0050 về sự tồn tại điểm bất động của một số lớp ánh xạ trong không gian với cấu trúc đều và ứng dụng luận văn tốt nghiệp

MINISTRY OFEDUCATION ANDTRAINING
VINHU N I V E R S I T Y

LEKHA NHHUNG

ONT H E E X I S T E N C E O F F I X E D P O I N
T FORS O M E M A P P I N G C L A S S E S
INSPACESWITH UN IFORMSTRUCT
UREANDA P P L I C A T I O N S

Speciality:M a t h e m a t i c a l A
nalysisCode:6 2 4 6 0 1 0 2

ASUMMARYOFMATHEMATICSDOCTORALTHESI
S

NGHEAN-2015


WorkiscompletedatVinhUniversity

Supervisors:
1. Assoc.P r o f . D r . T r a n VanAn
2. Dr.K i e u PhuongChi

Reviewer1:

Reviewer2 :

Reviewer3 :

ThesiswillbepresentedanddefendedatschoollevelthesisevaluatingCouncilatVinhUniversity
at...... h ...... d a t e ...... m o n t h ...... y e a r ......

Thesiscanbefoundat:
1. Nguyen Thuc Hao Library and InformationCenter
2. VietnamNationalLibrary


1

PREFACE

1

Rationale
1.1. The first result on fixed points of mappings was obtained in

1911.At thattime, L. Brouwer proved that:E v e r y

continuous

m a p p i n g f r o m a c o m p a c t c o n v e x set in a finite-dimensional
space into itself has at least one fixed point.In 1922, S.Banach
introduced a class of contractive mappings in metric spaces and proved
thefamous contraction mapping principle: Each contractive mapping
from

a

completemetricspace(X,

d)intoitselfhasauniquefixedpoint.ThebirthoftheBanachcontraction
principle

and

its

application

to

study

the

mapping

existence

of

solutionsofdifferentialequationsmarksanewdevelopmentofthestudyoffixedp
ointtheory.After that, many mathematicians have studied to extend the Banach
contractionmapping

principle

for

classes

of

maps

and

different

spaces.Expanding only contractivemappings, till 1977, was summarized and
compared with 25 typical formats by B.E.Rhoades.
1.2. The Banach contraction mapping principle associates with the
class of con-tractivemappingsT:X→Xin complete metric space (X, d) with
the contractivecondition
(B)d (Tx,Ty)≤kd(x,y),f o r a l l x , y∈ Xw h e r e 0 ≤k< 1.
There have been many mathematicians seeking to extend the Banach
contractionmapping principle for classes of mappings and different spaces. The first
extendingwasobtainedbyE.Rakotchbymitigatingacontractiveconditiono
¡
ftheform
¢
(R)d (Tx,Ty)≤ θ d(x,y)d
(x,y),f o r a l l x , y∈ X,w h e r e θ :  → [ 0 ,1)i s a
+

monotoned e c r e a s i n g f u n c t i o n .

In 1969, D. W. Boyd and S. W. Wong introduced an extended form of
the aboveresultbyconsideringacontractiveconditionoftheform

¡
(BW)d(Tx,Ty)≤ϕd(x,y), forall x,y∈ X,whereϕ: +→+isasemiright
¢
uppercontinuousfunctionandsatisfies
0≤ϕ(t)

2

In2001,B.E.Roades,whileimprovingandextendingaresultofY.I.Alber
and


S.Guerre-Delabriere,gaveacontractiveconditionoftheform
¡
(R1)d(Tx,Ty)≤d(x, y)−ϕ d(x, y) , for allx, y∈X,whereϕ:+→+is
¢
acontinuous,monotoneincreasingfunctionsuchthatϕ(t)=0ifandonlyift
=0.
Followingthewayofreducingcontractiveconditions,in2008,P.N.DuttaandB.
S.Choudhuryintroducedacontractiveconditionoftheform
¡
¢
¡
¢
¡
(DC)ψd(Tx,Ty)≤ψ
¢ d(x, y)−ϕ d(x, y) , for allx, y∈X,whereψ, ϕ:+→+is a
continuous, monotone non-decreasing functions such thatψ(t) = 0 =ϕ(t)
ifandonlyift=0.

In 2009, R. K. Bose and M. K. Roychowdhury introduced the notion of
new

gen-

eralizedweakcontractivemappingswiththefollowingcontractiveconditionin
¡
¢
¡
¢
¡
ordertostudycommonfixedpointsofmappings
¢
(BR)ψ d(Tx,Sy
≤ψd (x,y)− ϕd (x,y), forallx,y∈X,whereψ,ϕ: +→
+arecontinuousfunctionssucht h a t ψ (t)> 0 ,ϕ(t)> 0 f o r a l l t > 0 a n d ψ (0)= 0=
ϕ(0),moreover,ϕisamonotonenondecreasingfunctionandψisamonotone
increasingf u n c t i o n .
In 2012, B. Samet, C. Vetro and P. Vetro introduced the notion ofα-ψcontractivetypemappingsincompletemetricspaces,withacontractivecondi
¡
¢
≤
∈ →+
tionoftheform
(SVV)α (x,y)d(Tx,Ty)ψ d (x,y), f o rΣa l l x , yXw h e r e ψ :   isamonotoneno
+
+∞ψn
n-decreasingfunctionsatisfying
(t)<+∞forallt>0and
n=1

α:X×X→ +.
1.3. In recent years, many mathematicians have continued the trend of
generalizingcontractive conditions for mappings in partially ordered metric
spaces.Following thistrend, in 2006, T. G. Bhaskar and V. Lakshmikantham
introduced the notion ofcoupled fixed points of mappingsF: X×X→ Xw i t h
the

mixed

monotone

p r o p e r t y andobtainedsomeresultsfortheclassofthosemappingsinpartiallyor
deredmetricspacessatisfyingthecontractivecondition
¡
¢≤k
(BL)T h e r e e x i s t s k ∈ [0,1)s u c h t h a t d F(x,y),F(u,v)
³d(x,u)+d(y,v)´,
fora l l x , y,u,v∈ Xs u c h t h a t x ≥u,y≤ v.
2
In 2009, by continuing extending coupled fixed point theorems, V.
Lakshmikanthamand L. Ciric obtained some results for the class of
mappingsF:X×X→Xwithg-mixed monotone property, whereg:X→Xfrom a


partially
ordered
metric
spaceintoitselfandFs a t i s f i e s thefollowingcontractivecondition
¡
¢+ ¡

¢
³d g(x),g(u) d g(y),g(v) ´
¡
¢
(LC)d F(x,y),F(u,v) ≤ϕ
,
2


forall x ,y,u,v∈Xw i t h g(x)≥g(u),g(y)≤g(v)andF(X×X)⊂g(X).
In 2011, V. Berinde and M. Borcut introduced the notion of triple
fixed points fortheclassofmappingsF:X×X×X→Xand obtained some triple fixed
pointtheorems for mappings with mixed monotone property in partially
ordered metricspacessatisfyingthecontractivecondition
(BB)T h e r e e x i s t s c o n s t a n t s j , k , l ∈[ 0 ,1)s u c h t h a t j + k+ l
¡
<1s a t i s f y dF(x, y, z), F(u, v, w)≤jd(x, u) +kd(y, v) +ld(z, w),forallx, y, z, u, v,
w∈Xwithx≥u,y ≤v,z≥w.
¢
After that, in 2012, H. Aydi and E. Karapinar extended the above
result andobtained some triple fixed point theorems for the class of
mappingF:X×X×X→ Xwith mixed monotone property in partially ordered metric spaces and
satisfying thefollowingweakcontractivecondition
(AK)Thereexistsafunctionsuchthatforallxu,yv,zwwehave
Ă
Â
â
}.
d T F(x,y,z),T F(u,v,w)
max d(T x,T u),d(T y,T v),d(T z,T w)

1.4. The development of fixed point theory is motivated from its
popular ap-plications, especially in theory of differential and integral
equations, where the firstimpression is the application of the Banach
contraction

mapping

principle

to

studytheexistenceofsolutionsofdifferentialequations.
In the modern theory of differential and integral equations, proving the
existence ofsolutions or approximating the solutions are always reduced to
applying

appropriatelycertain

fixed

point

theorems.For

boundary

problems with bounded domain, fixedpoint theorems in metric spaces
are enough to do the above work well. However, forboundary problems with
unbounded

domain,

fixed

point

theorems

in

metric

spaces

arenotenoughtodothatwork.So,inthe70soflastcentury,alongwithseekingtoext
endtomappingclasses,onewaslookingtoextendtoclassesofwiderspaces.One of typicaldirections
of this expansion is seeking to extend results on fixed points of mappings
inmetricspacestotheclassoflocalconvexspaces,morebroadly,uniformspacesw
hichhasattractedtheattentionofmanymathematical, notablyV.G.An
gelov.
In 1987, V. G. Angelov considered the family of real functions Φ
={φα:α∈I}such that for eachα∈I,φα:+→+is a monotone increasing,
continuous,φα(0) = 0 and 0< φα(t)< tfor allt >0.Then he introduced the
notion
ofΦ¡
contractivemappings,whicharemappingsT:
M→ Xs a t i s f y i n g
(A)d α(Tx,Ty)≤φαd j(α)(x,y)forallx,y∈Ma n d forallα∈I,whereM⊂ X
¢

andobtainedsomeresultsonfixedpointsoftheclassofthosemappings.B y in
tro-


ducingthenotionofspaceswithjboundedproperty,V.G.Angelovobtainedsomeresultsontheuniqueexiste
nceofafixedpointoftheabovemappingclass.
Following the direction of extending results on fixed points to the
class of localconvex spaces, in 2005, B. C. Dhage obtained some fixed
point theorems in Banachalgebras by studying solutions of operator
equationsx=AxBxwhereA:X→X,B:S→Xare

two

operators

satisfying

thatAisD-Lipschitz,Bis completelycontinuous andx=AxByimpliesx∈Sfor
ally∈S, whereSis a closed, convexand bounded subset of the Banach
algebraX,such that it satisfies the contractivecondition
¡
(Dh)||T
x−Ty||≤φ||x−y||for
allx,
y∈X,whereφ:+→+is
¢
decreasingcontinuousfunction,
φ(0)=0.

a

non-

1.5. Recently, together with the appearance of classes of new contractive
mappings,and new types of fixed points in metric spaces, the study trend on
the fixed pointtheory has advanced steps of strong development.With
above reasons, in order toextend results in the fixed point theory for
classes of spaces with uniform structure,we chose the topic‘‘On the
existence

of

fixed

points

for

some

mapping

classesinspaces

withuniform structureandapplications”forourdoctoralthesis.

2

Objectiveo f t h e r e s e a r c h

The purpose of this thesis is to extend results on the existence of

fixed pointsin metric spaces to some classes of mappings in spaces with
uniform structure andapply to prove the existence of solutions of some
classes of integral equations withunboundeddeviation.

3

Subjecto f t h e r e s e a r c h
Study objects of this thesis are uniform spaces, generalized

contractive map-pings in uniform spaces, fixed points, coupled fixed points,
triple

fixed

points

of

somemappingc l a s s e s i n s p a c e s w i t h u n i f o r m s t r u c t u r e , s o m e c l a s s e s o
fintegralequations.

4

Scopeoftheresearch


The thesis is concerned with study fixed point theorems in uniform
spaces andapply to the problem of the solution existence of integral equations with

unboundeddeviationalfunction.


5

Methodologyo f t h e r e s e a r c h
We use the theoretical study method of functional analysis, the

method of thedifferential and integral equation theory and the fixed point
theory in process of study-ingthetopic.

6

Contributiono f t h e t h e s i s
The thesis is devoted to extend some results on the existence of

fixed points inmetric spaces to spaces with uniform structure. We also
considered the existence ofsolutions of some classes of integral
equations

with

unbounded

deviation,

which

wecannotapplyfixedpointtheoremsinmetricspaces.
The thesis can be a reference for under graduated students, master

students andPh.D students in analysis major in general,a n d t h e fi x e d
p o i n t t h e o r y a n d a p p l i c a t i o n s inparticular.

7

OverviewandOrganization ofther ese arch
Thecontentofthisthesisispresentedin3chapters.Inaddition,thethesisalso

consists Protestation, Acknowledgements, Table of Contents, Preface, Conclusionsand
Recommendations, List of scientific publications of the Ph.D. student
related tothethesis,andReferences.
In chapter 1, at first we recall some notions and known results about
uniformspaces which are needed for later contents.Then we introduce the
notion of (Ψ,Π)-)-contractive mapping, which is an extension of the notion of
(ψ,

of

ϕ)-contraction

P.

N.DuttaandB.S.Choudhuryinuniformspaces,andobtainedaresultontheexist
enceoffixedpoints of the (Ψ,Π)-)-contractive mapping inu n i f o r m s p a c e s . B y
i n t r o d u c i n g the notion of uniform spaces withj-monotone decreasing
property, we get a resulton the existence and uniqueness of a fixed point
of (Ψ,Π)-)-contractive mapping.Con-tinuously, by extending the notion ofα-ψcontractive mapping in metric spaces touniform spaces, we introduce the
notion of (β,Ψ1)-contractive mappings in uniformspaces and obtain some
fixed

point

theorems

for

the

class

of

those

mappings.

Theo-

rems,whichareobtainedinuniformspaces,areconsideredasextension
softheoremsincompletemetricspaces.Finally,applyingourtheoremsaboutfixedpointsoftheclass of
(β,Ψ1)-contractive mapping in uniform spaces, we prove the existence of solutions of a class of nonlinear integral equations with unbounded


deviations.Notethat, when we consider a class of integral equations with
unbounded

deviations,

wecannotapplyknownfixedpointtheoremsinmetricspaces.Mai n resultsof
Chapter

1 isTheorem1.2.6,Theorem1.2.9,Theorem1.3.11andTheorem1.4.3.
In Chapter 2, we consider extension problems in partially ordered
uniform spaces.Firstly,i n s e c t i o n 2 . 1 , w e o b t a i n r e s u l t s o n
couple

fi x e d

points

for

a

mapping

c l a s s in partially

ordered uniform spaces when we extend (LC)-contractive condition of
V.LakshmikanthamandL.Ciricformappingsinuniformspaces.In section 2.2, byextending
the contractive condition (AK) of H. Aydi and E.Karapinar for
mappingsin uniform spaces, we get results on triple fixed points of a
class

in

partially

ordereduniformspaces.Insection2.3,byintroducingnotionsofupper(lower)c

ouple,upper(lower) triple solution, and applying results in section 2.1, 2.2, we prove the
uniqueexistence of solution of some classes of non-linear integral equations
with unboundeddeviations.Main results of Chapter 2 are Theorem 2.1.5,
Corollary

2.1.6,

Theorem2.2.5,Corollary2.2.6,Theorem2.3.3andTheorem2.3.6.
In Chapter 3, at first we present systematically some basic notions
about locallyconvex algebras needed for later sections. After that, in
section

3.2,

by

extending

thenotionofD-

Lipschitzmapsformappingsinlocallyconvexalgebrasandbybasingonknownres
ultsinBanachalgebras,anduniformspaces,weproveafixed pointtheoremi
nlocallyconvexalgebraswhichisanextensionofanobtainedresultbyB.C.Dhage
.Finally,insection3.3,applyingobtainedtheorems,weprovetheexistenceofsol
utionofaclassofintegralequationsinlocallyconvexalgebraswithunboundeddevi
ations.MainresultsofChapter3areTheorem3.2.5,Theorem3.3.2.
In this thesis, we also introduce many examples in order to illustrate
our resultsandthemeaningofgivenextensiontheorems.

CHAPTER1
UNIFORMSPACES
ANDF I X E D P O I N T T H E O R E M S

In this chapter, firstly we present some basic knowledge about
uniform spacesand useful results for later parts.T h e n , w e g i v e s o m e fi x e d
p o i n t t h e o r e m s f o r t h e c l a s s of (Ψ,Π)-)-contractive mappings in
uniform

spaces.In

the

last

part

of

this

chapter,

weextendfixed pointtheoremsfortheclassof α-ψcontractivemappingsinmetricspacesto uniform spaces. After that, we apply these new
results

to

show

a

class

of

integralequationswithunboundeddeviationshavingauniquesolution.

1.1 Uniformspaces
In this section, we recall some knowledge about uniform spaces
needed for laterpresentations.
LetXb e a n onemptys e t, U, V⊂X×X.W e de n ote by 1)U −1={(x,y
)∈X×X: (y,x)∈U}.
2) U◦V={(x,z):∃y∈X,(x,y)∈U,(y,z)∈V}andU ◦Ui s re pla c e d b y U 2.
3) ∆(X)={(x,x):x∈X}iss a i d t o b e a di a g on a l ofX.
4) U[A]={y∈ X:∃x∈Asuchthat(x,y)∈U},w h e r e A ⊂XandU [{x}]i s repl
acedbyU[x].
Definition1.1.1.A n uniformityoruniformstructureonXisanon-emptyfamily
Uc o n s i s t i n g o f s u b s e t s o f X × Xwhichs a t i s f y t h e f o l l o w i n g c o n d i t i o n s
1) ∆(X)⊂Uf o r a l l U ∈ U.
2) IfU ∈ Ut h e n U −1∈U.
3) IfU∈ Ut h e n t h e r e e xi s t s V ∈Us u c h t h a t V 2⊂U.
4) IfU , V∈ Ut h e n U ∩ V∈ U.
5) IfU∈
U a n d U ⊂ V⊂ X×Xt h e n V ∈ U.Theo r d e r e d p
a i r ( X,U)i s c a l l e d a un i fo r m space.


Inthissection,wealsopresenttheconceptoftopologygeneratedbyunif
ormstructure,uniformspacewithuniformstructuregeneratedbyafamilyofpseudom

etrics,


Cauchys e q u e n c e , s e q u e n t i a l l y c o m p l e t e u n i f o r m s p a c e a n d t h e r e l
a t i o n s h i p b e t w e e n them.
Remark1 . 1 . 8 .
1)L e t X b e a u n i f o r m s p a c e . T h e n , u n i f o r m t o p o l o g y o n X i s genera
tedbythefamilyofuniformcontinuouspseudometricsonX
2) IfEi s locallyconvexspacewithasaturatedfamilyofseminorms {
pα}α∈I,
thenwecand e fi n e a f a m i l y o f a s s o c i a t e p s e u d o m e t r i c s ρα(x,
y)
=pα(x−y)f o r e v e r y x,
y∈E.Theuniformtopologygeneratedthefamilyofassociatepseudometricscoinci
deswiththeoriginaltopologyofthespaceE.T h e r e f o r e , asacorollaryo
four
results,weobtainfixedpointtheoremsinthelocallyconvexspace.
3) Letj:I→ Ib e anarbitrarymappingoftheindex Ii n t o itself.T h e itera
tionsofjcanbedefinedinductively
¡
¢
j 0 (α)=α,j k (α)=j j k−1 (α) ,k=1,2,...

1.2 Fixedp oi nt s of w ea k cont r ac ti vem ap pi n gs
In the next presentations, (X,P)orXwe mean a Hausdorff uniform space
whoseuniformity
is
generated
by
a

saturated
family
of
pseudometricsP={dα(x,
y)
:α∈I},whereI i s a n i n d e x s e t . N o t e t h a t ,
( X,P)isH a u s d o r ff i f o n l y i f d α(x, y) = 0f o r a l l α∈Ii m p l i e s x=y.
Definition1 . 2 . 2 . Au n i f o r m s p a c e ( X,P)i s s a i d t o b e j boundedi f f o r e v e r y
α∈ Ia n d x , y∈ Xt h e r e e x i s t s q = q (x,y,α)s u c h t h a t d jn(α)(x,y)≤ q(x,y,α)<
∞,foralln∈.
LetΨ={ψα:α∈I}beafamilyoffunctionsψ α:+→+whichismonotonenondecreasingandcontinuous,ψ α(t)=0ifonlyift=0,forallα∈I.
DenoteΠ)-={ϕα:α∈I}beafamilyoffunctionsϕ α:+→+,α∈Isuchthat
ϕαiscontinuous,ϕα(t)=0ifonlyift=0.
Definition 1.2.4.LetXbe a uniform space.
a(Ψ,Π)-)-contractiveonXi f

A mapT:X→Xis called

¡
¢
¡
¢
¡
¢
ψα dα(Tx,Ty) ≤ψ α dj(α)(x, y) −ϕα dj(α)(x,y) ,
forallx,y∈Xa n d forallψ α∈Ψ,ϕα∈Π,α∈I.
Definition1.2.5.A uniformspace(X,P)iscalledtohavethejmonotonedecreasing


propertyi ff d α(x,y)≥dj(α)(x,y)f o r a l l x , y∈ Xa n d a l l α ∈I.

Theorem1 . 2 . 6 .
LetXi s aHausdorffsequentially completeuniformspaceand
T: X→X.Supposethat


1) Tisa(ΨΨ,Π)-)-contractivemaponX.
2) Amapj:I→Iissurjectiveandthereexistsx0∈Xsuchthatthesequence
{xn}w i t h x n= T x n−1,n= 1 ,2,...s a t i s f y i n g d α(xm,xm+n)≥ d j(α)(xm,xm+n)f o r a l l
m,n≥0,a l l α∈I.
Then,Th a s atleastonefixedpoint.X.
Moreover,ifXh a s jmonotonedecreasingproperty,thenTh a s auniquefixed point.
©
}
=
Example1 . 2 . 7 . L e t X =  ∞ x = { xn}: x n∈  ,n= 1 ,2,.... F o r e v e r y n=
1 ,2, . . .w e d e n o t e b y P n: X → a m a p i s d e fi n e d b y P n(x)= x nf o r a l l
x={xn}
∈X.DenoteI=∗×+.For
every
(n,
pseudometricsd(n,r):X×X→,whichisgivenby

r)∈Iwe

define

a

. n(x)−Pn(y), fore v e r y ( x,y)∈X.
d(n,r)(x,y)=rP

.
Then,thec o l l e c t i o n o f p s e u d o m e t r i c s { d(n,r):
( n,r)∈ I}g e n e r a t e d a u n i f o r m i t y o n
X.
Nowf o r e v e r y ( n,r)∈ Iw e c o n s i d e r t h e f u n c t i o n s , w h i c h i s g i v e n
b y ψ (n,r)(t)=
2(n−1)
t,f o r a l l t ≥ 0,a n d p u t Ψ = Φ = { ψ
2n−1
³
´

(n,r)

:(n,r)∈I}.Denoteb y j : I→ I

,f o r a l l ( n,r)∈ Ia n d d e fi n e a m a p p i n
g
am a p i s d e fi n e d b y j (n,r)= ¡ 1 ¢2
n,r
1−
n
T: X→ Xw h i c h is de fin ed by
¡
¢
¡ 2¢
2
¡
2¢
),...,,

),1−
1−
(1−x
),...,1−
1−
(1−x
Tx=,1− 1− (1−x 1
2
n
3
3.
3
2
n
foreveryx={xn}∈X.
ApplyingT h e o r e m 1 . 2 . 6 , T h a s a u n i q u e fi x e d p o i n t , t h a t i s x ={1,1,...}.
Theorem1 . 2 . 9 .

LetXbeaHausdorffsequentiallycompleteuniformspaceand

T,S: X → Xbem a p p i n g s s a t i s f y i n g
¡
¢
¡
¢
¡
¢
ψα dα(Tx,Sy) ≤ ψ α dj(α)(x,y) −ϕ α dj(α)(x,y) ,
forallx, y∈ X,wher eψ α∈Ψ,ϕα∈Πf o r all α∈I.
Supposej:I→Ibe a

surjective
map andforsometix0∈Xsuch
that
thesequence{xn}withx2k+1=Tx 2k ,x 2k+2=Sx2k+1,
k≥0satisfiesdα(xm+n,
xm)≥dj(α)
(xm+n,xm)forallm,n≥0,α∈I.
Then,thereexistsu∈Xs u c h thatu=Tu=Su.
Moreover, ifXhas thej-monotone decreasing property, then there exists a
uniquepointu∈Xs u c h thatu=Tu=Su.


1.3 Fixedpoints of (β,Ψ1)-contractivetypemappings
DenoteΨ 1={ψα:α∈I}beafamilyoffunctions withtheproperties
(i) ψα:+→+ismonotonenon-decreasingandψ α(0)=0.
(ii) foreachα∈I,thereexistsψ α∈ 1suchthat

+

}
sup â
j n () (t):n=0,1,...

(t)a n d

n

(t)<+forallt>0.

n=1

Denotebya familyoffunctions={:XìX +,I}.
Definition1 . 3 . 7 . L e t ( X,P)bea u n i f o r m s p a c e ©
w i t h P =dα(x, }
y) :α∈IandT:X→Xbe a given mapping.We say thatTis an (β,Ψ1)-contractiveif for
everyfunctionβ α∈βa n d ψ α∈Ψ1wehave
¡
¢
βα(x,y).dα(Tx,Ty)≤ ψ α dj(α)(x,y) ,
fora l l x , y∈ X.
Definition1 . 3 . 8 . LetT : X→ X.W e s ay t h a t T i s a β -admissibleiff o r a ll x, y∈ X
andα ∈ I,β α(x,y)≥ 1i m p l i e s β α(Tx,Ty)≥ 1.

©
}
Theorem1.3.11.LetXbe a set andP=dα(x, y):α ∈ Ib e
a
family
o f pseudometrics onXsuch that(X,P)is a Hausdorff sequentially complete
uniformspace.LetT:X→Xbe
an(β,Ψ1)-contractive
mapping
satisfying
the
followingconditions
i) Ti s β-admissible.
ii) Theree x i s t s x 0∈ Xsucht h a t f o r e a c h α ∈ Iw e h a v e β α(x0,Tx 0 )≥ 1a n d
djn(α)(x0,Tx 0 )Also,assumeeither
a) Ti s continuous;or

b) fora l l α ∈ I,i f { xn}i s a s e q u e n c e i n X sucht h a t β α(xn,xn+1)≥ 1 f o r a l l n
andx n→x∈Xa s n→+∞,thenβ α(xn,x)≥1f o r alln∈∗.
Then,Th a s afixedpoint.
Moreover,i f X isj - b o u n d e d a n d f o r e v e r y x , y∈ X,t h e r e e x i s t s z ∈ Xs u c h t h a t
βα(x,z)≥1a n d β α(y,z)≥1f o r allα∈I,thenThasauniquefixedpoint.
Wealsogivesomeexamplestoillustrateforourresults.


1.4 Applicationstononlinearintegralequations
In this section, we wish to investigate the existence of a unique solution to
nonlinearintegralequations,asanapplicationtothefixedpointtheoremsprovedinth
eSection1.3.
Letu s c o n s i d e r t h e f o l l o w i n g i n t e g r a l e q u a t i o n s
(t)
Ă
Â
x(t)=
G(t,s)f s,x(s) ds,

(1.27)

0

where the functionsf:+ìandG:+ì++continuous.Thedeviation
:++is a continuous function, in general case, unbounded.Notethat, since
deviation ∆ :+→+is unbounded, we can not apply the known
fixedpointtheoremsinmetricspacefortheaboveintegralequations.
Assumption 1.4.1.A1) There exists a functionu:2→such that for
eachcompact subsetK⊂+, there exist a positive numberλandψK∈Ψ1such
that forallt∈+,foralla,b∈withu(a,b)≥0,wehave

∫
.f(t,a)−f(t,b).
∆(t)
¡
¢
G(t,s)ds≤1.
and λsup
≤λψK |a−b|
t∈K

0

A2)Thereexistsx 0∈C(+,)suchthatfor allt∈+,wehave
∫∆(t)
³
¡
¢ ´
u x0 (t)
G(t,s)f s,x0(s) d ≥0.
0
,
s
¡
¢
A3)F o r a l l t ∈+,x,y∈ C(+,),i f u x(t),y(t) ≥0,t h e n
¡
¢
¡
¢
∫ ∆(t)

³∫∆ ( t)
G(t,s)f s,x(s) ds
G(t,s)f s,y(s) ds´ ≥0.
u
0
0
,
A4)If{xn}isasequenceinC(+,)suchthatx n→ x∈ C(+,)andu(xn,xn+1)≥

0f o r a l l n ∈∗,t h e n u (xn,x)≥0f o r a l l n ∈∗.
A5)ForeachcompactsubsetK⊂+,thereexistsacompacts ˜ ⊂+such
etKthatforalln∈∗ ,wehave∆n (K)⊂K˜ .
Theorem1.4.3.S u p p o s e that Assumption1.4are fulfilled.T h e n , equation
Ă
(1.27)hasatleastonesolutioninC +,.
Â
Corollary1.4.4.Supposethat
1) f:+ì+iscontinuousandnon-decreasingaccordingtothesecondvariable.

2) ForeachcompactsubsetK +thereexistthepositivenumberand K Ψ1
sucht hat f o ra ll t ∈+,fo r a ll a ,b∈w i t h a ≤b,weh av e
∫
∆(t)
.f(t,a)−f(t,b).
¡
¢
G(t,s)ds≤1.
andλ sup
≤λψK |a−b|
t∈K

0