Generalization of Prolla's Theorem in Approximation

The principal factor in the proofs of many so-called intersection results and associated best approximation theorems—such as the famed Brouwer fixed-point theorem) is Knaster-Kuratowski-Mazurkiewicz theorem (KKM), which was extended from Rl to Hausdorff vector spaces by Ky Fan. This kind of theory necessitates introducing some form of abstract convex structure on a topological space. In both pure and applied mathematics, the theory of approximation plays a central role. The connection between best approximation and fixed-point theory provides a powerful tool for analyzing the nonlinear maps, the convergence of iterative processes, and the solving of variational problems. This paper is dedicated to presenting a new extension of the well-known Prolla’s theorem in the field of invariant approximation from a normed space to a kind of metric space. First, we will introduce the basic concepts of vector metric spaces, define the best approximation, and describe its relationship to fixed points. Second, we will substantiate the best approximation theorem using the state of KKM-map (where the convex combination of any finite elements in Θ⊂Π entirely belonged to the union of their images under the KKM-map,N,N:Θ→2Π). This theoretical construction has been applied to specific cases that were previously discussed as the best approximations in earlier papers.