ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN
MODULAR FUNCTION SPACES
T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
ABSTRACT
In his pape , we p o e ha i ρis a con ex, σ- ini e modula unc ion sa is ying
a ∆2- ype condi ion, Ca con ex, ρ-bounded, ρ-a.e. compac subse o Lρand
T:C→Caρ-asymp o ically nonexpansi e mapping, hen Thas a ixed poin .
In pa icula , any asymp o ically nonexpansi e sel -map de ined on a con ex
subse o L1(Ω, µ) which is compac o he opology o local con e gence in
measu e has a ixed poin .
1991 Ma hema ics subjec classi ica ion : P ima y 46E30; Seconda y 47H09,
47H10.
Key Wo ds: asymp o ically nonexpansi e mappings, ixed poin , modula unc-
ions.
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
Le (M, d) be a me ic space. A mapping, T:M→Mis said o be asymp-
o ically nonexpansi e i he e exis s a sequence {kn}o eal numbe s wi h
lim
n→∞ kn= 1 such ha
d(Tnx, Tny)≤knd(x, y)
o any x, y ∈Mand n∈N.In 1970 Goebel and Ki k [5] p o ed ha Thas a
ixed poin whene e Mis a con ex bounded closed subse o a Banach space X.
Fu he gene aliza ions o his esul we e p o ed by Yu and Dai [14] when Xis
2-uni o mly o und, by Ma ´ınez Ya˜nez [10] and Xu [12] when Xis k-uni o mly
o und o some k≥1,by Xu [13] when Xis nea ly uni o mly con ex and by Kim
and Xu [9] when Xhas uni o m no mal s uc u e. Some special s udies on he
heo y o he ixed poin o asymp o ically nonexpansi e mappings we e made
by many o he au ho s (see, o example, [2,11]).
The i s ixed poin esul s in modula unc ion spaces we e gi en by Khamsi,
Koz lowski and Reich [7]. 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. Fo ins ance,
ixed poin heo ems a e p o ed in [6,7] o nonexpansi e mappings, in [3] o
asymp o ically egula mappings and in [4] o uni o mly Lipschi zian mappings.
In his pape we will p o e he exis ence o ixed poin s o asymp o ically nonex-
pansi e mappings in modula unc ion spaces when he modula ρsa is ies some
con exi y and ∆2- ype p ope ies.
Ou esul s can be, in pa icula , applied o L1(Ω, µ), showing ha asymp o i-
cally nonexpansi e mappings ha e a ixed poin when hey a e de ined on a con ex
subse o L1(Ω, µ) which is compac wi h espec o he opology o con e gence
local in measu e.
1. PRELIMINARIES
We s a by e iewing some basic ac s abou modula spaces as o mula ed
by Koz lowski [8]. Fo mo e de ails he eade may consul [6,7].
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 NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 3
Ω = SKn. 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 →Rsuch 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 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)| o e e y ω∈Ω}.
De ini ion 1.2. 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.
Fo he sake o simplici y we w i e ρ( ) ins ead o ρ( , Ω).
De ini ion 1.3. A modula unc ion ρis called σ- ini e i he e exis s an in-
c easing sequence o se s Kn∈ P such ha 0 < ρ(Kn)<∞and Ω = SKn.
I is easy o see ha he unc ional ρ:M → [0,∞] is a modula and sa is ies
he ollowing p ope ies:
(i) ρ( ) = 0 i = 0 ρ-a.e.
4 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
(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}.
When ρis con ex, he o mula
|| ||ρ= in nα > 0; ρ
α≤1o
de ines a no m in he modula space Lρwhich is equen ly called he Luxembu g
no m. 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}.
De ini ion 1.4. 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 om [8] ha Eρ=Lρwhen ρsa is ies he ∆2-condi ion.
De ini ion 1.5. A unc ion modula is said o sa is y he ∆2- ype condi ion
i he e exis s K > 0 such ha o any ∈Lρwe ha e ρ(2 )≤Kρ( ).
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 on he modula space
Lρ.
De ini ion 1.6. Le Lρbe a modula space.
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 5
(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. 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. 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}<∞.
We ecall wo basic esul s (see [7]) in he heo y o modula spaces.
(i) I he e exis s a numbe α > 0 such ha ρ(α( n− )) →0, hen he e
exis s a subsequence {gn}no { n}nsuch ha gn→ ρ-a.e.
(ii) (Lebesgue’s Theo em) I n, ∈ M, n→ ρ-a.e. and he e exis s a
unc ion g∈Eρsuch ha | n| ≤ |g|ρ-a.e. o all n, hen || n− ||ρ→0.
We know, by [6,7] ha unde ∆2-condi ion he no m con e gence and modula
con e gence a e equi alen , which implies ha he no m and modula con e -
gence a e also he same when we deal wi h he ∆2- ype condi ion.
In he sequel we will assume ha he modula unc ion ρis con ex and sa is ies
he ∆2- ype condi ion.
De ini ion 1.7. Le ρbe as abo e. We de ine a g ow h unc ion ωby:
ω( ) = sup ρ( )
ρ( ), ∈Lρ {0} o all 0 ≤ < ∞.
We ha e he ollowing:
Lemma 1.1. [3] Le ρbe as abo e. Then he g ow h unc ion ωhas he ollowing
p ope ies:

6 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
(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.
Lemma 1.2. [3] 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 nex lemma will be o majo in e es h oughou his wo k.
Lemma 1.3. [6] Le ρbe a unc ion modula sa is ying he ∆2-condi ion and
{ n}nbe a sequence in Lρsuch ha n
ρ−a.e
→ ∈Lρand he e exis s k > 1such
ha sup
n
ρ(k( n− )) <∞. Then,
lim in
n→∞ ρ( n−g) = lim in
n→∞ ρ( n− ) + ρ( −g) o all g∈Lρ.
Mo eo e , we ha e
ρ( )≤lim in
n→∞ ρ( n).
2. AN EQUIVALENT TOPOLOGY
The concep o ρ-a.e. closed, compac se s ha e been s udied ex ensi ely in
he sequen ial case. One o he p oblem ha many au ho s ha e ound ha d o
ci cum en is whe he hese no ions a e ela ed o a opology. In his sec ion
we will discuss his p oblem. In pa icula , we will cons uc a opology τ o
which he ρ-a.e. compac ness is equi alen o he usual compac ness o τ. This
is c ucial when we y o use Zo n’s lemma.
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 7
F om now on, we assume ha he modula unc ion ρis, in addi ion, σ- ini e.
Se
d( , g) =
∞
X
k=1
1
2k
1
ρ(1Kk)ρ| −g|
1 + | −g|1Kk o any , g ∈Lρ.
Some basic p ope ies sa is ied by da e discussed in he ollowing p oposi ion.
P oposi ion 2.1. The unc ional dsa is ies he ollowing:
(1) d( , g) = 0 i and only i =g ρ-a.e.;
(2) d( , g) = d(g, );
(3) d( , g)≤ω(2)
2d( , h) + d(h, g);
o any , g and hin Lρ.
P oo . (1) and (2) a e ob ious. To p o e (3) we only need o ecall he inequali y
|a+b|
1 + |a+b|≤|a|
1 + |a|+|b|
1 + |b|
o all posi i e numbe s a, b and use he de ini ion o he g ow h unc ion ω.

Rema k 2.1. The unc ional dis no a dis ance because o (3). Bu he e a e
many ma hema ical objec s which ail he iangle inequali y bu a e e y use ul
ools. Tha is he case wi h d.
In he nex p oposi ion, we discuss he ela ionship be ween ρ-a.e. con e gence
and he con e gence o he unc ional d.
P oposi ion 2.2. Le ρbe a con ex, σ- ini e modula sa is ying he ∆2- ype
condi ion and { n}nbe a sequence o measu able unc ions. I { n}nis ρ-a.e.
con e gen o , hen
lim
n→∞ d( n, ) = 0.
Mo eo e , i
lim
n→∞ d( n, ) = 0,
8 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
hen he e exis s a subsequence { nk}kwhich con e ges ρ-a.e. o .
P oo . Assume ha { n}nρ-a.e. con e ges o . We will show ha lim
n→∞ d( n, ) =
0.Le ε > 0,and choose N∈Nsuch ha
∞
X
k=N+1
1
2k< ε. We ha e
lim
n→∞ d( n, )≤lim
n→∞
N
X
k=1
1
2k
1
ρ(1Kk)ρ| n− |
1 + | n− |1Kk+ε
=
N
X
k=1
lim
n→∞
1
2k
1
ρ(1Kk)ρ| n− |
1 + | n− |1Kk+ε.
Since
| n− |
1 + | n− |1Kk
ρ−a.e
−→ 0 as n→ ∞
o any k∈Nand | n− |
1 + | n− |1Kk≤1Kk, om Lebesgue’s Theo em we ob ain
lim
n→∞ ρ| n− |
1 + | n− |1Kk= 0 o e e y non null in ege k. Thus lim
n→∞ d( n, )≤ε
o each ε > 0 which means ha lim
n→∞ d( n, ) = 0.
Assume now ha lim
n→∞ d( n, ) = 0.Fo e e y non null in ege kwe ha e
lim
n→∞ ρ| n− |
1 + | n− |1Kk= 0.
Thus, he e exis s a subsequence { 1
n}no { n}nsuch ha | 1
n− |
1 + | 1
n− |1K1
ρ−a.e
−→ 0
and so 1
n
ρ−a.e
−→ in K1i.e. lim
n→∞ 1
n(x) = (x) whene e x∈K1 A1whe e
A1⊂K1and ρ(1A1) = 0.
By induc ion and using a diagonal a gumen we ob ain a subsequence o { n}n
which con e ges ρ-a.e. o . 
De ini ion 2.1. Le Cbe a subse o Lρ.
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 9
(a) Cis said o be d-closed i o any sequence { n}nin Cwhich d-con e ges
o , hen we ha e ∈C.
(b) Cis d-open i Lρ Cis d-closed.
(c) Cis said o be d-sequen ially compac i o each sequence { n}n he e
exis s a subsequence { nk}kwhich d- con e ges o a poin in C.
I is easily seen ha he amily o all d-open subse s o Lρ o m a opology on
Lρ.Fu he mo e, om p oposi ion (2.2) d-sequen ially compac se s and ρ-a.e.
compac se s a e iden ical. On he o he hand, e en hough dsa is ies (3) ins ead
o he iangula inequali y, he usual a gumen s which p o e ha sequen ial
compac ness and compac ness a e iden ical in me ic spaces hold in his se ing.
We also ha e d-sequen ial compac ness and d-compac ness a e iden ical.
3. TECHNICAL LEMMAS
In he sequel we assume ha ρis a con ex, σ- ini e modula unc ion sa is ying
he ∆2- ype condi ion, Cis a con ex, ρ-bounded and ρ-a.e. compac subse o he
modula unc ion space Lρand T:C→Cis a ρ-asymp o ically nonexpansi e
mapping, i.e. he e exis s a sequence o posi i e in ege s {kn}nwhich con e ge
o 1 such ha o e e y n∈Nand , g ∈Cwe ha e ρ(Tn −Tng)≤knρ( −g).
Lemma 3.1. Unde he abo e assump ions, le { n}nbe a sequence o elemen s
o C. Conside he unc ional Φ : C→Rde ined by Φ(g) = lim sup
n→∞
ρ( n−g).
Then, o any sequence {gm}min Cwhich ρ-a.e. con e ges o g∈Cwe ha e
Φ(g)≤lim in
m→∞ Φ(gm).
P oo . Since Cis ρ-a.e. compac , he e exis s a subsequence { φ(n)}no { n}n
such ha φ(n)
ρ−a.e
−→ ∈Cand lim
n→∞ ρ( φ(n)−g) = lim sup
n→∞
ρ( n−g).Hence
Φ(gm) = lim sup
n→∞
ρ( n−gm)
≥lim sup
n→∞
ρ( φ(n)−gm)
≥lim in
n→∞ ρ( φ(n)−gm).
16 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
Fix n≥n0. The e exis s k0≥1 such ha o all k≥k0, we ha e n(k)≥n+n0
and
ρ(Tn −Tn(k) ) = ρTn −Tn+(n(k)−n) =ρTn −Tn(Tn(k)−n )
≤knρ −Tn(k)−n < kn( +η).
No e ha i n
ρ−a.e
−→ and Sep{ n}n≥ε, hen by Lemma (1.3), we ha e
ε≤lim in
m→∞ lim in
n→∞ ρ( n− m)≤2 lim in
n→∞ ρ( n− ).
Combined wi h Lemma (1.3), we ge
lim in
n→∞ ρ( n) = lim in
n→∞ ρ( n− ) + ρ( )≥
2+ρ( ).
In pa icula , since {Tn(k) −Tn }kis ρ-a.e. con e gen o ∞−Tn as k→ ∞
and sa is ies Sep({Tn(k) −Tn }k)≥, we ge
ρ(Tn − ∞)≤lim in
k→∞ ρ(Tn(k) −Tn )−
2.
Hence
ρ( ∞−Tn )≤ +η−
2
which implies
= lim sup
n→∞
ρ( ∞−Tn )≤ +η−
2< .
This con adic ion comple es he p oo o Theo em 4.2.

Assume ha Lρ=Lp(Ω, µ) o a σ- ini e measu e µ. I Cis a con ex, bounded
and closed subse o Lp o 1 < p < ∞and T:C→Cis asymp o ically non-
expansi e, i is known ha Chas a ixed poin because Lpis uni o mly con ex.
Howe e he esul does no hold o p= 1 (e en o nonexpansi e mappings, see
[1]). Since L1is a modula space, Theo em (4.1) implies he exis ence o ixed
poin i p= 1 when Cis ρ-a.e. compac . Thus we can s a e.
Co olla y 4.1. Le (Ω, µ) be as abo e, C⊂L1(Ω, µ) a con ex bounded se
which is compac o he opology o local con e gence in measu e and T:C→C
asymp o ically nonexpansi e. Then, Thas a ixed poin .

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 17
P oo . Unde he abo e hypo hesis ρ-a.e. compac se s and compac se s in he
opology o local con e gence in measu e a e iden ical.

Re e ences
[1] D.E. Alspach. A ixed poin ee nonexpansi e map. P oc. Am. Ma h. Soc., (1981), 82,
423-424.
[2] S.C. Bose. Weak con e gence o he ixed poin o an asymp o ically nonexpansi e map.
P oc. Am. Ma h. Soc., (1978), 68, 305-308.
[3] T. Dominguez Bena ides, M.A. Khamsi, S. Samadi. Asymp o ically egula mappings in
modula unc ion spaces. P ep in .
[4] T.Dominguez Bena ides, M.A. Khamsi, S. Samadi. Uni o mly Lipschi zian mappings in
modula unc ion spaces. P ep in .
[5] K. Goebel and W.A. Ki k. A ixed poin heo em o asymp o ically nonexpansi e mappings.
P oc. Am. Ma h. Soc., (1972), 35, 171-174.
[6] M.A. Khamsi. Fixed poin heo y in modula unc ion spaces. Recen Ad ances on Me ic
Fixed Poin Theo y. Uni e sidad de Se illa, Se illa, (1996), 31-58.
[7] M.A. Khamsi, W.M. Koz lowski, S. Reich. Fixed poin heo y in modula unc ion spaces.
Nonlinea Anal., (1990), 14, 935-953.
[8] W.M. Kos lowski. Modula unc ion spaces. Dekke : New Yo k, Basel, (1988).
[9] T.-H. Kim and H.-K. Xu. Rema ks on asymp o ically nonexpansi e mappings. Nonlinea
Anal., o appea .
[10] C. Ma ´ınez Ya˜nez. A ixed poin heo em on k-uni o mly o und spaces. Nonlinea Anal.,
(1988), 13 857-861.
[11] G. Pass y. Cons uc ion o ixed poin s o asymp o ically nonexpansi e mappings. P oc.
Am. Ma h. Soc., (1982), 84, 213-216.
[12] H.-K. Xu. k-Uni o m o undi y and ixed poin s o mappings o asymp o ically nonexpansi e
ype. To appea in Chinese.
[13] H.-K. Xu. Exis ence and con e gence o ixed poin s o mappings o asymp o ically non-
expansi e ype. Nonlinea Anal., (1991) 16(12), 1139-1146.
[14] X.T. Yu and X. Dai. A ixed poin heo em o asymp o ically nonexpansi e mappings. J.
Ma h. (PRC), (1986), 6, 255-262.
18 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI
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]