Comments on some recent existence theorems of best proximity points for Kannan-type and Chatterjea-type mappings (Q1752021)

The author improves two theorems presented in the paper [\textit{S. S. Basha} et al., Optim. Lett. 7, No. 1, 145--155 (2013; Zbl 1283.90046)]. He shows that the results can be proved quite easily with the use of the original Kannan and Chatterjea fixed point theorems, even under slightly weaker assumptions. For instance, instead of the assumption that the pair \((S,T)\) of maps \(S:B\to A\), \(T:A\to B\) forms a weak \(K\)-cyclic contraction, the author assumes that the pair \((S,T)\) satisfies the so-called \textit{Condition} (b2b), i.e., \(d^*(x,Tx)=0\) implies \(d^*(STx,Tx)=d^*(x,Tx)\), \(d^*(x,Tx)>0\) implies \(d^*(STx,Tx)\neq d^*(x,Tx)\), for every \(x\in A\). Moreover, it is sufficient to assume that only one of the sets \(A,B\) is a complete space. In the last section of the paper the author gives several examples showing that numbers which appear in the statements of the main results are as good as possible.