Best proximity point theorems in the frameworks of fairly and proximally complete spaces

Let us contemplate the problem of solving the linear or non-linear equations of the form \(Tx=gx\) in the framework of metric space. When T is a non-self mapping and g is a self-mapping, it may cause the non-existence of a solution to the preceding equation. At this juncture, one is of course interested in computing an approximate solution \(x^*\) in the space such that \(Tx^*\) is as close to \(gx^*\) as possible. To be precise, if T is from A to B and g is from A to A, where A and B are subsets of a metric space, one is concerned with the computation of a global minimizer of the mapping \(x\longrightarrow d(gx, Tx)\) which serves as a measure of closeness between Tx and gx. This paper is concerned with the resolution of the aforesaid global minimization problem if T is a proximal contraction and g is an isometry in the frameworks of fairly and proximally complete spaces.