Best Simultaneous Approximation in Probabilistic Normed Spaces

In the present paper we define the concept of best simultaneous approximation on probabilistic normed spaces and study the existence and uniqueness problem of best simultaneous approximation in these spaces. Firstly, some definitions such as set of p-best simultaneous approximation, simultaneous p-proximinal and simultaneous p-Chebyshev, are generalized. Then some properties related to the p-best simultaneous approximation set is presented and indicated that the simultaneous p-proximinal set is invariant under the addition and multiplication. We also develop the theory of p-best simultaneous approximation in quotient of probabilistic normed spaces and discuss about the relationship between the simultaneous p-proximinal elements of a given space and its quotient space. We show that under what conditions, set of the p-best simultaneous approximation is transferred by the natural map to the quotient space, and conversely. Finally some useful theorems were obtained to characterization for simultaneous p-proximinality and simultaneous p-Chebyshevity of a given space and its quotient space.