AN ITERATIVE PROCESS FOR NONLINEAR LIPSCHITZIAN AND STRONGLY ACCRETIVE MAPPINGS IN UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH-SPACES

Suppose X is an s-uniformly smooth Banach space (s > 1). Let T: X --> X be a Lipschitzian and strongly accretive map with constant k is-an-element-of (0, 1) and Lipschitz constant L. Define S: X --> X by Sx = f - Tx + x. For arbitrary x0 is-an-element-of X, the sequence {x(n))n=1infinity is defined by x(n+1) = (1 - alpha(n))x(n) + alpha(n)Sy(n), y(n) = (1 - beta(n))x(n) + beta(n)Sx(n), n greater-than-or-equal-to 0, where {alpha(n)}n=0infinity, {beta(n)}n=0infinity are two real sequences satisfying: (i) 0 less-than-or-equal-to alpha(n)p-1 less-than-or-equal-to 2(-1) s(k + kbeta(n) - L2beta(n))(w + h)-1 for each n, (ii) 0 less-than-or-equal-to beta(n)p-1 less-than-or-equal-to min{k/L2, sk/(w + h) for each n, (iii) SIGMA(n) alpha(n) = infinity, where w = b(1 + L)s and b is the constant appearing in a characteristic inequality of X, h = max {1, s(s-1)/2}, p = min {2, s}. Then {x(n)}n=1infinity converges strongly to the unique solution of Tx = f. Moreover, if p = 2, alpha(n) = 2(-1)s(k + kbeta - L2beta)(w + h)-1, and beta(n) = beta for each n and some 0 less-than-or-equal-to beta less-than-or-equal min {k/L2, sk/(w + h)), then \\x(n+1) - q\\ less-than-or-equal-to rho(n/s)\\x1 - q\\, where q denotes the solution of Tx = f and rho = (1 - 4(-1)s2 (k + kbeta - L2beta)2(w + h)-1) is-an-element-of (0, 1). A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X. Suppose X is m-uniformly convex Banach spaces (m > 1) and c is the constant appearing in a characteristic inequality of X, two similar results are showed in the cases of L satisfying (1 - c2)(1 + L)m < 1 + c - cm(1 - k) or (1 - c2)L(m) < 1 + c - cm(1 - s).