Study of T-Contraction Mapping in Cone Metric Spaces

In this paper, the concept of metric space, replacing the set of real numbers by an ordered Branch space and defined a Cone metric space. The authors there described the convergence of sequences in cone metric spaces and introduced the completeness. Also, they proved some fixed point theorems of contractive mappings on complete cone metric Spaces. Since then, fixed point theorems for different (classic) classes of Mappings on these spaces have been appeared, see for instance the subject of "Functional Analysis" may be said to have started its development at the beginning of the present century. Functional Analysis born in the present century and growing very rapidly, is an area of mathematics based on linear algebra and topology combined together. The early growth of functional analysis was based on the nineteenth century function theory and was given a great impetus by the birth of lebesgue theory of measure and integration. Functional analysis is the study of certain topological-algebraic structures and of the methods by which knowledge of these structures can be applied to analytic problems. A topological space being simply a non-empty set in which there is given a class of subsets, called open sets.