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Extensions of Banach’s contraction principle. (English) Zbl 1200.54021
For a metric space $(X,d)$ and $A,B\subset X,$ let $d(A,B)=\inf\{d(a,b) :a\in A,B\in B\}$, $A_0=\{a\in A : \exists b\in B, d(a,b)=d(A,B)\}$, and $B_0=\{b\in B : \exists a\in A,\ d(a,b)=d(A,B)\}$. The set $B$ is called approximatively compact with respect to $A$ if for every $x\in A$, every sequence $(y_n)$ in $B$ such that $d(x,y_n)\to d(x,B)$ contains a subsequence converging to some point of $B$. A mapping $T:A\to B$ is called a proximal contraction if there exists $\alpha, \, 0\leq \alpha<1,$ such that for every $(x,y)\in A\times B$ for which there exists $(u,v)\in A\times B$ with $d(u,Tx)=d(v,Ty)=d(A,B),$ the following inequality holds $$ d(u,Tx)+d(Tx,Ty)+d(v,Ty)\leq \alpha d(x,y).$$ The author shows by an example (a mapping $T:[0,1]\to[2,3]$) that a proximal contraction need not be continuous. Let $(X,d)$ be a complete metric space, $\, A,B \subset X$ nonempty closed such that $B$ is approximatively compact with respect to $A$ and $A_0,B_0$ are nonempty. Then for every proximal contraction $T:A\to B$ with $T(A_0)\subset T(B_0),$ there is a unique point $x\in A$ such that $d(x,Tx)=d(A,B)$ (Theorem 3.1). In Theorem 3.3, a similar result is proved for two mappings $T:A\to B,\, S:B\to A,$ where $T$ is a proximal contraction and $S$ is nonexpansive.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
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