Relative Continuity, Proximal Boundedness and Best Proximity Point Theorems

This articles discusses some properties of relatively continuous mappings, a natural generalization of continuous mappings. Also, we introduce the notion of proximal boundedness and dilate upon a relationship among proximal boundedness, proximal completeness and proximal compactness. Finally, we utilize such results to elicit an extension of a Schauder's fixed point theorem, which states that every continuous self-mapping on a non-void compact convex subset of a normed linear space has a fixed point, to the case of non-self relatively continuous mappings. In fact, such an extension is proved in the form of a best proximity point theorem for relatively continuous mappings in the framework of a proximally compact space that has semi-sharp proximinality, thereby ascertaining the existence of an optimal approximate solution to some equations. Such an optimal approximate solution is known as a best proximity point and elicited as a result of approximation with the minimization of the error due to approximation. Further, an application of such a result is explored to elicit a common best proximity point theorem for a family of commuting affine mappings. Also, we furnish another application of our main result to find best proximity solution to an ordinary differential equation.