# Mixed Approximation for Nonexpansive Mappings in Banach Spaces

Authors: Qing-Bang Zhang; Fu-Quan Xia; Ming-Jie Liu
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.


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## Body

## 1. Introduction and Preliminaries

LetE be a real Banach space with norm ∥·∥, E* denote the dual space of E, and 〈·,·〉 denote the generalized duality pairing. Let J:E→2E* denote the normalized duality mapping defined by(1.1)J(x)={f*∈X*:〈x,f*〉=∥f*∥∥x∥,∥f*∥=∥x∥},∀x∈E.
It is well known that the following results: ∀x,y∈E, ∀j(x+y)∈J(x+y), ∀j(x)∈J(x),(1.2)∥x∥2+2〈y,j(x)〉≤∥x+y∥2≤∥x∥2+〈y,j(x+y)〉.LetU={x∈E:∥x∥=1} be the unit sphere of Banach space E, the norm of E is said to be Gâteaux differentiable if the limit(1.3)limt→0∥x+ty∥-∥x∥t
exists for each x,y∈U. Such an E is called a smooth Banach space. The norm of Banach space E is said to be uniformly Gâteaux differentiable if for each y∈U, the limit limt→0((∥x+ty∥-∥x∥)/t) is attained uniformly for x∈U. A Banach space E is said to be strictly convex if(1.4)∥x∥=∥y∥=1,x≠yimplies∥x+y∥2<1,
to be uniformly convex if for all ε∈[0,2], ∃δε>0 such that(1.5)∥x∥=∥y∥=1,∥x-y∥≥εimplies∥x+y∥2<1-δε.It is well known that (see [1, 2]): (1) if E has a uniformly Gâteaux differentiable norm, then J is norm-to-weak* continuous on bounded set of E. (2) If a Banach space E admits a sequentially continuous duality mapping J from weak topology to weak star topology, then the duality mapping J is single-valued. (3) Each uniformly convex Banach space E is reflexive and strictly convex and has fixed point property for nonexpansive self-mappings; every uniformly smooth Banach space E is a reflexive Banach space with a uniformly Gâteaux differentiable norm and has fixed point property for nonexpansive self-mappings.LetC be a nonempty closed convex subset of a Banach space E. If D is a nonempty subset of C, then a mapping P:C→D is said to be a retraction if Px=x for all x∈D. A mapping P:C→D is said to be a sunny if P(Px+t(x-Px))=Px,∀x∈C where t>0. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D. For more details, see [2]. Note that every closed convex subset of a Hilbert space is a nonexpansive retract. In the sequel, we always take P to denote the sunny nonexpansive retraction of E onto C.LetT:C→C be a nonexpansive mapping; for a sequence {αn}⊂(0,1) and a fixed contractive mapping f:C→C, the sequence {xn}, iteratively defined in C by(1.6)xn+1=αnf(xn)+(1-αn)Txn,n=0,1,2,…,
is said to be viscosity approximation. If f(xn)=y0 for a given y0∈C, it is called Halpern approximation which was first introduced by Halpern [3] in 1967. Under the following assumption:(1.7)(i)limn→∞αn=0,∑n=0∞αn=∞,(ii)∑n=0∞|αn+1-αn|<∞(orlimn→∞αn+1αn=1),
Xu [4] proved the strong convergence of {xn} to a fixed point of T in Hilbert spaces and in uniformly smooth Banach spaces in 2004. In [3], Halpern pointed out that the condition (i) is necessary for the convergence of the Halpern approximation to a fixed point of T. At the same time, he put forth the following open problem: is the condition (i) a sufficient condition for the convergence of the Halpern approximation to a fixed point ofT? which was put forward by Reich in [5] also. In order to answer the open question, many authors have done extensively some works; see [6–11] and the references therein. In [7–9], the strong convergence of the Halpern approximation depends on the convergence of the path xt=ty0+(1-t)Txt(t∈(0,1)). In [6], Song got rid of the dependence on the convergence of the path xt, and proved the convergence of the Halpern approximation under the assumptions for {αn} and T as follows:(1.8)(iii)∑n=0∞αn=∞,∑n=0∞|αn+1-αn|<∞(orlimn→∞αn+1αn=1),(iv)Kmin⋂F(T)≠∅,whereKmin={z∈C:𝒰n∥xn-z∥2=infy∈C𝒰n∥xn-y∥2} and 𝒰n is a Banach limit. Recently, many authors have studied extensively the problem of approximating a fixed point of nonexpansive nonself-mappings T:C→E in a Banach space, by using the Halpern type iteration (see [12–16]) and the viscosity type iteration (see [17–23]).In this work, on one hand, we will prove the strong convergence of the mixed viscosity iterative scheme, which is introduced as follows: for any chosenx0∈C,(1.9)yn=(1-θn)xn+θnTxn,xn+1=αnf(xn)+(1-αn)[(1-βn)xn+βnTyn]),∀n≥0,
where f:C→C is a fixed contractive mapping, {αn}, {βn}, and {θn} are the real number sequences in (0,1), to a fixed point of the nonexpansive mapping T:C→C in a real Banach space with uniformly Gâteaux differentiable norm under the following conditions:(C1)limn→∞αn=0,∑n=0∞αn=∞,0<lim infn→∞βn≤lim supn→∞βn<1,limn→∞θn=0,(1.10)(iv)′K̃min⋂F(T)≠∅,
where K̃min={z∈K:𝒰n∥xn-z∥2=infy∈C𝒰n∥xn-y∥2}, 𝒰n is a Banach limit, and {xn} is defined by (1.9). On the other hand, we will show that the condition (iv)′ is not necessary for proving the strong convergence of the mixed viscosity iterative scheme. As the applications, we will show some results about mixed viscosity type approximation and Halpern type approximation for nonexpansive nonself-mappings also. Our theorems complement and generalize the corresponding results in [8, 9, 16, 24–26].Now, we recall the following lemmas for proving our theorems firstly.Lemma 1.1 (see [27]).
Let{xn} and {yn} be bounded sequences in a Banach space E, {tn}⊂[0,1] a sequence satisfying
(1.11)0<liminfn→∞tn≤lim supn→∞tn<1.
Suppose that xn+1=tnyn+(1-tn)xn, for all n=0,1,2,…, and
(1.12)lim supn→∞(∥yn+1-yn∥-∥xn+1-xn∥)≤0.
Then limn→∞∥xn-yn∥=0.Lemma 1.2 (see [28]).
Let{an}, {bn} and, {cn} be sequences of nonnegative real numbers such that an+1≤(1-αn)an+αnbn+cn,∀n≥0. If {αn}⊂[0,1],
(1.13)∑n=1∞αn=∞,lim supn→∞bn≤0,∑n=1∞cn<∞,
then limn→∞an=0.


## 2. Main Results

In this section, the mixed viscosity iterations for a contractive self-mappingf for approximating to a fixed point of nonexpansive mapping T:C→C are studied in a real Banach space.Theorem 2.1.
LetC be a nonempty closed convex subset of a real Banach space E. Let T:C→C be a nonexpansive mapping with F(T)={x∈C:Tx=x}≠∅ and f:C→C a fixed contractive mapping with the contractive coefficient k∈(0,1). If the sequences {αn}, {βn}, and {θn} in (0,1) satisfy (C1) and for any x0∈C, the sequence {xn} is defined as follows:
(2.1)yn=(1-θn)xn+θnTxn,xn+1=αnf(xn)+(1-αn)[(1-βn)xn+βnTyn]),∀n≥0;
then we obtain the following:(1)
{xn} is bounded;(2)
limn→∞∥xn+1-xn∥=0 and limn→∞∥xn-Txn∥=0.Proof.
First we show that{xn} is bounded. Now let z∈F(T), then
(2.2)∥yn-z∥=∥(1-θn)(xn-z)+θn(Txn-Tz)∥≤∥xn-z∥;∥xn+1-z∥=∥αnf(xn)+(1-αn)[(1-βn)xn+βnTyn]-z∥,≤αn∥(f(xn)-f(z))+(f(z)-z)∥+(1-αn)∥(1-βn)(xn-z)+βn(Tyn-z)∥≤αnk∥xn-z∥+αn∥f(z)-z∥+(1-αn)βn∥Tyn-Tz∥+(1-αn)(1-βn)∥xn-z∥≤αnk∥xn-z∥+αn∥f(z)-z∥+(1-αn)βn∥yn-z∥+(1-αn)(1-βn)∥xn-z∥≤αn∥f(z)-z∥+(1-αn(1-k))∥xn-z∥≤max{11-k∥f(z)-z∥,∥xn-z∥}≤max{11-k∥f(z)-z∥,∥x0-z∥}.
Therefore, {xn} is bounded, so are {Txn}, {yn}, {Tyn}, and {f(xn)}. Next, we show that
(2.3)limn→∞∥xn+1-xn∥=0.
For alln≥0, let
(2.4)xn+1=γnxn+(1-γn)un,
where γn=(1-αn)(1-βn)∈(0,1), then
(2.5)0<liminfn→∞γn≤lim supn→∞γn<1.
Note that
(2.6)un+1-un=xn+2-γn+1xn+11-γn+1-xn+1-γnxn1-γn=αn+1f(xn+1)+(1-αn+1)[(1-βn+1)xn+1+βn+1Tyn+1]-γn+1xn+11-γn+1-αnf(xn)+(1-αn)[(1-βn)xn+βnTyn]-γnxn1-γn=αn+11-γn+1(f(xn+1)+Tyn+1)-αn1-γn(f(xn)+Tyn)-(Tyn+1-Tyn),(2.7)∥Tyn+1-Tyn∥≤∥yn+1-yn∥≤∥xn+1-xn∥+θn+1∥xn+1-Txn+1∥+θn∥xn-Txn∥.
Hence, we have that
(2.8)∥un+1-un∥≤αn+11-γn+1∥f(xn+1)+Tyn+1∥+αn1-γn∥f(xn)+Tyn∥+∥xn+1-xn∥+θn+1∥xn+1-Txn+1∥+θn∥xn-Txn∥.
Then it follows from the boundedness of {xn}, {Txn}, {Tyn}, and {f(xn)}, and (C1) that
(2.9)lim supn→∞(∥un+1-un∥-∥xn+1-xn∥)≤0.
It follow from (2.9) and Lemma 1.1 that
(2.10)limn→∞∥xn-un∥=0,(2.11)limn→∞∥xn+1-xn∥=0.
Since
(2.12)xn+1-xn=αn(f(xn)-xn)+(1-αn)βn(Tyn-xn),xn-yn=θn(xn-Txn),(2.13)(1-αn)βn∥Tyn-xn∥≤∥xn+1-xn∥+αn∥f(xn)-xn∥,
from (C1), (2.11) and, the boundedness of {xn}, {Txn}, and {f(xn)}, we have
(2.14)limn→∞∥xn-Tyn∥=0,limn→∞∥xn-yn∥=0.
From
(2.15)∥xn+1-Txn∥≤∥Tyn-Txn∥+∥xn-Tyn∥+∥xn+1-xn∥≤∥yn-xn∥+∥xn-Tyn∥+∥xn+1-xn∥,
we obtain
(2.16)limn→∞∥xn+1-Txn∥=0.
Therefore, limn→∞∥xn-Txn∥=0. This completes the proof.Proposition 2.2 (see [18]).
LetC be a nonempty closed convex subset of a real Banach space E which has uniformly Gâteaux differentiable norm. Suppose that {xn} is a bounded sequence of E such that limn→∞∥xn+1-xn∥=0 and 𝒰n is a Banach limit. If z∈C such that 𝒰n∥xn-z∥2=infy∈C𝒰n∥xn-y∥2, then
(2.17)lim supn→∞〈y-z,j(xn-z)〉≤0,∀y∈C.Let{xn} be defined by (2.1) and αn∈(0,1), it follows from Theorem 2.1 that {xn} is bounded. Let
(2.18)φ(y)=𝒰n∥xn-y∥2∀y∈C,
then φ(y) is convex and continuous. If E is reflexive, there exists z∈C such that φ(z)=infy∈Cφ(y) (see [2], Theorem 1.3.11). Let(2.19)K̃min={z∈C:φ(z)=infy∈Cφ(y)},
then K̃min≠∅ is a closed convex subset in a reflexive Banach space E.Theorem 2.3.
LetC be a nonempty closed convex subset of a real Banach space E which has uniformly Gâteaux differentiable norm. Suppose that T:C→C is a nonexpansive mapping with F(T)≠∅, f is a fixed contractive mapping with the contractive coefficient k∈(0,1), and the sequences {αn}, {βn} and {θn} in (0,1) satisfy (C1). If K̃min⋂F(T)≠∅, then the sequence {xn} defined by (2.1) converges strongly to a fixed point of T as n→∞.Proof.
Takez∈K̃min⋂F(T). It follows from Theorem 2.1 that {xn} is bounded and limn→∞∥xn+1-xn∥=0. For f(z)∈C, by Proposition 2.2, we have that
(2.20)lim supn→∞〈f(z)-z,j(xn-z)〉≤0.
Next, we show thatlimn→∞xn=z. Since
(2.21)xn+1-(αnf(xn)+(1-αn)z)=(xn+1-z)-αn(f(xn)-z),∥xn+1-(αnf(xn)+(1-αn)z)∥≤∥(1-αn)[(1-βn)xn+βnTyn]-(1-αn)z∥=∥(1-αn)[(1-βn)(xn-z)+βn(Tyn-Tz)]∥≤(1-αn)∥xn-z∥,
then
(2.22)∥xn+1-z∥2=〈xn+1-(αnf(xn)+(1-αn)z),j(xn+1-z)〉+αn〈f(xn)-z,j(xn+1-z)〉≤∥xn+1-(αnf(xn)+(1-αn)z)∥∥xn+1-z∥+αn(〈f(xn)-f(z),j(xn+1-z)〉+〈f(z)-z,j(xn+1-z)〉)≤12(1-αn)(∥xn-z∥2+∥xn+1-z∥2)+12αn(∥f(xn)-f(z)∥2+∥xn+1-z∥2)+αn〈f(z)-z,j(xn+1-z)〉≤12(1-αn(1-k2))∥xn-z∥2+12∥xn+1-z∥2+αn〈f(z)-z,j(xn+1-z)〉.
Hence,
(2.23)∥xn+1-z∥2≤(1-αn(1-k2))∥xn-z∥2+2αn〈f(z)-z,j(xn+1-z)〉.
Let an=∥xn-z∥2 and bn=(2/(1-k2))〈y-z,j(xn+1-z)〉; it follows from Lemma 1.2 that {xn} converges strongly to z∈C. This completes the proof.Proposition 2.4.
LetE be a real reflexive Banach space with uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, T:C→C a nonexpansive mapping with F(T)≠∅, and {xn} defined by (2.1). Then K̃min⋂F(T)≠∅.Proof.
From the reflexivity ofE and the definition of K̃min, it follows that K̃min is a nonempty closed convex subset of E. By Theorem 2.1, we know that limn→∞∥xn+1-Txn∥=0.
Claim thatT(K̃min)⊂K̃min. Indeed, for any x∈K̃min, we have
(2.24)φ(Tx)=𝒰n∥xn-Tx∥2=𝒰n∥xn+1-Tx∥2≤𝒰n(∥xn+1-Txn∥+∥Txn-Tx∥)2≤𝒰n∥xn-x∥2=φ(x).
Therefore, Tx∈K̃min and T(K̃min)⊂K̃min.
SinceF(T)≠∅, there exists unique u∈K̃min such that ∥z-u∥=infx∈K̃min∥z-x∥, for all z∈F(T). By Tz=z and Tu∈K̃min, we have
(2.25)∥z-Tu∥=∥Tz-Tu∥≤∥z-u∥.
Hence u=Tu by the uniqueness of u∈K̃min. Thus K̃min⋂F(T)≠∅.By the above results, we can obtain the following theorem.Theorem 2.5.
LetE be a real reflexive Banach space with uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, T:C→C a nonexpansive mapping with F(T)≠∅, and f a fixed contractive mapping with the contractive coefficient k∈(0,1). If the sequences {αn}, {βn}, and {θn} in (0,1) satisfy (C1), then the sequence {xn} defined by (2.1) converges strongly to a fixed point of T.Remark 2.6.
Theorem2.5 shakes off the assumption limn→∞∥Tyn-yn∥=0 in [26] and extends Theorem 1 in [24] shown in uniformly smooth Banach spaces.Theorem 2.7.
LetE be a real strictly convex Banach space E with uniformly Gâteaux differentiable norm and C a nonempty closed convex subset of E which is a sunny nonexpansive retract of E. Let T:C→E be a nonexpansive nonself-mapping with F(T)={x∈C:Tx=x}≠∅ and f:C→C a fixed contractive mapping with the contractive coefficient k∈(0,1). Suppose that the sequences {αn}, {βn}, and {θn} in (0,1) satisfy (C1), C is a sunny nonexpansive retract of E, and the sequence {xn} is defined as follows:
(2.26)foranyx0∈C,yn=(1-θn)xn+θnPTxn,xn+1=αnf(xn)+(1-αn)[(1-βn)xn+βnPTyn]),∀n≥0;
If {z∈C:𝒰n∥xn-z∥2=infy∈C𝒰n∥xn-y∥2}⋂F(T)≠∅, then the sequence {xn} converges strongly to a fixed point of T as n→∞.Proof.
It follows from [12, Lemmas 3.1 and 3.3] that F(T)=F(PT), and then F(PT)≠∅. By replacing T by PT in Theorem 2.3, we can show that the conclusion holds.Theorem 2.8.
LetE be a real reflexive and strictly convex Banach space with uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of a real Banach space E which is a sunny nonexpansive retract of E, T:C→E a nonexpansive nonself-mapping with F(T)≠∅, and f a fixed contractive mapping with the contractive coefficient k∈(0,1). If the sequences {αn}, {βn} and {θn} in (0,1) satisfy (C1), then the sequence {xn} defined by (2.26) converges strongly to a fixed point of T.Proof.
It follows from [12, Lemmas 3.1 and 3.3] that F(T)=F(PT), and then F(PT)≠∅. By replacing T by PT in Theorem 2.5, we can show that the conclusion holds.Corollary 2.9.
LetE be a real uniformly convex Banach space with uniformly Gâteaux differentiable norm, C, T, f, {αn}, and {βn} as Theorem 2.8. Then the sequence {xn} defined by (2.1) converges strongly to a fixed point of T.


## 3. Some Applications

In this section, we introduce the following Halpern type approximation: for the givenx0,u0∈C, the sequence {xn} is defined by(3.1)yn=(1-θn)xn+θnTxn,xn+1=αnu0+(1-αn)[(1-βn)xn+βnTyn]),∀n≥0,
and show some results about Halpern type approximation for nonexpansive mappings, which generalize and improve some known conclusions.DefineCmin={z∈C:φ(z)=infy∈Cφ(y)}, where φ(y)=𝒰n∥xn-y∥2 for all y∈C and {xn} is defined by (3.1).Theorem 3.1.
LetC be a nonempty closed convex subset of a real Banach space E which has uniformly Gâteaux differentiable norm. Suppose that T:C→C is a nonexpansive mapping with F(T)≠∅, f is a fixed contractive mapping with the contractive coefficient k∈(0,1), and the sequences {αn}, {βn}, and {θn} in (0,1) satisfy (C1). If Cmin⋂F(T)≠∅, then the sequence {xn} defined by (3.1) converges strongly to a fixed point of T as n→∞.Proof.
First we show that{xn} is bounded. Now let z∈F(T), then
(3.2)∥yn-z∥=∥(1-θn)(xn-z)+θn(Txn-Tz)∥≤∥xn-z∥,∥xn+1-z∥=∥(αnu0+(1-αn)[(1-βn)xn+βnTyn])-z∥≤αn∥u0-z∥+(1-αn)∥(1-βn)(xn-z)+βn(Tyn-z)∥≤αn∥u0-z∥+(1-αn)βn∥Tyn-Tz∥+(1-αn)(1-βn)∥xn-z∥≤αn∥u0-z∥+(1-αn)∥xn-z∥≤max{∥u0-z∥,∥x0-z∥}.
Therefore, {xn} is bounded. Then Cmin is well defined.
In the proofs of Theorems2.1 and 2.3, take f(x)=u0 for all x∈C; we can show similarly that the conclusion holds.Remark 3.2.
Ifθn=0 for all n=0,1,2,…, Theorem 3.1 weakens the condition of {βn} of Theorem 4.1 in [6].Theorem 3.3.
LetC be a nonempty closed convex subset of a real reflexive Banach space E which has uniformly Gâteaux differentiable norm. Suppose that T:C→C is a nonexpansive mapping with F(T)≠∅ and the sequences {αn}, {βn}, and {θn} in (0,1) satisfy (C1). Then the sequence {xn} defined by (3.1) converges strongly to a fixed point of T as n→∞.Proof.
Takef(x)=u0 for all x∈C in Theorem 2.5; it is easy to show that the conclusion holds.Remark 3.4.
Ifθn=0 for all n=0,1,2,…, Theorem 3.3 gets rid of the dependence on the implicit anchor-like continuous path zt=ty+(1-t)Tzt in Suzuki's Theorem 3 in [9] and Theorem 3.1 of C. E. Chidume and C. O. Chidume [8]. It also complements and generalizes [25, Theorem 1], which is proved in uniformly smooth Banach spaces.Theorem 3.5.
LetC be a nonempty closed convex subset of a real reflexive and strictly convex Banach space E which has uniformly Gâteaux differentiable norm. Suppose that C is a sunny nonexpansive retract of E, T:C→E is a nonexpansive nonself-mapping with F(T)≠∅, f is a fixed contractive mapping with the contractive coefficient k∈(0,1), the sequences {αn}, {βn}, and {θn} in (0,1) satisfy (C1), and the sequence {xn} is defined as follows,
(3.3)forthegivenx0,u0∈C,yn=(1-θn)xn+θnPTxn,xn+1=αnu0+(1-αn)[(1-βn)xn+βnPTyn]),∀n≥0.
Then the sequence {xn} converges strongly to a fixed point of T as n→∞.Remark 3.6.
Ifθn=0 for all n=0,1,2,…, Theorem 3.3 improves and generalizes Theorem 3.2 in [16]: it gets rid of the restriction of δ∈(0,1) and dependence on the implicit anchor-like continuous path zt=ty+(1-t)PTzt.



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