Common best proximity points: global minimal solutions
Given non-empty subsets \(A\) and \(B\) of a metric space, let \(S: A \to B\) and \(T: A \to B\) be non-self mappings. Taking into account the fact that, given any element \(x\) in \(A\), the distance between \(x\) and \(Sx,\) and the distance between \(x\) and \(Tx\) are at least \(d(A, B),\) a common best proximity point theorem affirms the existence of a global minimum of both functions \(x \mapsto d(x, Sx)\) and \(x \mapsto d(x, Tx)\) by imposing a common approximate solution of the equations \(Sx = x\) and \(T x = x\) to satisfy the constraint that \(d(x, Sx) = d(x, T x) = d( A, B )\). In this paper, the author discusses a common best proximity point theorem for a pair of nonself-mappings, one of which dominates the other proximally, thereby yielding common optimal approximate solutions of some fixed point equations when there is no common solution. A suitable example to demonstrate the validity of the hypotheses is given. The main result includes the common fixed point theorem due to \textit{G. Jungck} [Am. Math. Mon. 83, 261--263 (1976; Zbl 0321.54025)] for commuting continuous self-mappings, which in turn is an extension of Banach's contraction principle.
scientific article; zbMATH DE number 6156822
scientific article