Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}n⩾0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}n⩾0⊂[0,1] be such that ∑n⩾0αn=∞, and ∑n⩾0αn(kn−1)<∞. Suppose {xn}n⩾0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n⩾0, and suppose there exists a strictly increasing continuous function , ϕ(0)=0 such that 〈Tnx−x∗,j(x−x∗)〉⩽kn‖x−x∗‖2−ϕ(‖x−x∗‖), ∀x∈K. It is proved that {xn}n⩾0 converges strongly to x∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n⩾0 (where {un}n⩾0 is a bounded sequence in K and {an}n⩾0, {bn}n⩾0, {cn}n⩾0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.