Halpern type proximal point algorithm of accretive operators

The main objectives of this paper are to employ a new proof technique to prove the strong convergence of {x(n)} and {y(n)}, defined respectively by x(n+1) = alpha(n)u + beta(n)x(n) + (1 - alpha(n) - beta(n))J(rn)(A)x(n) and y(n+1) = beta(n)y(n) + (1 - beta(n))J(rn)(A) (alpha(n)u + (1 - alpha(n))y(n)), to some zero of accretive operator A in a uniformly convex Banach space E with a uniformly Gateaux differentiable norm (or with a weakly continuous duality mapping J(phi)) whenever {alpha(n)} and {beta(n)} are sequences in (0, 1) and {r(n)} subset of (0,+infinity) satisfying the conditions: lim(n ->infinity) alpha(n) = 0, Sigma(+infinity)(n=1)alpha(n) = +infinity, lim sup(n ->infinity) beta(n) < 1, lim inf(n ->infinity) r(n) > 0. Crown Copyright (C) 2011 Published by Elsevier Ltd. All rights reserved.