Iterative approximation with errors of fixed point for a class of nonlinear operator with a bounder range

Let X be a uniformly smooth real Banach space. Let T:X → X be continuos and strongly accretive operator. For a given f ε X, define S: X → X by Sx =f−Tx+x, for all x ε X. Let {a_n} _n=0 ^∞ , {β_n} _n=0 ^∞ be two real sequences in (0, 1) satisfying:

((i))

;

((ii))

Assume that {u_n} _n=0 ^∞ and {υ_n} _n=0 ^∞ are two sequences in X satisfying ‖u_n‖ = 0(α_n) and ‖υ_n‖ → 0 as n → ∞. For arbitrary x₀ ε X, the iteration sequence {x_n} is defined by

(1)

Moreover, suppose that {Sx_n} and {Sy_n} are bounded, then {x_n} converges strongly to the unique fixed point of S.