Weak Almost-Convergence Theorem without Opial's Condition

Abstract

Let E be Banach space with property (U,m,m+1,λ),λ∈R,m∈N, and a uniformly Gateaux differentiable norm; J: E→E* a duality mapping; D a nonempty closed convex bounded subset of E; and T: D→D a uniformly L-Lipschitzian asymptotically hemicontractive mapping with L<N(E)1/2 where N(E) is the normal structure coefficient of E satisfying the condition ‖x−Tny‖2≤〈x−Tny,J(x−y)〉 for all x,y∈D, n∈N∪{0}. Under the above conditions, the convergence of {J(xn−v)} for the sequence {xn} of the modified Ishikawa iteration process is established and then it is used to prove weak convergence of the process. The modified Ishikawa iteration process is defined as follows: For D a convex subset of a Banach space X and T a mapping D into itself, the sequence {xn}∞n=0 in D is defined by x0∈D,xn+1=(1−αn)xn+αnT((1−βn)xn+βnTxn),n≥0, where {αn} and {βn} satisfy 0≤αn,βn≤1 for all n and ∑∞n=0αn=∞.