CONE METRIC SPACES AND SOME FIXED POINT THEOREMS

Abstract

In this study, some topological concepts and definitions are generalized to cone metric spaces such as: sequentially closed set, bounded and totally bounded sets, c - net for sets, Lebesgue element, compact sets and continuous and sequentially continuous mappings. It is proved that every cone metric space is topological space, first countable topological space and 4 T - space. To prove Baire’s Category theorem in cone metric spaces, nowhere dense (Rare), Meager (first category) and Nonmeager (second category) sets are defined. Also, cone normed spaces, cone Banach spaces and the distance between two sets in cone metric spaces are defined. Furthermore, accompanied with some examples. To obtain some fixed point theorems in cone metric spaces, by assuming that the cone is a strongly minihedral cone, diametrically contractive and asymptotically diametrically contractive mappings are defined in cone metric spaces.

Finally, some fixed point theorems and common fixed points theorems of generalized contraction mappings are obtained by defining the scalar distance between two points in cone metric spaces.