| Let K be a subset of real normed linear space X, f : K (--->) X, f nonexpansive. Suppose there exists a set A (L-HOOK EQ) K such that, for each x(,0) (epsilon) A, {x(,n)} (L-HOOK EQ) A, where x(,n) is defined by x(,n+1) = (1 - C(,n))x(,n) + C(,n)f(x(,n)) and 0 0 such that, for some positive integer N and some sequence {x(,n)} (L-HOOK EQ) A,;it is proved that A is unbounded. This result is an extension of a theorem of M. Edelstein and R. C. O'Brian {J. London Math. Soc (2) 17 (1978), no. 3, 547-554; MR 80b:47074}. Using this result another proof of a theorem of S. Ishikawa {Proc. Amer. Math. Soc. 59 (1975), no. 1, 65-71; MR 54 ;An example is constructed to show that these results do not extend to the class of quasi-nonexpansive mappings. Then a sufficient condition is given under which the results extend.;Other theorems deal with the iterative solution of the equation x + Tx = f in a Banach space where the operator T is monotone. Assuming that a solution of the equation exists, iterative processes are constructed which converge to such a solution. Our theorems generalize important known results. In particular, a theorem of W. G. Dotson, Jr. {Math. Computation, Vol. 32 (141), 1978, 223-225} is generalized to include a much larger class of operators than was considered by Dotson. Also, the theorems of R. E. Bruck, Jr. {Bull. Amer. Math. Soc., 79 (1973), 1258-1262} are extended to more general Banach spaces than was considered in the above reference. Furthermore, our technique has led to the establishment of Lipschitz-like estimates for "the best approximation operator" in certain classes of Banach spaces that include both the L(,p) and l(,p) spaces for 1 < p < (INFIN). These estimates are of independent interest.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). |
