# A New Approach to the Approximation of Common Fixed Points of an Infinite Family of Relatively Quasinonexpansive Mappings with Applications

Authors: Wei-Qi Deng
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By using a specific way of choosing the indexes, we propose an iteration algorithm generated by the monotone CQ method for approximating common fixed points of an infinite family of relatively quasinonexpansive mappings. A strong convergence theorem without the stronger assumptions of the AKTT condition and the∗AKTT condition imposed on the involved mappings is established in the framework of Banach space. As application, an iterative solution to a system of equilibrium problems is studied. The result is more applicable than those of other authors with related interest.


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## Body

## 1. Introduction

LetC be a nonempty and closed convex subset of a real Banach space E. A mapping T:C→E is said to be nonexpansive if
(1.1)‖Tx-Ty‖≤‖x-y‖,∀x,y∈C.
A mapping T is said to be quasi-nonexpansive if F(T):={x∈C:x=Tx}≠∅ and
(1.2)‖Tx-p‖≤‖x-p‖,∀x∈C,p∈F(T).
It is easy to see that if T is nonexpansive with F(T)≠∅, then it is quasi-nonexpansive. There are many methods for approximating fixed points of quasi-nonexpansive mappings. In 1953, Mann [1] introduced the iteration as follows: a sequence {xn} is defined by
(1.3)xn+1=αnxn+(1-αn)Txn,
where the initial element x0∈C is arbitrary and {αn} is a sequence of real numbers in [0,1]. Approximation of fixed points of nonexpansive mappings via Mann's algorithm has extensively been investigated. One of the fundamental convergence results was proved by Reich [2]. In infinite-dimensional Hilbert spaces, Mann iteration can yield only weak convergence (see [3, 4]).Attempts to modify the Mann iteration method (1.3) for strong convergence have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.3) for a nonexpansive mapping T from C into itself in a Hilbert space: from an arbitrary x0∈C,
(1.4)yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
where PK denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. They proved that the sequence {xn} converges strongly to PF(T)x0.Recently, Su and Qin [6] introduced a monotone CQ method for nonexpansive mapping, defined as follows: from an arbitrary x0∈C,
(1.5)yn=αnxn+(1-αn)Txn,C0={z∈C:‖y0-z‖≤‖x0-z‖},Q0=C,Cn={z∈Cn-1∩Qn-1:∥yn-z∥≤∥xn-z∥},Qn={z∈Cn-1∩Qn-1:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,∀n≥0,
and it proved that the sequence {xn} converges strongly to PF(T)x0.We now recall some definitions concerning relatively quasi-nonexpansive mappings. LetE be a real smooth Banach space with norm ∥·∥ and let E* be the dual of E. The normalized duality mapping J from E to E* is defined by
(1.6)Jx={f∈E*:〈x,f〉=‖x‖2=‖f‖2},∀x∈E,
where 〈·,·〉 denotes the pairing between E and E*. Readers are directed to [7] (and its review [8]), where the properties of the duality mapping and several related topics are presented. The function ϕ:E×E→R+ is defined by
(1.7)ϕ(x,y)=‖x‖2-2〈x,Jy〉+‖y‖2,∀x,y∈E.LetT be a mapping from C into E. A point p in C is said to be an asymptotic fixed point [9] of T if C contains a sequence {xn} which converges weakly to p and limn→∞(xn-Txn)=0. The set of asymptotic fixed points of T is denoted by F^(T).We say that the mappingT is relatively nonexpansive (see [10]) if the following conditions are satisfied:(R1)
F(T)≠∅;(R2)
ϕ(p,Tx)≤ϕ(p,x), ∀p∈F(T);(R3)
F(T)=F^(T).If T satisfies (R1) and (R2), then T is called relatively quasi-nonexpansive.Several articles have provided methods for approximating fixed points of relatively quasi-nonexpansive mappings [11–16]. Employing the ideas of Su and Qin [6], and of Aoyama et al. [17], in 2008, Nilsrakoo and Saejung [18] used the following iterations to obtain strong convergence theorems for common fixed points of a countable family of relatively quasi-nonexpansive mappings in a Banach space
(1.8)x0∈C,C-1=Q-1=C;yn=J-1(αnJxn+(1-αn)JTnxn),Cn={z∈Cn-1∩Qn-1:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈Cn-1∩Qn-1:〈xn-z,Jx0-Jxn〉≥0},xn+1=Πnx0,∀n≥0.
However, the results were obtained under two stronger assumption conditions, namely, the AKTT-condition and the *AKTT-condition imposed on the involved mappings.Inspired and motivated by those studies mentioned above, in this paper, we use a modified type of the iteration scheme (1.8) for approximating common fixed points of an infinite family of relatively quasi-nonexpansive mappings; without stronger assumptions imposed on the involved mappings, a strong convergence theorem in Banach spaces is obtained for solving a system of equilibrium problems. The results improve those of other authors with related interest.


## 2. Preliminaries

Throughout the paper, letE be a real Banach space. We say that E is strictly convex if the following implication holds for x,y∈E:
(2.1)‖x‖=‖y‖=1,x≠y⟹‖x+y2‖<1.
It is also said to be uniformly convex if for any ϵ>0, there exists a δ>0 such that
(2.2)‖x‖=‖y‖=1,‖x-y‖≥ϵ⟹‖x+y2‖≤1-δ.
It is known that if E is uniformly convex Banach space, then E is reflexive and strictly convex. A Banach space E is said to be smooth if
(2.3)lim⁡t→0‖x+ty‖-‖x‖t
exists for each x,y∈S(E):={x∈E:∥x∥=1}. In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have uniformly Gâteaux differentiable norm if for each y∈S(E); the limit (2.3) is attained uniformly for x∈S(E). The norm of E is said to be Fréchet differentiable if for each x∈S(E); the limit (2.3) is attained uniformly for y∈S(E). The norm of E is said to be uniformly Fréchet differentiable (and E is said to be uniformly smooth) if the limit (2.3) is attained uniformly for x,y∈S(E).We also know the following properties (see, e.g., [19] for details). (1)
E (E*, resp.) is uniformly convex ⇔E* (E, resp.) is uniformly smooth.(2)
Jx≠∅ for each x∈E.(3)
IfE is reflexive, then J is a mapping from E onto E*.(4)
IfE is strictly convex, then Jx∩Jy=∅ as x≠y.(5)
IfE is smooth, then J is single-valued.(6)
IfE has a Fréchet differentiable norm, then J is norm-to-norm continuous.(7)
IfE is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.(8)
IfE is a Hilbert space, then J is the identity operator.LetE be a smooth Banach space. The function ϕ:E×E→R+ is defined by
(2.4)ϕ(x,y):=‖x‖2-2〈x,Jy〉+‖y‖2.
It is obvious from the definition of the function ϕ that
(2.5)(‖x‖-‖y‖)2≤ϕ(x,y)≤(‖x‖+‖y‖)2.
Moreover, we know the following results.Lemma 2.1 (see [13]).
LetE be a strictly convex and smooth Banach space, then ϕ(x,y)=0 if and only if x=y.Lemma 2.2 (see [11]).
LetE be a uniformly convex and smooth Banach space and let r>0. Then there exists a continuous, strictly increasing, and convex function g:[0,2r]→[0,∞) such that g(0)=0 and
(2.6)g(‖x‖-‖y‖)≤ϕ(x,y)
for all x,y∈Br:={z∈E:PzP≤r}.LetC be a nonempty and closed convex subset of E. Suppose that E is reflexive, strictly convex, and smooth. It is known in [20] that for any x∈E, there exists a unique point x*∈C such that
(2.7)ϕ(x*,x)=min⁡y∈Cϕ(y,x).
Following Alber [21], we denote such an x* by ΠCx. The mapping ΠC is called the generalized projection from E onto C. It is easy to see that in a Hilbert space, the mapping ΠC coincides with the metric projection PC. What follows are the well-known facts concerning the generalized projection.Lemma 2.3 (see [20]).
LetC be a nonempty closed convex subset of a smooth Banach space E and let x∈E. Then
(2.8)x*=ΠCx⟺〈x*-y,Jx-Jx*〉≥0,∀y∈C.Lemma 2.4 (see [20]).
LetE be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E, and let x∈E. Then
(2.9)ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.Dealing with the generalized projection fromE onto the fixed point set of a relatively quasi-nonexpansive mapping, we have the following result.Lemma 2.5 (see [18]).
LetE be a strictly convex and smooth Banach space, let C be a nonempty and closed convex subset of E, and let T be a relatively quasi-nonexpansive mapping from C into E. Then F(T) is closed and convex.LetC be a subset of a Banach space E and let {Tn} be a family of mappings from C into E. For a subset B of C, we say that (i)
({Tn},B) satisfies AKTT-condition if
(2.10)∑n=1∞‍sup⁡{‖Tn+1z-Tnz‖:z∈B}<∞;(ii)
({Tn},B) satisfies *AKTT-condition if
(2.11)∑n=1∞‍sup⁡{‖JTn+1z-JTnz‖:z∈B}<∞.


## 3. Main Results

Recall that an operatorT in a Banach space is closed if xn→x and Txn→y as n→∞, then Tx=y.Theorem 3.1.
LetE be a uniformly convex and uniformly smooth Banach space, C a nonempty and closed convex subset of E. Let {Ti}i=1∞:C→E be a sequence of closed and relatively quasi-nonexpansive mappings with F:=⋂i=1∞F(Ti)≠∅. Starting from an arbitrary x1∈C, the sequence {xn} is define by
(3.1)x1∈C,C0=Q0=C;yn=J-1(αnJxn+(1-αn)JTinxn),Cn={z∈Cn-1∩Qn-1:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈Cn-1∩Qn-1:〈xn-z,Jx1-Jxn〉≥0},xn+1=Πnx1,∀n≥1,
where Πn:=ΠCn⋂‍Qn and {αn} is a sequence in [0,1) with limsupn→∞αn<1; in is the solution to the positive integer equation: n=i+(m-1)m/2(m≥i,n=1,2,…), that is, for each n≥1, there exists a unique in such that
(3.2)i1=1,i2=1,i3=2,i4=1,i5=2,i6=3,i7=1,i8=2,i9=3,i10=4,i11=1,….
Then {xn} converges strongly to ΠFx1.Proof.
We first claim that bothCn and Qn are closed and convex. This follows from the fact that ϕ(z,yn)≤ϕ(z,xn) is equivalent to the following:
(3.3)2〈z,Jxn-Jyn〉≤‖xn‖2-‖yn‖2.
It is clear thatF⊂C=C0∩Q0. Next, we show that
(3.4)F⊂Cn∩Qn,∀n≥1.
Suppose that F⊂Ck-1∩Qk-1 for some k≥2. Letting p∈F, we then have
(3.5)ϕ(p,yk)=ϕ(p,J-1(αkJxk+(1-αk)JTik-xk))=‖p‖2-2〈p,αkJxk+(1-αk)JTikxk〉+‖αkJxk+(1-αk)JTikxk‖2≤‖p‖2-2αk〈p,Jxk〉-2(1-αk)〈p,JTikxk〉+αk‖xk‖2+(1-αk)‖Tikxk‖2=αk(‖p‖2-2〈p,Jxk〉+‖xk‖2)+(1-αk)(‖p‖2-2〈p,JTikxk〉+‖Tikxk‖2)=αkϕ(p,xk)+(1-αk)ϕ(p,Tikxk)≤αkϕ(p,xk)+(1-αk)ϕ(p,xk)=ϕ(p,xk).
This implies that F⊂Ck. It follows from xk=Πk-1x1 and Lemma 2.3 that
(3.6)〈xk-z,Jx1-Jxk〉≥0,∀z∈Ck-1∩Qk-1.
Particularly,
(3.7)〈xk-z,Jx1-Jxk〉≥0,∀p∈F
and hence F⊂Qk, which yields that
(3.8)F⊂Ck∩Qk.
By induction, (3.4) holds. This implies that {xn} is well defined. It follows from the definition of Qn and Lemma 2.3 that xn=ΠQnx1. Since xn+1=Πnx1∈Qn, we have
(3.9)ϕ(xn,x1)≤ϕ(xn+1,x1),∀n≥1.
Therefore, {ϕ(xn,x1)} is nondecreasing. Using xn=ΠQnx1 and Lemma 2.4, we have
(3.10)ϕ(xn,x1)=ϕ(ΠQnx1,x1)≤ϕ(p,x1)-ϕ(p,xn)≤ϕ(p,x1)
for all p∈F and for all n≥1, that is, {ϕ(xn,x1)} is bounded. Then
(3.11)lim⁡n→∞ϕ(xn,x1)exists.
In particular, by (2.5), the sequence {(∥xn∥-∥x1∥)2} is bounded. This implies that {xn} is bounded. Note again that xn=ΠQnx1 and for any positive integer k, xn+k∈Qn+k-1⊂Qn. By Lemma 2.4,
(3.12)ϕ(xn+k,xn)=ϕ(xn+k,ΠQnx1)≤ϕ(xn+k,x1)-ϕ(ΠQnx1,x1)=ϕ(xn+k,x1)-ϕ(xn,x1).
By Lemma 2.2, we have, for any positive integers m,n with m>n,
(3.13)g(‖xm-xn‖)≤ϕ(xm,xn)≤ϕ(xm,x1)-ϕ(xn,x1),
where g:[0,∞)→[0,∞) is a continuous, strictly increasing, and convex function with g(0)=0. Then the properties of the function g yield that {xn} is a Cauchy sequence in C, so there exists an x*∈C such that
(3.14)xn⟶x*(n⟶∞).
In view of xn+1=Πnx1∈Cn and the definition of Cn, we also have
(3.15)ϕ(xn+1,yn)≤ϕ(xn+1,xn),∀n≥1.
This implies that
(3.16)lim⁡n→∞ϕ(xn+1,yn)=lim⁡n→∞ϕ(xn+1,xn)=0.
It follows from Lemma 2.2 that
(3.17)lim⁡n→∞‖xn+1-yn‖=lim⁡n→∞‖xn+1-xn‖=0.
Since J is uniformly norm-to-norm continuous on bounded sets, we have
(3.18)lim⁡n→∞‖Jxn+1-Jyn‖=lim⁡n→∞‖Jxn+1-Jxn‖=0.
On the other hand, we have, for each n≥1,
(3.19)‖Jxn+1-Jyn‖=‖Jxn+1-(αnJxn+(1-αn)JTinxn)‖=‖(1-αn)(Jxn+1-JTinxn)-αn(Jxn-Jxn+1)‖≥(1-αn)‖Jxn+1-JTinxn‖-αn‖Jxn-Jxn+1‖,
and hence
(3.20)‖Jxn+1-JTinxn‖≤11-αn‖Jxn+1-Jyn‖+αn1-αn‖Jxn-Jxn+1‖.
From (3.18) and limsupn→∞αn<1, we obtain that
(3.21)lim⁡n→∞‖Jxn+1-JTinxn‖=0.
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
(3.22)lim⁡n→∞‖xn+1-Tinxn‖=lim⁡n→∞‖J-1(Jxn+1)-J-1(JTinxn)‖=0.
It follows from (3.17) that, as n→∞,
(3.23)‖xn-Tinxn‖≤‖xn-xn+1‖+‖xn+1-Tinxn‖⟶0.
Now, set 𝒦i={k≥1:k=i+(m-1)m/2,m≥i,m∈ℤ+} for each i≥1. Note that Tik=Ti whenever k∈𝒦i. For example, by the definition of 𝒦1, we have 𝒦1={1,2,4,7,11,16,…} and i1=i2=i4=i7=i11=i16=⋯=1. Then it follows from (3.23) that
(3.24)lim⁡Ki∋k→∞‖Tixk-xk‖=0,∀i≥1.
Since {xk}k∈𝒦i is a subsequence of {xn}, (3.14) implies that xk→x* as 𝒦i∋k→∞. It immediately follows from (3.24) and the closedness of Ti that x*∈F(Ti) for each i≥1, and hence x*∈F. Furthermore, by (3.10),
(3.25)ϕ(x*,x1)=lim⁡n→∞ϕ(xn,x1)≤ϕ(p,x1),∀p∈F.
This implies that x*=ΠFx1. The proof is completed.Remark 3.2.
Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. An example [22] of how to compute such a projection is given as follows.Dykstra's Algorithm
LetΩ1,Ω2,…,Ωp be closed and convex subsets of ℝn. For any i=1,2,…,p and x0∈ℝn, the sequences {xik} are defined by the following recursive formulae:
(3.26)x0k=xpk-1,xik=PΩi(xi-1k-yik-1),i=1,2,…,p,yik=xik-(xi-1k-yik-1),i=1,2,…,p,
for k=1,2,… with initial values xp0=x0 and yi0=0 for i=1,2,…,p. If Ω:=⋂i=1pΩi≠∅, then {xik} converges to x*=PΩ(x0), where PΩ(x):=arginfy∈Ω∥y-x∥2,forallx∈ℝn.


## 4. Applications

The so-calledconvex feasibility problem for a family of mappings {Ti}i=1∞ is to find a point in the nonempty intersection ⋂i=1∞F(Ti), which exactly illustrates the importance of finding fixed points of infinite families. The following example also clarifies the same thing.Example 4.1.
LetE be a smooth, strictly convex, and reflexive Banach space, C a nonempty and closed convex subset of E, and {fi}i=1∞:C→C a countable family of bifunctions satisfying the conditions: for each i≥1, (A1)
fi(x,x)=0;(A2)
fi is monotone, that is, fi(x,y)+fi(y,x)≤0;(A3)
limsupt↓0fi(x+t(z-x),y)≤fi(x,y);(A4)
the mappingy↦fi(x,y) is convex and lower semicontinuous.A system of equilibrium problems for {fi}i=1∞ is to find an x*∈C such that
(4.1)fi(x*,y)≥0,∀y∈C,i≥1,
whose set of common solutions is denoted by EP:=⋂i=1∞EP(fi), where EP(fi) denotes the set of solutions to the equilibrium problem for fi(i=1,2,…). It will be shown in Theorem 4.3 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonexpansive mappings.Example 4.2 (see [23]).
Letr>0. Define a countable family of mappings {Tr,i}i=1∞:E→C as follows:
(4.2)Tr,i(x)={z∈C:fi(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},∀i≥1.
Then we have that(1)
{Tr,i}i=1∞ is a sequence of single-valued mappings;(2)
{Tr,i}i=1∞ is a sequence of closed relatively quasi-nonexpansive mappings;(3)
F:=⋂i=1∞F(Tr,i)=EP.Now, we have the following result.Theorem 4.3.
LetC,E, and {αn} be the same as those in Theorem 3.1. Let {fi}i=1∞:C→C be a countable family of bifunctions satisfying the conditions (A1)–(A4). Let {Tr,i}i=1∞:E→C be a countable family of mappings defined by (4.2). Let {xn} be the sequence generated by
(4.3)x1∈C,C0=Q0=C;fin(un,y)+1r〈y-un,Jun-Jxn〉≥0,∀y∈C,yn=J-1(αnJxn+(1-αn)Jun),Cn={z∈Cn-1∩Qn-1:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈Cn-1∩Qn-1:〈xn-z,Jx1-Jxn〉≥0},xn+1=Πnx1,∀n≥1,
where in satisfies the positive integer equation: n=i+(m-1)m/2(m≥i,n=1,2,…). If F:=⋂i=1∞F(Tr,i)≠∅, then {xn} strongly converges to ΠFx1 which is a common solution of the system of equilibrium problems for {fi}i=1∞.Proof.
Since eachTr,i is single-valued, un=Tr,inxn for all n≥1. In addition, we have pointed out in Example 4.2 that F=EP and {Tr,i}i=1∞ is a sequence of closed relatively quasi-nonexpansive mappings. Hence, (4.3) can be rewritten as follows:
(4.4)x1∈C,C0=Q0=C;yn=J-1(αnJxn+(1-αn)JTr,inxn),Cn={z∈Cn-1∩Qn-1:ϕ(z,yn)≤ϕ(z,xn)},Qn={z∈Cn-1∩Qn-1:〈xn-z,Jx1-Jxn〉≥0},xn+1=Πnx1,∀n≥1.
Therefore, this conclusion can be obtained immediately from Theorem 3.1.



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