R E S E A R C H
Open Access
On common approximate fixed points of
monotone nonexpansive semigroups in
Banach spaces
Mostafa Bachar
1*and Mohamed A Khamsi
2,3*Correspondence:
1Department of Mathematics,
College of Sciences, King Saud University, Riyadh, Saudi Arabia Full list of author information is available at the end of the article
Abstract
In this paper, we investigate the common approximate fixed points of monotone nonexpansive semigroups of nonlinear mappings{T(t)}t≥0,i.e., a family such that T(0)x=x,T(s+t)x=T(s)◦T(t)x, where the domain is a Banach space. In particular we prove that under suitable conditions, the common approximate fixed points are the same as the common approximate fixed points set of two mappings from the family. Then we give an algorithm of how to construct an approximate fixed point sequence of the semigroup in the case of a uniformly convex Banach space.
MSC: Primary 47H09; secondary 46B20; 47H10; 47E10
Keywords: approximate fixed point; common approximate fixed point; fixed point; common fixed point; integral equation; monotone nonexpansive mapping;
semigroup
1 Introduction
Nonexpansive mappings are those maps which have Lipschitz constant equal to . The fixed point theory for such mappings is rich and varied. It finds many applications in non-linear functional analysis. The existence of fixed points for nonexpansive mappings in Ba-nach and metric spaces has been investigated since the early s; see,e.g., Belluce and Kirk [, ], Browder [], Bruck [], Lim [].
In recent years, a new direction has been very active essentially after the publication of Ran and Reurings fixed point theorem [] dealing with the extension of the Banach contraction principle to metric spaces endowed with a partial order. In particular, they show how this extension is useful when dealing with some special matrix equations. It is worth mentioning that similar results were discovered by Turinici [, ]. Another similar approach was carried out by Nieto and Rodríguez-López [] and used such arguments in solving some differential equations. In [] Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order.
mappings forms a semigroup ifT()x=xandT(s+t)x=T(s)T(t). Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the vector function space would define the state space and the mapping (t,x)→T(t)x
would represent the evolution function of a dynamical system. In the setting of this pa-per, the state space is a Banach space. Our approach is new and different from the ideas found in [–, ]. Indeed when one deals with a contraction, the focus is usually on the distance being complete. But when dealing with nonexpansive mappings, then geometric properties of the space are crucial.
For more on metric fixed point theory, the reader may consult the books [, ]. As for the semigroup theory, we suggest the references [–].
2 Preliminaries
Let (X,·) be a Banach vector space and suppose thatis a partial order onX. Through-out, we will assume that the partial orderand the liner structure ofXenjoys the follow-ing convexity property:
abandcd ⇒ αa+ ( –α)cαb+ ( –α)d
for anyα∈[, ] anda,b,c,d∈X. Moreover, we will assume that order intervals are closed. Recall that an order interval is any of the subsets
(i) [a,→) ={x∈X;ax}, (ii) (←,a] ={x∈X;xa},
for anya∈X. Next we give the definition of monotone nonexpansive mappings.
Definition . Let (X, · ,) be as above. LetC be a nonempty subset ofX. A map
T:B→Xis said to be
(a) monotone ifT(x)T(y)wheneverxy; (b) monotone nonexpansive ifTis monotone and
T(x) –T(y)≤ x–y
for anyx,y∈Csuch thatxy.
A fixed point ofT is any elementx∈Csuch thatT(x) =x. The set of all fixed points of
T is denoted byFix(T). A sequence{xn}is called an approximate fixed point sequence of
T iflimn→∞T(xn) –xn= . The set of approximate fixed point sequences ofT will be denoted byAFPS(T).
This definition is extended to the case of semigroup of mappings.
Definition . Let (X, · ,) be as above. LetC be a nonempty subset ofX. A one-parameter familyF={T(t);t≥}of mappings fromCintoCis said to be a monotone nonexpansive semigroup ifFsatisfies the following conditions:
(i) T()x=xforx∈C;
(ii) T(t+s) =T(t)◦T(s)fort,s∈[,∞);
Define then the set of all common fixed points ofFasFix(F) =t≥Fix(T(t)). Similarly, define the set of approximate point sequences ofF, denoted byAFPS(F), asAFPS(F) =
t≥AFPS(T(t)).
Next we give an example of such semigroup.
Example . Let (X, · ,) be as above. LetCbe a nonempty closed, bounded convex subset ofX. LetJ:C→Cbe a monotone nonexpansive mapping. Letx∈Cbe such that
xJ(x). Consider the recurrent sequence defined by
u(t) =x,
un+(t) =e–tx+
t
es–tJ(un(s))ds
(.)
for anyt∈[,A], whereAis a fixed positive number. Then the sequence{un(t)}is Cauchy for anyt∈[,A]. Indeed, let us suppose thatx,y: [,A]→Xare continuous functions. For eacht∈[,A], we have
e–ty(t) +
t
es–tx(s)ds≤e–ty∞+K(t)x∞, (.)
whereu∞=sup{u(s);t∈[,A]}andK(t) =tes–tds= –e–t. Indeed, lett∈[,A] be fixed, andτ ={ti;i= , , . . . ,n}be any subdivision of [,t]. Set
Sτ=e–ty(t) +
n–
i=
(ti+–ti)eti–tx(ti).
The family{Sτ}is norm-convergent to
S=e–ty(t) + t
es–tx(s)ds,
when|τ|=sup{|ti+–ti|;i= , , . . . , (n– )}goes to . Since
sup
s∈[,t]
x(s)≤ sup
s∈[,A]
x(s)=x∞
and
Sτ ≤e–ty(t)+
n–
i=
(ti+–ti)eti–tx∞≤e–ty(t)+K(t)x∞,
where we used
n–
i=
(ti+–ti)eti–t≤ t
es–tds=K(t),
then we haveS ≤e–ty(t)+K(t)x∞. Therefore we have
e–ty(t) + t
for anyt∈[,A]. Next we discuss the sequence{un(t)}. Since
u(t) =e–tx+
t
es–tJ(x)ds=e–tx+ –e–t J(x), t∈[,A],
we haveu(t)u(t) for anyt∈[,A]. Assume thatun(t)un+(t) for anyt∈[,A]. Then
we haveJ(un(t))J(un+(t)) for anyt∈[,A] becauseJis monotone. Using the properties
of the partial order, we get
un+(t) =e–tx+
t
es–tJun(s) dse–tx+ t
es–tJun+(s) ds=un+(t)
for anyt∈[,A]. By induction, we conclude that{un(t)}is monotone increasing for any
t∈[,A]. Moreover, we have
un+(t) –un+(t) =
t
es–tJun+(s) –J
un(s)
ds, t∈[,A],
which implies by using the monotone nonexpansive behavior ofJ
un+(t) –un+(t)≤
t
es–tJun+(s) –J
un(s) ds, t∈[,A], or
un+(t) –un+(t)≤
t
es–tun+(s) –un(s)ds, t∈[,A]
for anyn≥. From this inequality and by induction, we get un+(t) –un(t)≤
–e–A ndiam(C), t∈[,A],
wherediam(C) =sup{y–z;y,z∈C}<∞, sinceCis bounded. Hence un+h(t) –un(t)≤
( –e–A)n
– ( –e–A)diam(C) =e
A –e–A n
diam(C)
for anyt∈[,A] andn,h∈N. Clearly this implies that{un(t)}is Cauchy for anyt∈[,A]. Since order intervals are closed, we conclude that the limitu(t) of{un(t)}satisfiesun(t)
u(t) for anyn≥ andt∈[,A]. Using the monotone nonexpansive behavior ofJ, we get Jun(t) –J
u(t) ≤un(t) –u(t)≤eA
–e–A ndiam(C)
for anyt∈[,A] andn,h∈N. In particular, we havelimn→+∞J(un(t)) =J(u(t)), uniformly in [,A], which implies
u(t) =e–tx+ t
es–tJu(s) ds, t∈[,A]. (.)
F: [,∞)×C→Cby
T(t)x=u(t).
ThenFdefines a semigroup which is monotone nonexpansive. Indeed, we haveu(t) –
U(t)=x–y=x–y∞. Assumeun(t) –Un(t) ≤ x–y, then by using (.) we will have
un+(t) –Un+(t)≤e–tx–y+
t
es–tun(s) –Un(s)ds≤ x–y.
By induction,un(t) –Un(t) ≤ x–yholds for anyn∈Nandt∈[,A]. Hence T(t)x–T(t)y=u(t) –U(t)≤ x–y
for anyt∈[,A]. Moreover, we have
T(t+s)x=e–(t+s)x+ t+s
e–(t+s–σ)JT(σ)x dσ
=e–(t+s)x+
t
e–(t+s–σ)JT(σ)x dσ+
t+s
t
e–(t+s–σ)JT(σ)x dσ,
=e–s
e–tx+ t
e–(t–σ)JT(σ)x dσ
+
s
e–(s–σ)JT(t+σ)x dσ
=e–sT(t)x+ s
e–(s–σ)JT(t+σ)x dσ
=T(s)T(t)x
for anyt≥. Note that if we started by a pointx∈Csuch thatJ(x)x, then we would have found that the sequence{un(t)}is monotone decreasing for anyt≥. In other words, the conclusion would have been the same. Moreover, ifxis a fixed point ofJ, then we have
un(t) =xfor anyn∈Nandt≥. HenceT(t)x=xfor anyt≥,i.e.,x∈
t≥Fix(T(t)). On the other hand, it is easy to show that ifx∈Fix(F), thenx∈Fix(J), which means we have
Fix(J) = t≥
FixT(t) .
3 Common approximate fixed points of semigroups
Before we state our first result, we need the following definition.
Definition . Let (X, · ) be a Banach space andC⊂Xbe nonempty. A one-parameter familyF={T(t);t≥}of mappings fromCintoXis said to be:
(i) continuous onCif for anyx∈C, the mappingt→T(t)xis continuous,i.e., for any
t≥, we have lim
t→t
T(t)x–T(t)x=
(ii) strongly continuous onCif for any bounded nonempty subsetK⊂C, we have
lim
t→t
sup
x∈K
T(t)x–T(t)x = .
The following technical lemmas will be useful throughout.
Lemma . Let(X, · )be a Banach space and C⊂X be nonempty.Let J:C→C be a uniformly continuous mapping.Then we haveAFPS(J)⊂AFPS(Jm)for any m≥.
Proof Without loss of generality we may assume thatAFPS(J) is not empty. Let{xn} ∈
AFPS(J),i.e.,limn→∞J(xn) –xn= . Fixm≥, then we have Jm(xn) –xn≤
m–
k=
Jk+(xn) –Jk(xn)
for anyn≥. SinceTis uniformly continuous, then we must have
lim
n→∞J k+(x
n) –Jk(xn)=
for anyk≥. Sincemis fixed, we getlimn→∞Jm(xn) –xn= ,i.e.,{xn} ∈AFPS(Jm).
Lemma . Let(X, · )be a Banach space and C⊂X be nonempty.LetF={T(t);t≥}
be a one-parameter semigroup of uniformly continuous mappings from C into C.Letαand
βbe two positive real numbers.Then we have
AFPST(α) ∩AFPST(β) ⊂ t∈G+(α,β)
AFPST(t),
where G+(α,β) ={mα+kβ≥;m,k∈Z}.
Proof Without loss of generality we may assume thatAFPS(T(α))∩AFPS(T(β)) is not empty. Let{xn} ∈AFPS(T(α))∩AFPS(T(β)). Lett∈G+(α,β). Then we have two cases.
First assumet=mα+kβ, wherem,k≥. Then
T(t)xn=T(mα+kβ)xn=Tm(α)
Tk(β)xn ,
which implies
T(t)xn–xn≤Tm(α)
Tk(β)xn –Tm(α)xn+Tm(α)xn–xn
for any n ≥ . Using Lemma . and the uniform continuity of Tm(α), we get
limn→∞T(t)xn–xn= ,i.e.,{xn} ∈AFPS(T(t)). Next assume thatt=mα+kβ, where eithermorkis negative. Without loss of generality assumet=mα–kβ, wherem,k≥. We have
SinceT(mα) =T(mα–kβ)T(kβ), andT(mα–kβ) is uniformly continuous, Lemma . implieslimn→∞T(t)xn–xn= ,i.e.,{xn} ∈AFPS(T(t)). Hence
{xn} ∈
t∈G+(α,β)
AFPST(t) .
The following lemma, which can be found in any introductory course on real analysis, will be crucial to proving the first result on common approximate fixed point of semi-groups.
Lemma .[] Let G be a nonempty additive subgroup ofR.Then G is either dense in
Ror there exists a> such that G=a·Z={an,n∈Z}.Therefore ifαandβare two real numbers such that α
βis irrational,then the set
G(α,β) ={αn+βm;n,m∈Z}
is dense inR.In particular,the set G+(α,β) =G(α,β)∩[, +∞)is dense in[, +∞). Theorem . Let(X, · )be a Banach space and C⊂X be nonempty and bounded.Let
F={T(t);t≥}be a one-parameter semigroup of uniformly continuous mappings from C into C.Assume thatFis strongly continuous.Letαandβbe two positive real numbers such thatαβ is irrational.Then we have
AFPST(α) ∩AFPST(β) =AFPS(F).
Proof Since AFPS(F) ⊂ AFPS(T(α)) ∩ AFPS(T(β)), it is enough to just prove
AFPS(T(α)) ∩ AFPS(T(β)) ⊂ AFPS(F). Without loss of generality, assume that
AFPS(T(α))∩AFPS(T(β)) is not empty. Let{xn} ∈AFPS(T(α))∩AFPS(T(β)). Lemma . implies that
{xn} ∈
t∈G+(α,β)
AFPST(t) .
From Lemma ., we know thatG+(α,β) =G(α,β)∩[, +∞) is dense in [, +∞). Lett∈
[, +∞). Then there existstm∈G+(α,β),m≥, such thatlimm→∞tm=t. We have xn–T(t)xn≤xn–T(tm)xn+T(tm)xn–T(t)xn
≤xn–T(tm)xn+sup x∈C
T(t)x–T(t)x.
Letε> . SinceF is strongly continuous, there existsm≥ such that for anym≥m,
we have
sup
x∈C
T(tm)x–T(t)x<ε.
Since{xn} ∈AFPS(T(m)) from Lemma ., there existsn≥ such that
for anyn≥n. Hence
xn–T(t)xn≤xn–T(tm)(xn)+sup
x∈C
T(tm)x–T(t)x< ε
for anyn≥n. Sinceεwas arbitrarily positive, we conclude thatlimn→∞T(t)xn–xn= ,
i.e.,{xn} ∈AFPS(T(t)) for anyt≥. As a corollary to Theorem ., we get the following result.
Corollary . Let(X, · )be a Banach space and C⊂X be nonempty and bounded.Let
F={T(t);t≥}be a one-parameter semigroup of uniformly continuous mappings from C into C.Assume thatFis strongly continuous.Then we have
AFPS(F) =AFPST() ∩AFPST(π) =AFPST() ∩AFPST(√) .
All of the results obtained in this section may be easily stated in metric spaces. In the next section, we give an algorithm of how to construct an approximate fixed point sequence of two maps.
4 Common approximate fixed points of two monotone mappings
In this section we discuss a construction of a common approximate fixed point sequence of two monotone nonexpansive mappings defined on a Banach space (X, · ) endowed with a partial orderas described before. LetC⊂Xbe a nonempty convex subset. LetJ,H:
C→Cbe two mappings. Fixx∈C, Das and Debata [] studied the strong convergence
of Ishikawa iterates{xn}defined by
xn+=αnH
βnJ(xn) + ( –βn)xn + ( –αn)xn, (DD)
whereαn,βn∈[, ]. Under suitable assumptions, we will show that{xn}is an approximate fixed point sequence of bothJandH. Assume thatJandHare monotone nonexpansive. Assume that there existsx∈Csuch thatxJ(x) andxH(x). We will also assume
thatHandJhave a common fixed pointp∈Csuch thatxandpare comparable. Using
the convexity properties of the partial order, we will easily show thatxnandpare com-parable. SinceH andJare monotone nonexpansive, we getH(xn) –p ≤ xn–pand
J(xn) –p ≤ xn–pfor anyn≥. Hence
xn+–p=αnH(yn) + ( –αn)xn–p
≤αnH(yn) –p+ ( –αn)xn–p
≤αnyn–p+ ( –αn)xn–p
=αnβnJ(xn) + ( –βn)xn–p+ ( –αn)xn–p
≤αn
βnJ(xn) –p+ ( –βn)xn–p
+ ( –αn)xn–p
≤αn
βnxn–p+ ( –βn)xn–p
+ ( –αn)xn–p
whereyn=βnJ(xn) + ( –βn)xnfor anyn≥. This proves that{xn–p}is decreasing, which implies thatlimn→∞xn–pexists. Using the above inequalities, we get
lim
n→∞xn–p=nlim→∞αnH(yn) + ( –αn)xn–p = lim
n→∞
αnH(yn) –p+ ( –αn)xn–p
= lim
n→∞
αnyn–p+ ( –αn)xn–p
= lim
n→∞
αnβnJ(xn) + ( –βn)xn–p+ ( –αn)xn–p
= lim
n→∞
αn
βnJ(xn) –p+ ( –βn)xn–p + ( –αn)xn–p
.
Before we state the main theorem of this section, let us recall the definition of a uniformly convex Banach space.
Definition .[] Let (X, · ) be a Banach space.Xis said to be uniformly convex if and only if for anyε> , we haveδX(ε) > , where
δX(ε) =inf
–
x+y;x ≤,y ≤ andx–y ≥ε
.
It is well known that ifXis uniformly convex, thenXis reflexive []. Moreover, we have []
αx+ ( –α)y≤ –δX
min{α, –α}ε
for anyα∈(, ),ε> andx,y∈Xsuch thatx ≤,y ≤ andx–y ≥ε. The following lemma will be needed to prove the main result of this section.
Lemma .[] Let(X,·)be a uniformly convex Banach space.Let{xn}and{yn}be in X
such thatlim supn→∞xn ≤R,lim supn→∞yn ≤R,andlimn→∞αnxn+ ( –αn)yn=R,
whereαn∈[a,b],with <a≤b< ,and R≥.Then we have
lim
n→∞xn–yn= .
Next we state the main result of this section.
Theorem . Let C be a nonempty,closed and convex subset of a uniformly convex Ba-nach space(X, · ).Let H,J:C→C be monotone nonexpansive mappings such that H is uniformly continuous.Assume that there exists x∈C such that xJ(x)and xH(x).
We will also assume that H and J have a common fixed point p∈C such that x and p
are comparable.Consider the sequence{xn}defined by xand the recurrent formula(DD).
Assume thatαn,βn∈[α,β],with <α≤β< ,then
lim
n→∞xn–H(xn)= and nlim→∞xn–J(xn)= ,
Proof We have already seen that{xn–p}is decreasing. Setc=limn→∞xn–p. Ifc= , then all the conclusions are trivial. Therefore we will assume thatc> . We have already seen that
lim
n→∞xn–p=nlim→∞αnH(yn) + ( –αn)xn–p = lim
n→∞
αnH(yn) –p+ ( –αn)xn–p
.
We claim thatlimn→∞H(yn) –p=c. Indeed, letUbe a nontrivial ultrafilter overN. Then we havelimn,Uαn=α∞∈[α,β] andlimn,Uxn–p=c. Hence
c=α∞lim
n,U
H(yn) –p+ ( –α∞)c.
Sinceα∞= , we getlimn,UH(yn) –p=c. SinceUwas arbitrary, we getlimn→∞H(yn) –
p=cas claimed. Therefore we have
lim
n→∞xn–p=nlim→∞H(yn) –p=nlim→∞αnH(yn) + ( –αn)xn–p. Using Lemma ., we getlimn→∞H(yn) –xn= . Since we already proved
lim
n→∞xn–p=nlim→∞
αnyn–p+ ( –αn)xn–p
,
a similar argument will show thatlimn→∞yn–p=limn→∞xn–p. Moreover, we have
lim
n→∞xn–p=nlim→∞
αn
βnJ(xn) –p+ ( –βn)xn–p + ( –αn)xn–p
.
If we use again ultrafilters, one will easily prove thatlimn→∞J(xn) –p=limn→∞xn–p. Hence we have
lim
n→∞xn–p=nlim→∞J(xn) –p=nlim→∞βnJ(xn) + ( –βn)xn–p. Using again Lemma ., we getlimn→∞J(xn) –xn= . Since
lim
n→∞yn–xn=nlim→∞βnJ(xn) –xn= ,
and H is uniformly continuous, we get limn→∞H(xn) –H(yn)= . Combined with
limn→∞H(yn) –xn= , we get
lim
n→∞H(xn) –xn= .
Remark . If we assume thatαn=α andβn=β withα,β∈(, ), then under the as-sumptions of Theorem ., we can prove that
xn≤xn+ and yn≤yn+
convexity properties of the partial orderinX, we will conclude that in fact{xn}and{yn} are weakly convergent. It is not clear to us that the weak-limit of{xn}is a fixed point ofH andJ. This will be the case if we have a demi-closed property for monotone nonexpansive mappings. Otherwise, we may assume thatXsatisfies the Opial condition []. In this case the weak-limit of{xn}will be a fixed point ofTandS. As an example of a Banach space which satisfies all of the above assumptions, one may takeX= p, <p< +∞.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia.2Department of
Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA. 3Department of Mathematics and
Statistics, King Fahd University of Petroleum & Minerals, Dhahran, 31261, Saudi Arabia.
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No. RG-1435-079.
Received: 4 June 2015 Accepted: 13 August 2015 References
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