R E S E A R C H
Open Access
Asymptotic pointwise contractive type in
modular function spaces
Farhan Golkarmanesh
1*and Shahram Saeidi
2*Correspondence:
1Department of Mathematics,
Science and Research Branch, Islamic Azad University, Tehran, Iran Full list of author information is available at the end of the article
Abstract
In this paper, we introduce asymptotic pointwise contractive type conditions in modular function spaces and present fixed point results for mappings under such conditions.
MSC: 47H09; 47H10; 54H25
Keywords: asymptotic pointwise
ρ-contraction type; modular function space
1 Introduction
The notion of asymptotic pointwise contraction was introduced by Kirk []: Let (M,d) be a metric space. A mappingT :M→Mis called an asymptotic pointwise contraction if there exists a functionα:M→[, ) such that for each integern≥,
dTnx,Tny≤αn(x)d(x,y) for eachx,y∈M,
whereαn→αpointwise onM. Moreover, Kirk and Xu [] proved that ifCis a weakly
compact convex subset of a Banach spaceEandT:C→Can asymptotic pointwise con-traction, thenThas a unique fixed pointv∈C, and for eachx∈C, the sequence of Picard iterates{Tnx}converges in norm tov.
Very recently, Saeidi [] introduced the concept of (weak) asymptotic pointwise contrac-tion type: Let (M,d) be a metric space. A mappingT:M→Mis said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) ifTNis continuous for some integerN≥ and there exists a functionα:M→[, ) such that for eachxinM,
lim sup
n→∞ ysup∈M
dTnx,Tny–αn(x)d(x,y)
≤, (.)
resp. lim inf
n→∞ supy∈M
dTnx,Tny–αn(x)d(x,y)
≤
, (.)
whereαn→αpointwise onM.
It is easy to see that an asymptotic pointwise contraction is of asymptotic pointwise contraction type, but the converse is not true []. The following result was proved in [].
Theorem . []Let C be a nonempty weakly compact subset of a Banach space E,and let T:C→C be a mapping of weak asymptotic pointwise contraction type.Then T has a
unique fixed point v∈C and,for each x∈C,the sequence of Picard iterates{Tnx}converges
in norm to v.
On the other hand, Khamsi and Kozlowski [] studied the concept of asymptotic pointwise contractions in modular function spaces.
In this paper, motivated by Khamsi and Kozlowski [, ] and Saeidi [], we study the notion of asymptotic pointwise contraction type in a modular function space. Moreover, we present fixed results which extend the earlier results in [, ].
2 Preliminaries
Letbe a nonempty set, and letbe a nontrivialσ-algebra of subsets of. LetP be a δ-ring of subsets ofsuch thatE∩A∈P for anyE∈P andA∈. Let us assume that there exists an increasing sequence of setsKn∈Psuch that=
Kn. Byξwe denote the
linear space of all simple functions with supports fromP. ByM∞we denote the space
of all extended measurable function,i.e., all functionf :→[–∞, +∞] such that there exists a sequence{gn} ∈ξ,|gn| ≤ |f|andgn(ω)→f(ω) for allω∈.
By Awe denote the characteristic function of the setA.
Definition .[] Letρ:M∞→[,∞] be a nontrivial, convex and even function. We
say thatρis a regular convex function pseudomodular if: (a) ρ() = ;
(b) ρis monotone,i.e.,|f(ω)| ≤ |g(ω)|for allω∈impliesρ(f)≤ρ(g), where
f,g∈M∞;
(c) ρis orthogonally subadditive,i.e.,ρ(fA∪B)≤ρ(fA) +ρ(fB)for anyA,B∈such
thatA∩B=∅,f ∈M∞;
(d) ρhas the Fatou property,i.e.,|fn(ω)| ↑ |f(ω)|for allω∈impliesρ(fn)↑ρ(f),
wheref ∈M∞;
(e) ρis order continuous inξ,i.e.,gn∈ξand|gn(ω)| ↓impliesρ(gn)↓.
Similarly as in the case of measure spaces, we say that a setA∈isρ-null ifρ(gA) =
for everyg∈ξ. We say that a property holdsρ-almost everywhere if the exceptional set isρ-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 write Minstead of
M(,,P,ρ).
Definition .[, ] Letρbe a regular function pseudomodular;
(a) we say thatρis a regular convex function semimodular ifρ(αf) = for everyα> impliesf= ρ-a.e.;
(b) we say thatρis a regular convex function modular ifρ(f) = impliesf = ρ-a.e.
Definition .[–] Letρbe a convex 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 Luxemburg norm):
fρ=inf
α> ;ρ(f/α)≤.
In the following theorem, we recall some of the properties of modular function spaces that will be used later on in this paper.
Lemma .[–] Letρ∈R.Defining L
ρ={f ∈Lρ;ρ(f,·)is order continuous}and Eρ=
{f ∈Lρ;λf ∈Lρfor everyλ> },we have (i) Lρ⊃Lρ⊃Eρ;
(ii) Eρhas the Lebesgue property,i.e.,ρ(αf,Dk)→,forα> ,f ∈EρandDk↓∅;
(iii) Eρis the closure ofξ (in the sense of · ρ).
Definition .[, ] Letρ∈R.
(a) We say that{fn}isρ-convergent tof and writefn→f(ρ)if and only ifρ(fn–f)→.
(b) A sequence{fn}wherefn∈Lρis calledρ-Cauchy ifρ(fn–fm)→asm,n→ ∞.
(c) A setC⊂Lρis calledρ-closed if for any sequence{fn}inC, the convergence
fn→f(ρ)implies thatf belongs toC.
(d) A setC⊂Lρis calledρ-bounded ifsup{ρ(f–g);f ∈C,g∈C}<∞.
(e) For a setC⊂Lρ, the mappingT:C→Cis calledρ-continuous iffn→f(ρ), then
T(fn)→T(f)(ρ).
(f ) A setC⊂Lρis calledρ-a.e. closed if for any sequence{fn}inCwhichρ-a.e.
converges to somef, then we must havef ∈C.
(g) A setC⊂Lρis calledρ-a.e. compact if for any sequence{fn}inC, there exists a
subsequence{fnk}whichρ-a.e. converges to somef ∈C.
(h) Letf ∈LρandC⊂Lρ. Theρ-distance betweenf andCis defined as
dρ(f,C) =inf
ρ(f –g);g∈C.
Let us recall thatρ-convergence does not necessarily implyρ-Cauchy condition. Also,
fn→f does not imply in generalλfn→λf,λ> .
Definition .[] We say thatLρhas the property (R) if and only if every nonincreasing sequence {Cn} of nonempty,ρ-bounded,ρ-closed, convex subsets ofLρ has nonempty intersection.
Definition .[] We say that the function modularρis uniformly continuous if for every
> andL> , there existsδ> such that
Definition .[] A functionλ:C→[,∞], whereC⊂Lρis nonempty andρ-closed, is calledρ-lower semicontinuous if for anyα> , the setCα={f ∈C;λ(f)≤α}isρ-closed.
It can be proved thatρ-lower semicontinuity is equivalent to the condition
λ(f)≤lim inf
n→∞ λ(fn) providedf,fn∈Candρ(f –fn)→.
The following result plays an important role in the proof of the main results.
Lemma .[] Assume thatρ∈Rhas the property(R).Let C⊂Lρbe nonempty,convex, ρ-closed andρ-bounded.Ifϕ:C→[,∞)is aρ-lower semicontinuous convex function,
then there exists x∈C such that
ϕ(x) =inf
ϕ(x);x∈C.
Let us recall the notion ofρ-type.
Definition .[] LetC⊂Lρbe convex andρ-bounded. A functionτ:C→[,∞) is called a (ρ)-type (or shortly a type) if there exists a sequence{ym}of elements ofCsuch
that for anyz∈C, the following holds:
τ(z) =lim sup
m→∞
ρ(ym–z).
Lemma .[] Letρ∈Rbe uniformly continuous.Let C⊂Lρ be nonempty,convex,
ρ-closed andρ-bounded.Then anyρ-typeτ:C→[,∞)isρ-lower semicontinuous in C.
3 Asymptotic pointwise contractive type conditions in modular function spaces
Definition . [] Letρ∈RandC⊂Lρ be non-empty andρ-closed. A mapping T :
C→Cis called an asymptotic pointwise mapping if there exists a sequence of mappings αn:C→[, ] such that
ρTnf –Tng≤αn(f)ρ(f–g) for anyf,g∈C.
(a) If{αn}converges pointwise toα:C→[, ), thenT is called asymptotic pointwise
ρ-contraction.
(b) Iflim supn→∞αn(f)≤for anyf ∈C, thenTis called asymptotic pointwise
nonexpansive.
(c) Iflim supn→∞αn(f)≤kfor anyf ∈Cwith <k< , thenTis called strongly
asymptotic pointwiseρ-contraction.
Khamsi and Kozlowski proved the following results in modular function spaces.
Theorem .[] Let C⊂Lρbe nonempty,ρ-closed andρ-bounded.Let T:C→C be an
Theorem .[] Let us assume thatρ∈Ris uniformly continuous and has the prop-erty(R).Let C⊂Lρbe nonempty,convex,ρ-closed andρ-bounded.Let T:C→C be an
asymptotic pointwiseρ-contraction.Then T has a unique fixed point x∈C.Moreover,
the orbit{Tnx}isρ-convergent to x
for any x∈C.
Below, we introduce the notion of asymptotic pointwiseρ-contraction type in modular function spaces.
Definition . LetC⊂Lρbe nonempty,ρ-bounded andρ-closed. A mappingT:C→C is said to be of asymptotic pointwiseρ-contraction type (resp. of weak asymptotic point-wiseρ-contraction type) ifTN isρ-continuous for some integerN≥ and there exists a
functionα:C→[, ) such that, for eachxinC,
lim sup
n→∞
sup
y∈C
ρTnx–Tny–αn(x)ρ(x–y)
≤, (.)
resp. lim inf
n→∞ supy∈C
ρTnx–Tny–αn(x)ρ(x–y)
≤
, (.)
whereαn→αpointwise onM.
Taking
rn(x) =sup y∈M
ρTnx–Tny–αn(x)ρ(x–y)
∈R+∪ {∞},
it can be easily seen from (.) (resp. (.)) that
lim
n→∞rn(x) = , (.)
resp. lim inf
n→∞ rn(x)≤
(.)
for allx∈M, and
ρTnx–Tny≤αn(x)ρ(x–y) +rn(x). (.)
We will obtain fixed point results for these mappings in modular function spaces. First, it is worth mentioning that theρ-limit of any ρ-convergent sequence in Lρ is unique. This fact follows from the following reasoning: Assume thatρ(un–u)→ and
ρ(un–v)→. Then
ρ
u–v
≤
ρ(u–un) +
ρ(v–un)→, which implies thatu=v.
The following theorem is our main result.
Theorem . Letρ∈Rbe uniformly continuous and have the property(R).Let C⊂Lρbe
Proof Fix anx∈Cand define a functionτby
τ(u) =lim sup
n→∞ ρ
Tnx–u, u∈C.
By Lemma .,τisρ-lower semicontinuous inC. By Lemma ., then there existsx∈C such that
τ(x) =inf
τ(x);x∈C.
Let us prove thatτ(x) = . Indeed, for anyn,m≥, we have
τTmx
=lim sup
n→∞ ρ
Tnx–Tmx
=lim sup
n→∞ ρ
Tm+nx–Tmx
=lim sup
n→∞ ρ
TmTnx–Tmx
≤lim sup
n→∞ αm(x)ρ
Tnx–x
+rm(x) =αm(x)τ(x) +rm(x),
which implies
τ(x) =inf
τ(x);x∈C≤τTmx
≤αm(x)τ(x) +rm(x). (.)
Since T is of weak asymptotic pointwise ρ-contraction type, by (.) we have
lim infn→∞rm(x)≤. Thus, for a subsequence{rmk(x)}of{rm(x)}, we have
lim
k→∞rmk(x)≤. (.)
Now, by (.) and (.), we obtain
τ(x)≤lim inf
k→∞
αmk(x)τ(x) +rmk(x)
=α(x)τ(x),
which forcesτ(x) = asα(x) < . Henceρ(Tnx–x)→ asn→ ∞. From this and the continuity ofTN, for someN≥, it follows thatρ(TN+nx–TNx)→ asn→ ∞. Since theρ-limit of anyρ-convergent sequence is unique, we must haveTNx=x, namely,x is a fixed point ofTN. Now, repeating the above proof forx
instead ofx, we deduce that
Tnx
isρ-convergent to a membervofC;i.e.,ρ(Tnx–v)→. ButTkNx=xfor all
k≥. Hence,v=xand thenTnx→x(ρ).
We show thatTx=x; for this purpose, consider an arbitrary> . Then there exists a
k> such thatρ(Tnx–x) <for alln>k. So, by choosing a natural numberk>k/N, we obtain
ρ(Tx–x) =ρ
TTkNx
–x
=ρTkN+x–x
<.
It is easy to verify thatTcan have only one fixed point. Indeed, ifu,v∈Care fixed points ofT, then by (.), we have
ρ(u–v) =ρTnu–Tnv≤αn(u)ρ(u–v) +rn(u), ∀n≥.
Takinglim infin the above inequality, we obtain
ρ(u–v)≤α(u)ρ(u–v).
Sinceα(u) < andρ∈R, we immediately getu=v.
Next, using theρ-a.e. strong Opial property of the function modular, we prove a fixed point theorem which does not assume the uniform continuity ofρ.
Definition .[, ] We say thatLρsatisfies theρ-a.e. strong Opial property (or shortly SO-property) if for every{fn} ∈Lρwhich isρ-a.e. convergent to zero such that there exists aβ> for which
supρ(βfn)
<∞,
the following equality holds for anyg∈Lρ:
lim inf
n→∞(fn+g) =lim infn→∞ ρ(fn) +ρ(g).
Lemma .[] Letρ∈R.Assume that Lρhas theρ-a.e.strong Opial property.Let C⊂
Eρbe a nonempty,ρ-a.e.compact subset such that there existsβ> such thatδρ(βC) =
sup{ρ(β(x–y));x,y∈C}<∞.Let D⊂C be a nonemptyρ-a.e.closed subset.For any n≥,
letλn:D→[,∞)be such that for any y∈D,there exists a sequence{yn} ⊂C such that,
for every n≥,the following holds:
λn(y) –
n≤ρ(y–yn),
andρ(x–yn)≤λn(x)for every x∈D and n≥.Letλ(x) =lim supn→∞λn(x)for any x∈D.
Then there exists x∈D at whichλattains infimum,i.e.,
λ(x) =inf
λ(x);x∈D.
Theorem . Letρ∈R.Assume that Lρhas theρ-a.e.strong Opial property.Let C⊂Eρ
be a nonemptyρ-a.e.compact convex subset such thatδρ(βC) =sup{ρ(β(x–y));x,y∈C}<
∞for someβ> .Then any T:C→C of weak asymptotic pointwiseρ-contraction type has a unique fixed point x∈C.Moreover,the orbit{Tnx}isρ-convergent to xfor any
x∈C.
Proof Fix anx∈Cand define a functionτby
τ(u) =lim sup
n→∞
By Lemma . applied withλ(u) =τ(u),D=C,λn(u) =ρ(Tnx–u), and withyn=Tnx
chosen for allu∈C, there existsx∈Csuch that
τ(x) =inf
τ(x);x∈C.
The rest of the proof is like the one used for Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.2Department of
Mathematics, University of Kurdistan, Sanandaj, 416, Iran.
Acknowledgements
The authors are grateful to the referees for their careful reading of the paper and several helpful comments.
Received: 9 September 2012 Accepted: 11 March 2013 Published: 17 April 2013
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doi:10.1186/1687-1812-2013-101
Cite this article as:Golkarmanesh and Saeidi:Asymptotic pointwise contractive type in modular function spaces.
