A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

Let K be a nonempty closed convex subset of a real Banach space E. Let T:K→K be a generalized Lipschitz pseudo-contractive mapping such that F(T)≔{x∈K:Tx=x}≠0̸. Let {αn}n≥1,{λn}n≥1 and {θn}n≥1 be real sequences in (0,1) such that αn=o(θn),limn→∞λn=0 and λn(αn+θn)<1. From arbitrary x1∈K, let the sequence {xn}n≥1 be iteratively generated by xn+1=(1−λnαn)xn+λnαnTxn−λnθn(xn−x1),n≥1. Then, {xn}n≥1 is bounded. Moreover, if E is a reflexive Banach space with uniformly Gâteaux differentiable norm and if ∑n=1∞λnθn=∞ is additionally assumed, then, under mild conditions, {xn}n≥1 converges strongly to some x∗∈F(T).