FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN
MODULAR FUNCTION SPACES*
S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B.
PANYANAK
Abs ac . The pu pose o his pape is o s udy he exis ence o ixed poin s
o mul i alued nonexpansi e mappings in modula unc ion spaces. We apply
ou main esul o ob ain ixed poin heo ems o mul i alued mappings in he
Banach spaces L1and l1.
1. In oduc ion
The heo y o modula spaces was ini ia ed by Nakano [?] 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 [?] in 1959. 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 and sol ed in modula spaces (see, o
ins ance, [?,?,?,?]). In pa icula , some ixed poin heo ems o (single alued)
nonexpansi e mappings in modula unc ion spaces a e gi en in [?]. In 1969, Nadle
[?] es ablished he mul i alued e sion o Banach’s con ac ion p inciple in me ic
spaces. Since hen he me ic ixed poin heo y o mul i alued mappings has been
apidly de eloped and many o pape s ha e appea ed p o ing he exis ence o ixed
poin s o mul i alued nonexpansi e mappings in special classes o Banach spaces
(see, o ins ance, [?,?,?,?]). In his pape , we s udy simila p oblems in he
se ing o modula unc ion spaces. Namely, we p o e ha e e y ρ−con ac ion
T:C→Fρ(C) has a ixed poin whe e ρis a con ex unc ion modula sa is y-
ing he ∆2− ype condi ion and Cis a nonemp y ρ−bounded ρ−closed subse o
Lρ.By using his esul , we can asse he exis ence o ixed poin s o mul i al-
ued ρ−nonexpansi e mappings. Finally, we apply ou main esul o ob ain ixed
poin heo ems in he Banach space L1( esp. l1) o mul i alued mappings whose
domains a e compac in he opology o he con e gence locally in measu e ( esp.
w∗− opology).
Key wo ds and ph ases : Mul i alued mappings, ixed poin s, Modula unc ion spaces.
* Suppo ed by Thailand Resea ch Fund unde g an BRG4780013. The second au ho was
pa ially suppo ed by DGES, G an D.G.E.S. REF. PBMF2003-03893-C02-C01 and Jun a de
Andalucia, G an FQM-127. The hi d and ou h au ho s we e suppo ed by he Royal Golden
Jubilee p og am unde g an PHD/0250/2545 and PHD/0251/2545, espec i ely.
1
2 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK
2. P elimina ies
We s a by ecalling some basic concep abou modula unc ion spaces. Fo
mo e de ails he eade is e e ed o [?,?].
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 Ω, 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( o ins ance, Pcan be he se s o ini e measu e in a σ− ini e measu e
space). By Ewe deno e he linea space o all simple unc ions wi h suppo 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.
Le us ecall ha a se unc ion µ: Σ →[0,∞] is called a σ−subaddi i e
measu e i µ(∅)=0, µ(A)≤µ(B) o any A⊂Band µ(SAn)≤Pµ(An) o any
sequence o se s {An} ⊂ Σ.By χAwe deno e he cha ac e is ic unc ion o he se
A.
De ini ion 2.1. 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(αχA, 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 > 0, ρ(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∈Ωª.
De ini ion 2.2. A se Eis said o be ρ−null i ρ(α, E) = 0 o e e y 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 .
In he abo e condi ion, we de ine he unc ion ρ:M → [0,∞] by ρ( ) =
ρ( , Ω).We know om [?] ha ρsa is ies he ollowing p ope ies :
(i) ρ( ) = 0 i and only 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, β ≥0 and , g ∈ M.
In addi ion, i he ollowing p ope y is sa is ied
FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES3
(iii)’ ρ(α +βg)≤αρ( ) + βρ(g) i α+β= 1, α, β ≥0 and , g ∈ M,
we say ha ρis con ex modula .
The modula ρde ines a co esponding modula space Lρ,which is gi en by
Lρ={ ∈ M :ρ(λ )→0 as λ→0}.
In gene al he modula ρis no subaddi i e and he e o e does no beha e as
a no m o a dis ance. Bu one can associa e o a modula an F−no m. Recall ha
a unc ional k·k:X→[0,∞] de ines an F−no m i and only i
(1) kxk= 0 i and only i x= 0,
(2) kαxk=kxkwhene e |α|= 1,
(3) kx+yk ≤ kxk+kyk,
(4) kαnxn−αxk → 0 i αn→αand kxn−xk → 0.
The modula space Lρcan be equipped wi h an F−no m de ined by
k kρ= in nα > 0 : ρµ
α¶≤αo.
We know om [?] ha he linea space (Lρ,k·kρ) is a comple e me ic space.
I ρis con ex he o mula
k kρ= in nα > 0 : ρµ
α¶≤1o
de ines a no m which is equen ly called he Luxembu g no m. The o mula
k ka= in ½1
k(1 + ρ(k )) : k > 0¾
de ines a di e en no m which is called Amemiya no m. Mo eo e , k · kρand k · ka
a e equi alen no ms. We can also conside he space
Eρ={ ∈ M :ρ(α , ·) is o de con inuous o all α > 0}.
De ini ion 2.3. A unc ion modula ρis said o sa is y he ∆2−condi ion i
sup
n≥1
ρ(2 n, Dk)→0 as k→ ∞ whene e { n} ⊂ M, Dk∈Σ
dec eases o ∅and sup
n≥1
ρ( n, Dk)→0 as k→ ∞.
I is known ha he ∆2−condi ion is equi alen o Eρ=Lρ.
De ini ion 2.4. A unc ion modula ρis 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, he ∆2− ype condi ion and ∆2−condi ion a e no equi alen , e en
hough i is ob ious ha he ∆2− ype condi ion implies he ∆2−condi ion.
De ini ion 2.5. Le Lρbe a modula space.
4 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK
(1) The sequence { n} ⊂ Lρis said o be ρ−con e gen o ∈Lρi ρ( n− )→
0 as n→ ∞.
(2) The sequence { n} ⊂ Lρis said o be ρ−a.e. con e gen o ∈Lρi he
se {ω∈Ω : n(ω)9 (ω)}is ρ−null.
(3) A subse Co Lρis called ρ−closed i he ρ−limi o a ρ−con e gen se-
quence o Calways belongs o C.
(4) 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.
(5) A subse Co Lρis called ρ−compac i e e y sequence in Chas a ρ−con e gen
subsequence in 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
diamρ(C) = sup{ρ( −g) : , g ∈C}<∞.
We know by [?] ha unde he ∆2−condi ion he no m con e gence and mod-
ula 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 2.6. Le ρbe as abo e. We de ine a g ow h unc ion ωby
ω( ) = sup nρ( )
ρ( ): ∈Lρ,0< ρ( )<∞o o all 0 ≤ < ∞.
The ollowing p ope ies o he g ow h unc ion can be ound in [?].
Lemma 2.7. Le ρbe as abo e. Then he g ow h unc 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.
Lemma 2.8 (T. Dominguez Bena ides e al. [?]).Le ρbe as abo e. Then
k kρ≤1
ω−1(1/ρ( )) whene e ∈Lρ {0}.
The ollowing lemma is a echnical lemma which will be need because o lack
o he iangula inequali y.
FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES5
Lemma 2.9 (T. Dominguez Bena ides e al. [?]).Le ρbe as abo e, { 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)
and
lim
n→∞ ρ(gn) = 0 =⇒lim in
n→∞ ρ( n+gn) = lim in
n→∞ ρ( n)
In he same way as he Hausdo dis ance de ined on he amily o bounded
closed subse s o a me ic space, we can de ine he analogue o he Hausdo dis ance
o modula unc ion spaces. We will call ρ−Hausdo dis ance e en hough i is
no a me ic.
De ini ion 2.10. Le Cbe a nonemp y subse o Lρ.We shall deno e by Fρ(C) he
amily o nonemp y ρ−closed subse s o Cand by Kρ(C) he amily o nonemp y
ρ−compac subse s o C. Le Hρ(·,·) be he ρ−Hausdo dis ance on Fρ(Lρ),i.e.,
Hρ(A, B) = max nsup
∈A
dis ρ( , B),sup
g∈B
dis ρ(g, A)o, A, B ∈Fρ(Lρ).
whe e dis ρ( , B) = in {ρ( −g) : g∈B}is he ρ−dis ance be ween and B. A
mul i alued mapping T:C→Fρ(Lρ) is said o be a ρ−con ac ion i he e exis s
a cons an k∈[0,1) such ha
(2.1) Hρ(T , T g)≤kρ( −g), , g ∈C.
I (??) is alid when k= 1, hen Tis called ρ−nonexpansi e. A unc ion ∈Cis
called a ixed poin o a mul i alued mapping Ti ∈T .
3. Main esul s
We begin s a ing he Banach Con ac ion P inciple o mul i alued mappings
in modula unc ion spaces.
Theo em 3.1. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype con-
di ion, Ca nonemp y ρ−bounded ρ−closed subse o Lρ,and T:C→Fρ(C)a
ρ−con ac ion mapping, i.e., he e exis s a cons an k∈[0,1) such ha
(3.1) Hρ(T , T g)≤kρ( −g), , g ∈C.
Then Thas a ixed poin .
P oo . Le 0∈Cand α∈(k, 1).Since T 0is nonemp y, he e exis s 1∈T 0
such ha ρ( 0− 1)>0 (o he wise 0is a ixed poin o T). In iew o (??), we
ha e
dis ρ( 1, T 1)≤Hρ(T 0, T 1)≤kρ( 0− 1)< αρ( 0− 1).
Since dis ρ( 1, T 1) = in {ρ( 1−g) : g∈T 1},i ollows ha he e exis s 2∈T 1
such ha
ρ( 1− 2)< αρ( 0− 1).

6 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK
Simila ly, he e exis s 3∈T 2such ha
ρ( 2− 3)< αρ( 1− 2).
Con inuing in his way, he e exis s a sequence { n}in Csa is ying n+1 ∈T n
and
ρ( n− n+1)< αρ( n−1− n)
< α2(ρ( n−2− n−1))
< ...
< αn−1(ρ( 1− 2))
< αn(ρ( 0− 1))
≤αndiamρ(C),
Le M= diamρ(C), hen
1
αnM<1
ρ( n− n+1).
By Lemma ??, we ha e
³ω−1¡1
α¢´n
ω−1³1
M´< ω−1³1
ρ( n− n+1)´,
I ollows ha 1
ω−1³1
ρ( n− n+1)´<1
³ω−1¡1
α¢´n
ω−1³1
M´.
By Lemma ??, we ob ain
k n− n+1kρ<³1
ω−1¡1
α¢´n·1
ω−1³1
M´.
Since ω−1is s ic ly inc easing, we ha e 1
ω−1¡1
α¢<1.This implies ha { n}is
a Cauchy sequence in (Lρ,k · kρ).Since (Lρ,k·kρ) is a comple e me ic space,
he e exis s ∈Lρsuch ha { n}is k · kρ−con e gen o . Since unde he
∆2− ype condi ion, no m con e gence and modula con e gence a e iden ical, { n}
is ρ−con e gen o and ∈Cbecause Cis ρ−closed. Since n∈T n−1, we ha e
(3.2) dis ρ( n, T )≤Hρ(T n−1, T )≤kρ( n−1− )−→ 0.
We obse e ha , o each n, he e exis s gn∈T such ha
(3.3) ρ( n−gn)≤dis ρ( n, T ) + 1
n.
Thus, (??) and (??) imply ha lim
n→∞ ρ( n−gn) = 0.By Lemma ??,
lim sup
n→∞
ρ(gn− ) = lim sup
n→∞
ρ(gn− n+ n− ) = lim sup
n→∞
ρ( n− ) = 0.
Since T is ρ−closed, we can conclude ha ∈T . ¤
FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES7
The ollowing esul s will be e y use ul in he p oo o ou main heo em.
Theo em 3.2 (M. A. Khamsi [?]).Le { n} ⊂ Lρbe ρ−a.e. con e gen o 0.
Assume he e exis s k > 1such ha
sup
n≥1
ρ(k n) = M < ∞.
Le g∈Eρ, hen we ha e
lim in
n→∞ ρ( n+g) = lim in
n→∞ ρ( n) + ρ(g).
The ollowing lemma gua an ee ha e e y nonemp y ρ−compac subse o Lρ
a ains a nea es poin .
Lemma 3.3. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype condi ion,
∈Lρ,and Ka nonemp y ρ−compac subse o Lρ.Then he e exis s g0∈Ksuch
ha
ρ( −g0) = dis ρ( , K).
P oo . Le m= dis ρ( , K).Fo each n∈N, he e exis s gn∈Ksuch ha
m−1
n≤ρ( −gn)≤m+1
n.
By he ρ−compac ness o K, we can assume, by passing h ough a subsequence,
ha gn
ρ
−→ g0∈K. By Lemma ??, we ob ain
m= lim sup
n→∞
ρ(gn− ) = lim sup
n→∞
ρ(gn−g0+g0− )
= lim sup
n→∞
ρ(g0− )
=ρ(g0− ).
¤
We can now s a e ou main heo em.
Theo em 3.4. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype condi-
ion, Ca nonemp y ρ−a.e. compac ρ−bounded con ex subse o Lρ,and T:C→
Kρ(C)aρ−nonexpansi e mapping. Then Thas a ixed poin .
P oo . Fix 0∈C. Fo each n∈N, he ρ−con ac ion Tn:C→Fρ(C) is de ined
by
Tn( ) = 1
n 0+ (1 −1
n)T , ∈C.
By Theo em ??, we can conclude ha Tnhas a ixed poin , say n.I is easy o see
ha
dis ρ( n, T n)≤1
ndiamρ(C)−→ 0.
8 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK
Because o ρ−a.e. compac ness o C, we can assume, by passing h ough a subse-
quence, ha n
ρ−a.e.
−→ o some ∈C. By Lemma ??, o each n∈N, he e exis s
gn∈T nand hn∈T such ha
ρ( n−gn) = dis ρ( n, T n)
and
ρ(gn−hn) = dis ρ(gn, T )≤Hρ(T n, T )≤ρ( n− ).
Because o ρ−compac ness o T , we can assume, by passing h ough a subsequence,
ha hn
ρ
−→ h∈T . Since ρsa is ies he ∆2− ype condi ion, he e exis s K > 0
such ha ρ(2( n− )) ≤Kρ( n− ) o all n∈N.
This implies ha
sup
n≥1
ρ(2( n− )) ≤Ksup
n≥1
ρ( n− )<∞.
By Theo em ?? and Lemma ??, we ob ain
lim in
n→∞ ρ( n− ) + ρ( −h) = lim in
n→∞ ρ( n− + −h)
= lim in
n→∞ ρ( n−h)
= lim in
n→∞ ρ( n−gn+gn−hn+hn−h)
= lim in
n→∞ ρ(gn−hn)
≤lim in
n→∞ ρ( n− ).
I ollows ha ρ( −h) = 0 and hen we ha e =h∈T . ¤
Conside he space Lp(Ω, µ) o a σ− ini e measu e µwi h he usual no m.
Le Cbe a bounded closed con ex subse o Lp o 1 < p < ∞and T:C→K(C)
a mul i alued nonexpansi e mapping. Because o uni o m con exi y o Lp,i is
known ha Thas a ixed poin . Fo p= 1, T can ail o ha e a ixed poin e en in
he single alued case o a weakly compac con ex se C(see [?]). Howe e , since
L1is a modula space whe e ρ( ) = RΩ| |dµ =k k o all ∈L1,Theo em ??
implies he exis ence o a ixed poin when we de ine mappings on a ρ−a.e. compac
ρ−bounded con ex subse o L1.Thus we can s a e :
Co olla y 3.5. Le (Ω, µ)be as abo e, C⊂L1(Ω, µ)a nonemp y bounded con ex
se which is compac o he opology o he con e gence locally in measu e, and
T:C→K(C)a nonexpansi e mapping. Then Thas a ixed poin .
P oo . Unde he abo e hypo hesis ρ−a.e. compac se s and compac se s in he
opology o he con e gence locally in measu e a e iden ical (see [?]). Consequen ly,
Theo em ?? can be applied o ob ain a ixed poin o T. ¤
In he case o he space l1we also can ob ain a bounded closed con ex se C
and a nonexpansi e mapping T:C→Cwhich is ixed poin ee. Indeed, conside
FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES9
he ollowing easy and well known example :
Le
C=n{xn} ∈ l1: 0 ≤xn≤1 and
∞
X
n=1
xn= 1o.
De ine a nonexpansi e mapping T:C→Cby
T(x) = (0, x1, x2, x3, ...) whe e x={xn}.
Then Tis a ixed poin ee. Howe e , i we conside Lρ=l1whe e ρ(x) =
kxk,∀x∈l1.Then ρ−a.e. con e gence and w∗−con e gence a e iden ical on
bounded subse s o l1(see [?]). This ac leads us o ob ain he ollowing co ol-
la y :
Co olla y 3.6. Le Cbe a nonemp y w∗−compac con ex subse o l1and T:C→
K(C)a nonexpansi e mapping. Then Thas a ixed poin .
P oo . By he abo e a gumen , we know ha ρ−a.e. compac bounded se s
and w∗−compac se s a e iden ical. Then we can apply Theo em ?? o asse he
exis ence o a ixed poin o T. ¤
In ac ?? and ?? a e consequences o a gene al esul : Assume ha Xis a
linea no med space and τis a Hausdo opology on X. We say ha Xsa is ies
he s ic τ-Opial p ope y i
lim sup
n→∞
kxn−xk<lim sup
n→∞
kxn−yk
o each sequence {xn}in Xwhich con e ges o x o he opology τand each y6=x.
Following he same a gumen as in [?] i is easy o p o e he ollowing heo em:
Theo em 3.7. Le Xbe a Banach space, Ca con ex bounded sequen ially τ-
compac subse o X, and T:C→K(C)a nonexpansi e mapping. I Xsa is ies
he s ic τ-Opial p ope y, hen Thas a ixed poin .
When Xis a modula unc ion space equipped wi h ei he Luxembu g o
Amemiya no m, we can conside he opology τo con e gence ρ-a.e. In his case,
?? yields o he ollowing:
Theo em 3.8. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype condi-
ion. Assume ha Lρis equipped ei he wi h Luxembu g o Amemiya no m. Le C
be a nonemp y ρ−a.e. compac ρ−bounded con ex subse o Lρ,and T:C→K(C)
a nonexpansi e mapping. Then Thas a ixed poin .
P oo . F om [?] (Theo em 4.1 and 4.3), Xsa is ies he uni o m Opial p ope y
wi h espec o he opology o ρ-a.e. con e gence. Since ρ-a.e. compac se s and
ρ-a.e. sequen ially compac se s a e iden ical o his opology (see [?]), we can
deduce he esul om ?? ¤