ITERATION PROCESSES FOR NONLINEAR LIPSCHITZIAN STRONGLY ACCRETIVE MAPPINGS IN L(P) SPACES

Suppose X = L(p) (or l(p)), 1 < p less than or equal to 2. Let T: X --> X be a Lipschitzian and strongly accretive map with constant k is an element of (0, 1) and Lipschitz constant L satisfying p(p - 1)L(2) < p - 2(p - 1)(1 - k). Define S: X --> X by Sx = f - Tx + x. For arbitrary x(0) is an element of X, the sequence {x(n)}(n=1)(x) 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)}(x)(n=1), {beta(n)}(x)(n=0) are two real sequences satisfying (i) 0 less than or equal to alpha(n) less than or equal to 2(-1){p - 2(p - 1)(1 - k - k beta(n) + L(2) beta(n)) - p(2 - p)L(2)[1 - w beta(n) + (w + L(2) - 1)beta(n)(2)]}{p - 2(p - 1)(1 -k -k beta(n) + L(2) beta(n)) + (p - 1)(2)L(2)[1 - w beta(n) + (w + L(2) - 1)beta(n)(2)] - 1}(-1) for each n, (ii) 0 less than or equal to beta(n) less than or equal to min {w(w + L(2) - 1)(-1), w[4(p - 1)(L(2) - k) + (p - 1)(2)L(2)w]-1} for each n, (iii) Sigma(n) alpha(n) = infinity, where w = p - 2(p - 1)(1 - k) - p(2 - p)L(2). Then {x}(x)(n=1) convergences strongly to the unique solution of Tx = f. Moreover, if alpha(n) = 2(-1){p - 2(p - 1)(1 - k - k beta + L(2) beta) - p(2 p)L(2)[1 - w beta + (w + L(2) - 1)beta(2)]}{p - 2(p - 1)(1 - k - k beta + L(2) beta) + (p - 1)(2)L(2)[1 - w beta + (w + L(2) - 1)beta(2)] - 1}(-1) and beta(n) = beta for each n and some 0 less than or equal to beta less than or equal to min {w(w + L(2) - 1)(-1), w[4(p - 1)(L(2) - k) + (p - 1)(2)L(2)w](-1)} then \\x(n+1) - q\\ less than or equal to p (n/2)\\x(1) - q\\, where q denotes the solution of Tx = f and p = [1 - 4(-1){p - 2(p - 1)(1 - k - k beta + L(2) beta) - p(2 - p) x L(2)[1 - w beta + (w + L(2) - 1)beta(2)]}(2){p - 2(p - 1)(1 - k - k beta = L(2) beta) + (p - 1)(2)L(2)[1 - w beta + (w + L(2) - 1)beta(2)] - 1}(-1)] is an element of (0, 1). A related result details with the iterative approximation of Lipschitz strongly pseudocontractive maps in X. (C) 1994 Academic Press, Inc.