R E S E A R C H
Open Access
Fixed point theorems for a class of
generalized nonexpansive mappings
Fatemeh Lael
1*and Zohre Heidarpour
2*Correspondence:
1Department of Mathematics, Buein
Zahra Technical University, Buein Zahra, Qazvin, Iran
Full list of author information is available at the end of the article
Abstract
In this paper, we introduce a new class of generalized nonexpansive mappings. Some new fixed point theorems for these mappings are obtained.
MSC: 47H10
Keywords: monotone mapping; nonexpansive mapping; fixed point;Lp
1 Introduction and preliminaries
A nonexpansive mapping has a Lipschitz constant equal to . The fixed point theory for such mappings is very rich [–] and has many applications in nonlinear functional anal-ysis [].
We first commence some basic concepts about generalization of nonexpansive map-pings as formulated by Suzukiet al.[, ].
Definition [] LetCbe a nonempty subset of a Banach spaceX. We say that a mapping
T:C→Csatisfies condition (C) onCif x–T(x) ≤ x–yimpliesT(x) –T(y) ≤
x–y, forx,y∈C.
Of course, every nonexpansive mapping satisfies condition (C) but the converse is not correct and you can find some counterexamples for it in []. So the class of mappings which has condition (C) is broader than the class of nonexpansive mappings.
In [], condition (C) is generalized as follows.
Definition [] LetCbe a nonempty subset of a Banach spaceXandλ∈(, ). We say that a mappingT:C→Xsatisfies (Cλ)-condition onCifλx–T(x) ≤ x–yimplies T(x) –T(y) ≤ x–y, forx,y∈C.
So ifλ=, we will have condition (C). There are examples that show the converse is false; see [].
In [], monotone nonexpansive mappings are defined inL[, ].
We next review some notions inLp[, ]. All of them can be found in []. Consider the Riesz Banach spaceLp[, ], where
|f(x)|pdx< +∞andp∈(, +∞).
Also, we havef = when the set
x∈[, ] :f(x) = ,
has Lebesgue measure zero. In this case, we sayf = almost everywhere. An element of
Lp[, ] is therefore seen as a class of functions. The norm of anyf ∈Lp[, ] is given by
fp= (
|f(x)|pdx)
p. Throughout this paper, we will writeL
pinstead ofLp[a,b],a,b∈R and · instead of · p.
In this paper, we redefine Definition on a subset of Banach spaceLpand those
theo-rems which are proved in [] generalize to a wider class of monotone (Cλ)-condition with
preserving their fixed point property.
2 Main results
LetCbe a nonempty subset ofLpwhich is equipped with a vector order relation. A map
T:C→Cis called monotone if for allf gwe haveT(f)T(g). We generalize the (Cλ)-condition as follows.
Definition LetCbe a nonempty subset of a Banach spaceLp. Forλ∈(, ), we say that
a mappingT monotone (Cλ)-condition onC ifT is monotone and for allf g,λf –
T(f) ≤ g–fimpliesT(g) –T(f) ≤ g–f.
Note Definition is a generalization of the monotone nonexpansive mapping which is defined in [] as follows.
A mapTis said to be monotone nonexpansive ifTis monotone and forf g, we have
T(g) –T(f) ≤ g–f.
The next example is a direct generalization of monotone nonexpansive mapping.
Example LetC={f ∈Lp[, ] :f(x) =a}, where a∈[, ]. Forf,g∈C, consider the partial order relation
f g iff f(x)≤g(x).
LetT:C→Cbe defined by
T(f) =
, f = ,
, f = .
Then the mappingTsatisfies the monotone (C
)-condition but it fails monotone
nonex-pansiveness. Indeed, wheneverf g, if ≤f(x)≤g(x) < , thenT(f) –T(g) ≤ f–g. On the other hand, ≤f(x) < andg= , so if ≤f(x)≤ andg= , then we have again
T(f) –T(g) ≤ f–g, but if <f(x) < andg= , thenff– . Thus, the map-pingTsatisfying monotone (C
)-condition on [, ].
Letf = . andg= . Thenf gwhileT(f) –T(g)f–g. Thus,Tis not monotone nonexpansive.
The following lemmas will be crucial to prove the main result of this paper.
Lemma Let C be convex and T monotone.Assume that for some f∈C,fT(f).Then
the sequence fndefined by
λ∈(, ),satisfies
fnfn+T(fn)T(fn+).
for n≥.
Proof First, we prove thatfnT(fn). By assumption, we havefT(f). Assume thatfn
T(fn), forn≥. Then we have
fn=λfn+ ( –λ)fnλT(fn) + ( –λ)fn=fn+
i.e. fnfn+. SinceTis monotone,T(fn)T(fn+). We have
fn+=λT(fn) + ( –λ)fnλT(fn) + ( –λ)T(fn) =T(fn). Thus
fnfn+T(fn)T(fn+),
forn≥. The proof is closely modeled on Lemma . of [].
Note that under the assumption of Lemma , if we assumeT(f)f, then we have
T(fn+)T(fn)fn+fn for anyn≥.
A sequence{fn}inCis called an almost fixed point sequence forT, iffn–T(fn) → (a.f.p.s. in short).
Lemma Let T:C→Lp be a monotone(Cλ)-condition mapping and fnbe a bounded
a.f.p.s.for T.Then
lim inf
n fn–T(f)≤lim infnk fn–f,
for f ∈C which fnf andlim infnfn–f> ,for all n≥.
Proof Fixf ∈Csuch thatfnf. Sincefnis an a.f.p.s., for=lim infnfn–f, there isn
such thatfn–T(fn)<, for alln≥n. This implies that
λfn–T(fn)≤fn–T(fn)<<fn–f,
for alln≥n. SinceTsatisfies the monotone (Cλ)-condition, we have
T(fn) –T(f)≤ fn–f, ()
for alln≥n. So by the triangle inequality and (), we have
Thuslim infnfn–T(f) ≤lim infnfn–f. The proof is closely modeled on Lemma of
[].
Lemma [] If{fn}is a sequence of Lp-uniformly bounded functions on a measure space,
and fn→f almost everywhere,then lim inf
n fn
p=lim inf n fn–f
p+fp,
for all p∈(,∞).
In the following, letCbe a nonempty, convex, and bounded set andT :C→Cbe a
monotone (Cλ)-condition, for someλ∈(, ).
Theorem Let f∈C such that fT(f).Then fndefined in()is an a.f.p.s.
Proof Sincefn+=λT(fn) + ( –λ)fn, forn≥, we have
λfn–T(fn)=fn–fn+.
By Lemma , we havefnfn+. Therefore, monotone (Cλ)-condition implies thatT(fn) –
T(fn+) ≤ fn–fn+. Now, we can apply Lemma of [] to conclude thatlimnfn–T(fn)=
.
Example We show thatT, which is defined in Example , has an a.f.p.s. It is easy to see thatCis a nonempty, convex, and bounded subset ofLp. Also, we provedTobeys the monotone (C
)-condition. Moreover, T(). Thus, by Theorem ,Thas an a.f.p.s.
Now, we construct an a.f.p.s. according (). Letf= . Sofn= . Therefore fn–T(fn)= .
Thusfnis an a.f.p.s.
Theorem Let C be compact.Assume there exists f∈C such that fand T(f)are
com-parable.Then T has a fixed point.
Proof Letfnbe a sequence which is defined in (). By Theorem ,fnis an a.f.p.s. SinceC is compact,fnhas a convergent subsequencefnk tof. By triangle inequality, we get
lim inf nk
T(fnk) –T(f)≤lim nk
T(fnk) –fnk+lim inf nk
fnk–T(f).
Sincefnis an a.f.p.s., we have lim inf
nk T(fnk) –T(f)≤lim infnk fnk–T(f). ()
Again, by triangle inequality, we have
lim inf nk
fnk–T(f)≤lim nk
fnk–T(fnk)+lim inf nk
Therefore,
lim inf
nk fnk–T(f)≤lim infnk T(fnk) –T(f). ()
From equations () and (), we have
lim inf
nk fnk–T(f)=lim infnk T(fnk) –T(f). ()
By using the partially order and convergent propertiesfnk f. Lemma implies fnk
fnk+f. Sofnk+–fnk ≤ f –fnk. Sincefnk+–fnk=λ(fnk–T(fnk)), we get
λfnk–T(fnk)=fnk+–fnk. Therefore
λfnk–T(fnk)≤ f–fnk.
Thus the monotone (Cλ)-condition implies
T(fnk) –T(f)≤ fnk–f. ()
Sincefnk is bounded, Lemma implies lim inf
nk fnk–T(f)=lim infnk fnk–f+f–T(f). From equation (), we get
lim inf
nk fnk–f+f–T(f)=lim infnk T(fnk) –T(f). From equation (), we get
lim inf
nk fnk–f+f–T(f)≤lim infnk fnk–f.
This implies thatT(f) =f.
By Theorem , we can see thatTin Example , has a fixed point.
The following example shows that monotone (Cλ)-condition is a direct generalization
of (Cλ)-condition.
Example LetC=co{x,sin(x)}, wherex∈[–π ,
π
]. Define a partial order onCas follows:
f g iff f(x)≤g(x).
LetT:C→Cbe
T(f) =
sin(x) f =x,
SinceCis convex hull of a compact set{x,sin(x)}, so it is a nonempty, convex and compact subset ofLp. Putf =x. Thenf andT(f) are comparable. Also,T obeys the monotone (Cλ)-condition. Thus, by Theorem ,Thas a fixed point.
Note, forλ∈(, ),T does not obey the (Cλ)-condition. Because, forf =xandg=x+
sin(x), we haveλf –T(f) ≤ f –g, butT(g) –T(f)f–g.
Theorem Let C be a weakly compact subset of L.Assume,there is f ∈C such that
fT(f).Then T has a fixed point.
Proof By Theorem ,T has an a.f.p.s.fn. SinceCis weakly compact, there is a weakly convergent subsequencefnk to somef ∈C. Iflim infnkfnk–f= , thenfnk is convergent and we will have the same proof of Theorem . On the other hand, iflim infnkfnk–f> , then by Lemma ,
lim inf
nk fnk–T(f)≤lim infnk fnk–f. ()
We claim thatf =T(f). Because iff =T(f), sinceLsatisfies Opial condition, we have
lim inf nk
fnk–f<lim inf nk
fnk–T(f),
which is a contradiction with inequality ().
This result is a generalization of the original existence theorem in [, ] form monotone nonexpansive to monotone (Cλ)-condition. Therefore this class is bigger and is used to
answer the question asked by T Benavides []: DoesXalso satisfy the fixed point property for Suzuki-type mappings?
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, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran.2Department of Mathematics,
Payame Noor University, Tehran, 19395-3697, Iran.
Acknowledgements
The first author acknowledges Buein Zahra Technical University for supporting this research.
Received: 13 February 2016 Accepted: 20 July 2016 References
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