The Study Of Fixed Point Theorems In Cone Metric Space And N-Cone Metric Space

This paper mainly discusses the fixed point theorem in cone metric space and N-cone metric space under different contractive mapping.In the first chapter,we introduce the related concepts and its properties of cone metric space and N-cone metric space.In the second chapter,we discuss the fixed point theorem in cone metric space,We also obtain some conclusions:Let(X,d)be a complete cone metric space with regular cone P such that d(x,y)?int P,for all x,y?X with x?y.Let f,g,S and T be self-maps of a cone metric space X,satisfying:(?)f(X)(?)T(X),g(X)(?)S(X);(?)?(d(fx,gy))??(?d(Sx,Ty)+ ?d(fx,Sx)+ ?d(gy,Ty),+?d(fx,Ty)+ ?d(gy,Sx)),Where ?,?,?,?,??0 and ?+?+?+2?+2?<1,???If one of f(X),g(X),S(X)or T(X)is a complete subspace of X,then {f,S}and {g,T} have a common point of coincidence in X.Moreover if {f,S} and {g,T}are weakly compatible,then f,g,S and T have a unique common fixed point.In the third chapter,we introduce the concept of non-self mappings and its properties in cone metric space,and get the conclusion as follows:Let K be a nonempty complete subset of a cone metric space X with the convex property.{T:K?X} satisfies that there exists ??(0,(?)-1/2)such that?(d(Tx,Ty))???(d(x,y))-(?)(d(x,y)),for each x,y?K,where ???,(?)??.If T(X)?K for all x?(?)K,then T has a unique fixed point in K.In the fourth chapter,we also explore the fixed point theorems in N-cone metr-ic space and obtain some conclusions:Let(X,N)be an N-cone metric space,P be a normal cone with normal constant K and f,g:X?X be two mappings which satisfy the following condit-ions:(?)f(X)(?)g(X)(?)f(X)or g(X)is complete;(?)N(fx,fy,fz)?aN(gx,gy,gz)+ bN(gx,fx,fx)+ cN(gy,fy,fy)+dN(gz,fz,fz),for all x,y,z ?X,where a,b,c,d?0,a + 4b + 4c+2d<1.then f and g have a unique point of coincidence in X.Moreover if f and g are weakly compatible,then f and g have a unique common fixed point.