In this work, we introduce the concept of generalized α-ηrational proximal contraction of first and second kind. Then we establish some best proximity theorems for such kind of contraction in the framework of metric spaces. The presented results generalize and improve several existing results in the best proximity theory.

In this paper, we present the Hyers-Ulam-Rassias stability of quartic functional equation f(2x + y) + f(2xy) = 4.f(x + y) + 4f(xy) + 24f(x)  6f(y) in Random 2-Normed space .

A best proximity point for a non-selfmapping is that point whose distance from its image is as small as possible. In mathematical language, if X is any space, A and B are two subsets of X and T: A → B is a mapping. We can say that x is best proximity point if d(x, Tx) = d(A, B) and this best proximity point reduces to fixed point if mapping T is a selfmapping.The main objective in this paper is to prove the best proximity point theorem for the notion of Geraghty-contractions by using MT-function β which satisfies Mizoguchi-Takahashi's condition (equation (i)) in the context of metric space and we also provide an example to support our main result.

Here, we extend the notion of (E.A.) property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1–14, 2014) by introducing a new class of self-maps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of self-maps. Also, we guarantee the existence of common best proximity points for this class of maps satisfying generalized non-expansive type condition. We furnish an example in support of the proved results.