Common Coupled Coincidence Point Theorems In Cone Metric Space And Metric Space

In this paper, our research is about theorems of common coupled coincidence point and common coupled fixed point. For one thing, in metric space, we first introduce the concept of common coupled coincidence point for one single valued mapping and two multi-valued mappings, we also prove some common coupled coincidence point and common coupled fixed point theorems. For anther, in cone metric space, we obtain some common coupled coincidence point and common coupled fixed point theorems. Our results extend and generalize various known comparable results in the literature.In the first chapter, we introduced the concepts and properties of cone and cone metric space, concepts of common fixed point, coupled fixed point, coincidence point and so on. Also, we show some fixed point theorems in the literature in cone metric space. We obtain some common coupled coincidence point of the mappings F,G:X×X→X and f:X→Xsatisfy several generalized contractive conditions:Let (X,d) be a cone metric space, intP be nonempty. Let mappings F,G: X×X→X and f:X→X satisfy the following one of the three different types contractive conditions:for every x,y,u,v∈X, φ is a φ-map, a, b∈R+,a+b∈[0,1). Suppose mappings F,G,f satisfy:3) f(X) is a complete subspace of X. Then F, G, f have a common coupled coincidence point.Secondly, on the basis of the above three theorems, we obtain several corollaries of common coupled fixed point theorems:corollary1.3.2, corollary1.3.3, corollary1.3.5, corollary1.3.6, corollary1.3.8, corollary1.3.9.In the second chapter, we introduced the concepts of metric space and Hausdorff metric, researches about multi-valued fixed point theorems in metric space. We first introduce the concept of common coupled coincidence point of one single mapping.f:X→X and two mutil-valued mappings F,G:X×X→2X, if an element (x,y)∈X×X satisfies fx∈ F(x,y)∩G(x,y) and fy∈F(y, x)∩G(y, x). Also, we obtain some common coupled coincidence point of the hybrid mappings of{F,G,f} satisfy generalized contractive condition:Let (X, d) be a metric space, mappings F,G:X×J→CB(X) and f:X→X satisfy: for every x,y,u,v∈X, ai,i=1,2,3,4,5,6are nonnegative real numbers and satisfy:3) f(X) is a complete subspace of X. Then hybrid mappings F, G, f have a common coupled coincidence point.Secondly, on the basis of the theorem, we obtain several corollaries of common coupled fixed point theorems:corollary2.3.2, corollary2.3.3, corollary2.3.4, corollary2.3.5.