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On some nonself mappings. (English) Zbl 1024.47033
The authors obtain the following result (cf. Corollary 1.2) and present a slightly more general version of the same. Let \(X\) be a Banach space, \(K\) a nonempty closed subset of \(X\) and \(T\) a non-self mapping from \(K\) into \(X\) such that \[ d(Tx, Ty) \leq \lambda \max \{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)\} \] for all \(x, y\) in \(X\). If \(T\) maps the boundary of \(K\) into \(K\), then \(T\) has a unique fixed point in \(K\). Corollary 1.2 generalizes known results of B. E. Rhoades [Math. Jap. 23, 457–459 (1978; Zbl 0396.47038)] and others.

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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