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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.

##### MSC:
 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)
##### Keywords:
fixed point; non-self mapping
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