Fixed Points by a New Iteration Method

The following result is shown. If is a lipschitzian pseudo-contractive map of a compact convex subset of a Hilbert space into itself and is any point in , then a certain mean value sequence defined by <!-- MATH ${x_{n + 1}} = {\alpha _n}T[{\beta _n}T{x_n} + (1 - {\beta _n}){x_n}] + (1 - {\alpha _n}){x_n}$ --> converges strongly to a fixed point of , where <!-- MATH $\{ {\alpha _n}\}$ --> and <!-- MATH $\{ {\beta _n}\}$ --> are sequences of positive numbers that satisfy some conditions.