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On a fixed point theorem of Greguš. (English) Zbl 0597.47036
Let T and I be weakly-commuting mappings of a closed convex subset C of a Banach space X into C satisfying the inequality \(\| Tx-Ty\| \leq a\| Ix-Iy\| +(1-a)\max \{\| Tx-Ix\|\), \(\| Ty-Iy\| \}\) for all x,y in C, where \(0<a<1\). It is proved that if I is linear and non-expansive in C and such that the range of I contains the range of T, then T and I have a unique common fixed point in C.

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