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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:

54H25 Fixed-point and coincidence theorems (topological aspects)
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