# Hybrid Iteration Method for Common  Fixed Points of a

Authors: Lin Wang; Yi-Juan Chen; Rong-Chuan Du
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

LetE be a real uniformly convex Banach space, and let{Ti:i∈I} be N nonexpansive
mappings from E into itself with F={x∈E:Tix=x,i∈I}≠ϕ, where I={1,2,…,N}. From an arbitrary initial point x1∈E, hybrid iteration scheme {xn} is defined as follows: xn+1=αnxn+(1−αn)(Tnxn−λn+1μA(Tnxn)), n≥1, where A:E→E is an L-Lipschitzian
mapping, Tn=Ti, i=n(modN), 1≤i≤N, μ>0, {λn}⊂[0,1), and {αn}⊂[a,b] for some a,b∈(0,1). Under some suitable conditions, the strong and weak
convergence theorems of {xn} to a common fixed point of the mappings {Ti:i∈I} are obtained. The results presented in this paper extend and improve
the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro
(2007).


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## Body

## 1. Introduction

LetE be a Banach space endowed with the norm ∥·∥. A mapping T:E→E is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for any x,y∈E. F:E→E is said to be L-Lipschitzian if there exists constant L>0 such that ∥Fx-Fy∥≤L∥x-y∥ for any x,y∈E.LetH be a Hilbert space with inner product 〈·,·〉 and associated with norm ∥·∥, A:H→H is said to be η-strong monotone if there exists η>0 such that(1.1)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈H.The interest and importance of construction of fixed points of nonexpansive mappings stem mainly from the fact that it may be applied in many areas, such as imagine recovery and signal processing (see, e.g., [1–3]). Especially, numerous problems in physics, optimization, economics, traffic analysis, and mechanics reduce to find a solution of equilibrium problem. The equilibrium problem is to find(1.2)x*∈Csuchthatf(x*,y)≥0,∀y∈C,
where C is a nonempty closed convex subset of a Hilbert space H, f is a bifunction from C×C to R, and R is the set of real numbers.It has been shown by Blum and Oettli [4] and Noor and Oettli [5] that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Given a mapping T:C→H, let f(x,y)=〈Tx,y-x〉 for all x,y∈C. It is well-known that x*∈C is a solution of (1.2) if and only if 〈Tx*,y-x*〉≥0 for all y∈C. Very recently, Yao et al. [6] find a common element of the set of solutions of equilibrium problem (1.2) and the set of common fixed points of a finite family of nonexpansive mappings by using an iterative scheme of a finite family of nonexpansive mappings. See the references therein for more details.  Therefore, the topic on construction of fixed points of nonexpansive mappings is useful for equilibrium problems in physics, optimization, traffic analysis, and so forth.Motivated by earlier results of Xu and Kim [7] and Yamada [8], some authors [9–14] further extended hybrid iteration method used this method to approximate fixed points of nonexpansive mappings, and obtained some strong and weak convergence theorems for nonexpansive mappings.Recently, Wang [12] introduced an explicit hybrid iteration method for nonexpansive mappings and obtained the following convergence theorem.Theorem 1.1 ([12]).
LetH be a Hilbert space, let T:H→H be a nonexpansive mapping with F(T)={x∈H:Tx=x}≠ϕ, and let A:H→H be a η-strong monotone and L-Lipschizian mapping. For any given x1∈H, {xn} is defined by
(1.3)xn+1=αnxn+(1-αn)Tλn+1xn,n≥1,
where  Tλn+1xn=Txn-λn+1μA(Txn). If {αn}⊂[0,1) and {λn}⊂[0,1) satisfy the following conditions:  (1)   α≤αn≤β for some α,β∈(0,1); (2)    ∑n=2∞‍λn<∞; (3)    0<μ<2η/L2, then,(1)
{xn} converges weakly to a fixed point of T.(2)
{xn} converges strongly to a fixed point of T if only if lim infn→∞d(xn,F(T))=0.Very recently, Osilike et al. [11] extended Wang's results to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator and obtained the following result.Theorem 1.2 ([11]).
LetE be an arbitrary Banach space endowed with the norm ∥·∥, let T:E→E be a nonexpansive mapping with F(T)≠ϕ, and let A:E→E be an L-Lipschitzian mapping. Let {xn} be the sequence generated from an arbitrary x1∈E by
(1.4)xn+1=αnxn+(1-αn)Tλn+1xn,n≥1,
where Tλn+1xn=Txn-λn+1μA(Txn), μ>0, and {αn}⊂[0,1) and {λn}⊂[0,1) satisfy the following conditions: (1)    0<α≤αn≤1 for all n≥1 and some α∈(0,1); (2)    ∑n=1∞‍(1-αn)=∞; (3)    ∑n=2∞‍λn<∞, then,(1)
limn→∞∥xn-p∥ exists for each p∈F(T),(2)
limn→∞∥xn-Txn∥=0,(3)
{xn} converges strongly to a fixed point of T if and only if lim infn→∞d(xn,F(T))=0.Motivated by above work, we obtain the strong and weak convergence theorems for a finite family of nonexpansive mappings in uniformly convex Banach space by using hybrid iteration method. The results presented in this paper extend and improve the results of Wang [12] and partially improve the results of Osilike et al. [11].


## 2. Preliminaries

Throughout this paper, we denoteI={1,2,…,N}.A mappingT:K→E is said to be demicompact if, for any sequence {xn} in K such that ∥xn-Txn∥→0(n→∞), there exists subsequence {xnj} of {xn} such that {xnj} converges strongly to x*∈K.For studying the strong convergence of fixed points of a nonexpansive mapping, Senter and Dotson [15] introduced condition (A). Later on, Maiti and Ghosh [16], Tan and Xu [17] studied condition (A) and pointed out that Condition (A) is weaker than the requirement of demicompactness for nonexpansive mappings. A mapping T:K→K with F(T)={x∈K:Tx=x}≠ϕ is said to satisfy condition (A) if there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that ∥x-Tx∥≥f(d(x,F(T))) for all x∈K, where d(x,F(T))=inf{∥x-q∥:q∈F(T)}.A family of mappings{Ti:i∈I} from E into itself with F={x∈E:Tix=x,i∈I} is said to satisfy condition (B) if there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(t)>0 for all t∈(0,∞) such that ∑i=1N‍∥x-Tix∥/N≥f(d(x,F)) for all x∈E.A Banach spaceE is said to satisfy Opial's condition if, for any sequence {xn} in E, xn⇀x implies that lim supn→∞∥xn-x∥<lim supn→∞∥xn-y∥ for all y∈E with y≠x, where xn⇀x denotes that {xn} converges weakly to x.A mappingT with domain D(T) and range R(T) in E is said to be demi-closed at p if whenever {xn} is a sequence in D(T) such that {xn} converges weakly to x*∈D(T) and {Txn} converges strongly to p, then Tx*=p.In the coming Lemma we will use the following well-known results.Lemma 2.1 ([18]).
Let{αn} and {tn} be two nonnegative sequences satisfying
(2.1)αn+1≤(1+an)αn+bn,∀n≥1.
If ∑n=1∞‍an<∞ and ∑n=1∞‍bn<∞, then limn→∞αn exists.Lemma 2.2 (see [19]).
LetE be a real uniformly convex Banach space and let a, b be two constant with 0<a<b<1. Suppose that {tn}⊂[a,b] is a real sequence and {xn}, {yn} are two sequences in E. Then the conditions
(2.2)limn→∞∥tnxn+(1-tn)yn∥=d,lim supn→∞∥xn∥≤d,lim supn→∞∥yn∥≤d,
imply that limn→∞∥xn-yn∥=0, where d≥0 is a constant.Lemma 2.3 (see [20]).
LetE be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let T:K→K be a nonexpansive mapping. Then  I-T is demiclosed at zero.


## 3. Main Results

Theorem 3.1.
LetE be a real uniformly convex Banach space endowed with the norm ∥·∥, let I={1,2,…,N}, {T:i∈I} be N nonexpansive mappings from E into itself with F={x∈E:Tix=x,i∈I}≠ϕ, and let A:E→E be an L-Lipschitzian mapping. For any given x1∈E, {xn} is defined by
(3.1)xn+1=αnxn+(1-αn)Tλn+1xn,n≥1,
where Tλn+1xn=Tnxn-λn+1μA(Tnxn), μ>0, Tn=Ti, i=n(modN),1≤i≤N. If {αn} and {λn}⊂[0,1) satisfy the following conditions:(1)
a≤αn≤b for all n≥1 and some a,b∈(0,1),(2)
∑n=2∞‍λn<∞,then (1)
limn→∞∥xn-q∥ exists for each q∈F,(2)
limn→∞∥xn-Tixn∥=0 for each i∈I,(3)
{xn} converges strongly to a common fixed point of {Ti:i∈I} if and only if lim infn→∞d(xn,F)=0.Proof.
(1) For anyq∈F, we have
(3.2)∥xn+1-q∥=∥αn(xn-q)+(1-αn)(Tλn+1xn-q)∥≤αn∥xn-q∥+(1-αn)∥xn-q∥+λn+1(1-αn)μ∥A(Tnxn)∥≤∥xn-q∥+(1-αn)λn+1μ∥A(Tnxn)-A(q)∥+(1-αn)λn+1μ∥A(q)∥≤[1+(1-a)μLλn+1]∥xn-q∥+(1-a)λn+1μ∥A(q)∥.
Since ∑n=2∞‍λn<∞, it follows from Lemma 2.1 that limn→∞∥xn-q∥ exists.
(2) Sincelimn→∞∥xn-q∥ exists for any q∈F, {xn} is bounded. So are {A(Tnxn)} and {Tnxn}. Thus we may assume that limn→∞∥xn-q∥=d, that is,
(3.3)limn→∞∥αn(xn-q)+(1-αn)(Tλn+1xn-q)∥=d.
Since limn→∞∥xn-q∥=d, limn→∞λn=0 and
(3.4)∥Tλn+1xn-q∥=∥Tnxn-λn+1μA(Tnxn)-q∥≤∥xn-q∥+λn+1μ∥A(Tnxn)∥,
we have
(3.5)lim supn→∞∥Tλn+1xn-q∥≤d.
Thus, it follows from Lemma 2.2 that
(3.6)limn→∞∥xn-Tλn+1xn∥=0.
In addition,(3.7)∥xn-Tλn+1xn∥=∥xn-Tnxn+λn+1μA(Tnxn)∥≥||xn-Tnxn||-λn+1μ||A(Tnxn)||,
so
(3.8)∥xn-Tnxn∥≤∥xn-Tλn+1xn∥+λn+1μ∥A(Tnxn)∥.
Therefore, it follows from (3.6) that
(3.9)limn→∞∥xn-Tnxn∥=0.
On the other hand, since(3.10)xn+1=αnxn+(1-αn)[Tnxn-λn+1μA(Tnxn)],
we have
(3.11)xn+1-Tnxn=αn(xn-Tnxn)-(1-αn)λn+1μA(Tnxn).
Thus, it follows from (3.9) that
(3.12)limn→∞∥xn+1-Tnxn∥=0.
From (3.9) and (3.12), we can obtain
(3.13)limn→∞∥xn+1-xn∥=0.
Further, for any positive integer k, we also have
(3.14)limn→∞∥xn+k-xn∥=0.
For eachi∈I,
(3.15)∥xn-Tn+ixn∥=∥xn-xn+i+xn+i-Tn+ixn+i+Tn+ixn+i-Tn+ixn∥≤2||xn-xn+i||+||xn+i-Tn+ixn+i||.
It follows from (3.9) and (3.14) that limn→∞∥xn-Tn+ixn∥=0. This implies that limn→∞∥xn-Tixn∥=0 for each i∈I.
(3) Suppose that{xn} converges strongly to a common fixed point q of the mappings {Ti:i∈I}, then limn→∞∥xn-q∥=0. Since 0≤d(xn,F)≤∥xn-q∥, we have lim infn→∞d(xn,F)=0.
Conversely, suppose thatlim infn→∞d(xn,F)=0. Since {xn} is bounded, there exists constant M>0 such that ∥xn-q∥≤M. From (3.2), for any q∈F, we obtain
(3.16)∥xn+1-q∥≤[1+(1-a)μLλn+1]∥xn-q∥+(1-a)λn+1μ∥A(q)∥≤∥xn-q∥+λn+1δ,
where δ=(1-a)μ[LM+∥A(q)∥]. Furthermore, we have
(3.17)d(xn+1,F)≤d(xn,F)+λn+1δ.
It follows from Lemma 2.1 that limn→∞d(xn,F) exists. Since lim infn→∞d(xn,F)=0, we obtain that limn→∞d(xn,F)=0. We now show that {xn} is a Cauchy sequence.
For arbitraryε>0, there exists positive integer N1 such that d(xn,F)<ε/4  for all n≥N1. In addition, since ∑n=2∞‍λn<∞, there exists positive integer N2 such that ∑j=n∞‍λj<ε/4δ for all n≥N2. Taking N=max{N1,N2}, for any n,m≥N, from (3.16), we have
(3.18)∥xn-xm∥≤∥xn-q∥+∥xm-q∥≤2∥xN-q∥+2δ∑j=N∞‍λj+1.
Taking the infimum in above inequalities for all q∈F, we obtain
(3.19)∥xn-xm∥≤2d(xN,F)+2δ∑j=N∞‍λj+1<ε.
This implies that {xn} is a Cauchy sequence. Therefore there exists p∈E such that {xn} converges strongly to p. Since limn→∞∥xn-Tixn∥=0 for each i∈I, it follows from Lemma 2.3 that p∈F. This completes the proof.From Lemma2.3 and limn→∞∥xn-Tixn∥=0 for each i∈I, using routine method, we can easily show the following weak convergence theorem, whose proof is omitted.Theorem 3.2.
LetE be a real uniformly convex Banach space satisfying Opial's condition, let {Ti:i∈I} be N nonexpansive mappings from E into itself with F={x∈E:Tx=x}≠ϕ, and let A:H→H be an L-Lipschitzian mapping. For any given x1∈E, {xn} is defined as in Theorem 3.1, and {αn}⊂[0,1) and {λn}⊂[0,1) satisfy the conditions appeared in Theorem 3.1. Then {xn} converges weakly to a common fixed point of the mappings of {Ti:i∈I}.Example 3.3.
LetE=R be endowed with standard norm ∥·∥=|·|, where R is real number set. Define T1:[0,∞)→[0,∞) and T2:[0,∞)→[0,∞) by T1x=1/2+x/(1+x) and T2x=1/2+x2/(1+x) for all x∈[0,∞), respectively. Obviously, T1 and T2 are nonexpansive mappings, and 1 is a common fixed point of T1 and T2. Let A:[0,∞)→[0,∞) be defined by Ax=2x+1 for all x∈[0,∞). We now chose parameters {αn},  {λn} and μ as follows:
(3.20)αn=0.8-12n,λn=1(n+1)2,n≥1;μ=1.
It is easy to see that A is a 2-Lipschitzian mapping and {αn}, {λn}, and μ satisfy the conditions of Theorem 3.2. Then {xn} is generated by
(3.21)xn+1=(0.8-12n)xn+(0.2+12n)Tnxn-(1-αn)1(n+1)2A(Tnxn),n≥1,
where T2n-1=T1 and  T2n=T2. It follows from Theorem 3.2 that {xn} converges strongly to the common fixed point 1 of T1 and T2. As x1=2, by using Mathematical 5.0 to compute {xn}, we know that x10=0.82893, x20=0.935807, x50=0.994619, and x100=0.999272. This example shows that the algorithm is efficient for approximating common fixed points of nonexpansive mappings.Remark 3.4.
By using Theorem3.1 and Lemma 2.3, we can easily prove that {xn} converges strongly to a common fixed point of the mappings of {Ti:i∈I} if one of the mappings {Ti:i∈I} is demicompact or {Ti:i∈I} satisfies condition (B). Therefore the results presented in this paper improve and extend the results of Wang [12] and partially improve the results of Osilike et al. [11].Remark 3.5.
We do not know how to overcome the constraint condition∑n=1∞‍∥Tnxn-Tn+1xn∥<∞ when we try to extend Theorem 3.1 to arbitrary Banach spaces.



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