn(1)-αnβn(1))G(z,Jx0)+(1-βn(1)+αnβn(1))G(z,Jxn)},C0={z∈C:G(z,Jy0)≤G(z,Jx0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn⋂‍Qnf(x0),
with the conditions:(i)
βn(1)+βn(2)+βn(3)=1;(ii)
limn→∞βn(1)=0;(iii)
limsupn→∞βn(2)βn(3)>0;(iv)
0≤αn≤α<1forsomeα∈(0,1). Then {xn} converges strongly to x*, where x*=ΠFfx0.Proof.
Step1. We show that Cn and Qn are closed and convex for each n≥0.
From the definitions ofCn and Qn, it is obvious that Qn is closed and convex and Cn is closed for each n≥0. Moreover, since G(z,Jyn)≤(βn(1)-αnβn(1))G(z,Jx0)+(1-βn(1)+αnβn(1))G(z,Jxn) is equivalent to
(3.39)2〈z,(βn(1)-αnβn(1))Jx0+(1-βn(1)+αnβn(1))Jxn-Jyn〉≤(βn(1)-αnβn(1))∥x0∥2+(1-βn(1)+αnβn(1))∥xn∥2-∥yn∥2.
it follows that Cn is convex for each n≥0. So, Cn⋂‍Qn is a closed convex subset of E for all n∈N∪{0}.
Step2. We show that F⊂Cn⋂‍Qn for all n≥0. Observe that
(3.40)zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn).
Hence from the definition of G(x,Jy) and the convexity of ∥·∥2we have, for all p∈F, that
(3.41)G(p,Jzn)=G(p,βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn)=∥p∥2-2〈p,βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn〉+∥βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn∥2+2ρf(p)≤∥p∥2-2〈p,βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn〉+βn(1)∥Jx0∥2+βn(2)∥JTnxn∥2+βn(3)∥JSnxn∥2+2ρf(p)=βn(1)G(p,Jx0)+βn(2)G(p,JTnxn)+βn(3)G(p,JSnxn)≤βn(1)G(p,Jx0)+βn(2)G(p,Jxn)+βn(3)G(p,Jxn)=βn(1)G(p,Jx0)+(1-βn(1))G(p,Jxn).
By the similar reason we have, for all p∈F, that
(3.42)yn=J-1(αnJxn+(1-αn)Jzn),G(p,Jyn)=G(p,αnJxn+(1-αn)Jzn)=∥p∥2-2〈p,αnJxn+(1-αn)Jzn〉+∥αnJxn+(1-αn)Jzn∥2+2ρf(p)≤∥p∥2-2〈p,αnJxn+(1-αn)Jzn〉+αn∥Jxn∥2+(1-αn)∥Jzn∥2+2ρf(p)=αnG(p,Jxn)+(1-αn)G(p,Jzn)≤αnG(p,Jxn)+(1-αn)(βn(1)G(p,Jx0)+(1-βn(1))G(p,Jxn))=(1-αn)βn(1)G(p,Jx0)+(1-βn(1)+αnβn(1))G(p,Jxn).
That is, p∈Cn for all n≥0.
Next, we show thatF⊂Qn for all n≥0, we prove this by induction. For n=0, we have F⊂C=Q0. Assume that F⊂Qn. Since xn+1=ΠCn⋂‍Qnf(x0),   by Definition 2.15 we have
(3.43)G(xn+1,Jx0)≤G(z,Jx0),∀z∈Cn⋂‍Qn.
It is equivalent to
(3.44)ϕ(xn+1,Jx0)≤ϕ(z,Jx0),
and is equivalent to
(3.45)〈xn+1-z,Jx0-Jxn+1〉≥0,∀z∈Cn⋂‍Qn.
As F⊂Cn⋂‍Qn, by the induction assumptions, the last inequality holds, in particular, for all z∈F. This together with the definition of Qn+1 implies that F⊂Qn+1.
Step3. We show that xn→x* as n→∞, and x*∈F.
We now show thatlimn→∞G(xn,Jx0) exists. Since f:E→R is a convex and lower semi-continuous, applying Lemma 2.19, we see that there exists u*∈E* and α∈R such that
(3.46)f(y)≥〈y,u*〉+α,∀y∈E.
It follows that(3.47)G(xn+1,Jx0)=∥xn+1∥2-2〈xn+1,Jx0〉+∥x0∥2+2ρf(xn+1)≥∥xn+1∥2-2〈xn+1,Jx0〉+∥x0∥2+2ρ〈xn+1,u*〉+2ρα=∥xn+1∥2-2〈xn+1,Jx0-ρu*〉+∥x0∥2+2ρα≥∥xn+1∥2-2∥xn+1∥∥Jx0-ρu*∥+∥x0∥2+2ρα=(∥xn+1∥-∥Jx0-ρu*∥)2-∥Jx0-ρu*∥2+∥x0∥2+2ρα
Sincexn+1=ΠCn⋂‍Qnf(x0), for each p∈F, it follows from (3.47) that
(3.48)G(p,Jx0)≥G(xn+1,Jx0)≥(∥xn+1∥-∥Jx0-ρu*∥)2-∥Jx0-ρu*∥2+∥x0∥2+2ρα.
This implies that{xn}n=1∞ is bounded and so is {G(xn,Jx0)}n=0∞.
Sincexn+1=ΠCn⋂‍Qnf(x0)∈Cn⋂‍Qn⊂Cn-1⋂‍Qn-1, by Lemma 2.21, we have
(3.49)ϕ(xn+1,xn)+G(xn,Jx0)≤G(xn+1,Jx0).
It is obvious that
(3.50)G(xn,Jx0)≤G(xn+1,Jx0),
and so {G(xn,Jx0)}n=0∞ is nondecreasing. It follows that the limit of {G(xn,Jx0)}n=0∞ exists.
By the fact thatCn+m⋂‍Qn+m⊂Cn-1⋂‍Qn-1 and xn+1=ΠCn⋂‍Qnf(x0)∈Cn-1⋂‍Qn-1, by Lemma 2.21 we obtain
(3.51)ϕ(xn+m,xn)+G(xn,Jx0)≤G(xn+m,Jx0).
Taking the limit asm,n→∞ in (3.51), we obtain
(3.52)limn→∞ϕ(xn+m,xn)=0,
which holds uniformly for all m. By using Lemma 2.23, we get that
(3.53)limn→∞∥xn+m-xn∥=0,
which holds uniformly for all m. Then {xn} is a Cauchy sequence, therefore there exists a point x*∈C such that xn→x*. In particular, we have
(3.54)limn→∞∥xn+1-xn∥=0.
Since xn+1=ΠCn⋂‍Qnf(x0)∈Cn, from the definition of Cn, we know that
(3.55)G(xn+1,Jyn)≤G(z,Jyn)≤(βn(1)-αnβn(1))G(xn+1,Jx0)+(1-βn(1)+αnβn(1))G(xn+1,Jxn),
and is equivalent to
(3.56)ϕ(xn+1,yn)+2ρf(xn+1)≤(βn(1)-αnβn(1))(ϕ(xn+1,x0)+2ρf(xn+1))+(1-βn(1)+αnβn(1))(ϕ(xn+1,xn)+2ρf(xn+1)),
and is equivalent to
(3.57)ϕ(xn+1,yn)≤(βn(1)-αnβn(1))ϕ(xn+1,x0)+(1-βn(1)+αnβn(1))ϕ(xn+1,xn),
So,ϕ(xn+1,yn)→0.
By using Lemma2.23, we get that
(3.58)∥xn+1-yn∥→0,
as n→∞,
and henceyn→x* as n→∞. Since J is uniformly norm-to-norm continuous on bounded sets, we have
(3.59)limn→∞(1-αn)∥Jzn-Jxn∥=∥Jyn-Jxn∥=0.
Since0≤αn≤α<1, then
(3.60)limn→∞∥Jzn-Jxn∥=0.
SinceJ-1 is also uniformly norm-to-norm continuous on bounded sets, we have
(3.61)limn→∞∥zn-xn∥=0,
so that zn→x* as n→∞.
Since{xn} is convergent, then {xn} is bounded, so are {zn}, {JTnxn}, and {JSnxn}. From the definition of ϕ(x,y) and
(3.62)zn=J-1(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn),
we have, for all p∈F that
(3.63)ϕ(p,zn)=ϕ(p,J-1(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn))=∥p∥2-2〈p,βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn〉+∥βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn∥2.
Therefore by using Lemma 2.25 (inequality (2.21)), for all p∈F, we have
(3.64)ϕ(p,zn)≤∥p∥2-2〈p,βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn〉+βn(1)∥Jx0∥2+βn(2)∥JTnxn∥2+βn(3)∥JSnxn∥2-βn(2)βn(3)g(∥JTnxn-JSnxn∥)=βn(1)ϕ(p,x0)+βn(2)ϕ(p,Tnxn)+βn(3)ϕ(p,Snxn)-βn(2)βn(3)g(∥JTnxn-JSnxn∥)≤βn(1)ϕ(p,x0)+βn(2)ϕ(p,xn)+βn(3)ϕ(p,xn)-βn(2)βn(3)g(∥JTnxn-JSnxn∥)=βn(1)ϕ(p,x0)+(1-βn(1))ϕ(p,xn)-βn(2)βn(3)g(∥JTnxn-JSnxn∥),
and hence
(3.65)βn(2)βn(3)g(∥Jxn-JTnxn∥)≤βn(1)ϕ(p,x0)+(1-βn(1))ϕ(p,xn)-ϕ(p,zn).
Fromlimn→∞βn(1)=0 and xn→x*, zn→x*,we have
(3.66)βn(2)βn(3)g(∥JTnxn-JSnxn∥)→0,
as n→∞. From the properties of the mapping g, we have
(3.67)∥JTnxn-JSnxn∥→0,
as n→∞.
Since(3.68)zn=J-1(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn),
then we have
(3.69)Jzn=βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn.
Therefore
(3.70)∥Jxn-Jzn∥=∥Jxn-(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn)∥=∥βn(1)(Jxn-Jx0)+βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)∥≥∥βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)∥-∥βn(1)(Jxn-Jx0)∥,
which leads to
(3.71)∥βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)∥≤∥Jxn-Jzn∥+∥βn(1)(Jxn-Jx0)∥.
Since xn→x*, zn→x* and limn→∞βn(1)=0. Then from above inequality we obtain
(3.72)∥βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)∥→0.
On the other hand, by using the property of norm ∥.∥, we have
(3.73)∥βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)∥=∥βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)+βn(3)(Jxn-JTnxn)-βn(3)(Jxn-JTnxn)∥=∥(βn(2)+βn(3))(Jxn-JTnxn)+βn(3)(JTxn-JSnxn)∥≥∥(βn(2)+βn(3))(Jxn-JTnxn)∥-∥βn(3)(JTxn-JSnxn)∥,
which leads to the following inequality
(3.74)∥(βn(2)+βn(3))(Jxn-JTnxn)∥≤∥βn(2)(Jxn-JTnxn)+βn(3)(Jxn-JSnxn)∥+∥βn(3)(JTxn-JSnxn)∥.
Therefore, by using (3.67) and (3.72) we have
(3.75)∥(βn(2)+βn(3))(Jxn-JTnxn)∥→0.
This together with condition (ii) of Theorem3.3 implies that
(3.76)∥Jxn-JTnxn∥→0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, then we have
(3.77)∥xn-Tnxn∥→0,
as n→∞.  By using the same way, we can prove that
(3.78)∥xn-Snxn∥→0
as n→∞. Since xn→x* and Tn, Sn are uniformly closed, x*=Tnx*, and x*=Snx*, so that x*∈F.
Step4. we show that x*=ΠFfx0.
SinceF:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn)) is a closed and convex set, from Lemma 2.19, we know that ΠFfx0 is single valued and denote w=ΠFfx0. Since xn=ΠCn-1⋂‍Qn-1f(x0) and w∈F⊂Cn-1⋂‍Qn-1 we have
(3.79)G(xn,Jx0)≤G(w,Jx0),∀n≥0.
We know that G(ξ,Jφ) is convex and lower semicontinuous with respect to ξ when φ is fixed. This implies that
(3.80)G(x*,Jx0)≤liminfn→∞G(xn,Jx0)≤limsupn→∞G(xn,Jx0)≤G(w,Jx0).
From the definition of ΠFfx0 and x*∈F, we see that x*=w. This completes the proof.Based on Theorem3.3, we have the following.Corollary 3.4.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let {Tn}n=1∞,{Sn}n=1∞ be two countable families of weak relatively nonexpansive mappings of C into itself such that F:=(∩n=1∞F(Tn))∩(∩n=1∞F(Sn))≠∅. For any given gauss x0∈C, define a sequence {xn} in C by the following algorithm:
(3.81)zn=J-1(βn(1)Jx0+βn(2)JTnxn+βn(3)JSnxn),yn=J-1(αnJxn+(1-αn)Jzn),Cn={z∈Cn-1⋂‍Qn-1:ϕ(z,yn)≤(βn(1)-αnβn(1))ϕ(z,x0)+(1-βn(1)+αnβn(1))ϕ(z,xn)},C0={z∈C:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1⋂‍Qn-1:〈xn-z,Jx0-Jxn〉≥0},Q0=C,xn+1=ΠCn⋂‍Qn(x0),
with the conditions:(i)
βn(1)+βn(2)+βn(3)=1;(ii)
limn→∞βn(1)=0;(iii)
limsupn→∞βn(2)βn(3)>0;(iv)
0≤αn≤α<1forsomeα∈(0,1). Then {xn} converges strongly to x*, where x*=ΠFx0.Proof.
Puttingf(x)≡0, we can conclude from Theorem 3.3 the desired conclusion immediately.


## 4. Applications

LetE be a real Banach space and let E* be the dual space of E. Let C be a closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞). The equilibrium problem is to find x∈C such that
(4.1)f(x,y)≥0,∀y∈C.
The set of solutions of (4.1) is denoted by EP(f). Given a mapping T:C→E* let f(x,y)=〈Tx,y-x〉 for all x,y∈C. Then, p∈EP(f) if and only if 〈Tp,y-p〉≥0 for all y∈C, that is, p is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (4.1). Some methods have been proposed to solve the equilibrium problem in Hilbert spaces.For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:(A1)
f(x,x)=0,for allx∈E;(A2)
f is monotone, that is, f(x,y)+f(y,x)≤0,for allx,y∈E;(A3)
for allx,y,z∈E, limsupt↓0f(tz+(1-t)x,y)≤f(x,y);(A4)
for allx∈C, f(x,·) is convex and lower semi-continuous.Lemma 4.1 (see [34]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
(4.2)f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.Lemma 4.2 (see [34]).
LetC be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, and let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4). For r>0, define a mapping Tr:E→C as follows:
(4.3)Tr(x)={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈E. Then, the following hold(1)
Tr is single valued;(2)
Tr is a firmly nonexpansive-type mapping, that is, for all x,y∈E,
(4.4)〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;(3)
F(Tr)=EP(f);(4)
EP(f) is closed and convex;(5)
Tr is also a relatively nonexpansive mapping.Lemma 4.3 (see [34]).
LetC be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4), and let r>0 and x∈E, q∈F(Tr), then the following holds
(4.5)ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).Lemma 4.4.
LetE be a p-uniformly convex with p≥2 and uniformly smooth Banach space, and let C be a nonempty closed convex subset of E. Let f be a bifunction from C×C to R=(-∞,+∞) satisfying (A1)–(A4). Let {rn} be a positive real sequence such that limn→∞rn=r>0. Then the sequence of mappings {Trn} is uniformly closed.Proof.
(1) Let{xn} be a convergent sequence in C. Let zn=Trnxn for all n, then
(4.6)f(zn,y)+1rn〈y-zn,Jzn-Jxn〉≥0,∀y∈C,(4.7)f(zn+m,y)+1rn+m〈y-zn+m,Jzn+m-Jxn+m〉≥0,∀y∈C.
Puttingy=zn+m in (4.6) and y=zn in (4.7), we have
(4.8)f(zn,zn+m)+1rn〈zn+m-zn,Jzn-Jxn〉≥0,∀y∈C,f(zn+m,zn)+1rn+m〈zn-zn+m,Jzn+m-Jxn+m〉≥0,∀y∈C.
So, from (A2) we have(4.9)〈zn+m-zn,Jzn-Jxnrn-Jzn+m-Jxn+mrn+m〉≥0,
and hence
(4.10)〈zn+m-zn,Jzn-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0.
Thus, we have
(4.11)〈zn+m-zn,Jzn-Jzn+m+Jzn+m-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0,
which implies that
(4.12)〈zn+m-zn,Jzn+m-Jzn〉≤〈zn+m-zn,Jzn+m-Jxn-rnrn+m(Jzn+m-Jxn+m)〉≥0.
By using Lemma 2.25, we obtain
(4.13)cpcp-2p∥zn+m-zn∥p≤〈zn+m-zn,Jzn+m-Jxn-rnrn+m(Jzn+m-Jxn+m)〉=〈zn+m-zn,(1-rnrn+m)Jzn+m+rnrn+mJxn+m-Jxn〉.
Therefore, we get(4.14)cpcp-2p∥zn+m-zn∥p-1≤|1-rnrn+m|∥Jzn+m∥+∥rnrn+mJxn+m-Jxn∥.
On the other hand, for any p∈EP(f), from zn=Trnxn, we have
(4.15)∥zn-p∥=∥Trnxn-p∥≤∥xn-p∥,
so that {zn} is bounded. Since limn→∞rn=r>0, this together with (4.14) implies that {zn} is a Cauchy sequence. Hence Trnxn=zn is convergent.
(2) By using the Lemma4.2, we know that
(4.16)⋂n=1∞‍F(Trn)=EP(f)≠∅.
(3) From (1) we know that,limn→∞Trnx exists for all x∈C. So, we can define a mapping T from C into itself by
(4.17)Tx=limn→∞Trnx,∀x∈C.
It is obvious that, T is nonexpansive. It is easy to see that
(4.18)EP(f)=⋂n=1∞‍F(Trn)⊂F(T).
On the other hand, let w∈F(T),wn=Trnw, we have
(4.19)f(wn,y)+1rn〈y-wn,Jwn-Jw〉≥0,∀y∈C.
By (A2) we know(4.20)1rn〈y-wn,Jwn-Jw〉≥f(y,wn),∀y∈C.
Since wn→Tw=w and from (A4), we have f(y,w)≤0, for all y∈C. Then, for t∈(0,1] and y∈C,
(4.21)0=f(ty+(1-t)w,ty+(1-t)w)≤tf(ty+(1-t)w,y)+(1-t)f(ty+(1-t)w,w)≤tf(ty+(1-t)w,y).
Therefore, we have
(4.22)f(ty+(1-t)w,y)≥0.
Letting t↓0 and using (A3), we get
(4.23)f(w,y)≥0,∀y∈C
and hence w∈EP(f). From the above two respects, we know that F(T)=⋂n=0∞‍F(Trn).
Next we show{Trn} is uniformly closed. Assume xn→x and ∥xn-Trnxn∥→0, from the above results we know that Tx=limn→∞Trnx. On the other hand, from ∥xn-Trnxn∥→0, we also get limn→∞Trnx=x, so that x∈F(T)=⋂n=1∞‍F(Trn). That is, the sequence of mappings {Trn} is uniformly closed. This completes the proof.LetA be a multi-valued operator from E to E* with domain D(A)={z∈E:Az≠∅} and range R(A)={z∈E:z∈D(A)}. An operator A is said to be monotone if
(4.24)〈x1-x2,y1-y2〉≥0,
for each x1,x2∈D(A) and y1∈Ax1,y2∈Ax2. A monotone operator A is said to be maximal if it is graph G(A)={(x,y):y∈Ax} 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 is closed and convex. The following result is also well known.Theorem 4.5.
LetE be a reflexive, strictly convex, and smooth Banach space and let A be a monotone operator from E to E*. Then A is maximal if and only if R(J+rA)=E*, for all r>0.LetE be a reflexive, strictly convex, and smooth Banach space, and let A be a maximal monotone operator from E to E*. Using Theorem 4.5 and strict convexity of E, one obtains that for every r>0 and x∈E, there exists a unique xr such that
(4.25)Jx∈Jxr+rAxr.
Then one can defines a single-valued mapping Jr:E→D(A) by Jr=(J+rA)-1J and such a Jr is called the resolvent of A, one knows that A-10=F(Jr) for all r>0.Theorem 4.6.
LetE be a uniformly convex and a uniformly smooth Banach space, and let A be a maximal monotone operator from E to E*, let Jr be a resolvent of A for r>0. Then for any sequence {rn}n=1∞ such that liminfn→∞rn>0, {Jrn}n=1∞ is a uniformly closed sequence of relatively quasi nonexpansive mappings.Proof.
Firstly, we show that{Jrn}n=1∞ is uniformly closed. Let {zn}⊂E be a sequence such that zn→p and limn→∞∥zn-Jrnzn∥=0. Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(4.26)1rn(Jzn-JJrnzn)→0.
It follows from
(4.27)1rn(Jzn-JJrnzn)∈AJrnzn
and the monotonicity ofA that
(4.28)〈w-Jrnzn,w*-1rn(Jzn-JJrnzn)〉≥0
for all w∈D(A) and w*∈Aw. Letting n→∞, we have 〈w-p,w*〉≥0 for all w∈D(A) and w*∈Aw. Therefore from the maximality of A, we obtain p∈A-10=F(Jrn) for all n≥1, that is, p∈⋂n=1∞‍F(Jrn).
Next we showJrn is a relatively quasi nonexpansive mapping for all n≥1. For any w∈E and p∈F(Jrn)=A-10, from the monotonicity of A, we have
(4.29)ϕ(p,Jrnw)=∥p∥2-2〈p,JJrnw〉+∥Jrnw∥2=∥p∥2+2〈p,Jw-JJrnw-Jw〉+∥Jrnw∥2=∥p∥2+2〈p,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2=∥p∥2-2〈Jrw-p-Jrnw,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2=∥p∥2-2〈Jrnw-p,Jw-JJrnw〉+2〈Jrnw,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2≤∥p∥2+2〈Jrnw,Jw-JJrnw〉-2〈p,Jw〉+∥Jrnw∥2=∥p∥2-2〈p,Jw〉+∥w∥2-∥Jrnw∥2+2〈Jrnw,Jw〉-∥w∥2=ϕ(p,w)-ϕ(Jrnw,w)≤ϕ(p,w).
This implies that Jrn is a relatively quasi nonexpansive mapping for all n≥1. This completes the proof.



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*Source: 956950-2012-10-12.xml*