UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR
FUNCTION SPACES
T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI
ABSTRACT
Le ρbe a con ex modula unc ion sa is ying a ∆2- ype condi ion and Lρ he
co esponding modula space. Assume ha Cis a ρ-bounded and ρ-a.e compac
subse o Lρand T:C→Cis a k-uni o mly Lipschi zian mapping. We p o e
ha Thas a ixed poin i k < (˜
N(Lρ))−1/2whe e ˜
N(Lρ) is a geome ical coe i-
cien o no mal s uc u e. We also show ha ˜
N(Lρ)<1 in modula O licz spaces
o uni o mly con ex O licz unc ions.
1991 Ma hema ics subjec classi ica ion: P ima y 46E30; Seconda y 47H09,
47H10.
Key Wo ds: uni o mly Lipschi zian mappings, ixed poin , modula unc ions,
uni o m no mal s uc u e, uni o m con ex O licz unc ion, modulus o con exi y.
The i s au ho is pa ially suppo ed by PB-96-1338-C01-C02 and PAI-FMQ-0127.
1
2 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI
INTRODUCTION
The heo y o modula spaces was ini ia ed by Nakano [14] in 1950 in connec ion
wi h he heo y o o de spaces and ede ined and gene alized by Musielak and
O licz [13] in 1959. De ining a no m, pa icula Banach spaces o unc ions
can be conside ed. Me ic ixed heo y o hese Banach spaces o unc ions
has been widely s udied (see, o ins ance, [15]). Ano he di ec ion is based
on conside ing an abs ac ly gi en unc ional which con ols he g ow h o he
unc ions. E en hough a me ic is no de ined, many p oblems in ixed poin
heo y o nonexpansi e mappings can be e o mula ed in modula spaces (see,
o ins ance, [8] and e e ences he ein). In his pape , we s udy he exis ence
o ixed poin s o a mo e gene al class o mappings: uni o mly Lipschi zian
mappings. Fixed poin heo ems o his class o mappings in Banach spaces
ha e been s udied in [3,4] and in me ic spaces in [11,12] ( o u he in o ma ion
abou his subjec , see [2, chap e VIII] and e e ences he ein). The main ool
in ou app oach is he coe icien o no mal s uc u e ˜
N(Lρ). We p o e ha
unde sui able condi ions a k-uni o mly Lipschi zian mapping has a ixed poin
i k < (˜
N(Lρ))−1/2.In he las sec ion we show a class o modula spaces whe e
˜
N(Lρ)<1 and so, he abo e heo em can be success ully applied.
1. PRELIMINARIES
We s a by eco ding a b ie collec ion o basic concep s and ac s o modula
spaces as o mula ed by Kozlowski. Fo mo e de ails he eade is e e ed o [7],
[8], [10] and [13].
Le Ω be a nonemp y se and Σ be a non i ial σ-algeb a o subse s o Ω. Le
Pbe a δ- ing o subse s o Σ, such ha E∩A∈ P o any E∈ P and A∈Σ.
Le us assume ha he e exis s an inc easing sequence o se s Kn∈ P such ha
Ω = SKn. In o he wo ds, he amily Pplays he ole o he δ- ing o subse s
o ini e measu e. By Ewe deno e he linea space o all simple unc ions wi h
suppo s om P. By Mwe will deno e he space o all measu able unc ions,
i.e. all unc ions : Ω → < such ha he e exis s a sequence {gn} ∈ E,|gn| ≤ | |
and gn(ω)→ (ω) o all ω∈Ω. By 1Awe deno e he cha ac e is ic unc ion o
he se A.
UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 3
De ini ion 1.1. A unc ional ρ:E × Σ→[0,∞] is called a unc ion modu-
la i
(P1)ρ(0, E) = 0 o any E∈Σ,
(P2)ρ( , E)≤ρ(g, E) whene e | (ω)| ≤ |g(ω)| o any ω∈Ω, , g ∈ E and
E∈Σ,
(P3)ρ( , .) : Σ →[0,∞] is a σ-subaddi i e measu e o e e y ∈ E,
(P4)ρ(α, A)→0 as αdec eases o 0 o e e y A∈ P, whe e ρ(α, A) =
ρ(α1A, A),
(P5) i he e exis s α > 0 such ha ρ(α, A) = 0, hen ρ(β, A) = 0 o e e y
β > 0,
(P6) o any α > 0ρ(α, .) is o de con inuous on P, ha is ρ(α, An)→0 i
{An} ∈ P and dec eases o ∅.
The de ini ion o ρis hen ex ended o ∈ M by
ρ( , E) = sup{ρ(g, E); g∈ E,|g(ω)| ≤ | (ω)|ω∈Ω}.
A se Eis said o be ρ-null i ρ(α, E) = 0 o e e y α > 0.Fo he sake o
simplici y we w i e ρ( ) ins ead o ρ( , Ω).
I is easy o see ha he unc ional ρ:M → [0,∞] is a modula because i
sa is ies he ollowing p ope ies:
(i) ρ( ) = 0 i = 0 ρ-a.e.
(ii) ρ(α ) = ρ( ) o e e y scala αwi h |α|= 1 and ∈ M.
(iii) ρ(α +βg)≤ρ( ) + ρ(g) i α+β= 1, α≥0, β ≥0 and , g ∈ M.
In addi ion, i he ollowing p ope y is sa is ied
(iii)’ ρ(α +βg)≤αρ( ) + βρ(g) i α+β= 1 ; α≥0, β ≥0 and , g ∈ M,
we say ha ρis a con ex modula .
The modula ρde ines a co esponding modula space, i.e he ec o space Lρ
gi en by
Lρ={ ∈ M;ρ(λ )→0 as λ→0}.
We can also conside he space Eρ={ ∈ M;ρ(α , An)→0 as n→
∞ o e e y An∈Σ ha dec eases o ∅and α > 0}.
4 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI
A unc ion modula is said o sa is y he ∆2-condi ion i sup
n≥1
ρ(2 n, Dk)→
0 as k→ ∞ whene e { n}n≥1⊂ M, Dk∈Σ dec eases o ∅and
sup
n≥1
ρ( n, Dk)→0 as k→ ∞.We know (see [10]) ha Eρ=Lρwhen ρsa is ies
he ∆2-condi ion. When ρis con ex, he o mula
|| ||ρ= in nα > 0; ρ
α≤1o
de inies a no m in he modula space Lρwhich is equen ly called he Luxem-
bu g no m.
De ini ion 1.2.
(1) The sequence { n}n⊂Lρis said o be ρ-con e gen o ∈Lρi ρ( n−
)→0 as n→ ∞,
(2) The sequence { n}n⊂Lρis said o be ρ-a.e con e gen o ∈Lρi he
se {ω∈Ω; n(ω)6→ (ω)}is ρ-null.
(3) The sequence { n}n⊂Lρis said o be ρ-Cauchy i ρ( n− m)→0 as n
and mgo o ∞,
(4) A subse Co Lρis called ρ-closed i he ρ-limi o a ρ-con e gen sequence
o Calways belongs o C.
(5) A subse Co Lρis called ρ-a.e sequen ially closed i he ρ-a.e limi o a
ρ-a.e con e gen sequence o Calways belongs o C.
(6) A subse Co Lρis called ρ-a.e sequen ially compac i e e y sequence in
Chas a ρ-a.e con e gen subsequence in C.
(7) A subse Co Lρis called ρ-bounded i
δρ(C) = sup{ρ( −g); , g ∈C}<∞.
Le Bbe a bounded subse o Lρ.We de ine he ρ-ball o cen e ∈Lρand adius
> 0 by B( , ) = {g∈Lρ, ρ(g− )≤ }.We will deno e ( , B) = sup{ρ( −
g), g ∈B}, δ(B) = sup{ ( , B), ∈B}, R(B) = in { ( , B), ∈B}.We
de ine he admissible hull o Bas he in e sec ion o all ρ-ball con aining B, i.e:
ad(B) = {A:B⊂A⊂Lρ,whe e Ais a ρ-ball}.
UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 5
Bis said admissible i ad(B) = B. We de ine he no mal s uc u e coe icien
˜
N(Lρ) o Lρby
˜
N(Lρ) = sup R(B)
δ(B), B is admissible, ρ-bounded and ρ-a.e sequen ially compac .
The use ul ollowing p oposi ion is easily seen:
P oposi ion 1.1. Le Bbe a ρ-bounded subse o Lρand ∈Lρ.Then
(1) ( , ad(B)) = ( , B).
(2) δ(ad(B)) = δ(B).
We say ha ρsa is ies he ∆2- ype condi ion i he e exis s K > 0 such ha
ρ(2 )≤Kρ( ) o all ∈Lρ.In gene al, ∆2- ype condi ion and ∆2-condi ion
a e no equi alen , e en hough i is ob ious ha ∆2- ype condi ion implies ∆2-
condi ion. Assume ha ρis con ex and sa is ies he ∆2- ype condi ion. We
de ine a g ow h unc ion ωby
ω( ) = sup ρ( )
ρ( ),0< ρ( )<∞ o all 0 ≤ < ∞.
The ollowing p ope ies o he g ow h unc ion can be easily seen.
Lemma 1.1. Le ρbe a con ex unc ion modula sa is ying he ∆2- ype condi ion.
Then he g ow h un ion ωhas he ollowing p ope ies:
(1) ω( )<∞,∀ ∈[0,∞)
(2) ω: [0,∞)→[0,∞)is a con ex, s ic ly inc easing unc ion. So, i is
con inuous.
(3) ω(αβ)≤ω(α)ω(β); ∀α, β ∈[0,∞)
(4) ω−1(α)ω−1(β)≤ω−1(αβ);∀α, β ∈[0,∞),whe e ω−1is he unc ion in-
e se o ω.
The ollowing lemma shows ha he g ow h unc ion can be used o gi e an
uppe bound o he no m o a unc ion.

6 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI
Lemma 1.2. [5] Le ρbe a con ex unc ion modula sa is ying he ∆2- ype con-
di ion. Then
|| ||ρ≤1
ω−11
ρ( )whene e ∈Lρ.
The ollowing lemma can be ound in [7].
Lemma 1.3. Le ρbe a unc ion modula sa is ying he ∆2-condi ion and { n}n
be a sequence in Lρsuch ha n
ρ−a.e
→ ∈Lρand he e exis s k > 1such ha
supnρ(k( n− )) <∞. Then,
lim in
n→∞ ρ( n−g) = lim in
n→∞ ρ( n− ) + ρ( −g) o all g∈Lρ.
Lemma 1.4. Le ρbe a modula unc ion sa is ying he ∆2- ype condi ion. Le B
be a ρ-a.e sequen ially closed and ρ-bounded subse o Lρ. Le {gn}nbe a sequence
in Bsuch ha gn
ρ−a.e
→g. Then,
(1) ρ(g)≤lim in n→∞ ρ(gn).
(2) B(0, )∩Bis ρ-a.e sequen ially closed.
(3) ad(A)∩Bis ρ-a.e sequen ially closed, o all A⊂Lρ.
P oo . Condi ion (1) is a s aigh o wa d consequence o Lemma 1.3 applied o
he sequence gn
ρ−a.e
→gand he null unc ion. Condi ion (2) and (3) can be easily
deduced om (1).
2. FIXED POINT FOR UNIFORMLY LIPSCHITZIAN MAPPINGS
The ollowing lemma is he key o ou ixed poin esul .
Lemma 2.1. Le ρbe a modula unc ion sa is ying he ∆2- ype condi ion and
Baρ-bounded and ρ-a.e sequen ially compac subse o Lρ. Le { n}nand {gn}n
be sequences in B. Then, he e exis s g∈ ∩∞
n=1ad(gj, j ≥n)∩Bsuch ha
lim sup
n→∞
ρ(g− n)≤lim sup
j→∞
lim sup
n→∞
ρ(gj− n)
P oo . Le { n}nand {gn}nbe sequences in B. We de ine θ(h) = lim supn→∞ ρ(h−
n) o all h∈B. Since Bis ρ-sequen ially compac and ρ-bounded, he e ex-
is a subsequence {gφ(n)}n⊂ {gn}nsuch ha gφ(n)
ρ−a.e
→gand a subsequence
{ ψ(n)}n⊂ { n}nsuch ha limn→∞ ρ( ψ(n)−g) = lim supn→∞ ρ( n−g) and
UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 7
ψ(n)
ρ−a.e
→ ∈B. Since gφ(n)∈ad(gj, j ≥n)∩Bwhich is ρ-a.e sequen ially
closed (by p ope y (3) o Lemma 1.4) and gφ(n)
ρ−a.e
→g, we ob ain g∈ad(gj, j ≥
n)∩B o all n≥1.We will see ha θ(g)≤lim sup
j→∞
θ(gj).Indeed, om
Lemma 2.3 we ha e θ(gj) = lim supn→∞ ρ( n−gj)≥lim in n→∞ ρ( ψ(n)−gj) =
lim in n→∞ ρ( ψ(n)− ) + ρ( −gj).Thus, again using Lemma 2.3, we ob ain
lim sup
j→∞
θ(gj)≥lim in
n→∞ ρ( ψ(n)− ) + lim sup
j→∞
ρ( −gj)
≥lim in
n→∞ ρ( ψ(n)− ) + lim in
j→∞ ρ( −gφ(j))
= lim in
n→∞ ρ( ψ(n)− ) + lim in
j→∞ ρ(gφ(j)−g) + ρ( −g).
On he o he hand θ(g) = lim supn→∞ ρ( n−g) = lim in n→∞ ρ( ψ(n)−g) =
lim in n→∞ ρ( ψ(n)− ) + ρ( −g). The e o e, θ(g)≤lim supj→∞ θ(gj).
The ollowing lemma is inspi ed on [3] whe e a simila lemma is p o ed in
e lexi e Banach spaces (see also [12, Lemma 6] o a e sion in me ic spaces
wi h addi ional p ope ies).
Lemma 2.2. Le ρbe a unc ion modula sa is ying he ∆2- ype condi ion and
Ban be admissible, ρ-a.e sequen ially compac and ρ-bounded subse o Lρ.Le
{ n}be a sequence in Band ca cons an such ha c > ˜
N(Lρ). Then he e exis s
∈Bsuch ha
(1) lim sup
n→∞
ρ( − n)≤c δ({ n}n).
(2) ρ( −g)≤lim sup
n→∞
ρ( n−g) o all g∈B.
P oo . Le { n}nbe a sequence o B. Deno e Am=ad( j:j≥m)⊂Band
A=T∞
m=1 Am.Since Bis ρ-a.e sequen ially compac , he e exis s a subsequence
o { n}nρ−a.e con e gen , say o h. I is clea ha h∈Aand so A6=∅.
Fu he mo e, om P oposi ion 1.1 (2), we ha e δ(An)≤δ({ n}n).On he o he
hand, o any ∈Aand g∈Bwe ha e ρ(g− )≤ (g, A)≤ (g, An) = (g, { j:
j≥n}) = supj≥nρ(g− j).The e o e, ρ(g− )≤lim supn→∞ ρ(g− n) and (2)
holds o any ∈A. We will p o e ha he e exis s ∈Asa is ying (1). Wi hou
loss o gene ali y we may assume ha δ({ n}n)>0.Choose ε > 0 such ha
8 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI
˜
N(Lρ)δ({ n}n)+ε≤c δ({ n}n).By de ini ion o R(An), he e exis s gn∈Ansuch
ha (gn, An)< R(An)+ε≤˜
N(Lρ)δ(An)+ε≤˜
N(Lρ)δ({ n}n)+ε≤c δ({ n}n).
Since (gn, An) = (gn,{ j}j≥n) = supj≥nρ(gn− j),we ha e,
lim sup
j→∞
ρ(gn− j)≤c δ({ n}n) (A).
Using Lemma 2.5, he e exis s ∈ ∩∞
n=1ad(gi, i ≥n) such ha
lim sup
j→∞
ρ( − j)≤lim sup
n→∞
lim sup
j→∞
ρ(gn− j) (B).
We will check ha ∈A. Indeed, o all i, n in ege s such ha i≥nwe ha e
gi∈Ai⊂An.Thus, {gi}i≥n⊂Anwhich implies ad(gi, i ≥n)⊂Anand ∈A.
Using (B) and (A) i is clea ha lim supj→∞ ρ( − j)≤c δ({ n}n).
Theo em 2.1. Le ρbe a con ex unc ion modula sa is ying he ∆2-condi ion
and B an admissible, ρ-a.e sequen ially compac and ρ-bounded subse o Lρ.Sup-
pose ha ˜
N(Lρ)<1and le T:B→Bbe a k-uni o mly lipschi zian mapping
sa is ying k < (˜
N(Lρ))−1/2.Then, Thas a ixed poin .
P oo . We can assume ha k > 1; o he wise Twill be nonexpansi e and
he exis ence o a ixed poin is a consequence o [8, Theo em 3.5]. Choose a
cons an c,˜
N(Lρ)< c < 1 such ha 1 < k < c−1/2.Fix 0∈B. By Lemma 2.6,
we can induc i ely cons uc a sequence { j}j≥0⊂Bsuch ha o each j≥0
(1) lim supn→∞ ρ(Tn( j)− j+1)≤c δ({Tn( j)}n).
(2) ρ( j+1 −g)≤lim supn→∞ ρ(Tn( j)−g) o all g∈B.
Deno e Dj= lim supn→∞ ρ(Tn( j)− j+1) and h=ck2<1.Fo n≥m≥0,we
ha e
ρ(Tm j−Tn j)≤kρ( j−Tn−m j)
≤klim sup
i→∞
ρ(Ti j−1−Tn−m j)
≤k2lim sup
i→∞
ρ(Ti−(n−m) j−1− j)
≤k2Dj−1.
UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 9
Since Dj= lim supn→∞ ρ(Tn( j)− j+1)≤c δ({Tn( j)}n),we ob ain Dj≤
c k2Dj−1=hDj−1.Thus, Dj≤hjD0and we ha e
ρ( j+1 − j)≤ω(2)ρ( j+1 −Tn j) + ρ( j−Tn j)
≤ω(2)ρ( j+1 −Tn j) + lim sup
m→∞
ρ(Tm j−1−Tn j)
≤ω(2)ρ( j+1 −Tn j) + klim sup
m→∞
ρ(Tm−n j−1− j)
≤ω(2)ρ( j+1 −Tn j) + kDj−1.
Taking limsup as n→ ∞,we ob ain
ρ( j+1 − j)≤ω(2)(Dj+kDj−1)
≤ω(2)(hj+khj−1)D0
≤ω(2)(h+k)hj−1D0
≤Ahj,whe e A=ω(2)h+k
hD0.
Hence, he e exis s an in ege Nand some β < 1 such ha o j > N we ha e
ρ( j+1 − j)≤βj,which implies 1
βj≤1
ρ( j+1 − j).Using p ope ies (2) and (3)
o Lemma 1.1 we ob ain
ω−11
βj≤ω−11
ρ( j+1 − j)
and
ω−11
βj
≤ω−11
ρ( j+1 − j).
The e o e, by Lemma 2.2 we ha e
|| j+1 − j||ρ≤1
ω−11
ρ( j+1− j)≤1
ω−1(1
β)j.
Hence { j}is a Cauchy sequence in (Lρ,||.||ρ), he e exis s ∈Lρsuch ha
|| j− ||ρ→0, because (Lρ,||.||ρ) is comple e. Since unde ∆2-condi ion no m-
con e gence and modula -con e gence a e iden ical, { j}is modula con e gen
o . Thus, he e exis s a subsequence o { j}jρ-a.e con e gen o [1, Theo em
16 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI
ACKNOWLEDGEMENTS
The i s and hi d au ho s a e e y g a e ul o he Depa men o Ma hema -
ical Sciences a he Uni e si y o Texas a El Paso o hei hospi ali y while
comple ing his wo k.
Re e ences
[1] A.Kaminska. On uni o m con exi y o O licz spaces. Ma hema ics, P oceedings. Konink.
Nede l, Ak. We . Ams e dam, A 85 (1) (1982), 27- 36.
[2] J.M.Aye be, T.Dominguez Bena ides, G.Lopez Acedo. Measu es o Noncompac ness in
Me ic Fixed Poin Theo y. Bikhause : Basel (1997).
[3] E.Casini, E.Malu a. Fixed poin s o uni o mly Lipchi zian mappins in spaces wi h uni o mly
no mal s uc u e. Nonlinea Anal. 9(1985), 103-108.
[4] T.Dominguez Bena ides. Fixed poin heo ems o uni o mly Lipschi zian mappings and
asymp o ically egula mappings. Nonlinea Anal. 32 (1) (1998), 15-27.
[5] T.Dominguez Bena ides, M.A.Khamsi, S.Samadi. Asymp o ically egula mappings in mod-
ula unc ion spaces, (p ep in ).
[6] H.Hudzik, A.Kaminska, J.Musielak. On he con exi y coe icien o O licz spaces. Ma h. Z.
197 (1988), 291-295.
[7] M.A.Khamsi. Fixed poin heo y in modula unc ion spaces. Recen Ad ances on Me ic
Fixed Poin Theo y. 31-58. Uni e sidad de Se illa, Se illa (1996).
[8] M.A.Khamsi, W.M.Kozlowski, S.Reich. Fixed poin heo y in modula unc ion spaces. Non-
linea Anal. 14 (1990), 935-953.
[9] M.A.Khamsi, W.M.Kozlowski, C.Shu ao. Some geome ical p ope ies and ixed poin he-
o ems in O licz Spaces. J. Ma h. Anal. Appl. 155 (2) (1991), 393-412.
[10] W.M.Koslowski. Modula unc ion spaces. Dekke : New Yo k, Basel (1988).
[11] E.A.Li schi z. Fixed poin heo ems o ope a o s in s ongly con ex spaces (Russian)
Vo onez. Gos. Uni . T udy Ma . Fak. 16 (1975), 23-28.
[12] T.C.Lim, H.K.Xu. Uni o mly Lipschi zian mappings in me ic spaces wi h uni o m no mal
s uc u e. Nonlinea Anal. 25 (11) (1995), 1231-1235.
[13] J.Musielak, W.O licz. On modula spaces. S udia Ma h. 18 (1959), 591-597.
[14] H. Nakano. Modula ed semi-o de ed spaces, Tokyo (1950).
[15] D.Van Dul s , V.de Valk. (KK)-p ope ies, no mal s uc u e and ixed poin s o nonexpan-
si e mappings in O licz spaces. Canad. J. Ma h. 38 (1986), 728-750.

UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 17
Tomas Dominguez Bena ides, Depa men o Ma hema ical Analysis, Uni e -
si y o Se ille, P.O.Box 1160. 41080. Se ille (Spain).
E-mail add ess:[email p o ec ed]
Mohamed Amine Khamsi, Depa men o Ma hema ical Science, The Uni e -
si y o Texas a El Paso, El Paso, TX 79968, (U.S.A).
E-mail add ess:[email p o ec ed]
Sedki Samadi, Depa men o Ma hema ical Analysis, Uni e si y o Se ille,
p.o.box 1160. 41080. Se ille (Spain).
E-mail add ess:[email p o ec ed]