Abstract
Best proximity point theorems unravel the techniques for determining an optimal approximate solution, designated as a best proximity point, to the equation Tx = x which is likely to have no solution when T is a non-self mapping. This article presents best proximity point theorems for new classes of non-self mappings, known as generalized proximal contractions, in the setting of metric spaces. Further, the famous Banach's contraction principle and some of its generalizations and variants are realizable as special cases of the aforesaid best proximity point theorems.
Highlights
1 Introduction Fixed point theory focusses on the strategies for solving non-linear equations of the kind Tx = x in which T is a self mapping defined on a subset of a metric space, a normed linear space, a topological vector space or some pertinent framework
When T is not a self-mapping, it is plausible that Tx = x has no solution
Best approximation theorems and best proximity point theorems are suitable to be explored in this direction
Summary
Because of the fact that, for a non-self mapping T : A ® B, d(x, Tx) is at least d(A, B) for all x in A, a best proximity point theorem ensures global minimum of the error d(x, Tx) by confining an approximate solution x of the equation Tx = x to comply with the condition that d (x, Tx) = d(A, B). A mapping T : A ® B is said to be a generalized proximal contraction of the first kind if there exist non-negative numbers a, b, g, δ with a + b + g + 2δ < 1 such that the conditions d(u1, Tx1) = d(A, B) and d(u2, Tx2) = d(A, B)
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