Applications of Best Approximations in Soft Locally Convex Spaces

This paper is divided into two parts. In first part, we introduce a new definition of a soft set in certain type of soft topological space which is soft locally convex space, $\left(\widetilde{\mathbb{X}}_{A}, \tilde{\tau}_{\mathcal{P}}, A\right)$ and study its some basic properties. In second part, we obtain a results for a soft-non-expansive mappings on a best approximation soft set in $\left(\widetilde{\mathbb{X}}_{A}, \tilde{\tau}_{\mathcal{P}}, A\right)$ and we prove some fixed soft point theorems for these mappings. Most of the recommendations for studied were referring to the study of a new characteristics of the fixed soft point in the soft metric spaces. So, we expanded some fixed soft point theorems of soft-non-expansive on a best approximation soft set in $\left(\widetilde{\mathbb{X}}_{A}, \tilde{\tau}_{\mathcal{P}}, A\right)$ and establish several relationships between properties these results.