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:(ⅰ)f(X)(?)T(X),g(X)(?)S(X);(ⅱ)ψ(d(fx,gy))≤ψ(αd(Sx,Ty)+ βd(fx,Sx)+ γd(gy,Ty),+λd(fx,Ty)+ κd(gy,Sx)),Where α,β,γ,λ,κ≥0 and α+β+γ+2λ+2κ<1,ψ∈ΨIf 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 λ∈(0,(?)-1/2)such thatψ(d(Tx,Ty))≤λψ(d(x,y))-(?)(d(x,y)),for each x,y∈K,where ψ∈Ψ,(?)∈Φ.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:(ⅰ)f(X)(?)g(X)(ⅱ)f(X)or g(X)is complete;(ⅲ)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.