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 ASUMMARYOFMATHEMATICSDOCTORALTHESI S NGHEAN-2015 WorkiscompletedatVinhUniversity Supervisors: Assoc.P r o f D r T r a n VanAn 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: Nguyen Thuc Hao Library and InformationCenter VietnamNationalLibrary PREFACE 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 ≤k< 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 :  → [ ,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) ,ϕ(t)> f o r a l l t > 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)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 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 , 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 ∈[ ,1)s u c h t h a t j + k+ l ¡ 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 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 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 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