On a fixed point theorem of Greguš. (English) Zbl 0597.47036

Fisher, Brian; Sessa, Salvatore

Int. J. Math. Math. Sci. 9, 23-28 (1986).

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.

MSC:

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

Keywords:

weakly-commuting mappings of a closed convex subset; unique common fixed point