# Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive

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 present the notion of weak relatively
nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.


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## Body

## 1. Introduction and Preliminaries

Iterative methods for approximating fixed points of multivalued mappings in Banach spaces have been studied by some authors, see for instance [1–4]. Let C be a nonempty closed convex subset of a smooth Banach space E and T:C→C a multivalued mapping such that Tx is nonempty for all x∈C. In [4], Homaeipour and Razani have defined the relatively nonexpansive multivalued mapping and have proved some convergence theorems.Theorem SA 1  (see [4]).
LetE be a uniformly convex and uniformly smooth Banach space, and C a nonempty closed convex subset of E. Suppose T:C→C is a relatively nonexpansive multivalued mapping. Let {αn} be a sequence of real numbers such that 0≤αn≤1 and liminfn→∞αn(1-αn)>0. For a given x1∈C, let {xn} be the iterative sequence defined by
(1.1)xn+1=ΠCJ-1(αnJxn+(1-αn)Jzn),zn∈Txn.
If J is weakly sequentially continuous, then {xn} converges weakly to a fixed point of T.Theorem SA 2  (see [4]).
LetE be a uniformly convex and uniformly smooth Banach space, and C a nonempty closed convex subset of E. Suppose T:C→C is a relatively nonexpansive multivalued mapping. Let {αn} be a sequence of real numbers such that 0≤αn≤1 and liminfn→∞αn(1-αn)>0. For a given x1∈C, let {xn} be the iterative sequence defined by
(1.2)xn+1=ΠCJ-1(αnJxn+(1-αn)Jzn),zn∈Txn.
If the interior of F(T) is nonempty, then {xn} converges strongly to a fixed point of T.LetE be a Banach space with dual E*. We denote by J the normalized duality mapping from E to 2E* defined by
(1.3)Jx={f∈E*:〈x,f〉=∥x∥2=∥f∥2},
where 〈·,·〉 denotes the generalized duality pairing. It is well known that if E* is uniformly convex, then J is uniformly continuous on bounded subsets of E.As we all know that ifC is a nonempty closed convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces, and consequently it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.Next, we assume thatE is a smooth Banach space. Consider the functional defined as [5, 6] by
(1.4)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,forx,y∈E.
Observe that, in a Hilbert space H, (1.4) reduces to ϕ(x,y)=∥x-y∥2,x,y∈H.The generalized projectionΠC:E→C is a map that assigns to an arbitrary point x∈E the minimum point of the functional ϕ(x,y), that is, ΠCx=x¯, where x¯ is the solution to the following minimization problem:
(1.5)ϕ(x¯,x)=miny∈Cϕ(y,x),
existence and uniqueness of the operator ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, eg., [5–7]). In Hilbert space, ΠC=PC. It is obvious from the definition of function ϕ that
(1.6)(∥y∥-∥x∥)2≤ϕ(y,x)≤(∥y∥+∥x∥)2,∀x,y∈E.Remark 1.1.
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 (1.6), we have ∥x∥=∥y∥. This implies 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definitions of j, we have Jx=Jy, that is, x=y, see [8, 9] for more details.LetC be a nonempty closed convex subset of a smooth Banach space E and T:C→C a multivalued mapping such that Tx is nonempty for all x∈C. A point p is called a fixed point of T, if p∈Tp. The set of fixed points of T is represented by F(T). A point p∈C is called an asymptotic fixed point of T, if there exists a sequence {xn} in C which converges weakly to p and limn→∞d(xn,Txn)=0, where d(xn,Txn)=infu∈Txn∥xn-u∥. The set of asymptotic fixed points of T is represented by F^(T). Moreover, a multivalued mapping T:C→C is called relatively nonexpansive multivalued mapping, if the following conditions are satisfied:(1)
F(T)≠∅,(2)
ϕ(p,z)≤ϕ(p,x),  for  all   p∈F(T),  for  all  x∈C,  for  all z∈Tx,(3)
F^(T)=F(T).In [4], authors also give an example which is a relatively nonexpansive multivalued mapping but not a nonexpansive multivalued mapping.The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems for the weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. We first give the definition of weak relatively nonexpansive multivalued mapping as follows.LetC be a nonempty closed convex subset of a smooth Banach space E and T:C→C a multivalued mapping such that Tx is nonempty for all x∈C. A point p∈C is called an strong asymptotic fixed point of T, if there exists a sequence {xn} in C which converges strongly to p and limn→∞d(xn,Txn)=0, where d(xn,Txn)=infu∈Txn∥xn-u∥. The set of the strong asymptotic fixed points of T is represented by F~(T). Moreover, a multivalued mapping T:C→C is called weak relatively nonexpansive multivalued mapping, if the following conditions are satisfied:(I)
F(T)≠∅,(II)
ϕ(p,z)≤ϕ(p,x),  for  all  p∈F(T),  for  all  x∈C,  for  all  z∈Tx,(III)
F~(T)=F(T).We need the following Lemmas for the proof of our main results.Lemma 1.2 (see Kamimura and Takahashi [7]).
LetE be a uniformly convex and smooth 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 1.3 (see Alber [5]).
LetC be a nonempty closed convex subset of a smooth Banach space E and x∈E. Then, x0=ΠCx if and only if
(1.7)〈x0-y,Jx-Jx0〉≥0,fory∈C.Lemma 1.4 (see Alber [5]).
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
(1.8)ϕ(y,Πcx)+ϕ(Πcx,x)≤ϕ(y,x),∀y∈C.Lemma 1.5.
LetE be a uniformly convex and smooth Banach space, let C be a closed convex subset of E, and let T:C→C be a weak relatively nonexpansive multivalued mapping. Then F(T) is closed and convex.Proof.
First, we showF(T) is closed. Let {xn} be a sequence in F(T) such that xn→q. Since T is weak relatively nonexpansive multivalued mapping, we have
(1.9)ϕ(xn,z)≤ϕ(xn,q),∀z∈Tq,n=1,2,3,….
Therefore
(1.10)ϕ(q,z)=limn→∞ϕ(xn,z)≤limn→∞ϕ(xn,q)=ϕ(q,q)=0.
By Lemma 1.2, we have q=z, hence, Tq={q}, so q∈F(T). Therefore F(T) is closed. Next, we show F(T) is convex. Let x,y∈F(T), put z=tx+(1-t)y for any t∈(0,1). For w∈Tz, we have
(1.11)ϕ(z,w)=∥z∥2-2〈z,Jw〉+∥w∥2=∥z∥2-2〈tx+(1-t)y,Jw〉+∥w∥2=∥z∥2-2t〈x,Jw〉-2(1-t)〈y,Jw〉+∥w∥2=∥z∥2+tϕ(x,w)+(1-t)ϕ(y,w)-t∥x∥2-(1-t)∥y∥2≤∥z∥2+tϕ(x,z)+(1-t)ϕ(y,z)-t∥x∥2-(1-t)∥y∥2=∥z∥2-2〈tx+(1-t)y,Jz〉+∥z∥2=∥z∥2-2〈z,Jz〉+∥z∥2=ϕ(z,z)=0.
By Lemma 1.2, we have z=w, so z∈Tz, that is z∈F(T). Therefore, F(T) is convex. This completes the proof.Remark 1.6.
LetE be a strictly convex and smooth Banach space, and C a nonempty closed convex subset of E. Suppose T:C→C is a weak relatively nonexpansive multivalued mapping. If p∈F(T), then T(p)={p}.


## 2. An Example of Weak Relatively Nonexpansive Multivalued Mapping

Next, we give an example which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping.Example 2.1.
LetE=l2 and
(2.1)x0=(1,0,0,0,…)x1=(1,1,0,0,…)x2=(1,0,1,0,0,…)x3=(1,0,0,1,0,0,…)⋯xn=(1,0,1,0,0,…,0,1,0,0,0,…)⋯
It is obvious that {xn}n=1∞ converges weakly to x0. On the other hand, we have ∥xn-xm∥=2 for any n≠m. Define a mapping T:E→2E as follows:
(2.2)T(x)={{y=txn:nn+2≤t≤nn+1},ifx=xn(∃n≥1),-x,ifx≠xn(∀n≥1).
It is also obvious that F(T)={0} and
(2.3)∥0-y∥=∥y∥≤∥x∥=∥0-x∥,∀x∈E,y∈Tx.
Since E=L2 is a Hilbert space, we have
(2.4)ϕ(0,y)=∥0-y∥2=∥y∥2≤∥x∥2=∥0-x∥2=ϕ(0,x),∀x∈E,y∈Tx.
Next, we prove F^(T)=F(T). In fact, that for any strong convergent sequence {zn}⊂E such that zn→z0 and d(zn,Tzn)→0 as n→∞, then there exist sufficiently large nature number N such that zn≠xm, for any n,m>N. Then Tzn=-zn for n>N, it follows from
(2.5)∥zn-Tzn∥=d(zn,Tzn)→0
that 2zn→0 and hence zn→z0=0, this implies z0∈F(T), so that F^(T)=F(T). Then T is a weak relatively multivalued nonexpansive mapping.
We second claim thatT is not relatively multivalued nonexpansive mapping. In fact, that xn⇀x0 and
(2.6)d(xn,Txn)≤(nn+1-nn+2)→0
as n→∞ hold, but x0∈¯F(T).


## 3. Strong Convergence of Monotone Hybrid Algorithm

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 T:C→C be a weak relatively multivalued nonexpansive mapping such that F(T)≠∅. Assume that {αn}n=0∞,{βn}n=0∞,{γn}n=0∞ are three sequences in [0,1] such that αn+βn+γn=1, limn→∞αn=0 and 0<γ≤γn≤1 for some constant γ∈(0,1). Define a sequence {xn} in C by the following algorithm:
(3.1)x0∈Carbitrarily,yn=J-1(αnJx0+βnJxn+γnJzn),zn∈TxnCn={z∈Cn-1:ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)},n≥1,C0=C,xn+1=ΠCnx0,
then {xn} converges to q=ΠF(T)x0.Proof.
We first show thatCn is closed and convex for all n≥0. From the definitions of Cn, it is obvious that Cn is closed for all n≥0. Next, we prove that Cn is convex for all n≥0. Since
(3.2)ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)
is equivalent to
(3.3)2〈z,(1-αn)Jxn+αnJx0-Jyn〉≤(1-αn)∥xn∥2+αn∥x0∥2.
It is easy to get that Cn is convex for all n≥0.
Next, we show thatF(T)⊂Cn for all n≥0. Indeed, for each p∈F(T), we have
(3.4)ϕ(p,yn)=ϕ(p,J-1(αnJx0+βnJxn+γnJzn))=∥p∥2-2〈p,αnJx0+βnJxn+γnJzn〉+∥αnJx0+βnJxn+γnJzn)∥2≤∥p∥2-2αn〈p,Jx0〉-2βn〈p,Jxn〉-2γn〈p,Jzn〉+αn∥x0∥2+βn∥xn∥+γn∥zn∥2≤αnϕ(p,x0)+βnϕ(p,xn)+γnϕ(p,zn)≤αnϕ(p,x0)+βnϕ(p,xn)+γnϕ(p,xn)≤αnϕ(p,x0)+(1-αn)ϕ(p,xn).
So, p∈Cn, which implies that F(T)⊂Cn for all n≥0.
Sincexn+1=ΠCnx0 and Cn⊂Cn-1, then we get
(3.5)ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥0.
Therefore, {ϕ(xn,x0)} is nondecreasing. On the other hand, by Lemma 1.4 we have
(3.6)ϕ(xn,x0)=ϕ(ΠCn-1x0,x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),
for all p∈F(T)⊂Cn-1 and for all n≥1. Therefore, ϕ(xn,x0) is also bounded. This together with (3.5) implies that the limit of {ϕ(xn,x0)} exists. Put
(3.7)limn→∞ϕ(xn,x0)=d.
From Lemma 1.4, we have, for any positive integer m, that
(3.8)ϕ(xn+m,xn+1)=ϕ(xn+m,ΠCnx0)≤ϕ(xn+m,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+m,x0)-ϕ(xn+1,x0),
for all n≥0. This together with (3.7) implies that
(3.9)limn→∞ϕ(xn+m,xn+1)=0
uniformly for all m, holds. By using Lemma 1.2, we get that
(3.10)limn→∞∥xn+m-xn+1∥=0
uniformly for all m, holds. Then {xn} is a Cauchy sequence, therefore there exists a point p∈C such that xn→p.
Sincexn+1=ΠCnx0∈Cn, from the definition of Cn, we have
(3.11)ϕ(xn+1,yn)≤(1-αn)ϕ(xn+1,xn)+αnϕ(xn+1,x0).
This together with (3.9) and limn→∞αn=0 implies that
(3.12)limn→∞ϕ(xn+1,yn)=0.
Therefore, by using Lemma 1.2, we obtain
(3.13)limn→∞∥xn+1-yn∥=0.
Since j is uniformly norm-to-norm continuous on bounded sets, then we have
(3.14)limn→∞∥Jxn+1-Jyn∥=limn→∞∥Jxn+1-Jxn∥=0.
Noticing that
(3.15)∥Jxn+1-Jyn∥=∥Jxn+1-(αnJx0+βnJxn+γnJzn)∥=∥αn(Jxn+1-Jx0)+βn(Jxn+1-Jxn)+γn(Jxn+1-Jzn)∥≥γn∥Jxn+1-Jzn∥-αn∥Jxn+1-Jx0∥-βn∥Jxn+1-Jxn∥,
which leads to
(3.16)∥Jxn+1-Jzn∥≤1γn(∥Jxn+1-Jyn∥+αn∥Jx0-Jxn+1∥+βn∥Jxn+1-Jxn∥).
From (3.14) and limn→∞αn=0,0<γ≤γn≤1, we obtain
(3.17)limn→∞∥Jxn+1-Jzn∥=0.
Since J-1 is also uniformly norm-to-norm continuous on bounded sets, then we obtain
(3.18)limn→∞∥xn+1-zn∥=0.
Observe that
(3.19)∥xn-zn∥≤∥xn+1-xn∥+∥xn+1-zn∥.
It follows from (3.10) and (3.18) that
(3.20)∥zn-xn∥→0,asn→∞,(3.21)d(xn,Txn)→0,asn→∞.
Since we have proved that xn→p, which together with (3.21) and that, T is weak relatively multivalued nonexpansive mapping, implies that p∈F(T).
Finally, we prove thatp=ΠF(T)x0. From Lemma 1.4, we have
(3.22)ϕ(p,ΠF(T)x0)+ϕ(ΠF(T)x0,x0)≤ϕ(p,x0).
On the other hand, since xn+1=ΠCnx0 and F(T)⊂Cn, for all n. Also from Lemma 1.4, we have
(3.23)ϕ(ΠF(T)x0,xn+1)+ϕ(xn+1,x0)≤ϕ(ΠF(T)x0,x0).
By the definition of ϕ(x,y), we know that
(3.24)limn→∞ϕ(xn+1,x0)=ϕ(p,x0).
Combining (3.22), (3.23), and (3.24), we know that ϕ(p,x0)=ϕ(ΠF(T)x0,x0). Therefore, it follows from the uniqueness of ΠF(T)x0 that p=ΠF(T)x0. This completes the proof.Whenαn≡0 in Theorem 3.1, we obtain the following result.Theorem 3.2.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, and let T:C→C be a weak relatively multivalued nonexpansive mapping such that F(T)≠∅. Assume that {αn}n=0∞ is a sequences in [0,1] such that 0≤αn≤α<1 for some constant α∈(0,1). Define a sequence {xn} in C by the following algorithm:
(3.25)x0∈Carbitrarily,yn=J-1(αnJxn+(1-αn)Jzn),zn∈TxnCn={z∈Cn-1:ϕ(z,yn)≤ϕ(z,xn)},n≥1,C0=C,xn+1=ΠCnx0,
then {xn} converges to q=ΠF(T)x0.


## 4. Applications for Maximal Monotone Operators

In this section, we apply the above results to prove some strong convergence theorem concerning maximal monotone operators in a Banach spaceE.LetA be a multivalued 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
(4.1)〈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 its 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 4.1 (see Rockafellar [10]).
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 4.1 and strict convexity of E, we obtain that for every r>0 and x∈E, there exists a unique xr such that
(4.2)Jx∈Jxr+rAxr.
Then we can define a single valued mapping  Jr:E→D(A) by Jr=(J+rA)-1J and such a Jr is called the resolvent of A. We know that A-10=F(Jr) for all r>0, see [9, 11] for more details. Using Theorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space. Such a problem has been also studied in [1, 7, 10–20].Theorem 4.2.
LetE be a uniformly convex and uniformly smooth Banach space, let A be a maximal monotone operator from E to E* with A-10≠∅, and let JM:E→2E be a multivalued mapping defined as follows:
(4.3)JM:x↦{y∈E:y=Jrx,r∈M},x∈E,
where M is a set of real numbers such that infM>0. JM is called the multivalued resolvent of A. Then JM is a weak relatively nonexpansive multivalued mapping.Proof.
Since(4.4)F(JM)={p∈E:Jrp=p,r∈M}={p∈E:p∈A-10,r∈M}=A-10,
so that F(JM)≠∅. Next, we show
(4.5)ϕ(p,z)≤ϕ(p,x),∀p∈F(JM),∀x∈E,∀z∈JMx.
From the monotonicity of A, we have
(4.6)ϕ(p,z)=∥p∥2-2〈p,Jz〉+∥z∥2=∥p∥2+2〈p,Jx-Jz-Jx〉+∥Jrw∥2=∥p∥2+2〈p,Jx-Jz〉-2〈p,Jx〉+∥z∥2=∥p∥2-2〈z-p-z,Jx-Jz-Jx〉-2〈p,Jx〉+∥z∥2=∥p∥2-2〈z-p,Jx-Jz-Jx〉+2〈z,Jx-Jz〉-2〈p,Jx〉+∥z∥2≤∥p∥2+2〈z,Jx-Jz〉-2〈p,Jx〉+∥z∥2=∥p∥2-2〈p,Jx〉+∥x∥2-∥z∥2+2〈z,Jx〉-∥w∥2=ϕ(p,x)-ϕ(z,x)≤ϕ(p,x).
Finally, we show F~(JM)=F(JM). Observe that F~(JM)⊇F(JM) is obvious. Next, we show F~(JM)⊆F(JM). Let {zn}⊂E be a sequence such that zn→p and limn→∞d(zn,JMzn)=0. There exist sequences {yn} in E and {rn}⊆M such that
(4.7)limn→∞∥zn-yn∥=0,yn=Jrnzn.
Since J is uniformly norm-to-norm continuous on bounded sets, we obtain
(4.8)1rn(Jzn-Jyn)→0.
It follows from
(4.9)1rn(Jzn-Jyn)∈AJrnzn
and the monotonicity of A that
(4.10)〈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(JM). Hence JM is a weak relatively nonexpansive multivalued mapping. This completes the proof.By using Theorems3.1 and 4.2, we directly obtain the following result.Theorem 4.3.
LetE be a uniformly convex and uniformly smooth Banach space, let A be a maximal monotone operator from E to E* with A-10≠∅, let JM be a multivalued resolvent of A, where infM>0, and let {αn},{βn},{γn} be three sequences of real numbers such that αn→0 and 0<γ≤γn≤1 for some constant γ∈(0,1). Define a sequence {xn} of C as follows:
(4.11)x0∈Carbitrarily,yn=J-1(αnJx0+βnJxn+γnJzn),zn∈JMxnCn={z∈Cn-1:ϕ(z,yn)≤(1-αn)ϕ(z,xn)+αnϕ(z,x0)},n≥1,C0=C,xn+1=ΠCnx0,
where J is the duality mapping on E. Then {xn} converges strongly to ΠA-10x0, where ΠA-10 is the generalized projection from E onto A-10.



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