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

Let KK be a nonempty closed convex subset of a real Banach space EE. Let T:K→KT:K→K be a generalized Lipschitz pseudo-contractive mapping such that F(T)≔{x∈K:Tx=x}≠0̸F(T)≔{x∈K:Tx=x}≠0̸. Let View the MathML source{αn}n≥1,{λn}n≥1 and {θn}n≥1{θn}n≥1 be real sequences in (0,1)(0,1) such that View the MathML sourceαn=o(θn),limn→∞λn=0 and λn(αn+θn)<1λn(αn+θn)<1. From arbitrary x1∈Kx1∈K, let the sequence {xn}n≥1{xn}n≥1 be iteratively generated by View the MathML sourcexn+1=(1−λnαn)xn+λnαnTxn−λnθn(xn−x1),n≥1. Turn MathJax on Then, {xn}n≥1{xn}n≥1 is bounded. Moreover, if EE is a reflexive Banach space with uniformly Gâteaux differentiable norm and if View the MathML source∑n=1∞λnθn=∞ is additionally assumed, then, under mild conditions, {xn}n≥1{xn}n≥1 converges strongly to some x∗∈F(T)x∗∈F(T).