Introduction to soft cone metric spaces

Mathematical modeling attempts of Concepts that are relative to accuracy have increased the interest in the problems involving uncertainty. Different theories such as interval mathematics, probability theory, fuzzy set theory, approximated set theory, soft set theory have been developed for modeling and solving these problems. One of the most interesting of these theories is the fuzzy set theory. A fuzzy set can be defined by its membership function. Since a membership function is set up for each particular case it is extremely individual. For this reason, a set theory that is independent of the membership function construction was needed. In order to meet this need and to deal with uncertainty, soft set theory is used as a mathematical tool at the present time. In contrast to the fuzzy set and intuitive fuzzy set theory, the flexible set theory aims to remove ambiguity with a selection transformation instead of a real valued function. The flexible set theory has been successfully applied to many mathematical fields such as continuous differentiable functions, game theory, navigation research, Riemann integral, Peron integral, probability theory, measurement theory. In the recent past, many algebraic structures such as groups, rings and objects and spatial structures such as vectors, norms, metrics and topological have been successfully established on soft sets and very interesting results have been obtained. In this tesis, soft cone concept is defined in soft Banach spaces and soft cone metric structure is established on soft clusters with soft element. Relations between soft cone metric spaces and classical cone metric spaces are proved and examples are given. The soft open ball, soft closed ball, soft open set in soft cone metric spaces are defined and their basic properties are given. Under what conditions, it is proved that the elementary union and elementary intersection of soft open sets are open and a soft conic metric space is shown to be soft topological space relative to elementary operations. Also, the concepts of the convergence of sequences and Cauchy sequences in soft conic metric spaces and their properties are given. Finally, some important fixed point theorems in soft conic metric spaces are expressed and proved via soft element. The results obtained within the scope of this tesis will be the basis for further studies in this context.