ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR
FUNCTION SPACES
T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
1. ABSTRACT
Le ρbe a modula unc ion sa is ying a ∆2- ype condi ion and Lρ he co -
esponding modula space. The main esul in his pape s a es ha i Cis a
ρ-bounded and ρ-a.e sequen ially compac subse o Lρand T:C→Cis an
asymp o ically egula mapping such ha lim in
n→∞
|Tn|<2, whe e |S|deno es he
Lipschi z cons an o S, hen Thas a ixed poin . We show ha he es ima e
lim in
n→∞
|Tn|<2 canno be, in gene al, imp o ed.
1991 Ma hema ics subjec classi ica ion: P ima y 46E30; Seconda y 47H09,
47H10.
Key Wo ds: asymp o ically egula mappings, ixed poin , modula unc ions,
Opial p ope 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
2. INTRODUCTION
Le Xbe a me ic space. A mapping T:X→Xis said o be asymp o ically
egula i lim
nd(Tn+1x, Tnx) = 0 o each x∈X. This no ion was de ined by
B owde and Pe yshyn [2]. The exis ence o ixed poin s o asymp o ically
egula mappings has been widely s udied [3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17].
When Xis a con ex, closed, subse o a Banach space i is known [12] ha he
p oblem o he exis ence o ixed poin o a nonexpansi e mapping is equi alen
o he same p oblem o a nonexpansi e asymp o ically egula mapping. On
he o he hand, he heo y o modula spaces was ini ia ed by Nakano [21] 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 [20] in 1959. Besides he idea o de ining a no m and
conside ing pa icula Banach spaces o unc ions, ano he di ec ion is based on
conside ing an abs ac ly gi en un ional de ined on a linea space o unc ions
which con ols he g ow h o membe s o he space. E en hough a me ic is
no de ined, many p oblems in me ic ixed poin heo y can be e o mula ed in
modula spaces (see, o ins ance [13] and e e ences he ein). In his pape , we
s udy he exis ence o ixed poin s o asymp o ically egula mapping de ined
om a ρ-bounded and ρ-a.e sequen ially compac se Co a modula space Lρ
in o C. We ac ually p o e ha Thas a ixed poin i lim in
n|Tn|<2, |T|being
he exac Lipschi z cons an o T. I is wo hed o no ice he simplici y o his
s a emen in compa ison wi h simila esul s in Banach spaces, whe e a di e en
uppe bound o lim in
n|Tn|mus be conside ed o each space. E en in Hilbe
spaces (see [1, chap e IX]) a bes es ima e is unknown. We also gi e an example
showing ha his esul can no be, in gene al, imp o ed.
3. PRELIMINARIES
We s a by ecalling some 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 [13], [14], [15]
and [19].
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
ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 3
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.
De ini ion 3.1. A unc ional ρ:E × Σ→[0,∞] is called a unc ion modula 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}.
This will enable us o de ine ρ(α, E) o se s Eno in P; o he sake o simplici y,
we w i e ρ( ) ins ead o ρ( , Ω).
De ini ion 3.2. A se Eis said o be ρ-null i and only i ρ(α, E) = 0 o
α > 0. A p ope y p(ω) is said o hold ρ-almos e e ywhe e (ρ-a.e.) i he se
{ω∈Ω; p(ω) does no hold }is ρ-null. Fo example we will say equen ly
n→ ρ-a.e.
No e ha a coun able union o ρ-null se s is s ill ρ-null. In he sequel we will
iden i y se s Aand Bwhose symme ic di e ence A∆Bis ρ-null; simila ly we
will iden i y measu able unc ions which di e only on a ρ-null se .
4 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
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 Lρgi en by
Lρ={ ∈ M;ρ(λ )→0 as λ→0}.
The modula space Lρcan be equipped wi h an F-no m de ined by
|| ||ρ= in nα > 0; ρµ
α¶≤αo.
When ρis con ex he o mula
|| ||ρ= in nα > 0; ρµ
α¶≤1o
De ini ion 3.3.
(a) The sequence { n} ⊂ Lρis said o be ρ-con e gen o ∈Lρi ρ( n− )→
0 as n→ ∞,
(a’) The sequence { n} ⊂ Lρis said o be ρ-a.e con e gen o ∈Lρi he
se {ω∈Ω; n(ω)6→ (ω)}is ρ-null.
(b) The sequence { n} ⊂ Lρis said o be ρ-Cauchy i ρ( n− m)→0 as n
and mgo o ∞,
(b’) The sequence { n} ⊂ Lρis said o be ρ-a.e Cauchy i {ω∈Ω; { n(ω)}is
no a Cauchy sequence }is ρ-null.
(c) A subse Co Lρis called ρ-closed i he ρ-limi o a ρ-con e gen sequence
o Calways belongs o C.
(c’) 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.
ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 5
(d) A subse Co Lρis called ρ-sequen ially compac i e e y sequence in C
has a ρ-con e gen subsequence in C.
(d’) 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.
(e) A subse Co Lρis called ρ-bounded i
δρ(C) = sup{ρ( −g); , g ∈C}<∞.
De ini ion 3.4. Le ρbe a unc ion modula , we de ine a g ow h un ion ωρ:
[0,∞]→[0,∞] by
ωρ( ) = sup ½ρ( )
ρ( ); ∈Lρ,0< ρ( )<∞¾, ≥0.
The ollowing echnical esul [13] is undamen al o his wo k.
Lemma 3.1. Le { n}nbe a sequence in Eρsuch ha n
ρ−a.e
→ ∈Eρand he e
exis s k > 1 such ha supnρ(k( n− )) <∞. Then, we ha e
lim in
n→∞
ρ( n−g) = lim in
n→∞
ρ( n− ) + ρ( −g) o all g∈Eρ
and, he e o e,
lim in
n→∞
ρ( n− )≤lim in
n→∞
ρ( n−g) o all g∈Eρ.
¿F om his lemma a uni o m Opial p ope y- ype o Eρcan be de i ed. Re-
call [22] ha a Banach space is said o sa is y he uni o m Opial p ope y wi h
espec o an a bi a y opology τi o e e y c > 0 he e exis s > 0 such ha
lim in
n→∞
||xn+x|| ≥ 1 + , i {xn}nis a τ-null sequence wi h lim in
n→∞
||xn|| ≥ 1 and
||x|| ≥ c. F om lemma 3.1 i is clea ha lim in
n→∞
ρ( n+ )≥1+ci ρ( )≥cand
{ n}nis a ρ-a.e null sequence which sa is ies lim in
n→∞
ρ( n)≥1 and he e exis s
K > 1 such ha supnρ¡K( n− )¢<∞.

6 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
4. SOME TECHNICAL RESULTS
De ini ion 4.1. Le ρbe a unc ion modula . 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.
As examples o con ex un ion modula wi h ∆2- ype condi ion we men ion,
he usual lpspaces and unc ion modula o O licz spaces, whe e he measu e
space (Ω,Σ, µ) is σ- ini e, he measu e µis a omless and in ini e, and he O licz
unc ion ψis con ex sa is ying ∆2- ype condi ion, i.e.
lim sup
u→∞
ψ(2u)
ψ(u)<∞and lim sup
u→0
ψ(2u)
ψ(u)<∞.
No e ha he ∆2- ype condi ion implies he ∆2-condi ion. Fo his condi ion we
e e o [15] and [19]. In he sequel, we will assume ha ρis con ex and sa is ies
he ∆2-condi ion. In his case we ha e Lρ=Eρ.
The ollowing lemma can be easily p o ed.
Lemma 4.1. The g ow h un ion ωρhas he ollowing p ope ies:
(1) ωρ( )<∞,∀ ∈[0,∞)
(2) ωρ: [0,∞)→[0,∞) is a con ex, s ic ly inc easing unc ion. So, i is
con inuous.
(3) ωρ(αβ)≤ωρ(α)ωρ(β); ∀α, β ∈[0,∞)
(4) ω−1
ρ(α)ω−1
ρ(β)≤ω−1
ρ(αβ);∀α, β ∈[0,∞), whe e ω−1
ρis he in e se unc-
ion o ωρ.
The ollowing is a echnical lemma which will be needed because o lack o he
iangula inequali y.
Lemma 4.2. Le { n}and {gn}be wo sequences in Lρ. Then
lim
n→∞
ρ(gn) = 0 =⇒lim sup
n→∞
ρ( n+gn) = lim sup
n→∞
ρ( n).
ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 7
P oo . By p ope y (iii) o he modula ρ, we ha e
ρ( n+gn)≤ρµ n
1−ε¶+ρ³gn
ε´,∀ε∈(0,1)
Thus,
ρ( n+gn)≤ωµ1
1−ε¶ρ( n) + ωµ1
ε¶ρ(gn)
and
lim sup
n→∞
ρ( n+gn)≤ωµ1
1−ε¶lim sup
n→∞
ρ( n)
Since εis a bi a y and
ωµ1
1−ε¶→1 as ε→0+
we ob ain
lim sup
n→∞
ρ( n+gn)≤lim sup
n→∞
ρ( n).
Fu he mo e, he same a gumen p o es
lim sup
n→∞
ρ( n) = lim sup
n→∞
ρ( n+gn−gn)
≤lim sup
n→∞
ρ( n+gn).
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.
Lemma 4.3. Le Lρbe a unc ion modula space sa is ying he ∆2- ype condi-
ion. Then
|| ||ρ≤1
ω−1³1
ρ( )´
P oo . Assume, α < || ||ρ.We ha e 1 < ρ( /α) which implies
1
ρ( )< ω µ1
α¶
8 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
and so
ω−1µ1
ρ( )¶<1
α.
Le ing α→ || ||ρ
−, we ob ain || ||ρ≤1
ω−1³1
ρ( )´.
5. MAIN RESULT
In his sec ion we p esen he main esul o his wo k. Indeed, we will p o e a
ixed poin heo em o asymp o ically egula mappings. Ce ainly, one can also
conside mappings which a e asymp o ically egula wi h espec o he F-no m
induced by he modula unc ion. We should like o men ion ha , gene ally
speaking, he e is no na u al ela ion be ween hese wo kinds o asymp o ically
egula ness. Indeed all esul s exp essed in e ms o modula s a e mo e con e-
nien in he sense ha hei assump ions a e much easie o e i y.
Le Ca subse o Lρand T:C→C, we deno e by |T| he exac Lipschi z
cons an o T, i.e.
|T|= sup ½ρ(T −Tg)
ρ( −g): 6=g, , g ∈C¾and s(T) = lim in
n→∞
|Tn|.
Theo em 5.1. Le ρbe a con ex modula unc ion sa is ying he ∆2- ype con-
di ion, Caρ-bounded, ρ−a.e sequen ially compac subse o Lρ. Le T:C→C
be an asymp o ically egula mapping such ha s(T)<2. Then, T has a ixed
poin .
P oo . Choose a sequence {nk}o posi i e in ege s such ha s(T) = lim
k→∞
|Tnk|=
lim in
n→∞
|Tn|and de ine a unc ion on C by
( ) = in { > 0 : ∃g∈Csuch ha lim in
k→∞
ρ( −Tnkg)≤ }.
Since s(T)<2, he e exis s b∈(1,2) such ha s(T)< b < 2.Le ε∈(0,1) such
ha ωρµ1
ε¶<1
b−1and choose γ∈(0,1) such ha ωρµ1
ε¶<γ
b−1.Since
γ+ (1 −b)ωρ¡1
ε¢>0, we can choose δ∈(0,1) such ha
ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 9
δ < γ + (1 −b)ωρ¡1
ε¢. Now, we choose a numbe µ∈(0,1) such ha
µ < min (δ
ωρ¡1
1−ε¢,γ−δ+ωρ¡1
ε¢(1 −b)
ωρ¡1
ε¢b)
Finally,deno e
α= max (1 + µ−δ
ωρ¡1
1−ε¢, b(1 + µ)−γ−δ
ωρ¡1
ε¢).
Then 0 < α < 1.Since γ < 1, we can ind a posi i e in ege k0such ha
|Tnk0|< b and
ρ( −Tnk0 )> γ ( ).
Since µ > 0, we can also ind g∈Csuch ha
lim in
k→∞
ρ( −Tnkg)≤ ( )(1 + µ).
Conside a subsequence {nk0}o {nk}such ha {Tnk0g}is ρ-a.e con e gen in
C, say o h, and
lim
k0→∞
ρ( −Tnk0g) = lim in
k→∞
ρ( −Tnkg).
The e o e, using Lemma 4.2 and he asymp o ic egula i y o Twe ha e
lim sup
k0→∞
ρ(Tnk0 −Tnk0g)≤ |Tnk0|lim sup
k0→∞
ρ( −Tnk0−nk0g)
=|Tnk0|lim sup
k0→∞
ρ( −Tnk0g+Tnk0g−Tnk0−nk0g)
=|Tnk0|lim sup
k0→∞
ρ( −Tnk0g).
We spli he p oo in o wo cases:
Case 1. Assume ha
ρ( −h)≥δ
ωρ¡1
1−ε¢ ( )