# Strong Convergence of Viscosity Methods for Continuous Pseudocontractions in Banach Spaces

Authors: Filomena Cianciaruso; Giuseppe Marino; Luigi Muglia; Haiyun Zhou
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

We define a viscosity method for continuous pseudocontractive
mappings defined on closed and convex subsets of reflexive Banach spaces with
a uniformly Gâteaux differentiable norm. We prove the convergence of these
schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).


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## Body

## 1. Introduction

LetX be  a real Banach space  and let J be the normalized duality mapping from X into 2X* defined by(1.1)Jx={j(x)∈X*:〈x,j(x)〉=∥x∥2,∥j(x)∥=∥x∥},where X* denotes the dual space of X and 〈⋅,⋅〉 the generalized duality pairing between X and X*.Recall that if(1.2)limt→0∥x+th∥−∥x∥texists for each x and h on the unit sphere SX of X,
the norm of X is Gâteaux differentiable. Moreover, if
for each h∈SX the limit defined by (1.2) is uniformly
attained for x∈SX,
we say that the norm of X is uniformly Gâteaux differentiable.Definition 1.1.
A
mappingT:D(T)→X is said to be k-pseudocontractive (k∈ℝ) if, for every x,y∈D(T),
there exist some j(x−y)∈J(x−y) such that(1.3)〈Tx−Ty,j(x−y)〉≤k∥x−y∥2.In the inequality (1.3), if 0<k<1,
we say that T is strongly pseudocontractive. For k=1,T  is called pseudocontractive mapping.Among classes of nonlinear mappings, the class of
pseudocontractions is probably one of the most important classes of mappings.
This happens because of the corresponding relation between the classes of
pseudocontractions and accretive operators. In fact, a mappingA:D(A)→X is accretive (i.e., 〈Ax−Ay,j(x−y)〉≥0,
for all x,y∈D(A)) if and only if T:=I−A is pseudocontractive.LetT,V be two opportune mappings from C to C,
where C is a closed and convex subset of a Banach
space X.
Consider the variational inequality problem of
finding a fixed point x* of T, with respect to another mapping V,
to satisfy the inequality(1.4)〈x*−Vx*,j(y−x*)〉≥0,∀y∈Fix(T).A particular case occurs when V=f with f a ρ-contraction (i.e., ∥f(x)−f(y)∥≤ρ∥x−y∥ for all x,y∈C).
In this case, the method (implicit or explicit) that permits to solve the
variational inequality problem is known as viscosity approximation method. It
was first studied by Moudafi [1]
in Hilbert spaces and further developed by Xu [2] in more general setting.Next results, due to Morales [3] (2007), are the more general
results concerning the convergence of implicit viscosity methods for continuous
pseudocontractive mappings.In particular, the author studies the convergence of
the path defined as(1.5)xt=tf(xt)+(1−t)Txt,t∈(0,1),in more setting of Banach spaces
and in more large class of mappings f including the ρ-contraction mappings.Theorem 1.2 (see [3]).
LetC be a nonempty closed convex subset of a
reflexive Banach space X with a uniformly Gâteaux differentiable norm.
Let T and f:C→C be pseudocontractive and strongly
pseudocontractive continuous mappings, respectively. Suppose that every closed,
bounded, and convex subset of C has the fixed point property for nonexpansive
self-mappings. If the sets(1.6)E:={x∈C:Tx=λx+(1−λ)f(x)forsomeλ>1}and f(E) are bounded, then the path (xt)t∈(0,1) described by(1.7)xt=(1−t)Txt+tf(xt)strongly converges, as t→0,
to a fixed point q of T which is the unique solution of the
variational inequality(1.8)〈q−f(q),j(q−z)〉≥0∀z∈Fix(T).Corollary 1.3 (see [3]).
LetC be a nonempty closed convex subset of a
uniformly smooth Banach space X.
Let T,f:C→C be a pseudocontractive and strongly
pseudocontractive continuous mappings. If f is bounded and T admits at least a fixed point, then the path (xt)t∈(0,1) described by(1.9)xt=(1−t)Txt+tf(xt)strongly converges, as t→0,
to a fixed point q of T that is the unique solution of the variational
inequality(1.10)〈q−f(q),j(q−z)〉≥0,∀z∈Fix(T).Also in 2007, H. Zegeye et
al. in [4] proved a convergence theorem of viscosity
approximation methods for continuous pseudocontractive mappings in reflexive
and strictly convex Banach spaces.Theorem 1.4 (see [4]).
LetC be a nonempty closed and convex subset of a
real Banach space X reflexive, strictly convex that has uniformly
Gâteaux differentiable norm. Let T:C→C be a continuous pseudocontractive mapping and f:C→C be a ρ-contraction. Suppose further that Fix(T)≠∅.
Then, (xt)t∈(0,1) strongly converges, as t→0,
to a fixed point q of T that is the unique solution of the variational
inequality (1.8).Another interesting
implicit-type Halpern algorithm has been recently introduced by Yao, Liou, and
Chen in uniformly smooth Banach spaces.Theorem 1.5 (see [5]).
LetC be a closed and convex subset of a real
uniformly smooth Banach space X. Let T:C→C be a continuous pseudocontractive mapping. Let (αn)n∈ℕ,(βn)n∈ℕ,
and (γn)n∈ℕ be three real sequences in (0,1) satisfying the following conditions:
(i)
αn+βn+γn=1;(ii)
limn→+∞(αn/βn)=limn→+∞βn=0;(iii)
∑n∈ℕ(αn/(βn+αn))=+∞.Then, for
arbitrary initial value x0∈C and a fixed u∈C,
the sequence (xn)n∈ℕ defined by(1.11)xn=αnu+βnxn−1+γnTxnstrongly converges to a fixed
point of T.In our first result (Theorem2.1), we prove the
strong convergence of the viscosity implicit approximation method(1.12)xn=αnf(xn)+βnxn−1+γnTxn,where f:C→C is a continuous strongly pseudocontractive
mapping. This result has as particular case Theorem 1.5 when f is a constant mapping.On the other hand, on the idea of the implicit scheme
(1.11), Zhou in [6]
defines a Halpern explicit method for suitable continuous pseudocontractive
mappings. Fixing an element u∈C and an initial point x0=x00∈C,
he constructs elements (xnm)n∈N as follows:(1.13)xnm+1=αnu+βnxn+γnTxnm,m=0,1,2,….If for any n≥0 the continuous pseudocontractive mapping T admits an integer m that satisfies the following
condition:(1.14)∥Txnm+1−Txnm∥≤γn−1(1−γn)ϵn,then he defines iteratively a
sequence (xn)n≥0 as follows: called N(n) the least positive integer m satisfying (1.14),(1.15)xn+1=xn+10=xnN(n)+1=αnu+βnxn+γnTxnN(n),and he proves the convergence
for this explicit method. Of course if xnN(n)=xnN(n)+1,
one reobtains the implicit method (1.11).In our second theorem, we improve Zhou's result
[6] to the viscosity
setting. In both proofs, we use Morales's Theorem 1.2.Let us conclude this section by two lemmas that are
useful in many convergence results.Following the proof of Theorem 2.3 in [7], one can show
the following.Lemma 1.6.
LetC be a nonempty closed convex subset of a real
Banach space X with a uniformly Gâteaux differentiable norm,
let T,f:C→C be a pseudocontractive and strongly
pseudocontractive continuous mappings with Fix(T)≠∅.
Let (xn)n∈ℕ be a bounded sequence such that limn→+∞∥xn−Txn∥=0.
Define, for all t∈(0,1),(1.16)xt=tf(xt)+(1−t)Txt,and let us suppose that q=limt→0xt exists.
Then,(1.17)limsupn→+∞〈f(q)−q,j(xn−q)〉≤0.The following lemma on real sequences can be found in Liu [8].Lemma 1.7.
Let(an)n∈ℕ be a sequence of nonnegative real numbers
satisfying the following inequality:(1.18)an+1≤(1−tn)an+o(tn)+sn,∀n≥0,where (tn)n∈N is a sequence in ]0,1[ such that ∑n≥0tn=+∞ and (sn)n∈N is a summable sequence of positive numbers.
Then,(an)n∈ℕ converges to zero.


## 2. Convergence Results

In this
section, we prove the convergence's theorems on implicit and explicit viscosity
method.Theorem 2.1.
LetC be a nonempty closed convex subset of a
reflexive Banach space X with a uniformly Gâteaux differentiable norm.
Suppose that every closed, bounded, and convex subset of C has the fixed point property for nonexpansive
self-mappings. Let T:C→C be a continuous pseudocontractive mapping and
let f:C→C be a continuous strongly pseudocontractive
mapping (with constant 0<k<1) such that the sets(2.1)E:={x∈C:Tx=λx+(1−λ)f(x)forsomeλ>1}and f(E) are bounded.
Let(αn)n∈ℕ,(βn)n∈ℕ,
and (γn)n∈ℕ be three real sequences in (0,1) satisfying the following conditions:
(i)
αn+βn+γn=1;(ii)
limn→+∞αn=limn→+∞βn=0;(iii)
∑n∈ℕ(αn/(βn+(1−k)αn))=+∞.For arbitrary
initial point, x0∈C and a fixed n≥0,
we construct elements (xn)n∈ℕ as follows:(2.2)xn=αnf(xn)+βnxn−1+γnTxn.Then, (xn)n∈ℕ strongly converges to q,
where q∈Fix(T) is the unique solution of (1.8)(2.3)〈q−f(q),j(q−p)〉≥0,∀p∈Fix(T).Proof.
First of all, from [3], it follows that Fix(T)≠∅.
Now, we verify that the sequence(xn)n∈ℕ exists.
We prove that, for fixedα,β,γ∈(0,1) with α+β+γ=1 and z∈C,
the map(2.4)Sx=αf(x)+βz+γTxhas a unique fixed point. By
Deimling [9], it is
enough to show that S:C→C is strongly pseudocontractive and continuous.
Now,(2.5)〈Sx−Sy,j(x−y)〉=〈α(f(x)−f(y))+γ(Tx−Ty),j(x−y)〉=α〈f(x)−f(y),j(x−y)〉+γ〈Tx−Ty,j(x−y)〉≤αk∥x−y∥2+γ∥x−y∥2=(αk+γ)∥x−y∥2.Since (kα+γ)<1,
then
S is a strongly pseudocontractive. To prove the
claim of the theorem, we show firstly that (xn)n∈ℕ is bounded.
Pickingp∈Fix(T),
we have(2.6)∥xn−p∥2=αn〈f(xn)−p,j(xn−p)〉+βn〈xn−1−p,j(xn−p)〉+γn〈Txn−p,j(xn−p)〉=αn〈f(xn)−f(p),j(xn−p)〉+αn〈f(p)−p,j(xn−p)〉+βn〈xn−1−p,j(xn−p)〉+γn〈Txn−p,j(xn−p)〉≤[kαn+(1−αn−βn)]∥xn−p∥2+αn∥f(p)−p∥∥xn−p∥+βn∥xn−1−p∥∥xn−p∥which implies
that(2.7)∥xn−p∥≤αnαn(1−k)+βn∥f(p)−p∥+βnαn(1−k)+βn∥xn−1−p∥.By a simple induction, we get
that(2.8)∥xn−p∥≤max{∥f(p)−p∥1−k,∥x0−p∥}.Moreover, we have that limn→+∞∥xn−Txn∥=0.
In fact,(2.9)∥xn−Txn∥=∥αnf(xn)+βnxn−1−(αn+βn)Txn∥≤αn∥f(xn)−Txn∥+βn∥xn−1−Txn∥,and by boundedness of (xn)n∈ℕ and condition (ii), it follows the statement.
Let, for everyt∈(0,1),(2.10)xt=tf(xt)+(1−t)Txt.By Morales's Theorem 1.2, this
implicit method converges to a unique point q∈Fix(T) that is the unique solution of (1.8). Next, we
show that xn→q.
By Lemma1.6, we obtain(2.11)limsupn→∞〈f(q)−q,j(xn−q)〉≤0;then, if we define the real
sequence(2.12)σn:=max{0,〈f(q)−q,j(xn−q)〉},we can show that σn≥0 and limn→+∞σn=0.
So we  conclude(2.13)∥xn−q∥2=αn〈f(xn)−q,j(xn−q)〉+βn〈xn−1−q,j(xn−q)〉+γn〈Txn−q,j(xn−q)〉≤αn〈f(xn)−f(q),j(xn−q)〉+αn〈f(q)−q,j(xn−q)〉+βn∥xn−1−q∥∥xn−q∥++γn∥xn−q∥2≤(kαn+βn2+γn)∥xn−q∥2+βn2∥xn−1−q∥2+αnσn=[(k−1)αn−βn2+1]∥xn−q∥2+βn2∥xn−1−q∥2+αnσn≤[(k−1)αn2−βn2+1]∥xn−q∥2+βn2∥xn−1−q∥2+αnσn,which implies
that(2.14)∥xn−q∥2≤βn(1−k)αn+βn∥xn−1−q∥2+2αnσn(1−k)αn+βn=[1−(1−k)αn(1−k)αn+βn]∥xn−1−q∥2+2αnσn(1−k)αn+βn.By Liu's Lemma 1.7 and condition
(iii), we obtain that xn→q,
as n→∞.In the next theorem, we consider a viscosity
explicit method which extends (1.15) substituting the constant u with a ρ-contraction f,
and we establish a convergence's result for this scheme.Theorem 2.2.
LetC be a nonempty closed convex subset of a
reflexive Banach space X with a uniformly Gâteaux differentiable norm.
Suppose that every closed, bounded, and convex subset of C has the fixed point property for nonexpansive
self-mappings. Let T:C→C be a continuous pseudocontractive mapping and
let f:C→C be a ρ-contraction such that the set(2.15)E:={x∈C:Tx=λx+(1−λ)f(x)forsomeλ>1}is bounded.
Let(αn)n∈N,(βn)n∈N,
and (γn)n∈N be three real sequences in (0,1) satisfying the following conditions:
(i)
αn+βn+γn=1;(ii)
limn→+∞αn=limn→+∞βn=0;(iii)
∑n∈ℕ(αn/(1−γn))=+∞.Let (ϵn)n≥0 be a summable sequence of positive numbers.
For arbitrary initial pointx0=x00∈C and a fixed n≥0,
we construct elements (xnm)n∈N as follows: (2.16)xnm+1=αnf(xn)+βnxn+γnTxnm,m=0,1,2,….Suppose that there exists N(n),
the least positive integer satisfying the following condition:(2.17)∥TxnN(n)+1−TxnN(n)∥≤γn−1(1−γn)ϵn.Then, (xn)n∈ℕ defined as(2.18)xn+1=xn+10=xnN(n)+1=αnf(xn)+βnxn+γnTxnN(n)strongly converges to q,
where q∈Fix(T) is the unique solution of (1.8),(2.19)〈q−f(q),j(q−p)〉≥0∀p∈Fix(T).Proof.
We divide the proof into three
steps.
Step 1.
(xn)n∈ℕ is bounded.
Proof of Step1.
Pickingp∈Fix(T),
we have(2.20)∥xn+1−p∥2=αn〈f(xn)−p,j(xn+1−p)〉+βn〈xn−p,j(xn+1−p)〉+γn〈TxnN(n)−p,j(xn+1−p)〉=αn〈f(xn)−f(p),j(xn+1−p)〉+αn〈f(p)−p,j(xn+1−p)〉+βn〈xn−p,j(xn+1−p)〉+γn〈TxnN(n)−TxnN(n)+1,j(xn+1−p)〉+γn〈Txn+1−p,j(xn+1−p)〉≤(ραn+βn)∥xn−p∥∥xn+1−p∥+αn∥f(p)−p∥∥xn+1−p∥+(1−γn)ϵn∥xn+1−p∥+γn∥xn+1−p∥2,which implies that(2.21)∥xn+1−p∥≤ραn+βn1−γn∥xn−p∥+αn1−γn∥f(p)−p∥+ϵn.By a simple induction, we get
that(2.22)∥xn−p∥≤max{∥x0−p∥,∥f(p)−p∥1−ρ}+∑k=0n−1ϵk.
Step 2.
limn→+∞∥xn+1−Txn+1∥=0.
Proof of Step2.
Since(2.23)∥xn+1−Txn+1∥≤αn∥f(xn)−Txn+1∥+βn∥xn−Txn+1∥+γn∥TxnN(n)−TxnN(n)+1∥≤αn∥f(xn)−Txn+1∥+βn∥xn−Txn+1∥+ϵn,by the boundedness of (xn)n∈ℕ,
condition (ii), and the summability of (ϵn)n∈ℕ,
we obtain the claim.
Step 3.
limn→+∞xn=q.
Proof of Step .
As in Theorem2.1, set(2.24)q:=limt→0xt,where xt=tf(xt)+(1−t)Txt,
and(2.25)σn:=max{0,〈f(q)−q,j(xn+1−q)〉}.We known that limn→+∞σn=0;
now we show that xn→q.
In fact,(2.26)∥xn+1−q∥2=αn〈f(xn)−q,j(xn+1−q)〉+βn〈xn−q,j(xn+1−q)〉+γn〈TxnN(n)−q,j(xn+1−q)〉≤αn〈f(xn)−f(q),j(xn+1−q)〉+αn〈f(q)−q,j(xn+1−q)〉+βn〈xn−q,j(xn+1−q)〉+γn〈TxnN(n)−TxnN(n)+1,j(xn+1−q)〉+γn∥xn+1−q∥2≤(ραn+βn)∥xn−q∥∥xn+1−q∥+αnσn+(1−γn)ϵn∥xn+1−q∥+γn∥xn+1−q∥2≤ραn+βn2∥xn−q∥2+αn+βn2∥xn+1−q∥2+αnσn+(1−γn)ϵn∥xn+1−q∥+γn∥xn+1−q∥2=ραn+βn2∥xn−q∥2+1+γn2∥xn+1−q∥2+αnσn+(1−γn)ϵn∥xn+1−q∥2which implies
that(2.27)∥xn+1−q∥2≤(1−γn)−(1−ρ)αn1−γn∥xn−q∥2+2αnσn1−γn+2dϵn,d:=supn∈ℕ∥xn−q∥=(1−(1−ρ)αn1−γn)∥xn−q∥2+2αnσn1−γn+2dϵn.By Liu's Lemma 1.7, we obtain
that xn→q,
as n→∞.Remark 2.3.
We can prove that ifT is a nonexpansive mapping and xnm is defined as (2.32) of Theorem 2.2, then there
always exists a positive integer N(n) satisfying(2.28)∥TxnN(n)+1−TxnN(n)∥≤(1−γn)γnϵn.In fact, fixed n∈ℕ∪{0},
for every k∈ℕ∪{0},
we have(2.29)∥Txnk+1−Txnk∥≤γnk∥xn1−xn0∥.If xn1=xn0,
we are done. Otherwise, since 0<γn<1,
it follows that there exists a sufficiently large k=N(n)∈ℕ∪{0} such that(2.30)γnk≤(1−γn)ϵnγn∥xn1−xn0∥.It is also well known [3, 5, 10] that if T:C→C is a continuous pseudocontractive mapping,
defining the mapping g:C→C as g(x)=(2I−T)−1(x),
we can observe that the following hold:(1)
g is a nonexpansive mapping;(2)
Fix(T)=Fix(g);(3)
∥x−g(x)∥≤∥x−Tx∥,
for all x∈C.By Remark2.3 and Theorem 2.2, we have the
following.Corollary 2.4.
LetC be a nonempty closed convex subset of a real
reflexive Banach space X with a uniformly Gâteaux differentiable norm.
Suppose that every closed, bounded, and convex subset of C has the fixed point property for nonexpansive
self-mappings. Let T:C→C be a continuous pseudocontractive mapping and
let f:C→C be a ρ-contraction such that the set(2.31)E:={x∈C:Tx=λx+(1−λ)f(x)forsomeλ>1}is bounded.
Let(αn)n∈N,(βn)n∈N,
and (γn)n∈N be three real sequences in (0,1) satisfying the following conditions:
(i)
αn+βn+γn=1;(ii)
limn→+∞αn=limn→+∞βn=0;(iii)
∑n∈ℕ(αn/(1−γn))=+∞.
Let (ϵn)n≥0 be a summable sequence of positive numbers.
For arbitrary initial pointx0=x00∈C and a fixed n≥0,
we construct elements (xnm)n∈N as follows (here, as above, g=(2I−T)−1):(2.32)xnm+1=αnf(xn)+βnxn+γng(xnm),m=0,1,2,…,N(n),and we define (xn)n∈ℕ as(2.33)xn+1=xn+10=αnf(xn)+βnxn+γng(xnN(n)),where for every n∈ℕ,N(n)  is the positive integer such that(2.34)∥g(xnN(n)+1)−g(xnN(n))∥≤(1−γn)γnϵn.Then, (xn)n∈ℕ converges strongly to q∈Fix(T),
where q is the unique solution of (1.8),(2.35)〈q−f(q),j(q−p)〉≥0∀p∈Fix(T).



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