WEAK AND STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES

In a real HILBERT space H, starting from an arbitrary initial point x(0) is an element of H, an interative process is defined as follows: x(n+1) = a(n)x(n) + (1-a(n))T(f)(gimel n+1)yn, yn = b(n)x(n) + (1-b(n))T(g)(beta n)x(n), n >= 0, where T-f(gimel n+1) x = Tx - gimel(n+1)mu(f) f(Tx), T(g)(beta n)x = Tx - beta(n)mu(g)g(Tx), (for all x is an element of H), T : H -> H a nonexpansive mapping with F(T) not equal empty set and f (resp. g) : H -> H an eta(f) (resp. eta(g))-strongly monotone and k(f) (resp. k(g))-Lipschitzian mapping, {a(n)} subset of (0,1), {b(n)} subset of (0, 1) and {lambda(n)} subset of [0,1) {beta(n)} subset of [0,1). Under some suitable conditions, serveral convergence results of the sequence {x(n)} are shown.