MINISTRYOFEDUCATIONANDTRAINING VINH UNIVERSIT Y LEKHANHHUNG ONTHEEXISTENCEOFFIXEDPOINTFO R SOMEMAPPINGCLASSES INSPACESWITHUNIFORMSTRUCTUREAND APPLICATIONS Speciality Mathematical A n a l y s i s Code[.]
MINISTRYOF ED UC A TI ON AN D T R AININ G VINHU N I V E R S I T Y LEKHANHHUNG ONTHEEXISTENCEOFFIXEDPOINT FO RSOMEMAPPINGCLASSES INSPACESWITHUNIFORMSTRUCTUREAND APPLICATIONS Speciality:Mathematical A n a l y s i s Code:6 01 02 ASUMMARYOFMATHEMATICSDOCTORALTHESIS NGHEAN-2015 WorkiscompletedatVinhUniversity Supervisors: Assoc.P r o f D r T r a n VanA n Dr.K i e u PhuongChi Reviewer1 : Reviewer2 : Reviewer3 : Thesisw i l l b e p r e s e n t e d a n d d e f e n d e d a t s c h o o l l e v e l t h e s i s e v a l u a ti n g C o u n c i l a t VinhUn iversity at h d a t e m o nt h y e a r Thesisc an b ef ou n d a t : NguyenThucHaoLibraryandInformationCenter VietnamN ati onal L ib r ary PREFACE Rationale 1.1 The first result on fixed points of mappings was obtained in 1911.At thattime,L.B r o u w e r p r o v e d t h a t : E v e r y c o n ti n u o u s 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.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 ofdifferentialequationsmarksanew development of the study of fixed point theory.Afterthat,manymathematicianshavestudiedtoextendtheBanachcontractionmappi ng principle for classes of maps and different spaces Expanding only contractivemappings,till1977,wassummarizedandcomparedwith25typicalformatsbyB.E.Rhoades 1.2 TheBanachcontractionmappingprincipleassociateswiththeclassofcon-tractive mappingsT:X→Xin complete metric space (X, d) with the contractivecondition (B)d(Tx,Ty)≤kd(x,y),forall x, y∈Xw h e r e 0≤k f o r a l l t > a n d ψ (0)= =ϕ(0), moreover,ϕis a monotone non-decreasing function andψis a monotoneincreasingfunction In 2012, B Samet, C Vetro and P Vetro introduced the notion ofα-ψcontractivetypemappingsincompletemetricspaces,withacontractive condition ≤ oftheform ∈ → (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 Σ +∞ψ + 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 g (t)< + ∞fora llt>0and n=1 α:X×X→ R+ 1.3 Inrecentyears,manymathematicianshavecontinuedthetrendofgeneralizingcontractive conditionsformappingsinpartiallyorderedmetricspaces.Followingthistrend, 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 the class of those mappings in partially ordered metricspacessatisfyingthecontractivecondition F ≤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 (x,y),F(u,v) 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 LakshmikanthamandL.CiricobtainedsomeresultsfortheclassofmappingsF:X×X→Xwithg-mixed monotone property, whereg:X→Xfrom a partially ordered metric spaceintoitselfandFs a ti s fi e s thefollowingcontractive condition d (LC)dF (x,y),F(u,v)≤ϕ g + g (x),g(u) d (y),g(v) , 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 with mixed monotone property in partially ordered metricspacessatisfyingthe contractive condition (BB) There exists constantsj, k, l∈ [0,1) such thatj+k+l