## 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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### References:

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