Global Iterative Schemes for Accretive Operators

Author: Chidume C.E.   Zegeye H.  

Publisher: Academic Press

ISSN: 0022-247X

Source: Journal of Mathematical Analysis and Applications, Vol.257, Iss.2, 2001-05, pp. : 364-377

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Abstract

Let E be a real q-uniformly smooth Banach space and A : E → 2E be an m-accretive operator which satisfies a linear growth condition of the form ‖Ax‖ ≤ c(1 + ‖x‖) for some constant c > 0 and for all xE. It is proved that if two real sequences {λn} and {θn} satisfy appropriate conditions, the sequence {xn} generated from arbitrary x0E by xn + 1 = xn - λn(un + θn(xn - z)); unAxn n ≥ 0, converges strongly to some x* ∈ A- 1(0). Furthermore, if E is a reflexive Banach space with a uniformly Gâteaux differentiable norm, and if every weakly compact convex subset of E has the fixed-point property for nonexpansive mappings and A : D(A) ≔ E → 2E is m-accretive, then for arbitrary, z, x0E the sequence {xn} defined by xn + 1 + λn(un + 1 + θn(xn + 1 - z)) = xn + en, for unAxn, where enE is such that ∑‖en‖ < ∞ ∀n ≥ 0, converges strongly to some x* ∈ A- 1(0).