The Equivalence of Mann and Implicit Mann Iterations for Uniformly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces

In this paper, suppose E is an arbitrary uniformly smooth real Banach space, and D is a nonempty closed convex subset of E. Let T be a generalized Lipschitzian and uniformly pseudocontractive self-map with F(T) not equal 0. Suppose that {u(n)}(n=1)(infinity), {x(n)}(n=1)(infinity) are defined by Mann iteration and implicit Mann iteration respectively, with the iterative parameter {a(n)}(n=1)(infinity) satisfying certain conditions. Then the above two iterations that converge strongly to fixed point of T are equivalent.