Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings

Let K be a nonempty, closed and convex subset of a real Banach space E  . Let T:K→KT:K→K be a strictly pseudocontractive map in the sense of Browder and Petryshyn. For a fixed x0∈Kx0∈K, define a sequence {xn}{xn} byxn+1=(1-αn)xn+αnTxn,xn+1=(1-αn)xn+αnTxn,where {αn}{αn} is a real sequence defined in [0, 1] satisfying the following conditions (i) ∑n=1∞αn=∞, (ii) ∑n=1∞αn2<∞. Then liminfn→∞‖xn-Txn‖=0liminfn→∞‖xn-Txn‖=0. If, in addition, T   is demicompact, then {xn}{xn} converges strongly to some fixed point of T.