Fixed point theorems for non-self mappings (Q1057467)

Let K be a bounded closed convex subset of a Hilbert space X; a mapping \(T: K\to X\) is said to be an M-type mapping if it satisfies the condition \(\| Tx-Ty\| \leq \alpha (\| x-Tx\| +\| y-Ty\|)+\beta \| x-y\|\) for all x,y\(\in K\) where \(\alpha\geq 0\), \(\beta\geq 0\) and \(2\alpha +\beta \leq 1\). If an M-type mapping T satisfies the boundary condition T(Bdr K)\(\subset K\) then it has a fixed point. The author proves also a common fixed point theorem for an asymptotically commutative family of M-type mappings.