R E S E A R C H
Open Access
Viscosity approximations methods for
(
ψ
,
ϕ
)
-weakly contractive mappings
Swaminath Mishra
1, Rajendra Pant
2*and Rekha Panicker
1*Correspondence:
2Department of Mathematics,
Visvesvaraya National Institute of Technology, Nagpur, 440010, India Full list of author information is available at the end of the article
Abstract
In this paper, we study viscosity approximations with (ψ,ϕ)-weakly contractive mappings. We show that Moudafi’s viscosity approximations follow from Browder and Halpern type convergence theorems. Our results generalize a number of
convergence theorems including a strong convergence theorem of Song and Liu (Fixed Point Theory Appl. 2009:824374, 2009).
MSC: Primary 54H25; secondary 47H10
Keywords: (ψ,
ϕ)-weakly contractive mappings; Browder type convergence;
Halpern type convergence; Moudafi’s viscosity approximations1 Introduction and preliminaries
Let (M,d) be a metric space andf :M→Ma self-mapping. A pointz∈Mis said to be a fixed point off iff(z) =z. Throughout this paper,F(f) denotes the set of fixed points off,
Nthe set of natural numbers andMa metric space (M,d).
A mappingf:M→Mis a contraction if there existsr∈[, ) such that for allx,y∈M,
df(x),f(y)≤rd(x,y). (.)
The classicalBanach contraction principle(BCP) states that ‘Every contraction of a com-plete metric space has a unique fixed point.’ In , Boyd and Wong [] obtained the following interesting generalization of the BCP.
Theorem . Let f :M→M a self-mapping of a complete metric space M such that for all x,y∈M,
df(x),f(y)≤αd(x,y), (.)
whereα: [,∞)→[,∞)is upper semicontinuous from the right andα(t) <t for all t> .
Then f has a unique fixed point in M.
The mappingf:M→Msatisfying (.) is called a nonlinear contraction []. The mappingf:M→Mis called weakly contractive, if
df(x),f(y)≤d(x,y) –ϕd(x,y) (.)
for allx,y∈M, whereϕ: [,∞)→[,∞) is a continuous and nondecreasing function such thatϕ(t) = if and only ift= .
We note that (.) follows from Tasković [, ]. For an earlier work in this direction, we refer to Krasnosel’ski˘ıet al.[] and Dugundji and Granas []. Also, these mappings have been studied by Aĺber and Guerre-Delabriere [] and Rhoades [] as mentioned by Jachymski [] (see also []).
In this paper, we use the following class of mappings satisfying the so-called (ψ,ϕ )-condition (see for details [–]).
A mappingf :M→Mis called (ψ,ϕ)-weakly contractive if
ψdf(x),f(y)≤ψd(x,y)–ϕd(x,y) (.)
for allx,y∈M, whereψ,ϕ: [,∞)→[,∞) are both continuous and monotone nonde-creasing functions withψ(t) = =ϕ(t) if and only ift= .
Remark . We remark that ifϕ(t) = ( –r)twithr∈(, ), then (.) reduces to (.). Ifψ(t) =t, then (.) recovers (.). In fact, weakly contractive mappings are also related closely to nonlinear contractions. Ifαis continuous andϕ(t) =t–α(t) then (.) turns into (.). We have the following irreversible implications (see [], Example .).
(.)⇒(.)⇒(.)⇒(.).
Thus (ψ,ϕ)-weakly contractive mappings are more general than its predecessors as listed above.
Theorem .([], Theorem .) Every(ψ,ϕ)-weakly contractive mapping of a complete metric space has a unique fixed point.
It was observed by Ðorić [] that the continuity ofϕ can be relaxed to lower semi-continuity in Theorem ..
Definition . LetYbe a nonempty subset of a Banach spaceX. A mappingf :Y→Yis said to be nonexpansive if for allx,y∈Y,
f(x) –f(y)≤ x–y.
LetXbe a real Banach space with its dual spaceX∗andYbe a nonempty closed convex subset ofX. Let x,x∗be the dual pairing betweenx∈Xandx∗∈X∗, andJ:X→X∗be
the normalized duality mapping onXdefined by
J(x) =x∗∈X∗:x,x∗=x=x∗
for all x∈X. ThenX is said to besmoothor to have aGâteaux differentiable norm if
limt→x+ty–t xexists for eachx,y∈Xwithx=y= . A Banach spaceXis said to be
uniformly smooth whenever givenε> there existsδ> such that for allx,y∈Xwith
Definition .[] LetY be a nonempty closed convex subset of a Banach spaceXand
Za nonempty subset ofY. A retraction fromYtoZis a continuous mappingP:Y→Z
such thatP(x) =xforx∈Z. A retractionPfromY toZis sunny ifPsatisfies the prop-erty:P(P(x) +t(x–P(x))) =P(x) for allx∈Y andt> , wheneverP(x) +t(x–P(x))∈Y. A retractionPfromY toZis sunny nonexpansive ifPis both sunny and nonexpansive [–].
A well-known way to find a fixed point of a nonexpansive mapping is to use a contraction to approximate it (Browder [, ]). More precisely, fixz∈Yand define a mappingft:
Y→Ybyft(x) =tz+ ( –t)S(x) for allx∈Yand givent∈(, ). It is easy to see thatftis a
contraction onYand the BCP ensures thatfthas a unique fixed pointut∈Y, that is,
ut=tz+ ( –t)S(ut). (.)
In , Halpern [] introduced the following iteration for an arbitraryz∈Y and a se-quence{αn} ⊂(, ):
u∈Y, un+=αnz+ ( –αn)S(un) (.)
forn∈N, whereS:Y→Yis a nonexpansive mapping.
In the caseF(S)=∅, Browder [] (respectively, Halpern []) showed that{ut}
(respec-tively,{un}) converges strongly to the fixed point ofSthat is nearest tozin a Hilbert space.
A number of extensions and generalizations of their results have appeared in [, –] and elsewhere.
Theorem .[] Let Y be a bounded closed convex subset of a uniformly smooth Banach space X and S:Y→Y a nonexpansive mapping.Define a net{xα}in Y by
xα=αz+ ( –α)S(xα)
forα∈(, ),where z∈Y is fixed.Then{xα}converges strongly to P(z)asα→+,where P
is the unique sunny nonexpansive retraction from Y onto F(S).
Theorem .[, ] Let X,Y,S,P and z be as in Theorem..Define a sequence{un}in
Y by
u∈Y, un+=αnz+ ( –αn)S(un)
for n∈N,where{αn}is a real sequence in(, )satisfying
(C) lim
n→∞αn= , (C)
∞
n=
αn=∞, (C) lim n→∞
αn+
αn
= .
Then{un}converges strongly to P(z).
algorithm
un+=αnf(un) + ( –αn)S(un) (.)
forn∈N∪ {}, wheref:Y→Yis a contraction,S:Y→Ya nonexpansive mapping and
{αn} ⊆(, ), satisfying certain conditions, converges strongly to a fixed point ofSinY,
which is the unique solution to the following variational inequality:
(I–f)x∗,x∗–x≥, ∀x∈F(S).
Moudafi’s generalizations are called viscosity approximations. These methods can be ap-plied to convex optimization, linear programming, monotone inclusions, and elliptic dif-ferential equations []. In , Xu [] extended Moudafi’s results from Hilbert spaces to more general Banach spaces. Suzuki [] used Meir-Keeler type contractionsf in (.) to find fixed points ofSin Banach spaces. Recently, Song and Liu [] considered the fol-lowing viscosity approximations:
vn=αnf(vn) + ( –αn)Sn(vn);
un+=αnf(un) + ( –αn)Sn(un)
forn∈N, whereSn:Y→Y is a sequence of nonexpansive mappings andf :Y→Y is a
weakly contractive mapping.
In this paper, motivated by Moudafi [], Kopecká and Reich [], Suzuki [] and Song and Liu [], we study viscosity approximations with a more general class of weakly con-tractive mappings. We show that Moudafi’s viscosity approximations can be obtained from Browder and Halpern type convergence results.
2 Convergence results
Throughout this section,ψ,ϕ : [,∞)→[,∞) are continuous and strictly increasing functions such that
ψ(t) = =ϕ(t) if and only if t= .
Our main results are prefaced by the following lemmas and propositions.
Lemma .[, ] Let Y be a nonempty convex subset of a smooth Banach space X and Z a nonempty subset of Y.Let J be the duality mapping from X into X∗,and P:Y→Z a retraction.Then P is both sunny and nonexpansive if and only if
x–P(x),Jy–P(x)≤
for all x∈Y and y∈Z.
Lemma .[] Let{αn}be a sequence of positive reals and{βn}a sequence of nonnegative
reals such that
lim n→∞αn= ,
∞
n=
αn=∞ and lim
n→∞
βn
αn
Further,consider a sequence of nonnegative reals{n}and the recursive inequality
n+≤n–αnξ(n) +βn
for n∈N∪ {},whereξ()is continuous strictly increasing for≥andξ() = .Then
() limn→∞n= ;
() there exists a subsequence{nk}of{n}such that
nk≤ξ
–
nk
m=αm
+βnk
αnk
,
nk+≤ξ
–
nk
m=αm
+βnk
αnk
+βnk,
n≤nk+– n–
m=nk+
αm
θm
, nk+ <n<nk+,θm= m
i=
αi,
n+≤–
n
m=
αm
θm ≤
, ≤n<nk– ,
≤nk≤smax=max
s,
s
m=
αm
θm ≤
.
Definition .[] Let{Sn}be sequence of nonexpansive mappings on a closed convex
subsetYof a Banach spaceXandF=∞n=F(Sn)=∅.
(A) Let{αn}be a sequence in(, ]withlimn→∞αn= . Then(X,Y,{Sn},{αn})is said to satisfy Browder property if for eachz∈Y, a sequence{vn}defined by
vn=αnz+ ( –αn)Sn(vn), (.)
forn∈N, converges strongly.
(B) Let{αn}be a sequence in[, ]withlimn→∞αn= and∞n=αn=∞. Then
(X,Y,{Sn},{αn})is said to satisfy Halpern’s property if for eachz∈Y, a sequence {vn}defined by
v∈Y, vn+=αnz+ ( –αn)Sn(vn), (.)
forn∈N, converges strongly.
It is well known that if Xis a Hilbert space,Y is bounded and{Sn} is a constant
se-quenceS, then (X,Y,{Sn},{/n}) has both the Browder and the Halpern properties (cf.
[, , , ]).
Example . LetX= [,∞) equipped with the norm · defined byx=|x|andY= [, ] a closed convex subset ofX. Define a sequence of nonexpansive mappingsSn:Y→Y
bySn(x) =nxfor allx∈Yandn∈N. Let{αn}be a sequence in (, ] defined byαn=n+.
Then
and the sequence{vn}defined by
vn=αnz+ ( –αn)Sn(vn),
for eachz∈Yandn∈N, converges strongly to .
(ii) However, the quadruple(X,Y,{Sn},{αn})does not satisfy the Halpern property, as the series∞n=αnis a convergent series.
(iii) If we takeαn=n, then the quadruple(X,Y,{Sn},{αn})satisfies both the Browder
and the Halpern properties. We note that{Sn}is not a constant sequence here.
Proposition .([], Proposition ) Let the quadruple(X,Y,{Sn},{αn})satisfies
Brow-der’s property and{vn}is a sequence in Y,defined by(.).If P(z) =limn→∞vnfor each
z∈Y then P is a nonexpansive mapping on Y.
Proposition . ([], Proposition ) Let the quadruple(X,Y,{Sn},{αn})satisfies
Hal-pern’s property and{vn}is a sequence in Y,defined by(.).If P(z) =limn→∞vnfor each
z∈Y then
• P:Y→Y is a nonexpansive mapping; • P(z)does not depend on the initial pointv.
Proposition . Let Y be a closed convex subset of a smooth Banach space X and Z a nonempty subset of Y.Let S:Y →Y be a nonexpansive mapping, P:Y→Z a unique sunny nonexpansive retraction,and T:Y→Y a(ψ,ϕ)-weakly contractive mapping with
ψ convex.Then
(a) the composite mappingS◦Tis(ψ,ϕ)-weakly contractive onY;
(b) the mappingSt=tT+ ( –t)Sfort∈(, )is(ψ,φ)-weakly contractive onYandutis the unique solution of the fixed point equation
ut=tT(ut) + ( –t)S(ut),
whereφ(s) =tϕ(s)for each fixedt∈(, );
(c) P(T(z)) =zif and only ifz∈Yis the unique solution of the variational inequality
T(z) –z,J(y–z)≤ (.)
for ally∈Z.
Proof
(a) For anyx,y∈Y we have
ST(x)–ST(y)≤T(x) –T(y).
Sinceψis nondecreasing andTis a(ψ,ϕ)-weakly contractive, the above inequality reduces to
ψST(x)–ST(y)≤ψT(x) –T(y)
Therefore the mappingS◦Tis a(ψ,ϕ)-weakly contractive. (b) Letx,y∈Y. Then for each fixedt∈(, ), we have
St(x) –St(y)=tT(x) + ( –t)S(x)
–tT(y) + ( –t)S(y)
≤( –t)S(x) –S(y)+tT(x) –T(y)
≤( –t)x–y+tT(x) –T(y).
Sinceψis nondecreasing, the above inequality reduces to
ψSt(x) –St(y)≤ψ
( –t)x–y+tT(x) –T(y).
Convexity ofψimplies
ψSt(x) –St(y)≤( –t)ψ
x–y+tψT(x) –T(y).
SinceTis(ψ,ϕ)-weakly contractive, we have
ψSt(x) –St(y)≤( –t)ψ
x–y+tψx–y–ϕx–y
=ψx–y–tϕx–y.
Letφ(s) =tϕ(s). Then
ψSt(x) –St(y)≤ψ
x–y–φx–y.
Thus, the mappingsStis(ψ,φ)-weakly contractive and by Theorem .,Sthas a unique fixed pointutinY.
(c) By (a) and Proposition ., the mappingP◦Tis(ψ,φ)-weakly contractive. By Theorem .,P◦T has a unique fixed pointP(T(z)) =z∈Z. By Lemma ., such a
z∈Zsatisfies (.). Next, we show that the variational inequality (.) has a unique
solution. Letw∈Ybe another solution of (.). Then
T(w) –w,J(z–w)≤ (.)
and
T(z) –z,J(w–z)≤. (.)
Adding (.) and (.)
≥w–z–T(w) –T(z),J(w–z)
=w–z–T(w) –T(z)w–z
≥ w–z–T(w) –T(z)w–z
which implies that
w–z–T(w) –T(z)≤ or w–z ≤T(w) –T(z).
Sinceψis nondecreasing andTis a(ψ,ϕ)-weakly contractive, we have
ψw–z≤ψT(w) –T(z)
≤ψw–z–ϕw–z.
Thereforeϕ(w–z)≤, andw=z.
Now we present our first convergence result.
Theorem . Suppose the quadruple (X,Y,{Sn},{αn}) satisfies Browder’s property and
T :Y →Y is a(ψ,ϕ)-weakly contractive mapping withψ convex.For each z∈Y,put P(z) =limn→∞vn,where{vn}is a sequence in Y defined by(.).Then the sequence{un} ⊂Y
defined by
un=αnT(un) + ( –αn)Sn(un),
for n∈N,converges strongly to P(T(z)) =z∈Y.
Proof Proposition .(b) ensures the existence and uniqueness of{un}. It follows from
Proposition .(a) and Proposition . thatP◦T is a (ψ,ϕ)-weakly contractive mapping onY. Therefore, by Theorem . there exists a unique elementz∈Ysuch thatP(T(z)) =z. Define a sequence{vn}inYby
vn=αnT(z) + ( –αn)Sn(vn)
forn∈N. Then it is easy to see that{vn}converges strongly toP(T(z)).
Now, forn∈N,
un–vn ≤( –αn)Sn(un) –Sn(vn)+αnT(un) –T(z) ≤( –αn)un–vn+αnT(un) –T(z)
or
un–vn ≤T(un) –T(z).
Sinceψis nondecreasing, we have
ψun–vn
≤ψT(un) –T(z).
Further, (ψ,ϕ)-weak contractivity ofTimplies
ψun–vn
≤ψun–z
–ϕun–z
≤ψun–vn+vn–z
–ϕun–z
Sincevn→zasn→ ∞, we get
lim
n→∞ψ
un–vn
≤ lim n→∞ψ
un–vn+vn–z
– lim
n→∞
ϕun–z
= lim n→∞ψ
un–vn
– lim
n→∞ϕ
un–z
,
or
lim n→∞ϕ
un–z
≤.
The continuity ofϕandϕ() = imply that
lim
n→∞un–z= .
Therefore{un}converges strongly toz.
Corollary . Let X, Y,{Sn},{vn}, P,z and{αn} be as in Theorem.and T:Y→Y
a weakly contractive mapping.Then the sequence{un} ⊂Y defined by
un=αnT(un) + ( –αn)Sn(un),
for n∈N,converges strongly to P(T(z)) =z∈Y.
Proof This follows from Theorem . whenψ(t) =t.
Example . Let (X,·) andYbe as in Example .. Define the mappingsSn,T:Y→Y
by
Sn(x) =x for allx∈Yandn∈N and T(x) = –
x
for allx∈Y.
Letψ,ϕ: [,∞)→[,∞) be the functions defined by
ψ(t) =t and ϕ(t) = t .
Then the mappingSnis nonexpansive for eachn∈NandT is (ψ,ϕ)-weakly contractive.
Let{αn}be a sequence in (, ] defined byαn=n+. Then the quadruple (X,Y,{Sn},{αn})
satisfies the Browder property and for eachz∈Y, we have
vn=zαn+ ( –αn)Sn(vn)
=z n+ +
–
+n
vn,
or
By Theorem ., putP(z) =limn→∞vn=z. ThenPis an identity mapping. Now
un=αnT(un) + ( –αn)Sn(un)
=
n+
–un
+
–
+n
un,
or
un=
.
Nowlimn→∞un= =P(T()). Thus the quadruple (X,Y,{Sn},{αn}) satisfies all the
con-ditions of Theorem . and the sequence{un}strongly converges to .
The following theorem is our second convergence result.
Theorem . Suppose the quadruple(X,Y,{Sn},{αn})satisfies Halpern’s property and
T:Y→Y is a(ψ,ϕ)-weakly contractive mapping withψis convex.Put P(z) =limn→∞vn
for each z∈Y,where{vn}is defined by(.).Then the sequence{un} ⊂Y defined by
x∈Y and un+=αnT(un) + ( –αn)Sn(un)
for n∈N,converges strongly to a unique point P(T(z)) =z∈Y.
Proof By Propositions . and .(a), the mappingP◦Tis (ψ,ϕ)-weakly contractive onY. From Theorem ., there exists a uniquez∈Ysuch thatz=P(T(z)). Forn∈N, we define a sequence{vn}inY by
vn+=αnT(z) + ( –αn)Sn(vn).
Then by the assumption{vn}converges strongly toP(T(z)).
Now forn∈N, we have
un+–vn+=αn
T(un) –T(z)
+ ( –αn)
Sn(un) –Sn(vn).
SinceSnis nonexpansive andψis nondecreasing and convex, we have
ψun+–vn+
=ψαnT(un) –T(z)+ ( –αn)Sn(un) –Sn(vn)
≤αnψT(un) –T(z)+ ( –αn)ψSn(un) –Sn(vn) ≤αnψT(un) –T(z)+ ( –αn)ψ
un–vn
.
By (ψ,ϕ)-weak contractivity ofT, we get
ψun+–vn+
≤αn
ψun–z
–ϕun–z
+ ( –αn)ψ
un–vn
≤αn
ψun–vn+vn–z
–ϕun–vn+vn–z
+ ( –αn)ψ
un–vn
The continuity ofψ,ϕandvn→zasn→ ∞imply that
lim
n→∞ψ
un+–vn+
≤ lim n→∞αn
ψun–vn
–ϕun–vn
+ ( –αn)ψ
un–vn
= lim n→∞
ψun–vn
–αnϕ
un–vn
≤ lim
n→∞
ψun–vn
–αnϕ
un–vn
+αnvn–z
.
Thus, for some (sufficiently large)N≤n, we have
ψun+–vn+
≤ψun–vn
–αnϕ
un–vn
+αnvn–z.
Forn=ψ(un–vn), we get the following recursive inequality:
n+≤n–αnϕ(n) +βn,
whereεn=vn–zandβn=αnεn. Now, by Lemma .,
lim
n→∞ψ
un–vn
= .
The continuity ofψand the fact thatψ() = imply that
lim
n→∞un–vn= .
By the triangle inequality, we have
lim
n→∞un–z ≤nlim→∞un–vn+nlim→∞vn–z= .
Therefore{un}strong convergence of toz=P(T(z)).
Corollary .[] Let X,Y,{Sn},{vn},P,z and{αn}be as in Theorem.and T:Y→Y
a weakly contractive mapping.Then the sequence{un} ⊂Y defined by
x∈Y and un+=αnT(un) + ( –αn)Sn(un),
for n∈N,converges strongly to a unique point P(T(z)) =z∈Y.
Proof This follows from Theorem . whenψ(t) =t.
Competing interests
The authors declare that they do not have any competing interests.
Authors’ contributions
Each author equally contributed to this paper and read and approved the final manuscript.
Author details
1Department of Mathematical Sciences and Computing, Walter Sisulu University, Nelson Mandela Drive, Mthatha, 5117,
Acknowledgements
The authors are indebted to the referees and the Editor for their constructive comments and suggestions, which have been useful for the improvement of the paper.
Received: 5 April 2016 Accepted: 16 September 2016
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