R E S E A R C H
Open Access
A Pata-type fixed point theorem in modular
spaces with application
Mohadeseh Paknazar
1*, Madjid Eshaghi
2, Yeol Je Cho
3and Seyed Mansour Vaezpour
4*Correspondence: [email protected] 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 present a Pata-type fixed point theorem in modular spaces which generalizes and improves some old results. As an application, we study the existence of solutions of integral equations in modular function spaces.
MSC: Primary 47H10; secondary 46A80; 45G10
Keywords: fixed point; modular spaces; nonlinear integral equations
1 Introduction and preliminaries
In Nakano [] introduced the theory of modular spaces in connection with the the-ory of ordered spaces. Musielak and Orlicz [] in redefined and generalized it to obtain a generalization of the classical function spacesLp. Khamsiet al.[] investigated the fixed point results in modular function spaces. There exists an extensive literature on the topic of the fixed point theory in modular spaces (see, for instance, [–]) and the papers referenced there.
Recently, Pata [] improved the Banach principle. Using the idea of Pata, we prove a fixed point theorem in modular spaces. Then we show how our results generalize old ones. Also, we prepare an application of our main results to the existence of solutions of integral equations in Musielak-Orlicz spaces.
In the first place, we recall some basic notions and facts about modular spaces.
Definition . LetXbe an arbitrary vector space overK(=RorC).
(a) A functionρ:X→[, +∞]is called a modular if (i) ρ(x) = if and only ifx= ;
(ii) ρ(αx) =ρ(x)for every scalarαwith|α|= ;
(iii) ρ(αx+βy)≤ρ(x) +ρ(y)ifα+β= andα≥,β≥
for allx,y∈X. (b) If (iii) is replaced by
(iv) ρ(αx+βy)≤αρ(x) +βρ(y)ifα+β= andα≥,β≥, we say thatρis convex modular.
(c) A modularρdefines a corresponding modular space,i.e., the vector spaceXρgiven by
Xρ=
x∈X:ρ(λx)→asλ→.
Example . Let (X, · ) be a norm space, then · is a convex modular onX. But the converse is not true.
In general the modularρ does not behave as a norm or a distance because it is not subadditive. But one can associate to a modular theF-norm (see []).
Definition . The modular spaceXρcan be equipped with theF-norm defined by
|x|ρ=inf
α> ;ρ
x
α
≤α
.
Namely, ifρis convex, then the functional
xρ=inf
α> ;ρ
x
α
≤
,
is a norm called the Luxemburg norm inXρwhich is equivalent to theF-norm| · |ρ.
Definition . LetXρbe a modular space. (a) A sequence{xn}n∈NinXρis said to be:
(i) ρ-convergent toxifρ(xn–x)→asn→ ∞. (ii) ρ-Cauchy ifρ(xn–xm)→asn,m→ ∞.
(b) Xρisρ-complete if everyρ-Cauchy sequence isρ-convergent.
(c) A subsetB⊆Xρis said to beρ-closed if{xn}n∈N⊂Bwithxn→x, thenx∈B. (d) A subsetB⊆Xρis calledρ-bounded if
δρ(B) =sup
ρ(x–y) :x,y∈B<∞,
whereδρ(B)is called theρ-diameter ofB. (e) We say thatρhas the Fatou property if
ρ(x–y)≤lim infρ(xn–yn),
wheneverρ(xn–x)→,ρ(yn–y)→asn→ ∞. (f ) ρis said to satisfy the-condition if
ρ(xn)→ ⇒ ρ(xn)→ (asn→ ∞).
It is easy to check that for every modularρandx,y∈Xρ, () ρ(αx)≤ρ(βx)for eachα,β∈R+withα≤β, () ρ(x+y)≤ρ(x) +ρ(y).
Now we recall some basic concepts about modular function spaces as formulated by Kozlowski [].
Letbe a nonempty set and letbe a nontrivialσ-algebra of subsets of. LetPbe a
δ-ring of subsets ofsuch thatE∩A∈Pfor anyE∈P andA∈. Let us assume that there is an increasing sequence of setsKn∈P such that=
Kn.
In other words, the familyPplays the role ofδ-ring of subsets of finite measure. ByE we denote the linear space of all simple functions with supports fromP.
ByMwe denote the space of all measurable functions,i.e., all functionsf :→Rsuch that there exists a sequence{gn} ∈E,|gn| ≤ |f|andgn(w)→f(w) for allw∈. By Awe
Definition . A functionρ:E×→[, +∞] is called a function modular if
(i) ρ(,E) = for anyE∈;
(ii) ρ(f,E)≤ρ(g,E)whenever|f(w)| ≤ |g(w)|for anyw∈,f,g∈EandE∈; (iii) ρ(f,·) :→[, +∞]is aσ-sub-additive measure for everyf ∈E;
(iv) ρ(α,A)→asαdecreases to for everyA∈P, whereρ(α,A) =ρ(αA,A); (v) for anyα> ,ρ(α,·)is order continuous onP, that is,ρ(α,An)→if{An} ∈P and
decreases toφ.
The definition ofρis then extended tof ∈Mby
ρ(f,E) =supρ(g,E);g∈E, g(w) ≤ f(w) ,w∈. For simplicity, we writeρ(f) instead ofρ(f,).
One can verify that the functionalρ:M→[, +∞] is a modular in the sense of Defini-tion .. The modular space determined byρwill be called a modular function space and will be denoted byLρ. Recall that
Lρ=
f ∈M:lim
α→ρ(αf) =
.
Example . () The Orlicz modular is defined for every measurable real functionf by the formula
ρ(f) =
R
ϕ f(t) dμ(t),
whereμdenotes the Lebesgue measure inRandϕ:R→[,∞) is continuous. We also assume thatϕ(u) = if and only ifu= andϕ(t)→ ∞ast→ ∞.
The modular space induced by the Orlicz modular, is a modular function space and is called the Orlicz space. () The Musielak-Orlicz modular spaces (see []).
Let
ρ(f) =
ϕω, f(ω) dμ(ω),
whereμis aσ-finite measure onandϕ:×R→[,∞) satisfy the following:
(i) ϕ(ω,u)is a continuous even function ofu, which is non-decreasing foru> , such thatϕ(ω, ) = ,ϕ(ω,u) > foru= andϕ(ω,u)→ ∞asu→ ∞;
(ii) ϕ(ω,u)is a measurable function ofωfor eachu∈R; (iii) ϕ(ω,u)is a convex function ofufor eachω∈.
It is easy to check thatρ is a convex modular function and the corresponding modular space is called the Musielak-Orlicz space and is denoted byLϕ.
In the following we give some notions which will be used in the next sections.
Definition .(Khamsi []) LetCbe a subset of a modular function spaceLρ. A mapping
T:C→Cis calledρ-strict contraction if there existsλ< such that
Theorem .(Khamsi []) Let C be aρ-complete,ρ-bounded subset of Lρ and let T :
C→C be aρ-strict contraction.Then T has a unique fixed point z∈C.Moreover,z is the
ρ-limit of the iterate of any point in C under the action of T.
Definition .(Taleb and Hanebaly []) The functionu:I→Lϕ, whereI= [,A] for all
A> , is said to be continuous att∈Iif fortn∈Iandtn→t, thenρ(u(tn) –u(t))→ as
n→ ∞.
If we consider the Musielak-Orlicz modular with-condition, then the continuity ofu
attis equivalent to
(tn→t) ⇒ u(tn) –u(t)ρ→ (asn→ ∞).
LetCϕ=C(I,Lϕ) be the space of all continuous mappings fromI= [,A] intoLϕ.
Proposition .(Taleb and Hanebaly []) Suppose that the Musielak-Orlicz modular
ρ satisfies-condition and B⊂Lϕis aρ-closed and convex subset of Lϕ.For a≥,let
ρa(u) =sup{e–atρ(u(t)) :t∈I}for u∈Cϕ,then () (Cϕ,ρ
a)is a modular space,andρais a convex modular satisfying the Fatou property
and the-condition;
() Cϕisρ
a-complete;
() Cϕ=C(I,B)is aρa-closed,convex subset ofCϕ. 2 Main results
LetXρbe a modular function space,Cbe a nonempty,ρ-complete andρ-bounded subset
ofXρ,xbe an arbitrary point inCand letψ: [, +∞)→[, +∞) be an increasing function
vanishing with continuity at zero. Also, consider the vanishing sequence depending on
α≥,wn(α) = (αn)αnk=ψ(αk). LetT:C→Cbe a mapping. For notational purposes, we
defineTn(x),x∈X
ρandn∈ {, , , . . .}inductively byT(x) =xandTn+(x) =T(Tn(x)).
Theorem . Letα≥,β> and k≥be fixed constants.If the inequality
ρ(Tx–Ty)≤( –)ρ(x–y) +αψ()ρ(x–y) +kβ (.) is satisfied for every∈[, ]and every x,y∈C,then T has a unique fixed point z=T(z) which is theρ-limof the iterate of xunder the action of T.
Proof We first show existence. Let= in (.), thus we get
ρ(Tx–Ty)≤ρ(x–y) (.)
for allx,y∈C. We construct a sequence{xn}∞n=such thatxn=T(xn–) for alln∈N. Now
we claim{xn}isρ-Cauchy sequence inC. By (.), (.) for allm,n∈N, we have
ρ(xn+m–xn)≤( –)ρ(xn+m––xn–) +αψ()
ρ(xn+m––xn–) +k
β
. (.)
LetM:= (δρ(C) +k)β. SinceCisρ-bounded,Mis finite and from (.) we have
Letting= – (nn+)α, we have≤ α
n+. Keeping in mind thatψ is an increasing function,
ρ(xn+m+–xn+)≤
nα
(n+ )αρ(xn+m–xn) +
αα
(n+ )αψ
α
n+
M
⇒ (n+ )αρ(xn+m+–xn+)≤nαρ(xn+m–xn) +ααψ
α
n+
M. (.)
Lettingrn:=nαρ(xn+m–xn), we have from (.)
rn+≤rn+ααψ
α
n+
M
≤rn–+ααψ
α
n
M+ααψ
α
n+
M
.. .
≤r+ααM n+ k= ψ α k
=ααM
n+ k= ψ α k . Therefore
ρ(xn+m–xn)≤
α n α M n k= ψ α k
=Mwn(α). (.)
Taking limit asn→ ∞from both sides of (.), we getρ(xn+m–xn)→ asn→ ∞. Then
{xn}isρ-Cauchy sequence inC. SinceCisρ-complete, there existsz∈Csuch thatρ(xn–
z)→ asn→ ∞. From (.) we get
ρ
Tz–z
≤ρ(Tz–xn) +ρ(xn–z)
≤( –)ρ(z–xn–) +αψ()
ρ(z–xn–) +k
β
+ρ(xn–z).
Taking limit as→ afterwards asn→ ∞, we get
ρ
Tz–z
≤ρ(z–xn–) +ρ(xn–z)→.
ThenTz=z. On the other hand, by (.), we have
ρz–Tnx
=ρTz–Tnx
=ρ(z–xn+)
= lim
m→∞ρ(xm+n+–xn+) ≤Mwn(α)→ (asn→ ∞).
To show uniqueness, we suppose thatyis another fixed point ofT. Then from (.) we have
ρ(z–y) =ρ(Tz–Ty)≤( –)ρ(z–y) +αψ()ρ(z–y) +kβ. (.)
Thenρ(z–y)≤α–ψ()(ρ(z–y) +k)β→ as→, thereforez=y.
If for each∈(, ] strict inequality occurs in (.), then
–αρ(z–y) <ψ()ρ(z–y) +kβ.
Taking limit as→, we get contradiction unlessρ(z–y) = .
Remark . Theorem . is stronger than Theorem .. Indeed, with the hypothesis of Theorem ., if for eachf,g∈Candλ∈(, ), we have
ρ(Tf –Tg)≤λρ(f–g),
then byα=β= ,k= and
ψ() =
γγ
( +γ)+γ( –λ)γ
γ
for arbitraryγ > , we get
ρ(Tf –Tg)≤( –)ρ(f –g) +ψ()ρ(f–g)
is satisfied for every∈[, ]. Thus from Theorem .,Thas a unique fixed pointzwhich is theρ-limofTnf
for an arbitrary pointfinC.
3 Application
In this section, we study the existence of solution of the following integral equation:
u(t) =e–tf+
t
es–tTu(s)ds, (.)
where
(H) T:B→Bisρ-Lipschitz,i.e.,
∃κ> , ρ(Tu–Tv)≤κρ(u–v) (u,v∈B);
(H) Bis aρ-closed,ρ-bounded, convex subset of the Musielak-Orlicz spaceLϕsatisfying
the-condition;
(H) f∈Bis fixed.
Theorem . Under the conditions(H)-(H),for all A> ,integral equation(.)has a
Proof Define the operatorSonCϕby
Su(t) =e–tf+
t
es–tTu(s)ds
for allt∈I:= [,A].
st step. First we show thatS:Cϕ→Cϕ. Let u∈Cϕ andtn,t∈I for alln∈Nwith
tn→tasn→ ∞. We knowuisρ-continuous thusρ(u(tn) –u(t))→. From (H) we
getρ(Tu(tn) –Tu(t))→ asn→ ∞, thusTuisρ-continuous at t. By-condition
Tuis · ρ-continuous att, thereforeSuis · ρ-continuous attand consequently is
ρ-continuous att. Also, we have
t
es–tTu(s)ds∈
t
es–tds
coTu(s); ≤s≤t⊆ –e–tcoB,
wherecoBis a closed convex hull ofBin (Lϕ, · ρ).
ButBis convex andρ-closed, thencoB=B⊆Bρ=B, hence
Su(t)∈e–tB+ –e–tB⊆B (∀t∈I).
nd step. We show thatCϕisρa-complete andρa-bounded.
By Proposition ., Cϕ is aρa-closed subset of ρa-complete spaceCϕ, henceC ϕ
is
ρa-complete too.
Now letu,v∈Cϕ. By st stepu(t),v(t)∈Bfor allt∈I, then
ρa(u–v) =sup
e–atρu(t) –v(t);t∈I≤δρ(B) <∞,
therefore
δρa
Cϕ=supρa(u–v);u,v∈Cϕ
<∞.
rd step. Foru,v∈Cϕ, we have
ρa(Su–Sv)≤κ
–e–(+a)A
+a
ρa(u–v). (.)
Letw∈Cϕand{t
,t, . . . ,tn}be any division of [,t].
Now suppose
sup|ti+–ti|,i= , , . . . ,n–
→
asn→ ∞, then
n–
i=
(ti+–ti)eti–tw(ti) –
t
es–tw(s)ds
ρ
→.
By-condition,
ρ
n–
i=
(ti+–ti)eti–tw(ti) –
t
es–tw(s)ds
Using the Fatou property, we get
ρ
t
es–tw(s)ds
≤lim infρ
n–
i=
(ti+–ti)eti–tw(ti)
. (.) Furthermore, n– i=
(ti+–ti)eti–t≤
t
es–tds≤ –e–t≤ –e–A< .
By the convexity ofρ, we have
ρ
n–
i=
(ti+–ti)eti–tw(ti)
≤
n–
i=
(ti+–ti)eti–tρ
w(ti)
=
n–
i=
(ti+–ti)eti–teatie–atiρ
w(ti)
≤
n–
i=
(ti+–ti)e(+a)ti–tρa(w)
≤
t
e(+a)s–tds
ρa(w).
It follows from (.) that
ρ
t
es–tw(s)ds
≤
eat–e–t
+a
ρa(w). (.)
On the other hand,
ρSu(t) –Sv(t)=ρ
t
es–tTu(s) –Tv(s)ds.
Thus by (.), we have
ρSu(t) –Sv(t)≤
eat–e–t
+a
ρa(Tu–Tv),
sinceTisρ-Lipschitz, we have
ρSu(t) –Sv(t)≤
eat–e–t
+a
sup t∈I
e–atρTu(t) –Tv(t)
≤
eat–e–t
+a
κsup t∈I
e–atρu(t) –v(t)
=
eat–e–t +a
Therefore
e–atρSu(t) –Sv(t)≤κ
–e–(+a)t
+a
ρa(u–v)
≤κ
–e–(+a)A
+a
ρa(u–v)
for allt∈I, which implies (.).
th step. Letα=β= ,k= ,a> with
e–(+a)A>κ– ( +a)
κ .
If we have
κ( –e–(+a)A)
+a ≤( –) +
+γK
for allγ> ,∈[, ] and a constantK, then (.) implies that the inequality (.) is satis-fied byψ() =Kγ. To this end, we define
F() = ( –) ++γK–κ( –e
–(+a)A)
+a .
Now imposing the conditions onF, which implies ≤F() for all∈[, ], we obtain
K= γ
γ( +a)γ
(( +a)( +γ)+γ –κ( +γ)+γ( –e–(+a)A))γ
.
Therefore, from steps to and Theorem ., we conclude the existence of a fixed point ofSwhich is the solution of integral equation (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All 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, Semnan University, P.O. Box 35195-363, Semnan, Iran.3Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea. 4Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Hafez Ave., P.O. Box 15914, Tehran, Iran.
Received: 28 November 2012 Accepted: 17 May 2013 Published: 31 October 2013 References
1. Nakano, H: Modular Semi-Ordered Spaces. Tokyo Mathematical Book Series. Maruzen, Tokyo (1950) 2. Musielak, J, Orlicz, W: On modular spaces. Stud. Math.18, 49-65 (1959)
3. Khamsi, MA, Kozlowski, WM, Reich, S: Fixed point theory in modular function spaces. Nonlinear Anal.14(11), 935-953 (1990)
4. Taleb, A, Hanebaly, E: A fixed point theorem and its application to integral equations in modular function spaces. Proc. Am. Math. Soc.128(2), 419-426 (2000)
5. Dhompongsa, S, Domínguez Benavides, T, Kaewcharoen, A, Panyanak, B: Fixed point theorems for multivalued mappings in modular function spaces. Sci. Math. Jpn.63(2), 161-169 (2006)
7. Khamsi, MA, Latif, A, Al-Sulami, H: KKM and KY fan theorems in modular function spaces. Fixed Point Theory Appl. 2011, 57 (2011). doi:10.1186/1687-1812-2011-57
8. Farajzadeh, AP, Mohammadi, MB, Noor, MA: Fixed point theorems in modular spaces. Math. Commun.16, 13-20 (2011)
9. Hussain, N, Khamsi, MA, Latif, A: Banach operator pairs and common fixed points in modular function spaces. Fixed Point Theory Appl.2011, 75 (2011). doi:10.1186/1687-1812-2011-75
10. Abbas, M, Khan, SH, Razani, A: Fixed point theorems of expansive type mappings in modular function spaces. Fixed Point Theory12(2), 235-240 (2011)
11. Kozlowski, WM: Advancements in fixed point theory in modular function spaces. Arab. J. Math.1, 477-494 (2012). doi:10.1007/s40065-012-0051-0
12. Al-Mezel, SA, Al-Roqi, A, Khamsi, MA: One-local retract and common fixed point in modular function spaces. Fixed Point Theory Appl.2012, 109 (2012). doi:10.1186/1687-1812-2012-109
13. Mongkolkeha, C, Kumam, P: Some fixed point results for generalized weak contraction mappings in modular spaces. Int. J. Anal.2013, 247378 (2013). doi:10.1155/2013/247378
14. Bin Dehaish, BA, Kozlowski, WM: Fixed point iteration processes for asymptotic pointwise nonexpansive mapping in modular function spaces. Fixed Point Theory Appl.2012, 118 (2012). doi:10.1186/1687-1812-2012-118
15. Pata, V: A fixed point theorem in metric spaces. J. Fixed Point Theory Appl.10, 299-305 (2011). doi:10.1007/s11784-011-0060-1
16. Musielak, J: Orlicz Spaces and Modular Spaces. Lecture Notes in Math., vol. 1034. Springer, Berlin (1983) 17. Kozowski, WM: Modular Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 122.
Dekker, New York (1988)
18. Khamsi, MA: A convexity property in modular function spaces. Math. Jpn.44(2), 269-279 (1996)
doi:10.1186/1687-1812-2013-239
