The Study Of The Fixed Point Theorem In Metric Space

The article mainly aims at fixed point problem in three kinds of metric spaces. For one thing, in multiplicative metric space, we discuss the unique common fixed point of two pairs of weak commutative mappings on complete multiplicative metric space, where A and S are weak commutative, B and T also are weak commutative. Our results substantially generalize and extend the results of reference [12]. For another thing, we construct the common fixed point of multi-valued and single-valued mappings in complete metric space. Later we give some examples that multi-valued and single-valued mapping satisfy some given conditions, they have a unique fixed point. Lastly, we deal with the open question[1-3] of the reference[36].In the first chapter, we introduce the concepts and properties of multiplicative metric space and give the example to illustrate the rationality of the existence of multiplicative metric space. In refence [11], the author obtain that f has a unique fixed point when f is a multiplicative contractive mapping. Later, we introduce the concepts of commutative mapping and weak commutative mapping. The authors proved the common fixed point theorems of two pairs of weak commutative mappings on complete metric space in [12]. We generalize and extend the results to multiplicative metric space.Let S, T, A and B be self-mappings of a complete multiplicative metric space X they satisfies the following conditions:(1) SX(?)BX, TX(?)AX;(2) A and S are weak commutative, B and T also are weak commutative;(3) One of S,T,A and B is continuous;Then S, T, A and B have a unique common fixed point.The fourth condition of the above change tod(Spx, Tqy)≤{max{d(Ax, By), d(Ax, Spx),d(By,Tqy),d(Spx,By),d(Ax,Tqy)}}λ,S,T,A and B also have a unique common fixed point. Lastly, we give the example to illustrate that two pairs of weak commutative mapping have a unique fixed point when they satisfy all of the conditions of the theorem1.3.1.In the second chapter, we introduce the concepts and properties of metric space. We construct the existence of common fixed points of pairs of multi-valued and single-valued mappings when they are limit coincidentally commuting and point coincidentally commuting in complete metric space.Let (X, d) be a complete metric space, F:Xâ†'CB(X) and g:Xâ†'X are two mappings satisfying the following conditions: (2) FX(?)gX, gX is complete;(3) F and g are limit coincidentally commuting;(4)(?)∈Φ.Then F and g have a unique common fixed point u∈X such that Fu={u}=gu.When the third condition change to F and g are point coincidentally commuting, the result is right. i.e, theorem2.3.4. When F is changed to f, two single-valued mappings have a unique common fixed point. i.e, corollary2.3.3and2.3.5. We also give example to illustrate the rationality of the existence of the theorems.In the third chapter, we introduce the concepts and properties of b-metric space. Firstly, we introduce when a single-valued mapping f is a Meir-Keeler type operator on a complete b-metric space,f is a Picard operator is a family of multi-valued Meir-Keeler type operators. Consider the multi-valued fractal operator TF onis a Picard operator having aunique fixed point AF*∈Pcp(X). Lastly, we discuss the fixed point problem for TF is well-posed with respect to H.