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A note on existence and convergence of best proximity points for pointwise cyclic contractions. (English) Zbl 1237.54052

Let \((A,B)\) be a pair of nonempty, weakly compact, convex subsets of a metric space \(X\) and \(T: A\cup B\to A\cup B\) a map such that \(T(A)\subseteq B\) and \(T(B)\subseteq A\) satisfying that, for each \(u\in A\times B\), there exists \(\alpha: A\cup B\to (0,1)\) such that \(d(Tu,Tv)\leq \alpha(u) d(u,v)+(1- \alpha(u))\,\text{dist}(A,B)\) for all \(u\in A\), \(v\in B\) and for all \(u\in B\), \(v\in A\). Then there exist \(x\in A\), \(y\in B\) such that \(\| x-Tx\|=\| y-Ty\|= \text{dist}(A,B)\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
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