Fixed Point Theorems In Convex Metric Spaces

Fixed point problem is one of the main direction of research. It has been widely used in functional analysis, algebraic equation, differential equation, integral equation and implicit function. This article mainly aims at the fixed point theorem in convex metric space.The first chapter, we show some conceptitons of convex metric space and existing fixed point theorems in convex metric space.The second chapter, we firstly give the definition of common fixed point, then obtain some results of common fixed point for a two pair single-valued maps in convex metric spaces, the main content is as follows:One part, let (X, d)be a convex metric space with property I and C be a compact subset of X. Let T, G:C→X be commuting mappings such that T is G-nonexpansive where G2=G. If G is continuous and affine, C is G-starshaped respected to G, then T and G have a unique common fixed point in C.The other part is that (X, d) is a convex metric space with a convex structure W and K is a nonempty closed subset of X.Let the pair (/, g)be compatible on K such that for all x,y, y∈K, d{fx, fy)≤ad(gx, gy)+b max{d(gx, fx),d(gy, fy)}+max{d(gx, fx)+d(gy, fy), d(gx,fy)+d(gy, fx)}, where a, b, c are nonnegative real numbers such that a+b+2c=1.If f(K)(?) g(K) and g is W-affine and continuous, then there exists a unique common fixed point z of f and g. Moreover,f is continuous at z.The third chapter, we show the definition of muti-valued maps. Then we obtain the common fixed point theorems between muti-valued maps and single-value maps in convex metric space:Let(X,d)be a convex metric space. Let K be a nonempty closed subset of X. The mappings T,S:K→CB(X) is a pair of multi-valued maps and f, g:K→X is a pair of single-valued maps. Suppose that: H(Sx,Ty)≤ad(fx,gy)+β max{D(fx,Sx),D(gy,Ty)}+ymax{D(fx,Sx)+D(gy,Ty),D(fx,Ty)+D(gy,Sx)} for all x,y in X, where α,β,γ≥0satisfy: λ＝α+2β+3γ+αγ<1.(ⅰ)(?)K(?)fK∩gK;(ⅱ)Sk∩K(?)gK,TK∩(?)fK;(ⅲ) Jx∈(?)K implies Sx(?)K, gx(?)K implies Txc(?)K. f(K)and g(K) are complete, then there exist pointsu and win K such that fu∈Su,gw∈Tw,fu=gwand Su=Tw.The forth chapter, we show the definition of property (E.A). Then we obtain the fixed point theorems of mappings with property (E.A):Let K be a nonempty closed convex subset of a convex metric space (X,d). If the maps f and gare self-mappings defined on K which satisfy the inequality: d{fx,fy)≤ad(gx,gy)+b max{d(gx, fx), d{gy, fy)}+c max{d(gx,fx)+d(gy,fy),d(gx,fy)+d(gy,fx)}, where a, b and c are nonnegative real numbers such that a+b+2c=1. Ifg isW-affine, fK∈gK and gK (or fK)is a complete subspace of X, then(ⅰ)the maps f and g have a coincident point;(ⅱ) fv=u is a unique common fixed point of/and g provided the maps f and g are weakly compatible;(ⅲ)the mapping f is continuous at u provided g is continuous at u.