Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions

Let CC be a closed convex subset of a real Hilbert space HH and assume that TT is an asymptotically κκ-strict pseudo-contraction on CC with a fixed point, for some 0≤κ<10≤κ<1. Given an initial guess x0∈Cx0∈C and given also a real sequence {αn}{αn} in (0, 1), the modified Mann’s algorithm generates a sequence {xn}{xn} via the formula: xn+1=αnxn+(1−αn)Tnxnxn+1=αnxn+(1−αn)Tnxn, n≥0n≥0. It is proved that if the control sequence {αn}{αn} is chosen so that κ+δ<αn<1−δκ+δ<αn<1−δ for some δ∈(0,1)δ∈(0,1), then {xn}{xn} converges weakly to a fixed point of TT. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.