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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.

Relative Continuity, Proximal Boundedness and Best Proximity Point Theorems