APPROXIMATION OF FIXED-POINTS OF STRONGLY PSEUDOCONTRACTIVE MAPPINGS

Let E be a real Banach space with a uniformly convex dual, and let K be a nonempty closed convex and bounded subset of E. Let T: K --> K be a continuous strongly pseudocontractive mapping of K into itself. Let {c(n)}n=1infinity be a real sequence satisfying: (i) 0 < c(n) < 1 for all n greater-than-or-equal-to 1; (ii) SIGMA(n=1)infinity c(n) = infinity; and (iii) SIGMA(n=1)infinity c(n)b(c(n)) < infinity, where b: [0, infinity) --> [0, infinity) is some continuous nondecreasing function satisfying b(0) = 0 , b(ct) less-than-or-equal-to cb(t) for all c greater-than-or-equal-to 1 . Then the sequence {X(n)}n-1infinity generated by x1 is-an-element-of K, X(n+1) = (1 - c(n))x(n) + c(n)Tx(n), n greater-than-or-equal-to 1, converges strongly to the unique fixed point of T. A related result deals with the Ishikawa iteration scheme when T is Lipschitzian and strongly pseudocontractive.