The Study Of Fixed Point Theorems In Cone Metric Space And Partially Ordered Cone Metric Space

In this article, we mainly study fixed point problems in cone metric space and partially ordered cone metric space under different contractive mappings.In chapter 1, we introduce the related concepts and research status in cone metric space and partially ordered cone metric space.In chapter 2, we present common fixed point theorem under two single-valued mappings without appealing to the normal cone in cone metric space, that is, Let (X, d) be complete cone metric space, f,g:X→X satisfying: d(fx, gy)≤ad(x, y)+β[d(x, fx)+d(y, gy)]+γ[d(x, gy)+d(y, fx)] for any x,y∈X, α,β,γ≥0 and α+2β+2γ< 1, then f and g have a unique common fixed point in X. Moreover, any fixed point of f is that of g, and conversely.Then we study the common fixed point problems under a family of mappings{fλ}λ∈A . At last we discuss the fixed point in generalized cone metric space —— TVS-valued cone metric space, that is, Let (X, d) be TVS-valued cone metric space, f,g:X→X satisfying: d(fx, fy)≤ad(gx, gy)+β[d(fx, gx)+d(fy, gy)]+γ[d(fx, gy)+d(fy, gx)] for any x,y∈X, α,β,γ>0 and α+2β+2γ<1. If the range of g contains the range of f, and g(X) is a complete subspace of X, then f and ghave a unique coincidence point. Moreover if f and g are weakly compatible,f and g have a unique common fixed point.In chapter 3, we present triple coincidence fixed point theorem, that is, Let(X, d) be partially ordered cone metric space, F:X×X×X→X is continuous and has the mixed g-monotone property, gis self-map and g(X) is a complete subset of X, F(X×X×X)(?)g(X) such that d(F(x, y, z), F(u, v, w))≤jd(gx, gu)+kd(gy, gv)+ld(gz, gw) for all x,y,z,u,v,w∈X, gx≤gu, gy≥gv, gz≤gw with j,k,l∈[0,1), j+k+l∈[0,1). If there exist x0, y0, z0 ∈ X such that gx0≤F(x0,y0,z0),gy0≥F(y0,x0,y0),gz0≤F(z0,y0,x0). Then there exists x, y, z, ∈ X, such that gx= F(x, y, z), gy= F(y, x, y), gz= F(z, y, x).In the following, we omit the continuous condition of F, assuming X has the follow-ing properties:(1) if a non-decreasing sequence xn → x, then xn≤x, for all n.(2) if a non-increasing sequence yn → y, then yn≥y, for all n.We can also obtain the tripled coincidence fixed point theorem in partially ordered cone metric space.