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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 $d(Tu,Tv)\le \alpha(u) d(u,v)+(1- \alpha(u))\,\text{dist}(A,B)$ 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 $\Vert x-Tx\Vert=\Vert y-Ty\Vert= \text{dist}(A,B)$.

54H25Fixed-point and coincidence theorems in topological spaces
54E40Special maps on metric spaces
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