On The Existence Of Fixed Point For Some Mapping Classes In Spaces With Uniform Structure And Applications.docx

MINISTRYOFEDUCATIONANDTRAINING VINH UNIVERSIT Y LEKHANHHUNG ONTHEEXISTENCEOFFIXEDPOINTFO R SOMEMAPPINGCLASSES INSPACESWITHUNIFORMSTRUCTUREAND APPLICATIONS Speciality Mathematical A n a l y s i s Code[.]

d)intoitselfhasauniquefixedpoint.ThebirthoftheBanachcontractionm a p p i n g p r i n c i p l

e a n d i t s a p p l i c a ti o n t o s t u d y t h e e x i s t e n c e o f s o l u ti o n s of differential equations marks a new

(BW)d(Tx,Ty)≤ϕd (x,y), forallx,y∈X,whereϕ:R+→R+isasemiright

uppercon ti n u ous f u n cti o n a n d s a ti s fie s 0 ≤ϕ(t)<tf ora ll t ∈R+

In2001,B.E.Roades,whileimprovingandextendingaresultofY.I.Alberand

S.Guerre-Delabriere,gaveacontractive condition oftheform

(R1)d(Tx,Ty)≤d(x, y)−ϕd(x, y), for allx, y∈X,whereϕ:R+→R+is

acontinuous,monotoneincreasingfunctionsuchthatϕ(t)=0ifandonlyift=0.

Followingthewayofreducingcontractiveconditions,in2008,P.N.DuttaandB

S.Choudhuryint roduceda c ontra ctive conditionof t h efor m

(DC)ψd(Tx,Ty)≤ψ d(x, y)−ϕ d(x, y) , for allx, y∈X,whereψ, ϕ:R+→R+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 eralized weak contractive mappings with the following contractive condition inordertostudycommonfixedpoints ofmappings

(SVV)α (x,y)d(Tx,Ty)ψ d(x,y) ,f o r a l l x , yX whereψ :R R isam o n o t o n e n o n

-d e c r e a s i n g f u n c ti o n s a ti s f y i n gΣ+∞ψ

n (t)< + ∞fora l l t > 0 a n d α:X×X→ R+

1.3 Inrecentyears,manymathematicianshavecontinuedthetrendofgeneralizingcontractive

conditions for mappings in partially ordered metric spaces Following thistrend, in 2006, T G Bhaskar and V.Lakshmikantham introducedthe notion ofcoupled fixed points of

mappingsF:X×X→Xwith the mixed monotone propertyand obtained some results for

metricspacessatisfyingthecontractivecondition

(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) ≤k

d(x,u)+d(y,v), fora l l x , y,u,v∈ Xs u c h t h a t x ≥u,y≤ v.

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

spaceintoitselfandFs a ti s fi e s thefollowingcontractive condition

(LC)dF (x,y),F(u,v)≤ϕ

d g (x),g(u)+d g (y),g(v)

,

(AK)T h e r e e x i s t s a f u n c ti o n φs u c h thatforallx≤u,y≥v,z≤ww e hav

e

fora l l x , y,u,v∈ Xw i t h g (x)≥g(u),g(y)≤g(v)a n d F (X× X)⊂g(X).

In2011,V.BerindeandM.Borcutintroducedthenotionoftriplefixedpointsforthe class of

mappingsF:X×X×X→Xand obtained some triple fixed pointtheorems for mappings

metricspacessatisfyingthe contractive condition

(BB) There exists constantsj, k, l∈ [0,1) such thatj+k+l <1 satisfydF(x, y, z),F(u, v, w)≤jd(x, u)+kd(y, v)+ld(z, w),for allx, y, z, u, v, w∈Xwithx≥u,y ≤v,z≥w.

After that, in 2012, H Aydi and E Karapinar extended the above resultandobtained some triple fixed point theorems for the class of

mappingF:X×X×X→Xwith mixed monotone property in partially ordered metric spaces and satisfying thefollowingweakcontractive condition

d TF (x,y,z),TF(u,v,w)≤

φ maxd (Tx,Tu),d(Ty,Tv),d(Tz,Tw)}

1.4. The development of fixed point theory is motivated from its popular plications, especially in theory of differential and integral equations, where thefirstimpressioni s t h e a p p l i c a ti o n o ft h e B a n a c h c o n t r a c ti o n m a p p i n g p r i n

ap-c i p l e t o s t u d y theexistenap-ceofsolutions ofdifferential equations

Inthemoderntheoryofdifferentialandintegralequations,provingtheexistenceofsolutions orapproximating the solutions are always reduced to applying appropriatelycertain fixed point theorems.Forboundary problems with bounded domain, fixedpoint theorems in metric spaces are

forboundaryproblemswithunboundeddomain,fixedpointtheoremsinmetricspacesarenotenoughtodothatwork.S o , inthe70soflastcentury,alongwithseekingtoextendtomappingclasses,onewaslookingtoextendtoclassesofwiderspaces.O n e oftypicaldirections of this expansion is seeking toextend results on fixed points of mappings inmetric spaces to the class of local convex spaces, more

whichhasa tt r a ct e d t h e a tt e n ti o n o f ma n y m a t h e m a ti c a l , n ot a b l y V G A n g e l ov

In 1987, V G Angelov considered the family of real functions

Φ={φ α:α∈I}such that for eachα∈I,φα:R+→R+is a monotone increasing,

continuous,φ α (0)=0 and 0<φ α (t)<tfor allt>0.Then he introduced the notion contractivemappings,whicharemappingsT: M→ Xsatisfying

ofΦ-(A)d α (Tx,Ty)≤φ αdj(α) (x,y)forallx,y∈Ma n d forallα∈I,whereM⊂ X

ando b t a i n ed s o m er es u l t s o n fi x e d p o i n t s o f t h e c l a s s o f t h os e m a p p i n g s Byi n t r o-

ducingt h e n o ti o n o f s p a c e s w i t h j

-boundedp r o p e r t y , V G A n g e l o v o b t a i n e d s o m e resultsontheuniqueexistenceofafixed pointoftheabovemappingclass

Following the direction of extending results on fixed points to the class oflocalconvex spaces, in 2005, B C Dhage obtained some fixed point theorems in

Banachalgebrasb y s t u d y i n g s o l u ti o n s o f o p e r a t o r e q u a ti o n s x = Ax Bx w h e r e A :

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:R+→R+is a

non-decreasingcontinuous function, φ(0)=0.

1.5 Recently,togetherwiththeappearanceofclassesofnewcontractivemappings,and new

types of fixed points in metric spaces, the study trend on the fixed pointtheory has advanced steps of strong

toextendresu lt s in t h e fi x ed p oi n t t h e or y f o rc las ses of s pa c es wit h u n ifor m s t r u

c t u r e, wec h o s e t h e t o p i c ‘ ‘ O n t h e existenceof fixedp o i n t s f o r some mappingclassesinspaceswithuniformstructureandapplications” forourdoctor

map-4 Scopeo f t h e r e s e a r c h

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

The thesis can be a reference for under graduated students, master studentsandPh.D students in analysis major in general, and the fixed point theory andapplicationsinpar ticular.

The content of this thesis is presented in 3 chapters In addition, the thesisalsoconsistsProtestation,Acknowledgements,TableofContents, Preface,Conclusionsand Recommendations, List of scientific publications of the Ph.D studentrelated tothethesis,a ndReferences

In chapter 1, at first we recall some notions and known results aboutuniformspaces 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, ϕ)-contraction of

contractivemappingsinu n i f o r m spaces and obtain some fixed point theorems for theclass of those mappings Theo-rems, which are obtained in uniform spaces, areconsidered as extensions of theoremsin complete metric spaces.Finally, applying our theorems

about fixed points of theclasso f ( β,Ψ1)-contractivem a p p i n g i n u n i f o r m s p a c e s ,

w e p r o v e t h e e x i s t e n c e o f s o - lutions of a class of nonlinear integral equations with unbounded deviations.Notethat, when we consider a class of integral equations with unbounded deviations,

wecannotapplyknownfixed pointtheoremsin metricspaces.M a i n resultsofChapter

Theorem2.2.5,Corollary 2.2.6,Theorem2.3.3 andT heorem2.3.6.

In Chapter 3, at first we present systematically some basic notions aboutlocallyconvex algebras needed for later sections After that, in section 3.2, byextending thenotion ofD-Lipschitz maps for mappings in locally convex algebras and

by basing onknown results in Banach algebras, and uniform spaces, we prove a fixedpoint theoremin locally convex algebras which is an extension of an obtained result by B C Dhage.Finally, in section3.3, applying obtained theorems, we prove the existence of solutionof a class ofintegral equations in locally convex algebras with unboundeddeviations.MainresultsofChapter3ar eTheorem3.2.5, Theorem3.3.2

In this thesis, we also introduce many examples in order to illustrate ourresultsandthemeaningofgivenextensiontheorems

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 uniformspacesandusefulresultsforlaterparts.T h e n , wegivesomefixedpointtheoremsfortheclassof

(Ψ,Π)-)-contractive mappings in uniform spaces In the last part of this chapter, weextendfixedpointtheoremsfortheclassofα-ψ-contractivemappingsinmetricspacesto

uniform spaces After that, we apply these new results to show a class ofintegralequationswithunboundeddeviationshaving auniquesolution

Definition1.1.1.A n uniformityoruniformstructureonXis a non-empty family

Uco n sist i n g ofsubsetsofX×Xw h i c h satisfy thefollowingconditions

Cauchysequence,sequentially completeuniformspaceandtherelationship betweenthem.

Remark1.1.8.

1)LetXbeauniformspace.Then,uniformtopologyonXisgeneratedbythefamilyof uniformcontinuouspseudometricsonX

2) IfEislocallyconvexspacewithasaturatedfamilyofseminorms{p α}α∈I,

thenw e c a n d e fi n e a f a m i l y o f a s s oc i a t e p s e u d o m e t r ic s ρ α (x,y)=p α (x−y)f o r e v e

r y x,

y∈E.Theuniformtopologygeneratedthefamilyo f a s s o c i a t e p s e u d o m e t r i c s coin cideswiththeoriginaltopologyofthespaceE.T h e r e f o r e , asacorollaryofour

T: X→ X.S u p p o s e t h a t

Nowf o r e v e r y ( n,r)∈ Iwec o n s i d e r t h e f u n c ti 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 :(n,r)∈I}.D e n o t e b y j :I→I

ψαdα (Tx,Sy)≤ψ αdj(α) (x,y)−ϕ αdj(α) (x,y),

fora l l x , y∈X,w h e r e ψ α∈Ψ,ϕα∈Πf o r a l l α ∈I.

Supposej:I→Ibeasurjectivemap and for some ti x 0∈Xsuchthat thesequence{xn }with x 2k+1 =Tx 2k ,x 2k+2 =Sx 2k+1 , k≥0satisfies dα (x m+n, xm )≥d j(α) (x m+n,xm )fo rall m,n≥0,α∈I.

Then,t h e r e e x i s t s u ∈Xs u c h t h a t u =Tu=Su.

Moreover,ifXhasthej-monotonedecreasingproperty,thenthereexistsauniquepointu∈Xs u c h thatu=Tu=S u.

}

DenoteΨ1={ψ α: α∈I}bea f a m i l y o ff u n c ti o n s w i t h t h e p r o p e r ti e s

(i) ψ α: R+→R+ismonotonenon-decreas ingand ψ α(0)=0

(ii) foreachα∈I,thereexistsψ α ∈ Ψ1suchthat

Denotebyβa familyoffunctions β={β α:X×X→R+,α∈I}.

Definition 1.3.7.Let (X,P)be a uniform space withP=d α (x, y) :α∈IandT:X→Xbe a given mapping.We say thatTis an (β,Ψ1)-contractiveif for

everyfunctionβ α∈βa n d ψα∈Ψ1wehave

βα (x,y).d α (Tx,Ty)≤ψ αdj(α) (x,y), forallx,y∈X.

Definition1 3 8 L e t T: X→ X.W e saythatTi s aβ-admissibleifforallx,y∈ X

andα ∈I,β α (x,y)≥1i m p l i e s β α (Tx,Ty)≥1.

a) Ti s continuous;or

b) fora l l α ∈ I,i f { xn }isa s e q u e n c e i n X s u c h t h a t β α (x n,xn+1 )≥ 1f o r a l l n andx n→x∈Xa s n→+∞,th enβ α (x n,x)≥1f o r a l l n ∈N∗.

Then,T hasa fi x e d p o i n t

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 ∈ Xsucht h a t

βα (x,z)≥1a n d β α (y,z)≥1f o r a l l α ∈I,t h e n T h a s a u n i q u e fi x e d p o i n t

Wealsogivesomeexamplestoillustrateforourresults

0(s) d s

Inthissection, wewishtoinvestigatetheexistenceofauniquesolutiontononlinearintegralequations, as an application to the fixed point theorems proved in the Section1.3

Assumption1.4.1.A1) There exists a functionu:R2→Rsuch that for

eachcompacts u b s e t K ⊂R+,thereexistapositivenumberλ andψ K∈Ψ1suchthatforallt∈R

• Give and prove theorems which confirm the existence and unique existence of

afixed point for the classof (Ψ,Π)-)-contractive maps in uniform space (Theorem

1.2.6,1.2.9)

Theser e s u l t s a r e w r i tt e n i n t h e a r ti c l e : TranV a n A n , K i e u P h u o n g C

h i a n d Le Khanh Hung (2014),Some fixed point theorems in uniform spaces, submitted

toFilomat

• Give and prove a theorem which confirm the existence and unique existence of

afixed point for the class of (β,Ψ1)-contractive mappings in uniform spaces(Theorem1.3.11) And apply Theorem 1.3.11 to prove the unique existence of solution

of a classofintegralequationswit hunboundeddeviations

These results are written in the article: Kieu Phuong Chi, Tran Van An,

LeKhanh Hung (2014),Fixed point theorems for(α-Ψ)-contractive type mappings inuniformspacesandapplications ,Filomat(toappear).

˜

∫

2.1 Coupledfi x e d p o i n t s i n p a r ti a l l y o r d e r e d u

n i f o r m spaces

In 2006, T G Bhaskar and V Lakshmikantham introduced the notion of

coupledfixed points of mappingsF:X×X→Xwith mixed monotone property and

obtainedsomeresultsfortheclassofthosemappingsinpartiallyorderedmetricspaces

Definition2.1.1.Let(X,≤) be a partially ordered set andF:X×X→X.The

mappingFis said to have themixed monotone propertyifFis monotone

non-

decreasinginitsfirstargumentandismonotonenon-increasinginitssecondargument,thatis,foranyx,y∈X

ifx1,x2∈X,x1≤x2thenF (x1,y)≤F(x2,y)

and

ify1,y2∈X,y1≤y2thenF (x,y1)≥F(x,y2).

Inthissection, w e provesomecoupled fixedpoint theorems forgeneralized con-tractivemappingsinpa rtiallyordered un iform spaces

LetΦ1={φ α:R+→R+;α ∈I}beafamilyoffunctions withtheproperties:

i) φ αismonotonenon-decreasing.

ii) 0<φ α (t)<tfor allt>0 andφ α(0)=0

dαF (x,y),F(u,v)≤φα d

j(α) ( x, u ) + d j(α) ( y, v )

,

forallx ≤u,y≥v.

2) Foreachα∈I,thereexistsφ α ∈Φ1suchthatsup{φj n (α) (t):n=0,1, }≤

φ( t)and φ α (t)is non-decreasing.

α

t 3) Thereare x0,y0∈Xs u c h tha tx0≤F(x0,y0),y0≥F(y0,x0)a n d

Corollary2.1.6.InadditiontohypothesesofTheorem2.1.5,ifx0andy0are

compa-rablethenFh a s auniquefixedpoint, t hatis,ther eexistsx∈Xs u c h thatF(x,x)=x.

In 2011, V Berinde and M Borcut introduced the notion of triple fixed

pointsfor a classof mappingF:X×X×X→Xand obtained some triple fixed

pointtheorems for mappings with mixed monotone property in partially orderedmetricspaces.After that,in2012,H.AydiandE.Karapinarextendedtheaboveresultand

obtained some triple fixed point theorems for a class of mappingF:X×X×X→Xwith mixed monotone property in partially ordered metric spaces and satisfying thefollowingweakcontractive condition.

Let (X,≤) be a partially ordered set.Then, we define a partial order onX3inthefollowingway:

For( x,y,z),(u,v,w)∈X3then

(x,y,z)≤(u,v,w)ifa n d o n l y i f x≤u,y≥ va n d z ≤ w.

Wes a y t h a t ( x,y,z)a n d ( u,v,w)a r e c o m p a r a b l e i f

(x,y,z)≤(u,v,w) or (u,v,w)≤(x,y,z).

Also,wesay that ( x,y,z)isequal t o(u,v,w)i f andonlyi f x =u,y =va n d z=w.

Definition 2.2.4.LetXbe a uniform space A mappingT:X→Xis said to beICSifTis

injective, continuous and has the property: for every

sequence{x n }inX,ifsequence{Tx n }isconvergentthen{x n}isalsoconvergent

Theorem2 2 5 Let( X,≤)b e a p a r ti a l l y o r d e r e d s e t a n d P ={d α (x,y):α∈I}be

a family of pseudometrics on X such that(X,P)is a Hausdorff sequentially completeuniform space Let

mappinghavingthemixedmonotonepropertyonX.Supposethat