Convergence analysis of iterative sequences for a pair of mappings in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space E. Let S: C -> C be a quasi-nonexpansive mapping, let T : C -> C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F := {x is an element of C : Sx = x and Tx = x} not equal 0. Let {x(n)}(n >= 0) be the sequence generated from an arbitrary x(0) is an element of C by x(n+1)=(1-c(n))Sx(n)+c(n)T(n)x(n), n >= 0. We prove the necessary and sufficient conditions for the strong convergence of the iterative sequence {x(n)} to an element of F. These extend and improve the recent results of Moore and Nnoli.