R E S E A R C H
Open Access
On the fixed points of nonexpansive
mappings in modular metric spaces
Afrah AN Abdou
1*and Mohamed A Khamsi
2,3*Correspondence:
1Department of Mathematics, King
Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract
The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, have been recently introduced. In this paper we investigate the existence of fixed points of modular nonexpansive mappings. We also discuss some compactness properties of the family of admissible sets in modular metric spaces with uniform normal structure property.
MSC: Primary 47H09; secondary 46B20; 47H10; 47E10
Keywords: nonexpansive mapping;
2-condition; fixed point; modular function space; modular metric spaces; Orlicz spaces
1 Introduction
The purpose of this paper is to give an outline of fixed point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant ) on subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others. Modular metric spaces were introduced in [, ]. The main idea be-hind this new concept is the physical interpretation of the modular. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) ‘field of (generalized) velocities’: to each ‘time’λ> (the absolute value of ), an average velocity
ωλ(x,y) is associated in such a way that in order to cover the ‘distance’ between points x,y∈X, it takes timeλto move from x to y with velocityωλ(x,y). But the way we
ap-proach the concept of modular metric spaces is different. Indeed, we look at these spaces as a nonlinear version of the classical modular spaces, introduced by Nakano [], on vector spaces and modular function spaces, introduced by Musielak [] and Orlicz [].
In recent years, there was an increasing interest in the study of electrorheological fluids, sometimes referred to as ‘smart fluids’ (for instance, lithium polymetachrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces,Lp andW,p, wherepis a fixed constant, is not adequate, but rather the exponentpshould be able to vary [, ]. One of the most interesting problems in this setting is the famous Dirichlet energy problem [, ]. The classical technique used so far in studying this prob-lem is converting the energy functional, naturally defined by a modular, to a convoluted and complicated problem which involves a norm (the Luxemburg norm). The modular metric approach is more natural and has not been used extensively.
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. In recent years, there has been a great interest in the study of the fixed point property in modular func-tion spaces after the first paper [] was published in . More recently, the authors presented a fixed point result for pointwise nonexpansive and asymptotic pointwise non-expansive acting in modular functions spaces []. The theory of nonnon-expansive mappings defined on convex subsets of Banach spaces has been well developed since the s (see, e.g., Belluce and Kirk [], Browder [], Bruck [], and Lim []) and generalized to other metric spaces (see,e.g., [–]) and modular function spaces (see,e.g., []). The corre-sponding fixed point results were then extended to larger classes of mappings like wise contractions and asymptotic pointwise contractions [–], and asymptotic point-wise nonexpansive mappings []. In [], Penot presented an abstract version of Kirk’s fixed point theorem [] for nonexpansive mappings. Many results of a fixed point in met-ric spaces have been developed after Penot’s formulation. Using Penot’s work, the author in [] proved some results in metric spaces with uniform normal structure similar to the ones known in Banach spaces.
In this paper we investigate the existence of fixed points of modular nonexpansive map-pings defined in modular metric spaces. We also discuss some compactness properties of the family of admissible sets in modular metric spaces with uniform normal structure and prove similar results to the ones obtained in [].
For more on metric fixed point theory and on modular function spaces, the reader may consult the books [] and [], respectively.
2 Basic definitions and properties
Let X be a nonempty set. Throughout this paper, for a functionω: (,∞)×X×X→ [,∞], we write
ωλ(x,y) =ω(λ,x,y)
for allλ> andx,y∈X.
Definition .[, ] A functionω: (,∞)×X×X→[,∞] is said to be a modular metric on X if it satisfies the following axioms:
(i) x=yif and only ifωλ(x,y) = for allλ> ; (ii) ωλ(x,y) =ωλ(y,x)for allλ> andx,y∈X;
(iii) ωλ+μ(x,y)≤ωλ(x,z) +ωμ(z,y)for allλ,μ> andx,y,z∈X.
If instead of (i) we have only the condition (i)
ωλ(x,x) = for allλ> ,x∈X,
thenωis said to be a pseudomodular (metric) onX. A modular metricωonXis said to be regular if the following weaker version of (i) is satisfied:
Finally,ωis said to be convex if forλ,μ> andx,y,z∈X, it satisfies the inequality
ωλ+μ(x,y)≤ λ
λ+μωλ(x,z) + μ
λ+μωμ(z,y).
Note that for a metric pseudomodularωon a setX, and anyx,y∈X, the functionλ→ ωλ(x,y) is nonincreasing on (,∞). Indeed, if <μ<λ, then
ωλ(x,y)≤ωλ–μ(x,x) +ωμ(x,y) =ωμ(x,y).
Definition .[, ] Letωbe a pseudomodular onX. Fixx∈X. The two sets
Xω=Xω(x) =
x∈X:ωλ(x,x)→ asλ→ ∞
and
Xω∗=Xω∗(x) =
x∈X:∃λ=λ(x) > such thatωλ(x,x) <∞
are said to be modular spaces (aroundx).
It is clear thatXω⊂Xω∗, but this inclusion may be proper in general. It follows from [, ]
that ifωis a modular onX, then the modular spaceXωcan be equipped with a (nontrivial)
metric generated byωand given by
dω(x,y) =inf
λ> :ωλ(x,y)≤λ
for anyx,y∈Xω. Ifωis a convex modular onX, according to [, ] the two modular spaces
coincide,i.e.,X∗ω=Xω, and this common set can be endowed with the metricd∗ωgiven by
dω∗(x,y) =inf
λ> :ωλ(x,y)≤
for anyx,y∈Xω. These distances are called Luxemburg distances (see example below for
the justification).
Next we give the main example that motivated this paper.
Example . LetXbe a nonempty set andbe a nontrivialσ-algebra of subsets ofX. LetP be aδ-ring of subsets ofXsuch thatE∩A∈P for anyE∈P andA∈. Let us assume that there exists an increasing sequence of setsKn∈P such thatX=
Kn. By
E we denote the linear space of all simple functions with supports fromP. ByM∞ we denote the space of all extended measurable functions,i.e., all functionsf :X→[–∞,∞] such that there exists a sequence{gn} ⊂E,|gn| ≤ |f|andgn(x)→f(x) for allx∈X. By A we denote the characteristic function of the setA. Letρ:M∞→[,∞] be a nontrivial, convex and even function. We say thatρis a regular convex function pseudomodular if:
(i) ρ() = ;
(ii) ρis monotone,i.e.,|f(x)| ≤ |g(x)|for allx∈Ximpliesρ(f)≤ρ(g), where
f,g∈M∞;
(iv) ρhas the Fatou property,i.e.,|fn(x)| ↑ |f(x)|for allx∈Ximpliesρ(fn)↑ρ(f), where
f ∈M∞;
(v) ρis order continuous inE,i.e.,gn∈Eand|gn(x)| ↓impliesρ(gn)↓.
Similarly, as in 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 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(X,,P,ρ) =f ∈M∞;f(x)<∞ρ-a.e.,
where each f ∈M(X,,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(X,,P,ρ). Letρbe a regular function pseudomodular.
(a) We say thatρis a regular function semimodular ifρ(αf) = for everyα> implies
f = ρ-a.e.;
(b) We say thatρis a regular function modular ifρ(f) = impliesf = ρ-a.e.
The class of all nonzero regular convex function modulars defined onXis denoted by. Let us denoteρ(f,E) =ρ(fE) forf ∈M,E∈. It is easy to prove thatρ(f,E) is a function pseudomodular in the sense of Def. .. in [] (more precisely, it is a function pseudo-modular with the Fatou property). Therefore, we can use all results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [–]; see also Musielak [] for the basics of the general modular theory. Letρbe a convex function modular.
(a) The associated modular function space is the vector spaceLρ(X,), or brieflyLρ defined by
Lρ=
f∈M;ρ(λf)→asλ→.
(b) The following formula defines a norm inLρ(frequently called the Luxemburg norm):
fρ=infα> ;ρ(f/α)≤.
A modular function space furnishes a wonderful example of a modular metric space. In-deed, letLρbe a modular function space. Define the function modularωby
ωλ(f,g) =ρ
f –g
λ
for allλ> andf,g∈Lρ. Thenωis a modular metric onLρ. Note thatωis convex if and
only ifρis convex. Moreover, we have
f–gρ=d∗ω(f,g)
for anyf,g∈Lρ.
Definition . LetXωbe a modular metric space.
() The sequence(xn)n∈NinXωis said to beω-convergent tox∈Xωif and only if ω(xn,x)→asn→ ∞.xis called theω-limit of(xn).
() The sequence(xn)n∈N inXωis said to beω-Cauchy ifω(xm,xn)→asm,n→ ∞. () A subsetMofXωis said to beω-closed if theω-limit of anω-convergent sequence
ofMalways belongs toM.
() A subsetMofXωis said to beω-complete if anyω-Cauchy sequence inMis an ω-convergent sequence and itsω-limit is inM.
() A subsetMofXωis said to beω-bounded if we have
δω(M) =supω(x,y);x,y∈M
<∞.
In general, iflimn→∞ωλ(xn,x) = for someλ> , then we may not havelimn→∞ωλ(xn, x) = for allλ> . Therefore, as it is done in modular function spaces, we say thatω
satisfies-condition if this is the case,i.e.,limn→∞ωλ(xn,x) = for someλ> implies limn→∞ωλ(xn,x) = for allλ> . In [] and [], one can find a discussion about the con-nection betweenω-convergence and metric convergence with respect to the Luxemburg distances. In particular, we have
lim
n→∞dω(xn,x) = if and only if nlim→∞ωλ(xn,x) = for allλ>
for any{xn} ∈Xωandx∈Xω. And in particular we have thatω-convergence anddω
con-vergence are equivalent if and only if the modularωsatisfies the-condition. Moreover,
if the modularωis convex, then we know thatd∗ωanddωare equivalent, which implies
lim n→∞d
∗
ω(xn,x) = if and only if lim
n→∞ωλ(xn,x) = for allλ>
for any {xn} ∈Xω andx∈Xω [, ]. Another question that arises in this setting is the
uniqueness of theω-limit. Assume thatωis regular, and let{xn} ∈Xωbe a sequence such
that{xn}ω-converges toa∈Xωandb∈Xω. Then we have
ω(a,b)≤ω(a,xn) +ω(xn,b)
for anyn≥. Our assumptions implyω(a,b) = . Sinceωis regular, we geta=b,i.e., the ω-limit of a sequence is unique.
Let (X,ω) be a modular metric space. Throughout the rest of this work, we assume that
ωsatisfies the Fatou property,i.e., if{xn}ω-converges toxand{yn}ω-converges toy, then we must have
ω(x,y)≤lim inf
n→∞ ω(xn,yn).
For anyx∈Xωandr≥, we define the modular ball
Bω(x,r) =
y∈Xω;ω(x,y)≤r
.
of admissible subsets ofXω. Note thatAω(Xω) is stable by intersection. At this point we
need to define the concept of Chebyshev center and radius in modular metric spaces. Let A⊂Xbe a nonemptyω-bounded subset. For anyx∈A, define
rx(A) =sup
ω(x,y);y∈A
.
The Chebyshev radius ofAis defined by
Rω(A) =inf
rx(A);x∈A.
Obviously, we haveRω(A)≤rx(A)≤δω(A) for anyx∈A. The Chebyshev center ofAis
defined as
Cω(A) =x∈A;rx(A) =Rω(A)
.
Throughout the remainder of this work, for a subsetAof a modular metric spaceXω,
set
covω(A) =
{B:Bis a modular ball andA⊂B}.
Recall thatAisω-bounded ifδω(A) =sup{ω(x,y);x,y∈A}<∞.
Definition . Let (X,ω) be a modular metric space. LetCbe a nonempty subset ofXω. (i) We say thatAω(C)is compact if any family(Aα)α∈of elements ofAω(C)has a
nonempty intersection provided α∈FAα=∅for any finite subsetF⊂. (ii) We say thatAω(C)is countably compact or satisfies the property(R)if any
sequence(An)n≥of elements ofAω(C), which are nonempty and decreasing, has a nonempty intersection.
(iii) We say thatAω(C)is normal if for anyA∈Aω(C), not reduced to one point, ω-bounded, we haveRω(A) <δω(A).
(iv) We say thatAω(C)is uniformly normal if there existsc∈(, )such that for any
A∈Aω(C), not reduced to one point,ω-bounded, we haveRω(A)≤cδω(A).
Remark . Note that ifAω(Xω) is compact, thenXωisω-complete. Indeed, let{xn} ⊂Xω
be anω-Cauchy sequence. Set
rn= sup m,s≥n
ω(xm,xs)
for anyn≥. Since{xn}is anω-Cauchy sequence, thenlimn→∞rn= . By the definition ofrn, we getxm∈Bω(xn,rn) for anyn≥ andm≥n. Hence, for anyn,n, . . . ,np≥, we have
xm∈
≤i≤p
Bω(xni,rni)
for anym≥max{n,n, . . . ,np}. SinceAω(Xω) is compact, then
C= n≥
Ifz∈C, then we haveω(xn,z)≤rnfor anyn≥. Hence{xn} ω-converges toz, which
completes the proof of our statement.
3 Main results
Let us first start this section with the definition of nonexpansive mappings in the modular metric sense.
Definition . Let (X,ω) be a modular metric space. LetCbe a nonempty subset ofXω.
A mappingT:C→Cis said to beω-nonexpansive if
ω
T(x),T(y)≤ω(x,y) for anyx,y∈C.
For such a mapping, we denote byFix(T) the set of its fixed points, i.e.,Fix(T) ={x∈ C;T(x) =x}.
In [, ] the author defined Lipschitzian mappings in modular metric spaces and proved some fixed point theorems. Our definition is more general. Indeed, in the case of modular function spaces, it is proved in [] that
ωλT(x),T(y)≤ωλ(x,y) for anyλ>
if and only if dω(T(x),T(y))≤dω(x,y) for anyx,y∈C. Next we give an example, which
first appeared in [], of a mapping which isω-nonexpansive in our sense but fails to be nonexpansive with respect todω.
Example . LetX= (,∞). Define the Musielak-Orlicz function modular on the space of all Lebesgue measurable functions by
ρ(f) = e
∞
f(x)x+dm(x).
LetBbe the set of all measurable functionsf : (,∞)→Rsuch that ≤f(x)≤. Consider the map
T(f)(x) =
⎧ ⎨ ⎩
f(x– ) forx≥,
forx∈[, ].
Clearly,T(B)⊂B. In [], it was proved that for everyλ≤ and for allf,g∈B, we have
ρλT(f) –T(g)≤λρλ(f –g).
This inequality clearly implies thatT isω-nonexpansive. On the other hand, if we take f = [,], then
T(f)
ρ>e≥ fρ,
which clearly implies thatTis notdω-nonexpansive.
Theorem . Let (X,ω) be a modular metric space. Let C be a nonempty ω-closed
ω-bounded subset of Xω. Assume that the family Aω(C) is normal and compact. Let
T:C→C beω-nonexpansive.Then T has a fixed point.
Proof SinceAω(C) is compact, there exists a minimal nonemptyA∈Aω(C) such that T(A)⊂A. It is easy to check thatcovω(T(A)) =A. Let us prove thatδω(A) = ,i.e.,Ais
reduced to one point. Suppose thatδω(A)= . For anyx∈A, set
rx(A) =supω(x,y);y∈A
≤δω(A)≤δω(C) <∞.
SinceTisω-nonexpansive, we haveT(A)⊂Bω(T(x),rx(A)) for anyx∈A. Hence
covω
T(A)⊂Bω
T(x),rx(A).
So,rT(x)(A)≤rx(A) for anyx∈A. Next we fixa∈Aand define
Aa=x∈A;rx(A)≤ra(A).
Clearly,Aais not empty sincea∈Aa. Moreover, we have
Aa= x∈A
Bω
x,ra(A)∩A∈Aω(C).
And sincerT(x)(A)≤rx(A), for anyx∈A, we getT(Aa)⊂Aa. The minimality behavior
of AimpliesAa =A. In particular we haverx(A) =ra(A) for anyx∈A. Henceδω(A) = supx∈Arx(A) =ra(A) for anya∈A. SinceAω(C) is normal, we getδω(A) <δω(A), which is a
contradiction. Thus we must haveδω(A) = ,i.e.,Ais reduced to one point which is fixed
byT.
Next we give a constructive result discovered by Kirk [] which relaxes the compactness assumption in the above theorem. The main ingredient in Kirk’s constructive proof is a technical lemma due to Gillespie and Williams []. The next lemma is a modular version of the Gillespie and Williams result.
Lemma . Let(X,ω)be a modular metric space,and let C be a nonemptyω-bounded subset of Xω.Let T:C→C be anω-nonexpansive mapping.Assume thatAω(C)is normal.
Let A∈Aω(C)be nonempty and T -invariant,i.e.,T(A)⊂A.Then there exists a nonempty A∈Aω(C),which is T -invariant,such that
δω(A)≤
δω(A) +Rω(A)
.
Proof Setr=
(δω(A) +Rω(A)). We assume thatδω(A) > , otherwise we can take the set
A=A. SinceAω(C) is normal, we haveRω(A) <δω(A). HenceRω(A) <r, which implies
the existence ofa∈Asuch thatra(A) <r. Therefore, the set
D=a∈A:A⊂Bω(a,r)
=
x∈A
is a nonempty admissible subset ofC. Note that there is no reason forDto beT-invariant. Consider the family
F=M∈Aω(C) :D⊂MandT(M)⊂M.
Note thatFis nonempty sinceC∈F. SetL= M∈FM. The setLis an admissible subset ofCwhich containsD. Using the definition ofF, we deduce thatLisT-invariant. Consider B=D∪T(L) and observe thatcovω(B) =L. Indeed, sinceB⊂T(L)⊂LandL∈Aω(C), we
havecovω(B)⊂L. From this we obtain
Tcovω(B)
⊂T(L)⊂B⊂covω(B).
Hencecovω(B)∈FandL⊂covω(B). This gives the desired equality. Define
A=
x∈L:L⊂Bω(x,r)
.
We claim thatAis the desired set. Observe thatAis nonempty since it containsD(by the
definition ofD). Using the same argument, we can prove thatAis an admissible subset
ofC. On the other hand, it is clear thatδω(A)≤r. To complete the proof, we have to show
thatAisT-invariant. Letx∈A. By the definition ofA, we haveL⊂Bω(x,r). SinceTis ω-nonexpansive, we have
T(L)⊂Bω
T(x),r.
For anyy∈D, there holdsL⊂Bω(y,r). ButT(x)∈L, soT(x)∈Bω(y,r), which implies
y∈Bω(T(x),r). HenceD⊂Bω(T(x),r) holds. SinceB=D∪T(L), we getB⊂Bω(T(x),r).
Therefore, we must have
covω(B) =L⊂Bω
T(x),r.
By the definition ofA, it follows thatT(x)∈A. In other words,AisT-invariant. Let
x,y∈A, thenx,y∈L, which impliesω(x,y)≤rx(L)≤r,i.e.,
δω(A)≤r=
δω(A) +Rω(D)
.
Next we give the analogue of the main fixed point result in [].
Theorem . Let(X,ω)be anω-complete modular metric space,and let C be a nonempty
ω-closedω-bounded subset of Xω.Assume that the familyAω(C)is uniformly normal and
T:C→C isω-nonexpansive.Then T has a fixed point.
Proof SinceCisω-bounded, we haveC∈Aω(C) since
C=Bω
Now, let us takeC=A andT(C)⊂C. SinceAω(C) is uniformly normal, there exists
c∈(, ) such that
Rω(A)≤cδω(A) for anyA∈Aω(C).
By Lemma ., there existsA∈Aω(C) such thatA⊂A,T(A)⊂Aand satisfies
δω(A)≤
R(A) +δω(A)
.
Using the induction argument, we build a sequence{An} ⊂Aω(C) such thatAn+⊂An,
T(An+)⊂An+and
δω(An+)≤
R(An) +δω(An)
.
SinceRω(An)≤cδω(An), we get
δω(An+)≤
+c
δω(An),
which implies that
δω(An)≤
+c
n
δω(C).
Now, sinceδω(C) <∞, we get
lim
n→∞δω(An) = .
Letxn∈Anfor anyn≥. Then{xn}is anω-Cauchy sequence. SinceXωisω-complete
andCis anω-closed subset ofXω, thenCisω-complete. Thus{xn}ω-converges tox∈C. SinceAn+⊂AnandAnisω-closed for anyn≥, thenx∈Anfor anyn≥. Thus nAnis not empty and clearly is reduced to the single pointx. Indeed, lety∈ nAn, theny∈An for anyn≥. Hence
ω(x,y)≤δω(An).
Since limn→∞δω(An) = , we get ω(x,y) = . Since ω is regular, we get y= x, i.e.,
nAn={x}. SinceT(An)⊂Anfor anyn≥, we getT(x) =x.
The following technical proposition is needed to show an analogue to the main result in [].
Proposition . Let (X,ω) be an ω-complete modular metric space, and let C be a nonemptyω-closedω-bounded subset of Xω.Assume that the familyAω(C)is uniformly
normal.Consider the Cartesian product C∞=n≥C.Define: (,∞)×C×C→[,∞]
by
λ(xn), (yn)=sup n≥
Then:
(i) (C∞,)is an-complete modular metric space. (ii) C∞is-bounded withδ(C∞) =δω(C) <∞.
(iii) For any(xn)∈C∞andr∈[,∞],we have
B
(xn),r= n≥
Bω(xn,r),
where
B
(xn),r=(yn)∈C∞;
(xn), (yn)≤r,
andBω(xn,r) ={y∈C;ω(xn,y)≤r}.This implies that for anyA∈A(C∞),we have
A=n≥An,whereAn∈A(C). (iv) A(C∞)is uniformly normal.
Proof The proofs of (i), (ii), (iii) are easy and left to the reader. Let us prove (iv). Indeed, letA∈Aω(C∞) be nonempty and not reduced to one point. Then (iii) implies thatA=
n≥An, whereAn∈Aω(C). Letε> . SinceAω(C) is uniformly normal, there existsc∈
[, ) such that for anyn≥, for whichAnis not reduced to one point, there existsxn∈An such that
rxn(An)≤(c+ε)δω(An).
Hence
r(xn)(A) = sup
(yn)∈A
(xn), (yn)= sup
(yn)∈A
sup n≥
ω(xn,yn)
,
which implies
r(xn)(A)≤(c+ε)sup
n≥
δω(An) = (c+ε)δ(A).
So,R(A)≤(c+ε)δ(A). Sinceεwas arbitrary, we getR(A)≤cδ(A). This completes the
proof of (iv).
The following theorem shows that although we do not need compactness of the family of admissible sets in Theorem ., its assumptions imply a weaker form of compactness, mainly countable compactness.
Theorem . Let(X,ω)be anω-complete modular metric space,and let C be a nonempty
ω-closedω-bounded subset of Xω.Assume that the familyAω(C)is uniformly normal.Then Aω(C)has the property(R).
ThenA∈A(C∞). SinceA(C∞) is uniformly normal, thenA(A) is uniformly normal. Consider the shiftT:A→Adefined by
T(xn)= (xn+).
Obviously,Tisω-nonexpansive. Theorem . implies thatT has a fixed point,i.e., there exists (xn)∈Asuch thatT((xn)) = (xn). The definition ofT forces{xn}to be a constant sequence,i.e.,xn=x, for anyn≥. Obviously, we havex∈Anfor anyn≥, which implies
n≥An=∅.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.2Department of
Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, USA.3Department of Mathematics and Statistics,
King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.
Received: 2 June 2013 Accepted: 12 August 2013 Published: 28 August 2013 References
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Cite this article as:Abdou and Khamsi:On the fixed points of nonexpansive mappings in modular metric spaces.
