Fixed Point Theorems In Ultra Metric Space

Fixed point problem is always one of the the main direction of research in functional analysis, and it widely used in the algebraic equation, differential equation, integral equation, and implicit function theory. This article mainly aims at ultra metric space, research the common fixed point theorem, coupled fixed point theorem and coupled coincidence point theorem.The first chapter, we give the concept of ultra metric space, that is:Let(X,d)be a metric space, if the metric d satisfies strong triangle inequality d(x, y)≤max{d(x,z),d(z,y)}(?)x,y,z∈X, Then d is called an ultra metric on X and (X,d)is called an ultra metric space. Then introduce some of the existing fixed point theorems in ultra metric space.The second chapter, we firstly given the definitions of coincidence point and common fixed point, then obtain some results on coincidence point and common fixed point for a two pair of multi-valued and single-valued maps in ultra metric spaces, the main content is as follows:Let (X,d) be an ultra metric space. Let T,S:X→C(X) be a pair of multi-valued maps and f,g:X→X a pair of single-valued maps satisfying(ⅰ) fg(X)is spherically complete;(ⅱ)H(Sx,Ty)<max{d(fx,gy),d(fx,Sx),d(gy,Ty)} for all x,y→X,with fx≠gy(ⅲ)fS=Sf,jg=gf,fT=Tf,gS=Sg,gT=Tg,ST=TS;(ⅳ) S(X)(?)f(X)(?)(X)(?)g(X). Then there exist point u and v in X, such that fu∈Su, gv∈Tv, fu=gv, Su=Tv That is,f and S have a coincidence point u, g and T have a coincidence point v. Theorem2.2.1and theorem2.2.2are the focus of this chapter.The third chapter, we show the definition of couple fixed point and study the couple fixed point theorems in spherically complete ultra metric space(Theorem3.2.1, theorem3.2.2and Corollary3.2.3), namely:Let(X,d)be a spherically complete ultra metric space. If the map F:X×X→X satisfying any contractive conditions as follows, then F has a unique couple fixed point.(ⅰ)d(F(x,y),F(u,v))<max{d(x,u),d(y,v)};(ⅱ)d(F(x,y),F(u,v))<max{d(F(x,y),x),d(F(u,v),u)};(ⅲ) d(F(x,y), F(u,v))<max{d(F(x,y), u), d(F(u,v), x)}. On this basis, we further discusses the common coupled fixed point theorems in ultra metric space of a pair of maps and get the corresponding results:Let(X,d)be a spherically complete ultra metric space. If maps F:X×X→X and g:X→X satisfying F(X×X)(?)g(X), d(F(x,y),F(u,v))<max{d(gx,gu),d(gy,gv)},x,y,u,v(?)X,x≠y,u≠v Then there exist x, y∈X such that F(x, y)=gx and F(y, x)=gy, that is, F and g have a unique coupled coincidence point. Further if F and g commutes, then F and g have a unique common coupled fixed point, that is, there exits a unique pair(x,y)∈X×X such thatF(x,y)=gx=x and F(y,x)=gy=yFinally, introduced some connections and differences between couple fixed point and common coupled fixed point in ultra metric space. Theorem3.3.1, theorem3.3.2and corollary3.3.3are the main results of this chapter.