Common best proximity points and global optimal approximate solutions for new types of proximal contractions

Let $(\mathcal{X},d)$ be a metric space, $\mathcal{A}$ and $\mathcal{B}$ be two non-empty subsets of $\mathcal{X}$ and $\mathcal{S},\mathcal{T}: \mathcal{A} \to \mathcal{B}$ be two non-self mappings. In view of the fact that, given any point $x \in \mathcal{A}$, the distances between $x$ and $\mathcal{S}x$ and between $x$ and $\mathcal{T}x$ are at least $d(\mathcal{A}, \mathcal{B}),$ which is the absolute infimum of $d(x, \mathcal{S} x)$ and $d(x, \mathcal{T} x)$, a common best proximity point theorem affirms the global minimum of both the functions $x \to d(x, \mathcal{S}x)$ and $x \to d(x, \mathcal{T}x)$ by imposing the common approximate solution of the equations $\mathcal{S}x = x$ and $\mathcal{T} x = x$ to satisfy the condition $d(x, \mathcal{S}x) = d(x, \mathcal{T} x) = d(\mathcal{A, B}).$ In this paper, we present two new types of proximal contractions and develop a common best proximity point theorem for proximally commuting non-self mappings, thereby yielding the common optimal approximate solution of some fixed point equations when there is no common solution.