Hindawi Publishing Co po a ion
Fixed Poin Theo y and Applica ions
Volume 2009, A icle ID 809315, 8pages
doi:10.1155/2009/809315
Resea ch A icle
A Con inua ion Me hod o Weakly Con ac i e
Mappings unde he In e io Condi ion
Da id A iza-Ruiz and An onio Jim´
enez-Melado
Depa amen o de An´
alisis Ma em´
a ico, Facul ad de Ciencias, Uni e sidad de M´
alaga,
29071 M´
alaga, Spain
Co espondence should be add essed o An onio Jim´
enez-Melado, [email p o ec ed]
Recei ed 29 July 2009; Accep ed 8 Oc obe 2009
Recommended by Ma lene F igon
Recen ly, F igon p o ed ha , o weakly con ac i e maps, he p ope y o ha ing a ixed poin is
in a ian by a ce ain class o homo opies, ob aining as a consequence a Le ay-Schaude al e na i e
o his class o maps in a Banach space. We p o e he e ha he Le ay-Schaude condi ion in he
a o emen ioned esul can be eplaced by a modi ica ion o i , he in e io condi ion. We also show
ha ou a gumen s wo k o a ce ain class o gene alized con ac ions, hus complemen ing a
esul o Aga wal and O’Regan.
Copy igh q2009 D. A iza-Ruiz and A. Jim´
enez-Melado. This is an open access a icle dis ibu ed
unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
1. In oduc ion
Suppose ha Xis a Banach space, ha U⊂Xis an open bounded subse o X, con aining
he o igin, and ha :U→Xis a mapping. I is well known ha i sa is ies he Le ay-
Schaude condi ion de ined as
x/
λx, o x∈∂U, λ > 1L-S
and is a s ic se -con ac ion o , mo e gene ally, condensing, hen has a ixed poin in U
see, e.g., 1o 2. The i s con inua ion me hod in he se ing o a comple e me ic space
o con ac i e maps comes om he hands o G anas 3, in 1994, who ga e a homo opy
esul o con ac i e maps  o mo e in o ma ion on his opic see, e.g., 4,5o 6.
On he o he hand, i has been ecen ly shown in 7 ha , o condensing mappings,
he condi ion L-Scan be eplaced by a modi ica ion o i which we call he in e io condi ion,
2 Fixed Poin Theo y and Applica ions
and is de ined as ollows: a mapping :U→Xsa is ies he In e io Condi ion I-C, i he e
exis s δ>0 such ha
x/
λx, o x∈Uδ,λ>1,
x/
∈U, I-C
whe e Uδ{x∈U:dis x, ∂U<δ}some gene aliza ions o his esul can be ound in
8,9.
We ema k ha he condi ion I-Cby i sel canno be a subs i u e o he condi ion
L-S, and an addi ional assump ion on he domain o needs o be made in o de o
gua an ee he exis ence o a ixed poin o . The class o se s ha we need is de ined
as ollows: suppose ha U⊂Xis an open neighbo hood o he o igin. We say ha Uis
s ic ly s a shaped i o any x∈∂U we ha e ha {λx :λ>0}∩∂U {x}.I was
shown in 7 ha i Uis bounded and s ic ly s a shaped and :U→Xis a condensing
mapping sa is ying he condi ion I-C, hen has a ixed poin . O cou se, his esul includes
he case o a con ac i e map i.e., a map o which he e exis s k∈0,1such ha
d x, y ≤kdx, y o all x, y ∈U, bu ou aim in his no e is, ollowing he pa e n
o G anas 3and F igon e al. 10, o gi e a con inua ion me hod o weakly con ac i e
mappings, in he se ing o a comple e me ic space, unde some condi ions on he homo opy
which a e he coun e pa o he condi ion I-Cand he no ion o a s ic ly s a shaped se
in a space wi hou a ec o s uc u e. Finally, in he las sec ion we show ha ou a gumen s
also wo k o a class o gene alized con ac ions, hus complemen ing a esul o Aga wal
and O’Regan 11.
2. Weakly Con ac i e Maps
In his chap e we deal wi h he concep o weakly con ac i e maps, as i was in oduced by
Dugundji and G anas in 12.
De ini ion 2.1. Le X, dbe a comple e me ic space and Uan open subse o X.A unc ion
:U→Xis said o be weakly con ac i e i he e exis s ψ:X×X→0,∞compac ly
posi i e i.e., in {ψx, y:a≤dx, y≤b}θa, b>0 o e e y 0 <a≤bsuch ha
d x, y≤dx, y−ψx, y.2.1
I ψis a compac ly posi i e unc ion, we de ine o 0 <a≤b
γa, bmin{a, θa, b}.2.2
I was shown in 12 ha any weakly con ac i e map :X→Xde ined on a
comple e me ic space Xhas a unique ixed poin . Some yea s la e , F igon 5p o ed ha , o
weakly con ac i e maps, he p ope y o ha ing a ixed poin is in a ian by a ce ain class o
homo opies, ob aining as a consequence a Le ay-Schaude al e na i e o weakly con ac i e
maps in he se ing o a Banach space. We p o e he e ha he Le ay-Schaude condi ion in
he a o emen ioned esul can be eplaced by he condi ion I-C, and i will also be ob ained
as a consequence o a con inua ion me hod. The de ini ion o homo opy ha we need o ou
pu poses is he ollowing.
Fixed Poin Theo y and Applica ions 3
De ini ion 2.2. Le X, dbe a comple e me ic space, and Uan open subse o X.Le ,g :
U→Xbe wo weakly con ac i e maps. We say ha is I-C-homo opic o gi he e exis s
H:U×0,1→Xwi h he ollowing p ope ies:
P1Hx,1 xand Hx, 0gx o e e y x∈U;
P2 he e exis s δ>0 such ha x/
Hx,  o e e y x∈Uδ,wi h x/
∈U,and ∈0,1,
whe e Uδ{x∈U:dis x, ∂U<δ};
P3 he e exis s a compac ly posi i e unc ion ψ:X×X→0,∞such ha
dHx, ,Hy,  ≤dx, y−ψx, y o e e y x, y ∈U,and ∈0,1;
P4 he e exis s a con inuous unc ion φ:0,1→Rsuch ha , o e e y x∈Uand
, s ∈0,1,dHx, ,Hx, s ≤|φ −φs|;
P5i x∈∂U and 0 ≤λ<1, wi h Hx, λ∈∂U, hen Hx, 1/
∈U.
In he p oo o he main esul o his chap e we shall make use o he ollowing lemma
see F igon 5.
Lemma 2.3. Le x0∈X, >0, and h:Bx0, →Xweakly con ac i e. I dx0,hx0 <
γ /2, , henhhas a ixed poin .
Theo em 2.4. Le ,g :U→Xbe wo weakly con ac i e maps. Suppose ha is homo opic o g
and gUis bounded. I ghas a ixed poin in U, hen has a ixed poin in U.
P oo . We a gue by con adic ion. Suppose ha does no ha e any ixed poin in U,andle
Hbe a homo opy be ween and g, in he sense o De ini ion 2.1. Conside he se
A{λ∈0,1:xHx,λ o some x∈U},2.3
and no ice ha Ais nonemp y since ghas a ixed poin in U, ha is,0∈A. We will show ha
Ais bo h open and closed in 0,1, and hence, by connec edness, we will ha e ha A0,1.
As a esul , will ha e a ixed poin in U, which es ablishes a con adic ion.
To show ha Ais closed, suppose ha {λn}is a sequence in Acon e ging o λ∈0,1
and le us show ha λ∈A. Since λn∈A, he e exis s xn∈Uwi h xnHxn,λ
n.Fix
ε>0. Using ha gUis bounded and ha φis con inuous on he compac in e al 0,1,
i is easy o show ha he e exis s M>εsuch ha diam HU×0,1 ≤M, and hence
dxn,x
m≤M o all n, m ∈N. De ine μθε, Mand le n0∈Nbe such ha o all
n, m ≥n0,|φλn−φλm|<μ. Then dxn,x
m<ε o all n, m ≥n0because, o he wise, we
would ha e dxn,x
m≥ε o some n, m ≥n0, and hen
dxn,x
mdHxn,λ
n,Hxm,λ
m
≤dHxn,λ
n,Hxn,λ
m dHxn,λ
m,Hxm,λ
m
≤φλn−φλmdxn,x
m−ψxn,x
m
<μdxn,x
m−ψxn,x
m
≤dxn,x
m,
2.4
4 Fixed Poin Theo y and Applica ions
which is a con adic ion. Then {xn}is a Cauchy sequence and, since X, dis comple e, he e
exis s x0∈Usuch ha xn→x0as n→∞. In addi ion, x0Hx0,λsince o all n∈Nwe
ha e ha
dxn,Hx0,λ
 dHxn,λ
n,Hx0,λ

≤dHxn,λ
n,Hxn,λ
 dHxn,λ
,Hx0,λ

≤φλn−φλdxn,x
0−ψxn,x
0
≤φλn−φλdxn,x
0.
2.5
Obse e ha 0 ≤λ<1, because i λ1, hen x0Hx0,1 x0, which con adic s he ac
ha does no ha e any ixed poin in U.No ice ha x0∈U, because, o he wise, we would
ha e x0∈∂U, ha is,Hx0,λ∈∂U, and since 0 ≤λ<1, by P5, we ha e ha Hx0,1/
∈U.
Howe e , since x0∈∂U,{xn}→x0and xn∈U o all n∈N, he e exis s n0∈Nsuch ha
xn∈Uδ o all n≥n0. Hence, since xnHxn,λ
n o all n≥n0, applying P2, we ha e ha
xn∈U o all n≥n0, ha is,Hxn,1∈U o all n≥n0. Taking limi s, we a i e o he
con adic ion Hx0,1∈U.
The e o e, x0∈Uand, consequen ly, λ∈A.
Nex we show ha Ais open in 0,1.Le λ0∈A. Then he e exis s x0∈Uwi h x0
Hx0,λ
0.Le >0 be such ha Bx0, ⊂U,andle δ>0 such ha |φλ−φλ0|<γ /2, 
o e e y λ∈0,1wi h |λ0−λ|<δ. Then, i λ∈λ0−δ, λ0δ∩0,1,
dx0,Hx0,λ
 dHx0,λ
0,Hx0,λ

≤φλ0−φλ
<γ

2, .
2.6
Using Lemma 2.3,weob ain ha H·,λhas a ixed poin in U o e e y λ∈0,1such ha
|λ0−λ|<δ.Thusλ∈A o any λ∈λ0−δ, λ0δ∩0,1, and he e o e Ais open in 0,1.
As an immedia e consequence o he p e ious heo em, we ob ain he ollowing ixed
poin esul o he Le ay-Schaude ype o weakly con ac i e maps unde he condi ion
I-C.
Theo em 2.5. Suppose ha Uis an open and s ic ly s a shaped subse o a Banach space X, ·,
wi h 0∈U, and ha :U→Xis a weakly con ac i e map wi h Ubeing bounded. I sa is ies
he condi ion I-C, hen has a ixed poin in U.
P oo . Since sa is ies he condi ion I-C, he e exis s δ>0 such ha x/
λx o λ>1
and x∈Uδwi h x/
∈U. We may assume ha x/
 x o e e y x∈Uδ, because o he wise
we a e inished. De ine H:U×0,1→Xas Hx,  x,andle gbe he ze o map.
No ice ha ghas a ixed poin in U, ha is,0g0and also ha and ga e wo weakly
con ac i e mappings. So, he esul will ollow om Theo em 2.4 once we p o e ha is
I-C-homo opic o g. Le us check i .
Fixed Poin Theo y and Applica ions 5
P1Fo all x∈U,Hx,00· x0gxand Hx,11· x x.
P2Since sa is ies he condi ion I-C, we ha e ha x/
λx o x∈Uδwi h x/
∈U
and λ>1. Hence, x/
Hx,  o e e y x∈Uδ,wi h x/
∈U,and ∈0,1.
P3Since is weakly con ac i e, he e exis s a compac ly posi i e unc ion ψ:X×X→
0,∞such ha d x, y ≤dx,y−ψx, y o e e y x, y ∈U. Then, i x,y ∈U
and ∈0,1,
dHx, ,Hy,  
 x− y

≤d x, y
≤dx,y−ψx, y.
2.7
P4Since Uis bounded, he e exis s M≥0 such ha  x≤M o all x∈U.
Hence,
dHx, ,Hx, s 
 x
| −s|
≤M| −s|
φ −φs,
2.8
whe e φ:0,1→Ris he con inuous unc ion de ined as φ M .
P5Suppose ha o some x∈∂U and λ<1 we ha e ha Hx, λ∈∂U. Then, x/
0
since Hx,λλ x,0∈Uand Uis open. Le us see ha Hx, 1/
∈U: suppose,
on he con a y, ha Hx, 1∈U, ha is, x∈Uand de ine

λ:sup ≥1: x∈U.2.9
Then, i is easy o see ha 
λ x∈∂U, which con adic s ha Uis s ic ly s a
shaped, since we also ha e ha λ x∈∂U.
3. A Class o Gene alized Con ac ions
A mul i ude o gene aliza ions and a ian s o Banach’s con ac i e condi ion ha e been
gi en a e Banach’s heo em see, e.g., Rhoades 13 and, ecen ly, Aga wal and O’Regan
11ha e gi en a homo opy esul  hus gene alizing a ixed poin heo em o Ha dy and
Roge s 14 unde he ollowing gene alized con ac i e condi ion: he e exis s a∈0,1
such ha o all x, y ∈X
d x, y≤amaxdx, y,dx, x,dy, y,1
2dx, ydy, x.3.1

6 Fixed Poin Theo y and Applica ions
In his sec ion we gi e a homo opy esul o his class o mappings unde he
condi ion I-C. In he p oo o ou heo em we shall use he ollowing esul 11.
Lemma 3.1. Le X, dbe a comple e me ic space, x0∈X, >0, and h:Bx0, →X. Suppose
ha he e exis s a∈0,1such ha o x,y ∈Bx0, one has
dhx,h
y≤amaxdx, y,dx, hx,dy, hy,1
2dx,hydy,hx,
dx0,h
x0 <1−a .
3.2
Then he e exis s x∈Bx0, wi h xhx.
The p oo o he ollowing heo em is e y simila o he p oo o Theo em 2.4, and we
gi e a ske ch o i .
Theo em 3.2. Le X, dbe a comple e me ic space, and Uan open subse o X.Le ,g :U→X
be wo maps such ha he e exis s H:U×0,1→Xwi h he ollowing p ope ies:
P1Hx,1 xand Hx, 0gx o e e y x∈U;
P2 he e exis s δ>0such ha x/
Hx,  o e e y x∈Uδ,wi h x/
∈U, and ∈0,1,
whe e Uδ{x∈U:dis x, ∂U<δ};
P3 he e exis s a∈0,1such ha o all x, y ∈Uand λ∈0,1one has
dHx,λ,Hy,λ
≤amaxdx, y,dx,Hx, λ,dy,Hy,λ,1
2dx,Hy,λdy,Hx, λ;
3.3
P4 he e exis s a con inuos unc ion φ:0,1→Rsuch ha , o e e y x∈Uand , s ∈0,1,
dHx, ,Hx, s ≤|φ −φs|;
P5i x∈∂U and 0≤λ<1,wi hHx, λ∈∂U, henHx, 1/
∈U.
I ghas a ixed poin in U, hen has a ixed poin in U.
P oo . Suppose ha does no ha e any ixed poin in Uand conside he nonemp y se
A{λ∈0,1:Hx,λx o some x∈U}.3.4
We will a i e o a con adic ion by showing ha A0,1, and o his we only need p o e
ha Ais closed and open in 0,1.
To show ha Ais closed in 0,1, conside a sequence {λn}in A,wi hλn→λ∈0,1
as n→∞, and show ha λ∈A; ha is, ha he e exis s x0∈Uwi h Hx0,λx0. To p o e
ha x0exis s, ake any sequence {xn}in Uwi h xnHxn,λ
n, p o e ha {xn}is Cauchy,
and de ine x0as he limi o {xn},asn→∞.
Tha {xn}is a Cauchy sequence, as well as x0Hx0,λ, ollows om s anda d
a gumen s which can be seen in 11, Theo em 3.1. I emains o show ha x0∈U.
Fixed Poin Theo y and Applica ions 7
To p o e his, suppose ha i is no ue and a i e o a con adic ion as ollows: we ha e
ha Hx0,λx0∈U U∂U, and also ha 0 ≤λ<1, because does no ha e any
ixed poin in U. Then, by P5 x0/
∈∂U. On he o he hand, x0lim xn∈Ubecause
xn∈U o nla ge enough. To be con inced o i , jus apply P2:sincex0∈∂U,{xn}→x0
and xn∈U o all n∈N, he e exis s n0∈Nsuch ha xn∈Uδ o all n≥n0. Then, xn∈U
o all n≥n0since xnHxn,λ
n.
To p o e ha Ais open a gue as in Theo em 2.4,useLemma 3.1 ins ead o Lemma 2.3.
As an immedia e consequence, we ob ain he ollowing esul , whose p oo is omi ed
because i is analogous o he p oo o Theo em 2.5.
Theo em 3.3. Suppose ha Uis an open and s ic ly s a shaped subse o a Banach space X, ·,
wi h 0∈U, and ha :U→Xis map wi h Ubeing bounded. Assume also ha he e exis s
a∈0,1such ha o all x, y ∈Uand λ∈0,1one has
dλ x,λ y
≤amaxdx, y,dx, λ x,dy,λ y,1
2dx,λ ydy,λ x.
3.5
I sa is ies he condi ion I-C, hen has a ixed poin in U.
Acknowledgmen
This esea ch is pa ially suppo ed by he Spanish G an MTM2007-60854and egional
Andalusian G an s FQM210, FQM1504Go e nmen s.
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