Weak almost-convergence theorem without-Opial's condition

Let E be Banach space with property (U, m, m + 1, lambda), lambda is an ele ment of R, m is an element of N, and a uniformly Gateaux differentiable nor m; J: E -->E* a duality mapping; D a nonempty closed convex bounded subset of E; and T: D --> I) a uniformly L-Lipschitzian asymptotically hemicontrac tive mapping with L < N(E)(1/2) where N(E) is the normal structure coeffici ent of E satisfying the condition \\x - T(n)y\\(2) <less than or equal to> <x - T(n)y, J(x - y)> for all x, y is an element of D, n is an element of N boolean OR {0}. Under the above conditions, the convergence of {J(x(n) - v )} for the sequence {x(n)} of the modified Ishikawa iteration process is es tablished 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 {x(n )}(n=0)(infinity) in D is defined by x(0) is an element of D, x(n+1) = (1- alpha (n))x(n) + alpha T-n((1 - beta (n))x(n) + beta (n)Tx(n)), n less than or equal to 0, where {alpha (n)} and {beta (n)} satisfy 0 less than or equ al to alpha (n), beta (n) less than or equal to 1 for all n and (n=0)Sigma (infinity)alpha (n) = infinity. (C) 2001 Academic Press.