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On a generalization of a Greguš fixed point theorem. (English) Zbl 1079.47509
Summary: Let \(C\) be a closed convex subset of a complete convex metric space \(X\). In this paper, a class of selfmappings on \(C\), which satisfy the nonexpansive type condition \[ d(Tx,Ty)\leq a\max \{d(x,y),c[d(x,Ty)+d(y,Tx)]\}+b\max \{d(x,Tx),d(y,Ty)\} \] (where \(0<a<1,\;a+b=1,\;c\leq \frac {4-a}{8-a}\)), is introduced and investigated. The main result is that such mappings have a unique fixed point.

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