R E S E A R C H
Open Access
On monotone contraction mappings in
modular function spaces
Monther R Alfuraidan
1, Mostafa Bachar
2*and Mohamed A Khamsi
3*Correspondence:
mbachar@ksu.edu.sa
2Department 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
We prove the existence of fixed points of monotone-contraction mappings in modular function spaces. This is the modular version of the Ran and Reurings fixed point theorem. We also discus the extension of these results to the case of pointwise monotone-contraction mappings in modular function spaces.
MSC: Primary 47H09; secondary 47H10
Keywords: fixed point; modular function space; monotone mappings; pointwise contraction
1 Introduction
The purpose of this paper is to give an outline of a fixed point theory for monotone-contraction mappings defined on some subsets of modular function spaces which are nat-ural generalizations of both function and sequence variants of many important spaces like the Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and many others []. Recently, the authors in [] presented a series of fixed point results for pointwise contractions and asymptotic pointwise contractions acting in mod-ular functions spaces. The current paper operates within the same framework.
The importance for applications of mappings defined within modular function spaces consists in the richness of the structure of modular function spaces, which, besides be-ing Banach spaces, are equipped with modular equivalents of norm or metric notions and also are equipped with almost everywhere convergence and convergence in sub-measure. In many cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural as modular type assump-tions can be more easily verified than their metric or norm counterparts; see for exam-ple []. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in Banach linear spaces and in metric spaces.
The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well developed since the s, see for example [–], and gener-alized to other metric spaces [–], and modular function spaces [, ]. The correspond-ing fixed point results were then extended to larger classes of mappcorrespond-ings like asymptotic mappings [, ], pointwise contractions [], and asymptotic pointwise contractions and nonexpansive mappings [, –].
In recent years, a version of the Banach contraction principle [] was given in partially ordered metric spaces [, ] and in metric spaces with a graph []. In this work, we discuss some of these extensions to modular function spaces.
2 Preliminaries
Let be a nonempty set and be a nontrivialσ-algebra of subsets of. Let P be a
δ-ring of subsets of, such thatE∩A∈P for anyE∈P andA∈. Let us assume that there exists an increasing sequence of setsKn∈P such that=
Kn. ByE we denote the linear space of all simple functions with supports fromP. ByM∞we will denote the space of all extended measurable functions,i.e.all functionsf :→[–∞,∞] such that there exist a sequence{gn} ⊂E,|gn| ≤ |f|andgn(ω)→f(ω) for allω∈. By Awe denote the characteristic function of the setA.
Definition . Letρ:M∞→[,∞] be a nontrivial even function. We say that the regu-lar functionρis pseudomodular if:
(i) ρ() = ;
(ii) ρis monotone,i.e.|f(ω)| ≤ |g(ω)|for allω∈impliesρ(f)≤ρ(g), where
f,g∈M∞;
(iii) ρis orthogonally additive,i.e.ρ(fA∪B) =ρ(fA) +ρ(fB), for anyA,B∈such that A∩B=∅,f ∈M;
(iv) ρhas the Fatou property,i.e.|fn(ω)| ↑ |f(ω)|for allω∈impliesρ(fn)↑ρ(f), wheref ∈M∞;
(v) ρis order continuous inE,i.e.gn∈Eand|gn(ω)| ↓impliesρ(gn)↓.
Similar to the case of measure spaces, we say that a setA∈isρ-null ifρ(gA) = for everyg∈E. We say that a property holdsρ-almost everywhere if the exceptional set is
ρ-null. As usual, we identify any pair of measurable sets whose symmetric difference is
ρ-null as well as any pair of measurable functions differing only on aρ-null set. With this in mind we define
M(,,P,ρ) =f ∈M∞;f(ω)<∞ρ-a.e., (.)
where eachf ∈M(,,P,ρ) is actually an equivalence class of functions equalρ-a.e. rather than an individual function. Where no confusion exists we will writeMinstead of
M(,,P,ρ).
Definition . Letρbe a regular function pseudomodular.
() We say thatρis a regular function semimodular ifρ(αf) = for everyα> implies
f= ρ-a.e.;
() we say thatρis a regular function modular ifρ(f) = impliesf = ρ-a.e. The class of all nonzero regular function modulars defined onwill be denoted by.
Definition .[, , ] Letρbe a function modular.
(a) A modular function space is the vector spaceLρ(,), or brieflyLρ, defined by
Lρ=
f∈M;ρ(λf)→asλ→.
(b) The following formula defines a norm inLρ(frequently called the Luxemburg
norm):
fρ=inf
α> ;ρ(f/α)≤.
In the following theorem we recall some of the properties of modular spaces.
Theorem .[, , ] Letρ∈ .
() (Lρ, · ρ)is complete and the norm · ρis monotone w.r.t.the natural order inM.
() fnρ→if and only ifρ(αfn)→for everyα> .
() Ifρ(αfn)→for anα> then there exists a subsequence{gn}of{fn}such that gn→ρ-a.e.
() ρ(f)≤lim infρ(fn)wheneverfn→f ρ-a.e. (Note:this property is equivalent to the
Fatou property.)
The following definition plays an important role in the theory of modular function spaces [].
Definition . Letρ∈ . We say thatρhas the-property ifsupnρ(fn,Dk)→, when-ever{Dk} ∈is decreasing,k≥Dk=∅, andlimk→+∞supnρ(fn,Dk) = .
Theorem . Letρ∈ .The following conditions are equivalent: (a) ρhas,
(b) ifρ(fn)→thenρ(fn)→,
(c) ifρ(αfn)→for anα> thenfnρ→,i.e. the modular convergence is equivalent
to the norm convergence.
We will also use another type of convergence which is situated between norm and mod-ular convergence. It is defined, among other important terms, in the following.
Definition . Letρ∈ .
(a) We say that{fn}isρ-convergent tofand writefn→ (ρ)if and only ifρ(fn–f)→. (b) A sequence{fn}wherefn∈Lρis calledρ-Cauchy ifρ(fn–fm)→asn,m→ ∞. (c) A setB⊂Lρis calledρ-closed if for any sequence offn∈B, the convergence
fn→f (ρ)implies thatf belongs toB.
(d) A setB⊂Lρis calledρ-bounded ifsup{ρ(f–g);f ∈B,g∈B}<∞.
(e) A setB⊂Lρis called stronglyρ-bounded if there existsβ> such that
sup{ρ(β(f–g));f∈B,g∈B}<∞.
(f ) A setC⊂Lρis calledρ-a.e. closed if for any{fn}inCwhichρ-a.e. converges to somef, then we must havef ∈C.
(g) A setC⊂Lρis calledρ-a.e. compact if for any{fn}inC, there exists a subsequence
Let us note thatρ-convergence does not necessarily implyρ-Cauchy condition. Also, fn→f does not imply in generalλfn→λf,λ> . Using Theorem . it is not difficult to prove the following.
Proposition . Letρ∈ .
(i) Lρisρ-complete.
(ii) Lρis a lattice,i.e.,for anyf,g∈Lρ,we havemax{f,g} ∈Lρandmin{f,g} ∈Lρ.
(iii) ρ-ballsBρ(x,r) ={y∈Lρ;ρ(x–y)≤r}areρ-closed andρ-a.e.closed.
Using the property () of Theorem ., we get the following result.
Theorem . Letρ∈ .Let{fn}be aρ-Cauchy sequence in Lρ.Assume that{fn}is mono-tone increasing,i.e.,fn≤fn+ρ-a.e. (resp.decreasing,i.e.,fn+≤fnρ-a.e.),for any n≥. Then there exists f ∈Lρsuch thatρ(fn–f)→and fn≤f ρ-a.e. (resp.f ≤fnρ-a.e.),for any n≥.
Let us finish this section with the modular definitions of some special mappings, which will be studied throughout. The definitions are straightforward generalizations of their norm and metric equivalents [, –].
Definition . Letρ∈ and letC⊂Lρbe nonempty andρ-closed. A mappingT:C→
Cis said to be monotone increasing (resp. decreasing) ifT(f)≤T(g)ρ-a.e. (resp.T(g)≤ T(f)ρ-a.e.) wheneverf ≤gρ-a.e., for anyf,g∈C.Tis said to be monotone wheneverT is either monotone increasing or decreasing. Moreover,Tis called
(i) a monotone contraction ifT is monotone and there existsk< such that
ρT(f) –T(g)≤kρ(f–g)
wheneverf,g∈Candf ≤gρ-a.e.;
(ii) a pointwise monotone contraction ifTis monotone and for anyf ∈Cthere exists
kf< such that
ρT(f) –T(g)≤kf ρ(f–g)
wheneverg∈Csuch thatf ≤gorg≤f ρ-a.e.;
(iii) an asymptotic pointwise monotone contraction ifTis monotone and for anyf ∈C
andn≥, there exists a constantknf such that
ρTn(f) –Tn(g)≤kfnρ(f –g)
withkf =lim supn→∞kfn< , wheneverg∈Csuch thatf≤gorg≤f ρ-a.e. We shall call a pointf∈Ca fixed point ofTif and only ifT(f) =f.
Example . Let us consider a pharmacokinetics sub-model of ascorbic acid (in shortAA) absorption in the intestines in healthy subjects developed in []; such a model can be written as
dAgut(t)
dt = –
γAgut(t) β+Agut(t)
–KgutAgut(t), t=τn, (.)
Agut
τn+=Agut(τn) +DAAn. (.)
Heren≥ andAgutrepresent the amount ofAAin the intestines (inμmol) and the initial
conditionAgut() =D(inμmol) is given or estimated by a physician at given steady state,
and =τ<τ<τ<· · ·<τn<τn+<· · ·, where τn represent the time episode of oral intake,Agut(τn+) represents the right limit ofAgut(t) at the pointτn. For practical reasons
and without loss of any generality, we will assume that the given dose of oral intakeDAAn
isωperiodic and equal to the constant Dover [, +∞). The description of the model parametersγ,β, andKgut with their units are given in []. Moreover, we find that the
solution of (.)-(.) exists fort≥, and it is positive and bounded from above. In fact (.)-(.) with given initial conditions have a unique solution for allt≥. Now letAgut(t)
be a solution of (.). If there exists at=inf{t> ,t=τn:Agut(t) < }such thatAgut(t) = ,
then we will have dAgutdt(t)|t< and also from (.), we get
>dAgut(t) dt
t
= – γAgut(t)
β+Agut(t)
–KgutAgut(t) = ,
which is a contradiction. Therefore we haveAgut(t)≥, for allt=τn, and by using the impulsive equation (.) we getAgut(t)≥, for allt> .
Now we will show the boundedness ofAgut(t), for allt> . We have from (.)
dAgut(t)
dt ≤–KgutAgut(t), t=τn, A
τn+=Agut(τn) +DAAn,
for anyn≥. Then we have by successive methods
Agut(t)≤
k
i=
DAAi–e
–Kgut(hi+hi++···+hk)+D
AAk
e–Kgut(t–τk–), τ
k–≤t<τk,
Agut(t)≤DAA, ≤t<τ,
wherek≥,hi=τi–τi–> ,∀i= , , . . . , and as we haveDAAiis a finite number, then
Agut(t) is bounded from above. It is easy to check that the solution of (.)-(.) for all
t>τn,n≥, can be written as
Agut(t) =
≤τn<t
DAAne–Kgut(t–τn)–
t
γAgut(s) β+Agut(s)
e–Kgut(t–s)ds
=D
≤τn<t
e–Kgut(t–τn)–
t
γAgut(s) β+Agut(s)
e–Kgut(t–s)ds.
In order to find anω-periodic solution, where the dose of the treatment will be fixed and equal toD, the amount ofAAin the intestines needs to satisfyAgut(t) =Agut(t+ω), for all
Note that by evaluating the periodic conditionAgut(t) =Agut(t+ω), we conclude that
Agut(t) is an ω-periodic solution of (.)-(.) if and only ifAgut is a fixed point of the
following operator:
T(Agut)(t) =D
t≤τn<t+ω
G(t,τn) – t+ω
t
G(t,s) γAgut(s)
β+Agut(s)
ds
=D
t≤τn<t+ω
G(t,τn) – t+ω
t
G(t,s)fAgut(s)
ds,
whereG(t,s) =e–Kgut(t+ω–s)
–e–Kgutω andf(x) =
γx
β+x. Let us consider the following function modular:
ρ(g) = ω
t+ω
t
G(t,s)g(s)ds
dt.
Using the mean-value theorem, we get
≤–T(Agut)(t) –T(Agut)(t)
=
t+ω
t
G(t,s)fAgut(s)
–fAgut(s)
ds
≤γ β
t+ω
t
G(t,s)Agut(s) –Agut(s)
ds,
wheneverAgut≤Agut. HenceT(Agut)≤T(Agut) and
ρT(Agut) –T(Agut)
=
ω
t+ω
t
G(t,s)T(Agut)(s) –T(Agut)(s)
ds dt
≤ωγ
β ρ(Agut–Agut).
Note thatTis continuous. Ifω γ<β, thenT is a monotone decreasing contraction map-ping.
3 Fixed point of monotone contraction mappings
In this section we discuss an analogue of the Ran and Reurings fixed point theorem [] in modular function spaces. The key feature in this fixed point theorem is that the Lip-schitzian condition on the nonlinear map is only assumed to hold on elements that are comparable in the partial order.
Theorem . Letρ∈ .Let C⊂Lρbe nonempty,ρ-closed andρ-bounded.Let T:C→C
be a monotone increasing contraction mapping.Assume that there exists f∈C which is
comparable to T(f),i.e.,f≤T(f)or T(f)≤fρ-a.e.Then{Tn(f)}isρ-Cauchy and ρ-converges tof a fixed point of T¯ .Moreover, if f ∈C is comparable tof¯,then{Tn(f)}
ρ-converges to¯f.
Proof SinceTis a monotone increasing contraction, there existsα< such that
Letf∈Csuch thatfandT(f) are comparable. Assume thatf≤T(f)ρ-a.e. SinceTis
monotone increasing, we getTn(f
)≤Tn+k(f), for anyn≥ andk≥, which implies
ρTn+k(f) –Tn(f)
≤α ρTn+k–(f) –Tn–(f)
≤αnρTk(f) –f
≤αnδρ(C)→,
where δρ(C) =sup{ρ(f –g) :f,g ∈C}<∞, because C is ρ-bounded. Since α < and
δρ(C) <∞, we conclude that{Tn(f)}isρ-Cauchy. Theρ-completeness ofLρimplies the
existence off¯∈Lρsuch thatlimn→∞ρ(Tn(f) –f¯) = . BecauseCisρ-closed, we getf¯∈C.
Since{Tn(f
)}is monotone increasing, the assumptions of Theorem . are satisfied and
will imply thatTn(f
)≤ ¯f ρ-a.e., for anyn≥. In particular, we have
ρTn(f) –T(f¯)
≤α ρTn–(f) –¯f
.
Since
ρ
¯ f–T(f¯)
≤ρ¯f–Tn(f)
+ρTn(f) –T(f¯)
≤ρ¯f–Tn(f)
+α ρTn–(f) –f¯
→, asn→ ∞,
T(f¯) =f¯, which means thatf¯is a fixed point ofT. Finally letf ∈Csuch thatf andf¯are comparable. SinceTis monotone,Tn(f) andTn(f¯) =f¯are comparable. Hence
ρTn(f) –f¯≤αnρ(f–f¯)≤αnδρ(C), for anyn≥.
Sinceα< , we conclude that{Tn(f)}ρ-converges tof¯.
The missing information in Theorem . is the uniqueness of the fixed point. In fact we do have a partial positive answer to this question. Indeed ifh¯andg¯are two fixed points of T such thath¯≤ ¯gρ-a.e., then we must haveh¯=g¯. In generalT may have more than one fixed point.
Remark . The conclusion of Theorem . does not extend easily to the case of
mono-tone decreasing contractions. Note that the mapping T in Example . is monotone
decreasing. But if we assume that there existsf∈C such thatf≤T(f)≤T(f) or
T(f)≤T(f)≤fρ-a.e., then{Tn(f)}isρ-Cauchy andρ-converges tof¯, a fixed point
ofT. Moreover, iff∈Cis comparable tof¯, then{Tn(f)}ρ-converges tof¯.
Next we discuss the case of monotone pointwise Lipschitzian mappings and extend the recent results of [].
Theorem . Let us assume that ρ ∈ . Let C ⊂Lρ be nonempty, ρ-closed and ρ
-bounded. Let T:C→C be a pointwise monotone increasing contraction or asymptotic pointwise monotone increasing contraction.Then any two comparable fixed points of T are equal.Moreover,if fis a fixed point of T,then the orbit{Tn(f)}ρ-converges to f,for
Proof Since every pointwise monotone increasing contraction is an asymptotic pointwise monotone increasing contraction, we can assume thatTis an asymptotic pointwise mono-tone increasing contraction,i.e.
ρTn(f) –Tn(g)≤kfnρ(f –g),
for any f ∈C and n ≥, and any g ∈ C comparable to f, where {kfn} is such that lim supn→∞kfn=kf < . Letf,f∈Cbe two fixed points ofTsuch thatf≤fρ-a.e. Then
we have
ρ(f–f) =ρ
Tn(f) –Tn(f)
≤kfnρ(f–f),
for anyn≥. If we letn→ ∞, we will get
ρ(f–f)≤kfρ(f–f).
Sincekf< andCisρ-bounded, we conclude thatρ(f–f) = ,i.e. f=f. This proves the first part of Theorem .. To prove the convergence, assume thatfis a fixed point ofT. Fix
an arbitraryf ∈Csuch thatf ≤forf≤f ρ-a.e. Let us prove that{Tn(f)}ρ-converges
tof. Indeed sinceTis monotone increasing, thenTn(f) andTh(f) =fare comparable
for anyn,h≥. Hence
ρTn+m(f) –f
=ρTn+m(f) –Tn(f)
≤kfnρTm(f) –f
,
for anyn,m≥. Hence
lim sup m→∞ ρ
Tn+m(f) –f
≤lim sup m→∞ k
n fρ
Tm(f) –f
.
Sincelim supm→∞ρ(Tn+m(f) –f) =lim supm→∞ρ(Tm(f) –f), we get
lim sup m→∞ ρ
Tm(f) –f
≤knf lim sup m→∞ ρ
Tm(f) –f
,
for anyn≥. If we letn→ ∞, we obtain
lim sup m→∞
ρTm(f) –f
≤kf lim sup m→∞
ρTm(f) –f
.
Sincekf< , we getlim supn→∞ρ(Tn(f) –f) = , which implies the desired conclusion
limn→∞ρ(Tn(f) –f) = .
Theorem . does not discuss the existence part of the fixed point. Next we investigate this problem. We will start our discussion with the existence of fixed point results in the case of uniformly continuous function modulars.
Definition . Letρ∈ . We will say that the function modularρis uniformly continuous if for everyε> andL> there existsδ> such that
Let us mention that uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [] or in sequence Orlicz spaces [] the uniform continuity of the Orlicz modular is equivalent to theproperty.
Lemma .[] Letρ∈ be uniformly continuous.Let K⊂Lρ be nonempty,convex,ρ
-closed,andρ-bounded.Then anyρ-typeτ :K→[,∞)isρ-lower semicontinuous in K, where
τ(g) =lim sup
n→∞ ρ(fn–g), for all g∈K,
where{fn} ∈K.
Before we state our next result, we need the following definition of a property that plays in the theory of modular function spaces a role similar to the reflexivity in Banach spaces (seee.g.[]).
Definition .[] We say thatLρ has property (R) if and only if every decreasing
se-quence{Cn}of nonempty,ρ-bounded,ρ-closed, and convex subsets ofLρhas a nonempty
intersection.
Theorem . Assume thatρ∈ is uniformly continuous and has property(R).Assume thatρsatisfies the-property.Let K⊂Lρbe nonempty,convex,ρ-closed,andρ-bounded.
Let T :K→K be a pointwise monotone increasing contraction or asymptotic pointwise monotone increasing contraction.Assume there exists f ∈K such that f and T(f)are com-parable.Then T has a fixed point f∈K.
Proof Since every pointwise monotone increasing contraction is an asymptotic pointwise monotone increasing contraction, we can assume thatTis an asymptotic pointwise mono-tone increasing contraction. In view of Theorem ., it is enough to show thatThas a fixed point. Let us assume thatf ≤T(f)ρ-a.e. The case whenT(f)≤f is done in a similar way and will be omitted. Set
C={g∈K;f≤gρ-a.e.}.
Sinceρ satisfies the-property, Cis nonempty, ρ-closed, and convex. It is clear that
T(C)⊂CsinceTis monotone increasing. Define the functionτ :C→[, +∞) by
τ(g) =lim sup n→∞ ρ
Tn(f) –g.
Note thatτ(g)≤δρ(K), for anyg∈C. Lemma . implies thatτisρ-lower semicontinuous
inC. Sinceρhas property (R), there existsf∈Csuch that
τ(f) =inf
τ(g) :g∈C.
Let us prove thatτ(f) = . Indeed, sincef ≤fρ-a.e., for anyn,m≥ we have
ρTn+m(f) –Tm(f)
≤kfmρTn(f) –f
If we letngo to infinity, we getτ(Tm(f
))≤kfmτ(f), which implies
τ(f) =inf
τ(g) :g∈K≤τTm(f)
≤kfm τ(f).
Passing withmto infinity, we getτ(f)≤kfτ(f), which forcesτ(f) = , askf< . Hence, lim supn→∞ρ(Tn(f) –f
) = holds,i.e.
lim n→∞ρ
Tn(f) –f
= .
Theρ-continuity ofT impliesfis a fixed point ofT.
Remark . As we see in the proof of Theorem ., the lower semicontinuity ofρ-types is crucial. In [], it is proved that if ρ is orthogonally additive, and{fn} ⊂Lρ isρ-a.e.
convergent to such that there existsβ> with
sup n
ρ(βfn) =M<∞,
then for anyg∈Eρ, we havelimn→∞(ρ(fn+g) –ρ(fn) –ρ(g)) = , which implies lim inf
n→∞ ρ(fn+g) =lim infn→∞ ρ(fn) +ρ(g).
We say thatLρhas theρ-a.e. strong Opial property whenever this conclusion is satisfied.
In this case, anyρ-type is lower semicontinuous in the sense of convergenceρ-a.e. This will help prove that ifLρ has theρ-a.e. strong Opial property,K⊂Eρ be a nonempty, ρ-a.e. compact, stronglyρ-bounded, and convex subset, then anyT :K→Kwhich is a pointwise monotone increasing contraction or asymptotic pointwise monotone increas-ing contraction has a fixed point provided there exists f ∈K such thatf andT(f) are comparable.
Indeed since every pointwise monotone increasing contraction is an asymptotic
point-wise monotone increasing contraction we can assume that T is an asymptotic
point-wise monotone increasing contraction. Let us assume thatf ≤T(f)ρ-a.e. The case when T(f)≤f is done in a similar way and will be omitted. Set
C={g∈K;f≤gρ-a.e.}.
ThenCis a nonempty,ρ-a.e. compact, stronglyρ-bounded, and convex subset ofK. Since T is monotone increasing, we haveT(C)⊂C. Defineτ:C→[, +∞) by
τ(h) =lim sup n→∞
ρTn(f) –h.
SinceCisρ-bounded, we haveτ(h) < +∞, for anyh∈C. Using the properties ofρ-types under these assumptions, there existsf∈Csuch that
τ(f) =inf
τ(h);h∈C.
Using the same argument as in the proof of Theorem ., we will get τ(f) = . Hence,
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 & Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia. 2Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia.3Department of
Mathematical Science, The University of Texas at El Paso, El Paso, TX 79968, USA.
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: 21 September 2014 Accepted: 28 January 2015 References
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